Introduction

A class of ionic hydrate solids comprising alkaline and alkaline earth cations and anions with glass-forming ability exhibits a specific physico-geometrical reaction pathway during thermal dehydration in forming its anhydride. These hydrates include alkaline and alkaline earth acetate, borate, phosphate, thiosulfate, and others [1,2,3,4,5,6]. The thermal dehydration process is characterized by a multistep mass loss process that includes a step accompanied by liquefaction and the subsequent dehydration of liquefied reactant to form an amorphous anhydride. In general, the thermal dehydration/decomposition of solids accompanied by liquefaction exhibits complex characteristics in the reactions of individual hydrated compounds, as has been recently demonstrated for the thermal dehydration of organic hydrates [7,8,9,10]. Among these reactions, the process focused on in this study is characteristic of anhydride glass formation, which resembles the sol–gel process for glass preparation [11]. Therefore, the details of the physico-geometrical kinetic features of the multistep process are worth investigating from the perspective of physical chemistry to reveal the kinetic behavior of the specific reaction class, accompanied by liquefaction, and engineering to control the sol–gel process for glass preparation.

The thermal dehydration of potassium tetraborate tetrahydrate (PTB-TH) is a typical example of the class of multistep reactions accompanied by liquefaction and the subsequent dehydration of liquefied reactant to form an anhydride glass. In our previous study [2], the multistep mass loss process of thermal dehydration comprised of three mass loss steps was described as: (1) a surface reaction in the solid-state accompanied by crack formation; (2) a rapid mass loss process accompanied by liquefaction to form the aggregate of the gel powders; and (3) the dehydration of gel powder aggregate and glass formation. The recent advancement in the kinetic analysis of multistep thermal dehydration/decomposition reactions using thermoanalytical (TA) measurements [12,13,14] is expected to reveal the complex kinetic features of the physico-geometrical multistep thermal dehydration of PTB-TH. However, there are some difficulties in applying the multistep kinetic analysis using the so-called mathematical deconvolution analysis (MDA) [14,15,16,17,18] and kinetic deconvolution analysis (KDA) [13,14,15, 18, 19]. During the thermal dehydration of PTB-TH, the contributions of each component reaction step to the overall process vary according to the heating conditions, such as the heating rate (β). For such a process observed frequently for the thermal decomposition of various materials [5, 20, 21], the conventional isoconversional analysis is not applicable to a series of the kinetic curves of the specific reaction step at different β, which were obtained by separating the partially overlapping component steps using MDA, because the isoconversional method assumes a single step reaction exhibiting the constant value of total mass loss irrespective of β. In the case of KDA based on a cumulative kinetic equation, a simultaneous optimization of kinetic parameters for a series of overall kinetic curves at different β is not possible because of the changes in the contributions of component reaction steps to be optimized as the parameters through KDA. Variation among the contributions is also expected for PTB-TH samples having different particle sizes. Thus, one of the challenges in this study is to extend the well-established MDA and KDA to a process that involves varying contributions from the reaction steps.

In this study, the thermal dehydration of PTB-TH having different particle sizes was systematically traced using simultaneous thermogravimetry (TG)–differential thermal analysis (DTA). The contributions of the component reaction steps under different heating conditions were determined using MDA and the trend in the variation of the contributions depending on the heating conditions and the particle size was interpreted by considering the physico-geometrical reaction characteristics of each reaction step. In addition, a modulated temperature TG technique was used to estimate the apparent activation energy (Ea) at each step [22,23,24,25,26,27,28]. Using the contribution and apparent Ea value of each reaction step as the initial value, the multistep thermal dehydration kinetics was further analyzed using KDA. The kinetic features of individual reaction steps were interpreted with the aid of other physico-chemical techniques, including high temperature X-ray diffractometry (XRD) and Fourier transform infrared spectroscopy (FTIR), as well as in-situ and ex-situ microscopic observations. Specific kinetic features of the multistep thermal dehydration of PTB-TH, including the self-induced sol–gel process of glass formation, are discussed in light of a detailed kinetic analysis.

Materials and method

Sample and its characterization

A reagent-grade potassium tetraborate tetrahydrate (K2B4O7∙4H2O, PTB-TH; special grade, Nacalai Tesque Inc.) was used as the sample without further purification. The sample was characterized by powder XRD and FTIR. The XRD pattern of the sample after it was press-fitted to a sample holder plate was recorded using a diffractometer (RINT 2200 V, Rigaku) by irradiating with monochrome Cu-Kα (40 kV, 20 mA) and scanning at 2θ from 5° to 60° in steps of 0.02° at a scan speed of 4° min−1. The sample was diluted with KBr, and the FTIR spectrum was obtained by the diffuse reflectance method using a spectrometer (FTIR 8400S, Shimadzu). The sample particles were sieved to different fractions of particle sizes using stainless-steel sieves (53–90, 90–150, 150–300, 300–500, and 500–1000 µm). The sample particles in different sieve fractions were observed using a digital microscope (Hozan) and photographed. The sample particles in the 150–300 μm size fraction were subjected to TG–DTA measurement using a DTG-60 instrument (Shimadzu). The sample of 10.0 mg, which was weighed in a Pt pan (diameter: 6.0 mm; depth: 2.5 mm), was heated from 303 to 1123 K at a heating rate (β) of 5 K min−1 under a stream of dry N2 gas at a flow rate (qv) of 100 cm3 min−1.

Thermal dehydration behavior

The sample particles in the 150–300 μm fraction were press-fitted to a platinum sample-holder plate, and changes in the XRD pattern of the sample during heating, which was done in accordance with two stepwise isothermal heating programs under a stream of dry N2 gas (qv = 100 cm3 min−1), were recorded using the RINT 2200 V diffractometer that was equipped with a programmable heating chamber. In one stepwise isothermal heating program, the sample was heated from 300 to 1023 K at a β of 5 K min−1, during which isothermal holding intervals of 15 min each were inserted at every 50 K from 323 to 1023 K. In the other heating program, the sample was heated from 300 to 523 K at a β of 2 K min−1, during which isothermal holding intervals of 15 min each were inserted at every 10 K from 333 to 523 K. The XRD pattern of the sample was recorded in each isothermal holding interval under the same conditions that were used for the sample characterization at room temperature.

The sample particles of each size fraction (53–90, 90–150, 150–300, 300–500, and 500–1000 µm) were weighed out in portions of 5.0 mg in a Pt sample pan (diameter: 6.0 mm; depth: 2.5 mm). Simultaneous TG–DTA measurements (DTG-60, Shimadzu) were performed by heating the sample from 300 to 1123 K at different values of β (1 ≤ β/K min−1 ≤ 10) under a stream of N2 gas at a qv of 100 cm3 min−1. In addition, TG–DTA curves were recorded under modulated temperature conditions by heating the sample from 300 to 798 K at a β of 2 K min−1 with a saw-like modulation characterized by an amplitude of 20 K and a period of 5 min.

Changes in the surface morphology of the sample (150–300 μm) with the progress of thermal dehydration were observed using an optical microscope (BH-20, Olympus) when the sample was heated from 300 to 523 K at a β of 5 K min−1 under a stream of N2 gas (qv = 100 cm3 min−1) using a heating stage (STH-M100, S.T. Japan) and recorded as a movie.

Under the same measurement conditions as the aforementioned TG–DTA measurements and at a β of 5 K min−1, partially dehydrated samples (150–300 μm) were prepared by heating them to different temperatures. After they were cooled to room temperature under a stream of N2 gas, the preheated sample particles were recovered from the TG–DTA instrument. The sample particles were then coated with a thin layer of Pt by sputtering (JFC-1600, JEOL, 30 mA, 30 s). The surfaces of the Pt-coated sample particles were observed using a scanning electron microscope (SEM; JSM-6510, JEOL). The FTIR spectra of the preheated sample particles were also recorded using the aforementioned spectrometer.

Results and discussion

Sample characterization

Figure S1 in ESI shows the XRD pattern and FTIR spectrum of the sample. The XRD pattern agreed with the pattern reported for K2B4O5(OH)4∙2H2O (orthorhombic, S.G. = P212121(19), a = 11.7850, b = 12.9170, c = 6.8650, α = β = γ = 90.000, PDF 00-029-0987) [29, 30]. All FTIR peaks were attributed to K2B4O5(OH)4·2H2O [31]. The assignment of the absorption peaks is shown in Table S1. The particle morphologies of the sample in different size fractions are shown in Figure S2. Figure 1 shows TG–derivative TG (DTG)–DTA curves recorded, while heating the 150–300 μm sample from 303 to 1123 K under a stream of dry N2. The total mass loss observed until 750 K was 23.8%, which agreed with the value calculated for the thermal dehydration to form the anhydride (23.59%).

$${\text{K}}_{{2}} {\text{B}}_{{4}} {\text{O}}_{{7}} \cdot {\text{4H}}_{{2}} {\text{O}} \to {\text{K}}_{{2}} {\text{B}}_{{4}} {\text{O}}_{{7}} + {\text{ 4H}}_{{2}} {\text{O}}$$
(1)
Fig. 1
figure 1

TG–DTG–DTA curves recorded by linearly heating the sample (150–300 μm; m0 = 10.02 mg) at a β of 5 K min−1 under a stream of dry N2 gas (qv = 100 cm3 min−1)

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Thermal behavior

Under linear nonisothermal conditions (Fig. 1), the mass loss process exhibited a multistep behavior comprised of three distinguishable mass loss steps accompanied by the corresponding endothermic DTA peaks. The thermal dehydration began at approximately 345 K with gradual mass loss behavior (first step). Subsequently, a rapid mass loss occurred midway through the reaction at approximately 405 K (second step). On further heating, a gradual mass loss process was observed over a large temperature range until approximately 750 K (third step). At the end of mass loss process, an exothermic DTA peak was observed at approximately 720 K, and this was followed by another exothermic DTA peak at approximately 860 K. These exothermic phenomena were attributed to the crystallization of the dehydration product [2]. The solid product melted at 1068 K with a sharp endothermic DTA peak, which corresponded to the melting point of K2B4O7 [2].

Figures 2 and S3 show changes in the XRD pattern of the sample during stepwise isothermal heating. The original XRD pattern completely disappeared in the temperature range of 373–423 K (Figs. 2a and S3); this temperature range corresponded to the second mass loss step observed in the TG–DTG–DTA curves (Fig. 1). The XRD pattern immediately after the second mass loss step exhibited hallows with two maximum intensities at approximately 29° and 44° but changed to an XRD pattern of poorly crystallized material in the temperature range of 433–523 K (Fig. 2b) that corresponded to the first half of the third mass loss step. In the second half of the third mass loss step, the XRD pattern with hallow observed at 423 K was reproduced at 573 K. Therefore, at the end of the mass loss process, an amorphous K2B4O7 was produced. The crystallization of the amorphous K2B4O7 occurred in the temperature range of 723–773 K, which corresponded to the first exothermic DTA peak. The XRD pattern apparently remained unchanged during further heating to 1023 K and exhibited the gradual growth of each diffraction peak with the increase in temperature. The crystallization product had an XRD pattern that corresponded to the K2B4O7 crystal (Fig. 2c, triclinic, S.G. = P − 1(2), a = 6.4840, b = 9.6040, c = 10.4130, α = 89.280, β = 102.750, γ = 101.250, PDF 01–070-1494) [32]. Notably, no identifiable changes in the XRD pattern were observed in the temperature range corresponding to the second exothermic DTA peak. The FTIR spectra of the samples that were preliminary heated to different temperatures indicated a gradual attenuation of the absorption peaks that were attributed to the H–O–H and O–H vibrations as the temperature increased (Figure S4). In addition, a broadening of the absorption peaks attributed to the B–O and O–B–O vibrations was observed at temperatures higher than 428 K, which corresponds to the second mass loss step in the TG–DTG–DTA curves and the formation of the amorphous phase in the XRD.

Fig. 2
figure 2

Changes in the XRD pattern of the sample during heating under a stream of dry N2 gas (qv = 100 cm3 min−1): a an overview of the changes during heating from 323 to 1023 K, b changes in the temperature range from 423 to 723 K, and c XRD pattern at 1023 K

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Figure 3 shows SEM images of the surface texture of the partially dehydrated sample particles as characterized by different degrees of reaction (α) with reference to the total mass loss value of dehydration. The flat and smooth particle surface of the original sample particles (Fig. 3a) is maintained until thermal dehydration is initiated (Fig. 3b). As soon as the thermal dehydration is initiated, cracks begin to grow on the surface of the particles along a specific direction (Fig. 3c). The in-situ optical microscopic observation of the surface texture during the first mass loss step also revealed the growth of cracks on the particle surfaces (Figure S5). The cracks further developed during the second mass loss step (Fig. 3d). Notably, at this stage, the sample particles were sticky after they were recovered from the sample pan of the TG measurement, which indicated that the second mass loss step was accompanied by the gelation or melting of the sample particles. Notably, in our previous study [2], formation of aggregated fine powders has been observed in the particle interior as the product of the second mass loss step. The surface morphology of the sample particles did not significantly change during the third mass loss step (Fig. 3e, f).

Fig. 3
figure 3

SEM images of the partially dehydrated sample particles (150–300 μm) obtained by preliminarily heating the particles to different temperatures at a β of 5 K min−1 under a stream of dry N2: a original sample; b 343 K (α = 0.00); c 403 K (α = 0.17); d 428 K (α = 0.46); e 463 K (α = 0.62); and f 703 K (α = 1.00)

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Figures 4 and S6 show the TG–DTG–DTA curves for the thermal dehydration of PTB-TH with individual particle size fractions at different β values under a stream of dry N2 gas. For the sample of 150–300 μm (Fig. 4), the TA curves shifted to higher temperatures with an increase in β as a general kinetic behavior. However, changes in the contribution of the individual mass loss steps were observed depending on the value of β, and were typically characterized by a decrease in the mass loss value in the first mass loss step. The same behaviors were observed for all samples with different particle size fractions (Figure S6). The shift in the TA curves to higher temperatures and the decrease in the mass loss value in the first step were also observed with an increase in the sample particle size (Figure S7).

Fig. 4
figure 4

TG–DTG–DTA curves for the thermal dehydration of PTB-TH (150–300 μm, m0 = 5.00 ± 0.04 mg) at different β values under a stream of dry N2 gas (qv = 100 cm3 min−1)

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Mathematical deconvolution analysis

To parameterize the multistep mass loss behavior of the thermal dehydration, each DTG curve was deconvoluted using a statistical function (Fi(t)) into three mass loss steps based on Eq. (2) (MDA) [14,15,16,17,18].

$$\frac{{{\text{d}}m}}{{{\text{d}}t}} = \mathop \sum \limits_{i = 1}^{3} F_{{\text{i}}} \left( t \right)$$
(2)

Various Fi(t) functions that are capable of fitting peaks having an asymmetric shape were preliminarily tested, and a log normal four-parameter function (Eq. (3)) was selected as the most appropriate function for obtaining the best empirical fitting for all three DTG peaks.

$$F\left( t \right) = a_{0} {\text{exp}}\left[ { - \frac{{{\text{ln}}\left( 2 \right){\text{ln}}\left( {\frac{{\left( {t - a_{1} } \right)\left( {a_{3}^{2} - 1} \right)}}{{a_{2} a_{3} }} + 1} \right)^{2} }}{{{\text{ln}}\left( {a_{3} } \right)^{2} }}} \right],$$
(3)

where a0–a3 are the parameters that describe the amplitude, center, width, and shape, respectively. Figure 5 shows the results of the MDA for the thermal dehydration of PTB-TH with 150–300 μm particle sizes. The results for the other samples with different particle sizes are shown in Figures S8–S11. The results of the MDA provided statistically significant empirical fittings to the overall DTG curve with three Fi(t) functions, irrespective of β and particle size (Fig. 5a, Figures S8(a)–S11(a)). The contributions (ci) of the individual mass loss steps were determined from the ratio of the peak areas of the empirically deconvoluted peaks (Fig. 5b, Figures S8(b)–S11(b)). In all samples having different particle sizes, the contribution of the first mass loss step decreased with the increase in β, which was accompanied by a compensative increase in the contribution of the second mass loss step. Meanwhile, the contribution of the third mass loss step was approximately constant, irrespective of β.

Fig. 5
figure 5

Results of the MDA for the thermal dehydration of PTB-TH (150–300 μm): a a typical fitting result for the reaction at a β of 3 K min−1 and b contributions of individual mass loss steps and those changes with β

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This phenomenon can be explained by the difference in the shift of the reaction to higher temperatures with an increase in β between the first and second mass loss steps. Namely, the first mass loss step was initiated on the particle surface as a solid-state reaction that exhibited normal kinetic behavior by shifting to higher temperatures with an increase in β. However, the reaction temperature of the second mass loss step accompanied by gelation was less sensitive to the increase in β. Therefore, the overall α reached by solid-state thermal dehydration as assigned by the first mass loss step at the gelation temperature decreased with an increase in β. After the reactant gelled, the rapid mass loss occurred in the second mass loss step until the viscosity of the reactant attained at a certain value, which was roughly corresponded to the α range of 0.5–0.6. The residual water molecules were evolved in the third mass loss step with a nearly constant contribution. The phenomenon observed in the third mass loss step is resembles the sol–gel process in glass formation [11].

Kinetic analysis under modulated temperature conditions

For an ideal multistep process characterized by independent component steps and the constant contributions irrespective of β, a series of kinetic curves for the individual mass loss steps can be obtained from an MDA. However, in the present case, the contributions of the component reaction steps vary with β, and the conventional isoconversional kinetic calculation cannot be applied to the series of kinetic curves for the individual mass loss steps, separated by the MDA. To estimate the apparent Ea value for each mass loss step, we apply the modulated temperature TG as a possible method. Figure 6 shows the results of the modulated temperature TG measurement and the kinetic calculation for the thermal dehydration of PTB-TH (150–300 μm), and the results for the other samples having different particle sizes were shown in Figures S12–S15. The DTG signal responds to the temperature modulation (Fig. 6a). After the TG–DTG curves were converted to kinetic data comprising (t, T, α, dα/dt), the series of kinetic data points at the maximum and minimum temperatures of the modulated temperature profile were extracted. Using the data points at maximum and minimum temperatures, we calculated the two series of kinetic data by systematically varying α from 0.01 to 0.99 in steps of 0.01 using the spline interpolation method. The Ea value was roughly calculated using the two data points at the maximum and minimum temperatures at various α values, as was done in the jump method of kinetic calculation [28].

$$E_{{{\text{a,}}\alpha }} = \left[ {\frac{{RT_{{{\text{max}}}} T_{{{\text{min}}}} }}{{T_{{{\text{max}}}} {-}T_{{{\text{min}}}} }}{\text{ln}}\frac{{\left( {{\text{d}}\alpha /{\text{d}}t} \right)_{{{\text{max}}}} }}{{\left( {{\text{d}}\alpha /{\text{d}}t} \right)_{{{\text{min}}}} }}} \right]_{\alpha } ,$$
(4)

where R is the gas constant. The subscripts, max and min, denote the data points at maximum and minimum temperatures, respectively. As a result, three distinct α ranges characterized by individual constant Ea values, i.e., (1)–(3) in Fig. 6b, were found irrespective of the samples with different particle sizes (Figures S12(b)–S15(b)). In addition, the Ea values in each α range determined for the sample having different particle sizes were approximately coincident as shown in Fig. 7. The average Ea values for each α range were (1) 69.0 ± 9.9, (2) 141.9 ± 5.2, and (3) 98.9 ± 4.7 kJ mol−1 for the first to the third α ranges, respectively. Notably, large fluctuations in Ea values observed at the ending stage of the overall reaction (α > 0.7) are owing to the slow mass loss rate in this stage. By comparing, the contributions of each mass loss step and the α ranges exhibiting individual constant Ea values, the apparent Ea values can be interpreted as corresponding to the Ea values for the first to third mass loss steps, respectively.

Fig. 6
figure 6

Results of kinetic analysis for the thermal dehydration of PTB-TH (150–300 μm, m0 = 5.04 mg) at a β of 2 K min−1 with a saw-like modulation (amplitude: 20 K; period: 5 min) under a stream of dry N2 gas (qv = 100 cm3 min−1): a TG–DTG–Temperature profile curves and b Ea values at different α values

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Fig. 7
figure 7

Comparison of Ea values at different α values determined using modulated temperature TG and jump method of kinetic calculation for the thermal dehydration of PTB-TH having different particle sizes

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Kinetic deconvolution analysis

To empirically describe the kinetics of the individual mass loss steps during the thermal dehydration of PTB-TH, the overall process was assumed to comprise three independent subprocesses that partially overlapped. Under such an assumption, the overall rate behavior is expressed using a cumulative kinetic equation [13,14,15, 18, 19].

$$\begin{aligned} & \frac{{{\text{d}}\alpha }}{{{\text{d}}t}} = \mathop \sum \limits_{i = 1}^{3} c_{{\text{i}}} A_{{\text{i}}} {\text{exp}}\left( { - \frac{{E_{{{\text{a}},{\text{i}}}} }}{RT}} \right)f_{{\text{i}}} \left( {\alpha_{{\text{i}}} } \right) \\ & {\text{with}}\;\;\mathop \sum \limits_{i = 1}^{3} c_{{\text{i}}} = 1\;\;{\text{and}}\;\;\mathop \sum \limits_{i = 1}^{3} c_{{\text{i}}} \alpha_{{\text{i}}} = \alpha , \\ \end{aligned}$$
(5)

where Ai, αi, and fi(αi) are the Arrhenius preexponential factor, degree of reaction, and kinetic model function of the mass loss step i, respectively. For analytical purposes, an empirical kinetic model function known as the Šesták–Berggren model with three kinetic exponents (SB(m, n, p)) [33,34,35] was used for all the mass loss steps because of the significant flexibility of the function.

$$f\left( \alpha \right) = \alpha^\text{m} \left( {1 - \alpha } \right)^\text{n} \left[ { - {\text{ln}}\left( {1 - \alpha } \right)} \right]^\text{p}$$
(6)

Using the cumulative kinetic equation (Eq. (5)) with the empirical kinetic model function (Eq. (6)), the kinetics of all the mass loss steps are characterized by simultaneously optimizing all the kinetic parameters, including ci, Ai, Ea,i, mi, ni, and pi to fit the experimental kinetic curves through a nonlinear least-squares analysis, which is called KDA. Because of the variations in the ci values according to β and the sample particle size, the KDA was carried out for the individual overall kinetic curves obtained for the samples having different particle sizes at different β values. To prevent incorrectly optimizing the 18 parameters, the initial ci and Ea,i values in Eqs. (5) and (6) were set using the values preliminarily determined by the MDA and from the modulated temperature TG measurements, respectively. Furthermore, the SB(0, 1, 0) function equivalent to the first-order reaction model was used for the initial setting. After that, the order of Ai values for setting the initial values was graphically determined. Thereafter, all kinetic parameters were simultaneously optimized by nonlinear least-squares analysis to minimize the square sum of the residuals of the fitting to the differential kinetic curve.

$$F = \mathop \sum \limits_{j = 1}^{M} \left[ {\left( {\frac{{{\text{d}}\alpha }}{{{\text{d}}t}}} \right)_{{{\text{exp}},{\text{j}}}} - \left( {\frac{{{\text{d}}\alpha }}{{{\text{d}}t}}} \right)_{{{\text{cal}},{\text{j}}}} } \right]^{2} ,$$
(7)

where M is the number of data points in a kinetic curve. The subscripts, exp and cal, denote the experimental and calculated values, respectively.

Figure 8 depicts a typical result of KDA for the thermal dehydration of PTB-TH (150–300 μm) at a β of 3 K min−1. The KDA results for the reactions of the samples having other particle sizes at the same β are shown in Figure S16. The KDA was performed for individual kinetic curves recorded at different β values of 1–10 K min−1 for the 53–90 and 90–150 μm samples, whereas for the larger particle size samples the range of β was limited to lower values because of the additional mass loss step that occurred during the third reaction step at high β values: 1–7 K min−1 for the 150–300 μm sample, 1–5 K min−1 for the 300–500 μm sample, and 1–3 K min−1 for the 500–1000 μm sample. All the kinetic curves for the reactions of different particle size samples at different β values were fitted as the sum of three independent component reaction steps exhibiting statistically significant correlations as characterized by the determination coefficient (R2 > 0.97). Table 1 summarizes the optimized kinetic parameters for the reaction of the individual samples of different particle sizes as values averaged over different β values.

Fig. 8
figure 8

Graphic representation of the KDA result for the thermal dehydration of PTB-TH (150–300 μm) at a β of 3 K min−1 under a stream of dry N2 (qv = 100 cm3 min−1)

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Table 1 Kinetic parameters for the multistep thermal dehydration of PTB-TH to form amorphous anhydride, optimized by MDA and KDA
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On reviewing the optimized kinetic parameters, we noted that the variation in ci values depending on the β observed for the first and second reaction steps was accompanied by a variation in the Ai value of the corresponding reaction step. The correlation between the ci and Ai values can be empirically explained by the changes in the physico-geometric meaning of αi due to the different ci values and thus the total mass loss value of the reaction step. Namely, a change in c1 and c2 values implies there are changes in the thickness of the surface product layer produced during the first reaction step and in the volume of the residual PTB-TH in the particle interior for the second reaction step. The size factor of the reactant is absorbed in the rate constant for formulating the kinetic model function at a constant temperature; therefore, the Ai value changes with a variation in ci. Despite the variations in ci and Ai with variations in β and sample particle size during the first and second reaction steps, the optimized Ea,i and the kinetic exponents in SB(mi, ni, pi) were practically invariant between the reactions under different β and sample particle size conditions, which indicates the mechanism of the individual reactions in the first and second reaction steps remained unchanged with the change in ci values related to the values of β and particle size. Unlikely for the first and second reaction steps, the optimized kinetic parameters for the third reaction step were practically invariant, irrespective of β and sample particle size.

Kinetic interpretation of each component reaction step

The rate behavior of individual reaction steps under isothermal conditions was simulated by drawing an experimental master plot using the optimized Ai and SB(mi, ni, pi). The hypothetical reaction rate (dαi/dθi) at an infinite temperature is expressed using Ozawa’s generalized time (θi) for each reaction step i [36,37,38,39,40,41,42].

$$\frac{{{\text{d}}\alpha_{{\text{i}}} }}{{{\text{d}}\theta_{{\text{i}}} }} = \left( {\frac{{{\text{d}}\alpha_{{\text{i}}} }}{{{\text{d}}t}}} \right){\text{exp}}\left( {\frac{{E_{{{\text{a}},{\text{i}}}} }}{RT}} \right) = A_{{\text{i}}} f_{{\text{i}}} \left( {\alpha_{{\text{i}}} } \right)\;\;{\text{with}}\;\;\theta_{{\text{i}}} = \mathop \smallint \limits_{0}^{t} {\text{exp}}\left( { - \frac{{E_{{{\text{a}},{\text{i}}}} }}{RT}} \right){\text{d}}t$$
(8)

Because, the rate behavior of the first and third reaction steps was characterized as a deceleration process, the (dαi/dθi) values calculated at various αi values were divided by that at αi = 0.5 to draw the experimental master plot of (dαi/dθi)α(i)/(dαi/dθi)0.5 versus αi. However, the second reaction step (i = 2) exhibited the maximum reaction rate midway through the reaction step. Therefore, the normalization of the experimental master plot was performed by dividing (dα2/dθ2) values at various α2 by that at the maximum reaction rate. Figure 9 shows the experimental master plots for the individual reaction steps. In the individual reaction steps, the experimental master plots were practically identical for the different particle size samples.

Fig. 9
figure 9

Experimental master plots at an infinite temperature for the individual mass loss steps of the thermal dehydration of PTB-TH, calculated using the SB(mi, ni, pi) optimized via KDA: a first, b second, and c third mass loss steps

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The experimental master plot for the first reaction step exhibited approximately linear deceleration behavior, which corresponds to the first-order rate raw (F(1)): f(α) = 1 – α (Fig. 9a). Because the first reaction step was interpreted as the surface reaction, the rate behavior of F(1) describes the reaction of the original particle surfaces to form the surface product layer, which is controlled by random nucleation. The rate behavior of the second reaction step originating from the gelation of the internal reactant is characterized by the occurrence of a maximum reaction rate midway through the reaction step at α2 = 0.23, irrespective of sample particle size (Fig. 9b). The acceleration and deceleration stage can be interpreted as accompanying the liquefaction of the reactant and the removal of water molecules from the molten reactant in the scheme of the core–shell structure of the molten hydrate and surface product layer. A detectable concaved shape of the experimental master plot during the deceleration process is indicative of the water molecules diffusing through the surface product layer. Conventionally, thermal dehydration/decomposition accompanied by liquefaction has been described as a simple conjunction of two kinetic equations that describe the acceleration and deceleration stages [8, 43,44,45,46], which switch at the point of the completion of liquification.

$$\frac{{{\text{d}}\alpha_{2} }}{{{\text{d}}\theta_{2} }} = A_{2\left( 1 \right)} f_{2\left( 1 \right)} \left( {\alpha_{2} } \right) + A_{2\left( 2 \right)} f_{2\left( 2 \right)} \left( {\alpha_{2} } \right),$$
(9)

where 1 and 2 in parentheses denote the acceleration and deceleration stages, respectively. We attempted to fit the experimental master plot in the individual α2 ranges of acceleration (0.01 ≤ α2 ≤ 0.23) and deceleration (0.23 ≤ α2 ≤ 0.99) stages using the truncated Šesták–Berggren model, SB(m,n) (p = 0 in Eq. (6)) [9, 10], which corresponds to an empirical form of an autocatalytic reaction model known as the Prout–Tompkins model [47]. The results showed that the practically the same SB(m, n) models were obtained for both reaction stages, i.e., SB(0.59, 1.94) and SB(0.58, 1.92) for the acceleration and deceleration stages, respectively. This may indicate that the liquefaction of the internal reactant and the diffusional removal of the water molecules occur simultaneously with a geometrical constraint. As in the contracting geometry of the reaction interface for the diffusional removal of water molecules, an advance of the liquefaction interface from the reaction interface toward the center of the particle can be expected for explaining the overall rate behavior of the second reaction step. In this geometrical model, a liquefied layer appears inside the reaction interface, which is the source of the water molecules to be removed from the reacting particles through the surface product layer. The third reaction step is apparently the diffusional removal of the residual water molecules from the gelatinated particles to form a glassy anhydride, as is expected from the deceleration rate behavior of the experimental master plot, which has a concave shape (Fig. 9c). However, the rate behavior is slightly different from that of the well-defined diffusion-controlled models, including the one-dimensional growth controlled by diffusion shown by the Johnson–Mehl–Avrami model, JMA(0.5) [48,49,50,51,52,53], and contracting geometry reactions controlled by diffusion, D(1), D(2), and D(4) [54, 55]. An interaction between the second and third reaction steps should be considered in interpreting the rate behavior of the third reaction step. In the superimposed region of the second and third reaction step, the removal of the residual water molecules is initiated in the surface product layer under the conditions of a continuous supply of water molecules to the surface product layer from the internal reaction interface of the second reaction step.

Conclusions

On linearly heating crystalline particles of PTB-TH, thermal dehydration occurs exhibiting multistep mass loss behavior comprising three partially overlapping steps. The first step begins at approximately 345 K, irrespective of β and particle size. The first step reaction, which is characterized as occurring on the particle surface to form a surface product layer, can be interpreted as a typical physico-geometrical kinetic process that exhibits variation in the mass loss rate, depending on β and particle size. Before the end of the first mass loss step, the mass loss rate dramatically increases at approximately 405 K, and the second mass loss step begins. This phenomenon can be explained by the liquefaction of the solid reactant at 405 K, accompanied by the sudden disappearance of the sharp XRD peaks attributed to PTB-TH and appearance of an XRD pattern of amorphous phase with two hollows having a maximum intensity at approximately 29° and 44°. The crack formation in each particle along a specific direction to slice the particle is also responsible for the rapid mass loss behavior of the second mass loss step. The rapid mass loss during the second step terminates in a limited temperature width of approximately 40 K, and the third mass loss step, characterized by a gradual mass loss behavior, continues until approximately 750 K to form a glassy anhydride. At the midpoint of the second mass loss step, the XRD hollows of the initially formed amorphous phase disappear. The XRD pattern changes to one that is characterized by poor diffraction without no hollows indicated, and this pattern continues to the midpoint of the third mass loss step. The XRD pattern characteristic of amorphous phase reappears at 573 K. It is expected, that the reacting particle contains a large number of water molecules in this temperature region and exhibits poor XRD diffraction. The regeneration of the amorphous phase, characterized by the XRD pattern having two hollows, through the evolution of water molecules resembles the sol–gel process of glass formation.

The three-step mass loss process exhibits a complex kinetic behavior, i.e., it is characterized by variation in the contributions in the individual mass loss steps to the overall dehydration relative to β and particle size. The observed trend of lowering in the contribution of the first mass loss step with an increase in β and particle size can be explained by the change in the mass loss within the limited temperature range from the initiation of the mass loss process to the gelation of the reacting particles at 405 K, because the reaction initiation temperature shifts to higher temperatures with an increase in β and particle size. For an analysis of the multistep kinetics accompanied by changes in the contributions of the component reaction steps with changes in β, a technique of modulated temperature TG can be used for the first estimation of the Ea values of the individual reaction steps. Using the Ea values as initial values, the multistep mass loss process of thermal dehydration was successfully described using the cumulative kinetic equation (Eq. (5)) with SB(m, n, p) model (Eq. (6)) as the empirical model function by optimizing all the kinetic parameters, from which the Ea values for each reaction step were determined to be Ea,1 = 67.9 ± 1.4, Ea,2 = 122.7 ± 0.9, and Ea,3 = 81.8 ± 1.5 kJ mol−1 for the thermal dehydration of the 150–300 μm sample. These values were practically invariant between the reactions of different particle size samples, regardless of the significant variations in the contributions of each reaction step according to β and particle size. In addition, the individual reaction steps exhibited characteristic rate behavior at a constant temperature, as shown in overview with an experimental master plot of each reaction step (Fig. 9). The first reaction step was characterized by a linear deceleration process described by first-order kinetics, which described by the consumption of the original reactant surface to form the surface product layer. The second reaction step showed a maximum rate midway through the step, which is a specific behavior of the reaction accompanied by liquefaction. However, unlikely from a simple process accompanied by liquefaction as expressed by a combination of kinetic equations for the acceleration and deceleration stages, the acceleration and deceleration stages could be described by the same kinetic equation as an autocatalytic model. The third process, corresponded to the self-generated sol–gel process of glass formation, was apparently controlled by diffusion. The kinetic features of the individual component reaction steps revealed by the advanced kinetic analysis for the physico-geometrical multistep process well illustrated the role of the individual reaction step to the overall process of the self-induced sol–gel process of glass formation observed during the thermal dehydration of PTB-TH.