Summary of Investigations in Regard to the Kinetics of Absorbed Water Dehydration from Different Hydrogels

Abstract

A review of novel kinetics models of dehydration (DH) of equilibrium swollen hydrogels: poly(acrylic acid) hydrogel (PAAH), poly(acrylic-co-methacrylic acid) (PAMAH), and poly(acrylic acid)-g-gelatin (PAAGH), is presented. Kinetic curves of isothermal and non-isothermal dehydration of hydrogels were measured using thermogravimetric methods. The kinetic complexity of the dehydration process was analyzed by different methods: integral, differential, Kissinger-Assakura-Sanura (KAS), and Vyzovkin’s method. The complex kinetics of dehydration of hydrogels was described by a series of new kinetic models: distribution apparent energy activation model (DAEM), Webull’s distribution of reaction times, the dependence of the degree of conversion (α) on the temperature which is defined by the logistic function, coupled single step-approximation and iso conversion curve. Procedures were developed for calculating the function of the density distribution of probability (g(Ea)) of apparent activation energy (Ea). The relationship between the phase state of the absorbed water in hydrogel and the form of function of distribution of apparent Ea and kinetic parameters of dehydration was analyzed.

1. Introduction

The existence of hydrogels dates from the 1960s when Wichterle and Lim first hypothesized the possibility of using the cross-linked hydrophilic polymer poly(2-hydroxyethyl methacrylate) (PHEMA) for contact lenses [1]. Hydrogels are three-dimensional (3D) cross-linked structures that are mainly composed of hydrophilic homopolymers or copolymers connected by chemical or physical bonds, which have the ability to absorb a significant amount of water or other biological fluids, without dissolving or losing their structural integrity (swelling) [2]. They represent a unique class of macromolecular materials with specific physicochemical properties: the ability to retain a large amount of water solution in their structure (from 20% to several 10,000% in relation to the weight in the dry state), the ability to possess a high degree of flexibility similar to natural tissue and exhibit physical, chemical and mechanical stability in the swollen state [3]. The amount of water that these materials can absorb and thus increase their initial volume is fascinating and exceeds by several orders of magnitude the amount of aqueous solution that other gels can. They are also called “smart”, “intelligent”, “stimuli-responsive” or “environmental-sensitive” and attracted great attention in recent years [4].

Due to the large amount of water that they can absorb, hydrogels are often called superabsorbent materials. The hydrogels’ ability to absorb huge amounts of water results from the presence of side hydrophilic functional groups on their polymer chains. On the contrary, their resistance to dissolution is a consequence of crosslinking between the polymers’ chains. The water inside the hydrogel allows free diffusion of dissolved molecules, while the polymer acts as a matrix that holds the water [5].

Hydrogel in the dry state is called xerogel, and it is a solid and brittle material that shows the typical properties of solid substances that are the result of cross-linking. Hydrogels are called permanent or chemical gels when they are covalently crosslinked, or physical gels when entanglements, weak associations of the van der Waals type, or hydrogen bonds formed a network. Crosslinks between polymer chains create the structure of the hydrogel network and give them physical integrity [6, 7]. Hydrogels can be natural, synthetic, semi-synthetic, or their combinations. Hydrogels can give a response i.e. react to external stimuli (temperature, nature of the solvent, pH value, ionic strength, electric or magnetic field action, light, biological agents, radiation, etc. through notable changes in their macroscopic properties, which are most frequently manifested through the changes in volume [8, 9]. Depending on the design of the polymer network, these volume changes can occur continuously or discontinuously over a certain range of stimulus changes or at a certain critical value of the stimulus.

Due to their specific properties, hydrogels become one of the upcoming classes of materials that have found wide applications in various fields. Among the numerous areas of application of hydrogels, the ones in the field of medicine and pharmacy are particularly significant [10, 11], especially in controlled and targeted drug release [12, 13], regenerative medicine and tissue engineering, contact lenses, biosensors, etc. [14]. Beyond their biomedical applications [15, 16], hydrogel represents an ideal basis for new materials for applications in biotechnology, environmental protection, agriculture, agrochemistry, horticulture, cosmetics, as superabsorbent in hygiene products, packaging materials for storing food, textile materials for special purposes, in sensor materials, etc. In recent times, the applications of hydrogels and hydrogel-derived materials (HDM) presents emerging novel materials platform in electrochemical energy conversion systems, including metal-air batteries, fuel cells, and water-splitting electrolyzers, due to their specific and tailorable physicochemical properties [17].

The broadest functional applications of hydrogels are founded on their distinguishing capability to reversibly absorb (swell) and release (dehydrate) water. Exactly because of that, and since water removal and uptake includes fundamental principles of physics, knowledge of the mechanism and kinetics of both swelling and dehydration of hydrogels is of the utmost importance, in order to be able to optimize their efficiency in particular applications. Despite the fact that the phenomenon of swelling of hydrogels, including its mechanism and kinetics, is among the most studied from a fundamental aspect in the field of hydrogels [18, 19], on the contrary, however, the same cannot be said for the dehydration of hydrogels. Hydrogel dehydration itself has been much less studied, since the analysis of the dehydration process in hydrogel materials is an extremely difficult task requiring complex approaches [20], and in particular there is a very limited number of published papers in the literature that deal with the issue of the mechanism and kinetics of dehydration.

Since the discovery and first application of hydrogels are related to contact lenses, and the state of hydration, that is dehydration, is very important for this application, there are a number of works dealing with the dehydration of contact lenses. However, on the one hand, they mostly relate to bio-medical and physiological aspects, [21, 22, 23] and they primarily relate to silicone lenses [24], while physicochemical studies of the kinetics of dehydration are much rarer.

In vitro dehydration of new and worn contact lenses by using different types of hydrogels including one silicone-hydrogel was investigated. It was found that contact lenses based on hydrogels exhibited different dehydration behavior, the different behavior during dehydration that was manifested corresponds to different rates of dehydration and stages of dehydration [21].

The dehydration process in hydrogels used in ophthalmology as intraocular lenses were investigated by Chamerski et al., using equilibrium-swelled hydrogels in deionized water and saline solution. Studies of the dehydration process were carried out by use of gravimetric analysis, Fourier-Transform Infrared, and Positron Annihilation Lifetime Spectroscopy. Obtained results revealed changes in hydrogen bonding structure and free volume holes induced by saline solution ingredients. Observation of the process at the molecular level has given information about water transport in the free volume holes on the basis of changes in hydrogen bonds and demonstrated a more filled and hydrogen-bonded structure in the case of fluid containing inorganic compounds. More stable network formation can be explained by the influence of such compounds on changes in water binding, and thus in internal structure transformation toward its improvement [20].

The dehydration kinetics of the poly(vinyl alcohol) (PVA) hydrogel aimed at wound dressing materials has been investigated. The effects of the thickness and initial water content amount on the dehydration process were evaluated. The results showed that the dehydration rate of the PVA hydrogel wound dressing has an inverse dependency on the hydrogel’s thickness while the initial water content has no significant effect. The authors developed a mathematical model on the basis of the diffusion mechanism to predict the dehydration process of the wound dressings and the obtained results confirmed that the main phenomenon governing the dehydration of the wound dressings is diffusion [25].

Hawlader et al. [26] used the one-dimensional diffusion model to describe the transfer of heat and mass from the wet to the dry region of the hydrogel during dehydration. Water diffusion during the dehydration of polyacrylamide (PAAm) hydrogel was investigated by Roques et al. [27]. Based on the obtained results, they suggested a mathematical model that was able to well describe the diffusion kinetics of water during hydrogel dehydration. Kept et al. examined the applicability of different kinetic models for mathematically describing the kinetics of hydrogel dehydration/ drying [28]. The research group of Peckan developed a fluorescence technique for in situ monitoring gelation, swelling, and dehydration processes of various hydrogel based on determining changes of fluorescent spectra of gels formed by solution-free copolymerization, [29], dehydration of κ-carrageenan gels at different temperatures [30], and dehydration of PAAm hydrogels with various cross-linking degrees [31].

The kinetics of non-isothermal dehydration (NIT) of polyacrylic acid hydrogels (PAAH) has been investigated using various kinetic methods such as Kissinger, Coats-Redfern, Van-Krevelen, and Horowitz-Metzger [32, 33]. Kinetic of non-isothermal dehydration of a silver nanocomposite hydrogel of poly(acrylic acid) grafted onto salep which was not possible to describe the complete dehydration process by a single mechanism have been investigated [34].

Dehydration of chitosan fibers-enhanced gellan gum hydrogel and chitosan fibers-enhanced polysaccharide hydrogels have been investigated and established two distinct kinetics stages: diffusion and nucleation [35, 36]. Ma et al. investigated dehydration kinetics of poly(vinyl alcohol)/poly(vinyl pyrrolidone)/hydroxyapatite composite hydrogel and found that consists of water diffusion through hydrogel network and evaporation [37].

Non-isothermal dehydration of equilibrium swollen PAAH [38, 39], poly(acrylic-co-methacrylic acid) (PAMAH) [38], and poly(acrylic acid)-g-gelatin (PAAGH) [33] hydrogel has been investigated. The kinetics of isothermal dehydration of equilibrium swollen PAAH [40] and PAMAH [41] was presented. The comparative kinetic study of NIT and isothermal dehydration (IT) of PAAH was performed [42, 43], as well as on IT kinetics of water evaporation and PAAGH [44]. The fluctuating (changing) structure of hydrogels during dehydration has been found [42].

Belich et al. extensively investigated the dehydration of alginate hydrogels including various approaches. The effects of operational procedures and other parameters, calcium, and alginate concentration, and the addition of biopolymer co-solutes on water evaporation from alginate gel beads have been investigated [45]. The non-isothermal water evaporation for a series of alginate-based gel beads was performed aimed at understanding the state of water. The observed shoulders at high temperatures of thermogravimetric curves (TG) have been ascribed to evaporation of water molecules [45]. The investigations of water evaporation from alginate gel beads showed that calorimetric approach to hydrogel matrix release properties can be used as the predicting tool for the diffusion of solvents [46]. The quasi-isothermal dehydration of thin films of pure water and aqueous sugar solutions was investigated. The effect of sugar on the dehydration process was evaluated. It was established that the trehalose molecules slow down the diffusion of water molecules through the substrate [47].

Jovanovic and Adnadjevic et al. investigated NIT of Ca-alginate hydrogel for the first time. The dependence of apparent activation energy on the degree of dehydration was determined by Friedman’s differential iso conversion method. It was shown for the first time that the kinetics of NIT of Ca-alginate hydrogel can be successfully described entirely by the statistical model of hydrogel dehydration. The existence of two-phase states of water absorbed on the Ca-alginate hydrogel was confirmed and related to the observed changes in the values of kinetic parameters, at a constant heating rate, with temperature [48].

The effects of e microwave irradiation on hydrogel DH were investigated by Adnadjevic and Jovanovic and co-workers. The isothermal kinetic of water evaporation and PAAGH dehydration were investigated under microwave heating conditions (MWH). The IT kinetic curves of water evaporation and hydrogel DH have been mathematically described complete by the Polanyi–Winger equation. The resonant transfer of a certain energy amount from the reaction system to the libration vibration of molecules of water is suggested as the mechanism of water molecules’ activation, both for evaporation and dehydration [49].

The main goal of this chapter is to get a deeper insight into the essence of newly established kinetic models of hydrogel dehydration in order to expand knowledge about the mechanism and kinetics of hydrogel dehydration.

2. Experimental part

2.1 Hydrogel synthesis

Poly(acrylic acid) hydrogel (PAAH), which has been applied for this investigation was synthesized by a procedure based on the simultaneous radical polymerization of acrylic acid and cross-linking of the formed poly(acrylic acid), according to the general procedure described in details [50]. Poly(acrylic acid-co-methacrylic acid) hydrogel (PAMAH) was synthesized by a procedure of radical co-polymerization of acrylic acid and methacrylic acid (1:1 mol ratio) and cross-linking of the polymers formed, using the thoroughly described procedure [51]. Synthesis of poly(acrylic acid)-g-gelatin hydrogel (PAAGH) was described in detail [44].

2.2 Preparation of equilibrium swollen hydrogels

Synthesized hydrogels have been washed-out, as described in hydrogel synthesis procedures, and subsequently air-dried in laboratory oven under a defined temperature regime until constant mass. The obtained products (xerogels) were stored in a vacuum exicator until use. With the aim to evaluating dehydration kinetics, the xerogels samples were grounded and allowed to swell (24 h) in bidistilled water at ambient temperature to ensure to reach the equilibrium state. The equilibrium swollen samples were undertaken, and excess water was drained and wiped with tissue paper to remove surface water immediately before the dehydration experiment.

2.3 Thermogravimetric measurements

2.3.1 Non-isothermal thermogravimetric measurements

NIT curves were recorded by a Du Pont thermogravimetric analyzer TGA model 9510. The analyses were performed with 20–25 ± 1 mg samples of equilibrium swollen hydrogel in platinum pans under nitrogen atmosphere, N₂ purity 5.0, and gas flow rate of 10 mL min⁻¹. Samples were heated in the temperature range from ambient temperature to 500 K with different heating rates from 5 to 30 K min⁻¹.

2.3.2 Isothermal thermogravimetric measurements

The isothermal mass loss experiments were carried out using a TA Instruments-SDT simultaneous TGA-DSC thermal analyzer model 2960. The analysis was performed with 20 ± 1 mg samples of equilibrium swollen hydrogel in platinum pans under a nitrogen atmosphere at a gas flow rate of 10 ml min⁻¹. Isothermal runs were performed at nominal temperatures of 306 K, 324 K, 345 K, and 361 K. The samples were heated from the start to the selected dehydration temperature at a heating rate of 300 K min⁻¹ and then held at that temperature for a given reaction time.

2.4 Calculation of the dehydration degree

The degree of dehydration was calculated as:

$$\alpha =\frac{m_0-m}{m_o-m_t}$$E1

where m₀, m, m_f refers to the initial, actual, and final masses of the sample at a thermogravimetric curve.

2.5 Mathematical consideration

All the experimental data fitting has been performed by using Origin Program and Levenberg-Marquardt method. The coefficient of determination R² is used as an error function to minimize the error distribution between the experimental data and model.

3. Results and discussion

3.1 Kinetics of non-isothermal dehydration of equilibrium swollen PAAH

Kinetics of non-isothermal dehydration of equilibrium swollen PAAH have been investigated in detail in the work of Adnadjevic et al. [39]. Figure 1 shows: (a) TG curves and (b) conversion curves (α = α (T)_vh) of NITD of PAAH experimentally obtained at different heating rates (v_h).

Figure 1.

TG curves (a) and conversion curves (b) of NIT of PAAH at different heating rates: (black line)—5^o/min; (red line)—10^o/min; (blue line)—20^o/min.

The conversion curves, at all of the investigated heating rates, are asymmetric by shape. The increase in the heating rates leads to the increase in the values of the inflection temperature (T_p), the final temperature (T_f), as well as in the degree of asymmetry.

Since the conversion curves exhibited complex shape, Friedman’s differential iso-conversional method [52] was applied to determine the dependencies of E_a,α and ln[A_αf(α)] from the degree of dehydration. The dependency E_a,α on α is shown in Figure 2, whereas Figure 3 shows the dependency ln[A_αf(α)] on α.

Figure 2.

The dependence of apparent E_a on α [39].

Figure 3.

The dependence of ln[A_αf(α)] on α [39].

The values of kinetics’ parameters decrease in a complex manner with increasing α. The complex change in the value of the kinetic parameters with the α indicates that the NIT of poly(acrylic acid) hydrogel is a kinetically complex process. By analyzing the shape of dependence E_a,α and ln[A_αf(α)] on α, it can be concluded that there are two characteristic shapes of change on the dependence curves E_a,α and ln[A_αf(α)] on α. For α ≤ 0.2 the values of E_a,α, and ln[Aαf(α)] decrease abruptly and almost linearly with an increase in α, while for α ≥ 0.2 the increase in α leads to a slow decrease in the values of E_a,α and ln[A_αf(α)]. Between the values of ln[A_αf(α)] and E_a,α there is a relationship, the so-called compensation effect defined by the relation:

$$\mathit{ln}\left[A_{\alpha }f\left(\alpha \right)\right]=-4.14+0.38E_{a,\alpha }$$E2

The existence of characteristic shape of changes in the value of kinetic parameters with α indicates that the phase state of the absorbed water in the hydrogel change with α, and that absorbed water exists in two different phase states. Most frequently for the mathematical description of complex chemical reactions, the so-called distributed activation energy model (DAEM) is used. DAEM is based on the assumption that a complex chemical reaction can be modeled as an infinite number of irreversible first-order paralleled reactions with the different values of Ea and A [53]. According to DAEM, the degree of dehydration can be calculated based on the eq. [54] (3):

$$\propto =1-\underset{0}{\overset{\infty }{\int }}\mathit{exp}\left[-\underset{0}{\overset{t}{\int }}\mathit{\text{Aexp}}\left(-\frac{\mathit{Ea}}{\mathit{Rt}}\right)\mathit{dt}\right]g\left(E_a\right)\mathit{dE}$$E3

where g(E_a) is the density distribution probability of apparent activation energies. The density distribution probability of apparent activation energies is defined by the expression:

$$g\left(E_a\right)=\frac{\mathit{d\alpha }}{d{E}_a}$$E4

The expression (4) can be written in the form (5):

$$g\left(E_a\right)=\left(\frac{\mathit{d\alpha }}{\mathit{dT}}\right)\left(\frac{\mathit{dT}}{{\mathit{dE}}_a}\right)$$E5

using which, based on the knowledge of the dependence ${\left(\frac{\mathit{d\alpha }}{\mathit{dT}}\right)}_{v_h}$and $\left(\frac{\mathit{dT}}{{\mathit{dE}}_a}\right)$at a certain heating rate, g(E_a) can be calculated at that heating rate. Figure 4 shows the calculated values $g\left(E_a\right)$.

Figure 4.

The effect of v_h on the shape of g(E_a) [39].

The density function of the probability distribution of apparent activation energies is in the shape of a narrow symmetrical peak with a maximum probability density, g(E_a)_max = 0.067 at E_a = 23.9 kJ/mol. The shape of g(E_a) is independent of the heating rate of the system. Therefore, the kinetic complexity of NITD of absorbed water on PAAH is a consequence of the distribution of reactivity of water molecules in the hydrogel, due to which there is a change in E_a and lnA with a change in the degree of dehydration.

3.2 Kinetics of isothermal dehydration of equilibrium swollen PAAH

Kinetics of IT of PAAH has been investigated in detail [40]. TG and conversion curves of IT of PAAH at different temperatures are shown in Figure 5a and b.

Figure 5.

(a) TG curves of ITD of PAAH at different temperatures (black line—306 K; (red line)—324 K; (blue line)—346 K, green line—361 K) and (b) conversion curves of ITD of PAAH at different temperatures [40].

The conversion curves, at all of the investigated temperatures, have a similar asymmetric sigmoidal shape. With the increase in temperature, the degree of asymmetry of the curves increases, while the duration of dehydration shortens. In order to determine the degree of kinetic complexity of ITD using the integral iso conversion method [55] the shape of dependence of E_a, and ln[A/g(α)] was determined. Figure 6a and b show the dependence of E_a,_α and ln[A/g(α)].

Figure 6.

Dependence of apparent E_a (a) and (b) ln[A/g(α)] on α.

The values of kinetics parameters of isothermal dehydration of adsorbed water in PAAH increase complexly with an increase in the value of α. The complex change in values of kinetics parameters indicates that dehydration is a kinetically complex process.

Assuming that: (a) the degree of IT at a certain moment of time (α(t)_T) is proportional to the probability of dehydration of the dehydrating centers at that moment of time (p(t)_T) = (α(t)_T) Eq. (6); (b) the probability of dehydration of the dehydrating centers at that moment of time can be described by the Weibull function of distribution of the probability of the reaction times of the dehydrating centers [56], given with Eq. (6) at follows (7):

$$p{\left(t\right)}_T=1-\mathit{exp}\left[-{\left(\frac{t}{{\eta }_T}\right)}^{{\beta }_T}\right]$$E6

$$a{\left(t\right)}_T=1-\mathit{exp}\left[-{\left(\frac{t}{{\eta }_T}\right)}^{{\beta }_T}\right]$$E7

Figure 7 shows a comparison of experimental data (symbols) and calculated values of α (solid line) at T = 306 K and 361 K as an example.

Figure 7.

Experimental conversion curves of PAAH at: (□) 306 K and (○) 361 K.

Based on the results shown in Figure 7, it can be concluded that the conversion curves of IT absorbed water in PAAH can be completely described mathematically by Eq. (8)

$$a{\left(t\right)}_T=1-\mathit{exp}\left[-{\left(\frac{t}{{\eta }_T}\right)}^{{\beta }_T}\right]$$E8

Table 1 shows the effects of temperature on parameters β_T and η_T of the Weibull distribution function.

Temperature, (K)	βT	ηT (min)	R^2
306	1.41	79.37	0.9960
324	1.49	23.42	0.9964
345	1.58	8.37	0.9989
361	1.94	5.23	0.9993

Table 1.

The effects of temperature on parameters β_T and η_T of the Weibull distribution function.

The values of the shape parameter increase with increasing temperature in accordance with Eq. (9):

$$\beta \left(T\right)=10.85\mathit{exp}\left[-\left(\frac{635}{T}\right)\right]$$E9

In contrast, the values of the scale parameter ŋ decrease with the increase in temperature in accordance with Eq. (10):

$$\eta \left(T\right)=3.70x{10}^8\mathit{exp}\left[-\left(\frac{T}{20}\right)\right]$$E10

Complete description of isothermal conversion curves by Eq. (8) and knowledge of the functional dependence of t on E_a,α (Eq. (5)) enables the calculation of g(E) (Eq. (11)).

$$g\left(E_a\right)=\left|\left(\frac{\mathit{d\alpha }}{\mathit{dt}}\right)\left(\frac{\mathit{dt}}{{\mathit{dE}}_a}\right)\right|$$E11

Based on the mathematical dependencies: α = f(t)_T the dependence Ea,α = Ea,α (α)_T can be transformed into the dependence E_a,α = E_a,α (t)_T, i.e. the dependence t = t(E_a,α)_T. Numerical analysis revealed that the t = (E_a,α)_T can be mathematically described by Eq. (12).

$$t=t_{o,T}\mathit{exp}\left({\epsilon }_{o,T}{E}_{a,\alpha }\right)$$E12

where t_o,T and ε_o,T are fitting coefficients. The effect of temperature on the value of fitting coefficients is shown in Table 2.

Temperature, T (K)	t0,T (min)	ε0,T (mol kJ^−1)	R^2
306	0.00139	0.23925	0.9999
324	0.00074	0.22634	0.9999
345	0.00048	0.21340	0.9999
361	0.00184	0.17372	0.9999

Table 2.

Effect of temperature on the values of fitting coefficients t_0,T and ε_0,T for the ITD of PAAH.

Since:

$${\left(\frac{\mathit{d\alpha }}{\mathit{dt}}\right)}_{v_h}=\left[\left(\frac{{\beta }_T}{v_h{\eta }_T}\right)\right]{\left[\left(\frac{T-T_o}{v_h{\eta }_T}\right)\right]}^{{\beta }_T-1}\mathit{exp}{\left[-\left(\frac{T-T_o}{v_h{\eta }_T}\right)\right]}^{{\beta }_T}$$E13

and

$$\frac{\mathit{dt}}{{\mathit{dE}}_a}=t_{o,T}{\epsilon }_{o,T}\mathit{exp}\left({\epsilon }_{o,T}{E}_{a,\alpha }\right)$$E14

by applying Eq. (11) p(E_a) was calculated.

Figure 8 shows calculated values of p(E_a).

Figure 8.

Effect of temperatures on the shape of the calculated density distribution function of activation energies (p(E_a)_T [40].

The g(E_a) under IT has an asymmetric bell shape, invariant with temperature, which clearly shows a maximum of g(E_a) = 0.122 at E_a = 45.8 kJ/mol. The determined temperature dependence of the g(Ea) function indicates that it is right related to the distribution of reactivity of dehydration species in absorbed water at a given type of their activation. The calculated values of g(E_a), according to the shape and the values of the characteristic quantities [40], differ significantly from the values of g(E_a) obtained when calculating the NIT of absorbed water in PAAH. The established difference in the shape and values of the characteristic quantities g(E_a) during non-isothermal and isothermal heating clearly indicates a different mechanism of activation of reaction species during different modes of heating the system.

3.3 Kinetics of nonisothermal dehydration of PAMAH

Kinetics of NIT of PAMAH has been investigated in detail [38]. Figure 9 shows non-isothermal conversion curves of absorbed water in PAMH at different heating rates.

Figure 9.

Conversion curves of NIT of PAMAH at different heating rates (dots) and their mathematical fittings using logistic function (line): (a) v_h = 5 K min⁻¹; (b) v_h = 10 K min⁻¹; (c) v_h = 20 K min⁻¹; (d) v_h = 30 K min⁻¹ [38].

As in the case of NIT PAAH conversion curves, at all investigated heating rates, they have a complex asymmetric shape. As the heating rate increases, the degree of asymmetry of the conversion curves increases and also the temperature interval within which DH takes place is diminished. Since the conversion curves of DH PAMH cannot be successfully fitted by the most commonly used reaction models characteristic of reactions with the participation of a solid phase [57] and the models of kinetics of DH PAAH shown above. Naya et al. [58] and Cao [59] used the logistic function (Eq. (15)) for the mathematical description of the complex non-isothermal kinetics of polymer degradation:

$$\alpha {\left(T\right)}_{v_h}=1-\left[\frac{w}{1+\mathit{exp}\left(a+\mathit{bT}\right)}\right]$$E15

where w is experimentally achieved the maximum degree of conversion, and a and b are the parameters of the logistic function.

Bearing that in mind, the experimental DH conversion curves obtained at different heating rates were fitted by Eq. (15). Figure 9 (solid lines) shows the calculated conversion curves at different heating rates. As can be seen from Figure 9, the DH conversion curves of PAMH fitted with the logistic function completely mathematically describe the experimental DH curves. Table 3 shows the effect of the heating rate on the parameters of the logistic function.

Heating rate (K min^−1)	W	a	b (K^−1)	R^2
5	0.98	−33.59	0.10	0.9992
10	0.98	−25.08	0.07	0.9988
20	0.97	−23.70	0.06	0.9984
30	0.96	−24.62	0.06	0.9981

Table 3.

Effect of heating rate on logistic function parameter values and corresponding R² values between experimental data and the logistic regression model prediction.

The values of parameters w and b decrease with the increase in heating rate, while the value of parameter a changes complexly with the decrease in heating rate. There is no data in the literature about the physical meaning of applying the logistic function to describe the kinetics of dehydration. Bearing in mind that the logistic function is mathematically very similar to the Prout-Tomkins’ eq. [60, 61], which was developed to describe the isothermal degradation of KMnO₄. A comparative analysis of the connection between the logistic function and the Prout-Tomkins equation was performed. Based on that analysis, it was concluded that there are functional connections between the parameter of the logistic function (w,b) and the rate constants of nucleus branching (k_b) and nucleus termination (k_t), which are described by expressions (16) and (17).

$$w_{v_h}=\frac{k_{b,v_h}}{k_{T,v_h}}$$E16

$$b_{v_h}=\frac{k_{b,v_h}}{v_h}$$E17

Table 4 shows the effect of the heating rate at k_b and k_t.

vh (K min^−1)	kb (min^−1)	kT (min^−1)
5	0.505	0.515
10	0.716	0.731
20	1.316	1.356
30	1.974	2.047

Table 4.

Effect of heating rate at k_b and k_t.

At all of the investigated heating rates, the value of k_t and k_b increases with the increase in heating rate the values of both k_t and k_b increase. The established functional relationships between the parameters of the logistic function and value of k_t indicate that the logistic function can describe kinetically complex reactions with 3 elementary kinetic stages: nucleation, autocatalysis, and termination.

The established possibility of a mathematical description of the kinetics of NITof PAAGH allows assuming a new model of the mechanism of dehydration of adsorbed water from the hydrogel. According to that model, the DH of absorbed water does not take place by the immediate release of individual water molecules from the absorbed phase but takes place in three well-defined stages: nucleation (formation of clusters of water molecules of critical dimensions), autocatalytic growth of the formed nuclei and termination (decrease in the concentration of nuclei due to the completion process). Bearing in mind the fluctuating structure of the hydrogel, formation of critical water-dehydrating nuclei and their dehydration leads to the collapse of the existing fluctuating structure of the hydrogel. The newly formed fluctuating structure of the hydrogel enables the formation of a large number of dehydration nuclei (autocatalytic growth), which leads to an abrupt acceleration of the dehydration process. At high degrees of dehydration, the rate of dehydration decreases due to the decrease in the number of dehydration nuclei and the transformation of the hydrogel into a xerogel.

3.4 Kinetics of non-isothermal dehydration of PAAGH

Kinetics of non-isothermal dehydration of PAAGH has been investigated by using distributed activation energy model [33]. The TG curves of the NIT of PAAGH are shown in Figure 10.

Figure 10.

Thermogravimetric curves of NIT of PAAGH at v_h: 5 K min⁻¹ (thick line), 10 K min⁻¹ (spaced line), 15 K min⁻¹ (dotted line), and 20 K min⁻¹ (spaced dotted line) [33].

TG curves NIT of PAGH have a complex asymmetric shape (concave down). With an increase in v_h, there is an increase in the degree of symmetry and a widening of the temperature interval in which the DH process takes place. In order to determine the degree of kinetic complexity of dehydration using Vyzovkin’s method [62] and Kissinger-Akahira and Sunozze (KAS) method [63, 64], the dependences of E_a and lnA on α were determined. The dependence of E_a on α is shown in Figure 11 and lnA on α in Figure 12.

Figure 11.

Dependence of E_a,α on α calculated by Vyazovkin’s (circle) and KAS (square) methods [33].

Figure 12.

Dependence of ln[A_α/min⁻¹] on α [33].

The results shown in Figures 11 and 12 indicate that both methods of calculating kinetic’s parameters lead to identical dependences of E_a and lnA on α (within the limits of experimental error). The complex shape of dependence of E_a and lnA on α confirms the kinetic’s complexity of the DH process. On the Figures 11 and 12, the 2 characteristic shapes of change of E_a and lnA from α can be noticeably observed within the range α ≤ 0.16 values of E_a,α and lnA_α decrease almost linearly with increasing α. On the contrary, at α ≥ 0.15, increasing the value of α leads to a slow decrease of E_a from 40.5 to 24 kJ/mol, that is, lnA from 13 to 5 min⁻¹. Between the values E_a,α and lnA_α, there is a linear correlation relationship (compensation effect) which is mathematically described by the Eq. (18)

$$\mathit{ln}\left[A_{\alpha }\right]=-6.43+0.47E_{a,\alpha }$$E18

In the work of Šimon [65], the so-called single-step approximation for describing the kinetics of chemical reactions and physical-chemical processes that take place in a solid state. In accordance with it, the rate of solid-state reaction can be described by Eq. (19) where k(T) is the reaction rate constant and f(α) is the function that describes the reaction model of the reaction.

$$\frac{\mathit{d\alpha }}{\mathit{dt}}=k\left(T\right)f\left(\alpha \right)$$E19

In the case of kinetically complex reactions, Eq. (19) transforms into Eq. (20):

$$\frac{\mathit{d\alpha }}{\mathit{dt}}=A_{\alpha ,T}\mathit{exp}\left(-\frac{E_{a,\alpha }}{{\mathit{RT}}_{\alpha }}\right)f{\left(\alpha \right)}_T$$E20

If it takes place in NIT conditions, Eq. (20) is transformed into Eq. (21):

$${\left(\frac{\mathit{d\alpha }}{\mathit{dT}}\right)}_{v_v}=\frac{A_{\alpha }\mathit{exp}\left(-\frac{E_{a,\alpha }}{{\mathit{RT}}_{\alpha }}\right)}{v_v}f{\left(\alpha \right)}_T$$E21

Eq. (21) enables to calculate the dependence of k(α) and f(α) on temperature based on the dependence of Ea and lnA on α and on the experimentally calculated dependence using the expressions (22) and (23).

$$k{\left(\alpha \right)}_{v_h}=A_{\alpha ,v_h}\mathit{exp}\left(-\frac{E_{a,\alpha ,v_h}}{{\mathit{RT}}_{\alpha ,v_h}}\right)$$E22

$$f{\left(\alpha \right)}_{v_h}=\frac{v_h{\left(\frac{\mathit{d\alpha }}{\mathit{dT}}\right)}_{v_h}}{k{\left(\alpha \right)}_{v_h}}$$E23

Figure 13 shows the dependence of k(α) on α.

Figure 13.

Dependence of kvhα on α at different v _h: 5 K mol⁻¹ (filled square), 10 K mol⁻¹ (circle), 15 K mol⁻¹ (times symbol), 20 K mol⁻¹ (empty square) [33].

The values of k(α) nonlinearly decrease with the increase in α, for each of the investigated v_h. At a particular value of α (iso conversion), k(α) increases with an increase in v_h, which designates that k(α) decreases with the increase in temperature. The functional dependence of the reaction model of dehydration on α is shown in Figure 14.

Figure 14.

Function f(α) at different v_h: 5 K mol⁻¹ (filled square), 10 K mol⁻¹ (circle), 15 K mol⁻¹ (times symbol), 20 K mol⁻¹ (empty square).

On all of the investigated v_h f(α) has a similar shape and gradually increases with the increase of α up to α = 0.85. A further increase in α leads to a sharp decrease in the value of f(α). The functional relationship between f(α) and α can be mathematically described by the expression (24).

$$f{\left(\alpha \right)}_{v_h}=m_{v_h}{\alpha }^n$$E24

Table 5 shows the effects of v_h on m and n, where n and m are the parameters of expression (24).

vh (K min^−1)	n	m
5	1.97	1.04
10	1.93	1.02
15	1.94	1.02
20	1.84	1.09

Table 5.

Effect of v_h on values of m and n.

An increase in v_h leads to a decrease in the value of the parameter n, while the parameter m changes slightly. The independence of the shape of f is f(α) from v_h indicates that the basic cause of the complex nature of dehydration kinetics is the change in k(α) caused by changes in the structure of the absorbed phase caused by DH. The established shape of the dependence of k(α) and f(α) also enables explained the phenomenon of the decrease in the value of v_h DH with T, that is, α. That happens during the investigated non-isothermal dehydration experiments in all cases when the rate of decrease in the value of k(α) is higher than the rate of increase in the value of f(α).

4. Conclusion

The kinetics of DH of all of the investigated hydrogels is of a complex nature. The complex kinetics of dehydration of hydrogels is a consequence of the fluctuating structure of the hydrogel, the phase state of the absorbed water, and the thermal activation of the hydrogel. By applying DAEM, isothermal and non-isothermal kinetics of PAAH dehydration were fully described. Procedures for determining the shape of g(E_a) have been developed. It was shown that the method of activation (isothermal and non-isothermal heating) of the system affects the shape of g(E_a) and the value of the kinetic parameters. The logistic model fully describes the kinetics of NIT dehydration of PAMAH. The functional relationship between the parameters of the logistic function and the constants k_b and k_t was determined. It was shown that the rate of nucleation of dehydration centers is the kinetically limiting stage of PAMAH dehydration rate. By applying the coupled single step-approximation and calculating the iso conversion dependence of Ea and lnA on α, a new kinetic model was developed to describe the complex kinetics of PAAGH dehydration. It was found that the rate constant of dehydration at all investigated v_h decreases with the increase in temperature, while the values of the function parameters of the reaction model increase. The decrease in the value of the dehydration rate constant with temperature is associated with the change in the rate of nucleation.

Acknowledgments

The present investigations were supported by The Ministry of Education, Science and Technological Development of the Republic of Serbia, under Contracts No: 451-03-68/2022-14/200051; 451-03-68/2022-14/200146; 451-03-47/2023-01/200051.