**6. Prospects for new asymmetric reaction using crystal**

New aspects of chemistry are deployed by effectively utilizing the unique nature of crystals. In this chapter, a new approach to asymmetric chemistry using chiral crystals found in our research groups is introduced. We succeeded in transferring the chirality expressed by chiral crystallization of achiral substrate to products in a variety of reactions in homogeneous systems. Stereoselective reaction using chiral crystals was limited to the topochemical solidstate photoreaction, and the reactions expanded in a homogeneous system were able to achieve the highly stereoselective asymmetric reaction. Some readers may think that chiral molecules possessing memory effects introduced in this chapter are like acrobats using a very narrow energy band. Figure 19 shows an energy diagram for racemization of chiral molecules that express chirality by spontaneous crystallization or by Crystallization-Induced Enantiomer Transformation (CIET). We can effectively use molecules with

Fig. 19. Energy diagram for racemization of axially chiral and achiral materials available for asymmetric synthesis via spontaneous chiral crystallization

activation free energy of racemization from 10 to 30 kcal mol-1 for chiral memory. It is not an unusual technique, but a rather typical approach that can be widely utilized in asymmetric reactions. We have found many more applications to this method using chiral memory; we will present them in future work. In addition, this method could be applied to the diastereoselective reaction using chiral memory derived from CIDT. These findings suggest that the molecular information in the crystal can be widely applied to a variety of chemistry, in addition to organic asymmetric reaction.


**5. Conclusions**

Our applied research demonstrates that National Park planning needs to be changed, facing the inconsistency of many aspects, both in national and in local framework:

In comparison with traditional approach of separation between nature sanctuaries and human activities, the new plan produces a paradigm shift, to induce a fertile relationship among all the components of local communities and to define a positive environmental restoration.

Based on the plan scenario, we will wait for long-term consequences in planning and actions in both regional land use management and environment protection and raise the awareness of the advantages of a protection model based local resource development.

**Author Contributions:** All authors have contributed to the whole work. In particular, for writing—original draft, review and editing, the paragraphs were written as follow: 1. Introduction, M.C. and D.R.; 2.1. Circular Development and Circular Metamorphosis: A Theoretical Framework for the Plan of the National Park, M.C.; 2.2. The National Park of Lucania Apennines, Valdagri and Lagonegrese: A Case Study, M.C.; 2.3. Evidence in Local Context, D.R.; 2.4. The Structural Analysis: Current Status and Ongoing Priority Projects, D.R.; 2.5. The SWOT Matrix as Synthesis of Targeted Analysis, M.C. and D.R.; 3.1. The Inner Area Policy as a Beacon for Development, M.C.; 3.2. Interpretation for Drawing a Regeneration Flagship Project, D.R.; 4. Discussion: How to Balance Environment Preservation and Community Development? A 'Cluster Approach' for Circular Regeneration, M.C. and D.R.; 5. Conclusions, M.C. and D.R. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** Authors' research field activities and results were produced in a multisectoral project group composed of many experts and stakeholders, as follow: National Park Authority's management group and technical o fficers; the research group, composed by RPA spa with Salvatore Corlianò and Dino Bonadies for general aspects, Sandro Amorosino for law aspects, Evagreen Ltd. with prof. Giuseppe Bazan and Dino Erdfeld for environmental aspects, Emanuela Coppola, Remo Votta, Valeria Mauro for anthropic aspects.

**Conflicts of Interest:** The authors declare no conflict of interest.


*2.2. Electric Vehicle Charging Model*

The probability of an individual EV traveling a distance *d* can be represented by the logarithmic normal distribution function [20]

$$h(d,\mu,\sigma) = \frac{1}{d\sqrt{2\pi\sigma^2}}e^{-\frac{\left(\ln d - \mu\right)^2}{2\sigma^2}}\tag{2}$$

where μ and σ are the mean and standard deviation, respectively.

According to the traveling distance *d* of the EV, the remaining capacity can be calculated:

$$SOC = (1 - \frac{d}{d\_m}) \times 100\% \tag{3}$$

where *dm* is the maximum travel distance.

According to [21], the charging start time can be modeled as a normal distribution function with specified parameters, and we represent the time when the EV starts charging as a set.

$$T\_{start}^{k} = \begin{bmatrix} \underline{T\_{start}^{k}} \, \overline{T\_{start}^{k}} \end{bmatrix} \tag{4}$$

where *Tk start* is the charging start time of the *n*th EV, and *Tk start* and *Tk start* are the lower and upper limits, respectively.

In this study, we considered that all EV batteries have the same capacity *E* and the same charging power *Pc*. Thus, the charging end time *Tk end* can be obtained:

$$T\_{cud}^k = T\_{start}^k + \frac{(1 - SOC^k) \times E}{P\_\varepsilon} \tag{5}$$

The total charging power of EVs at each moment is the sum of the charging power of an individual EV:

$$PEV\_t^G = \sum\_{k=1}^K PEV\_{k,t}^G \tag{6}$$

where *PEVG k*,*t* is the charging power of the *k*th EV at period *t*. In addition, *K* is the number of EVs dispatched.

The uncertainty of the charging power of the EV is still expressed in the form of a set:

$$\begin{aligned} PEV\_t^G &= \overline{PEV\_t} + PEV\_t\\ \text{s.t.} & \underline{PEV\_t} \le PEV\_t \le \overline{PEV\_t} \end{aligned} \tag{7}$$

where *PEVG t* is the total charging power of EVs at period *t*, *PEVt* is the forecasted value of the charging power of EVs at period *t*, *PEV*ˆ *t* is the deviation, and *PEV*ˆ *t*, *PEV*ˆ *t* are its lower and upper limits, respectively.


**Author details**

Christine Deasy1 \* and Patricia Mannix‐McNamara2

\*Address all correspondence to<EMAIL_ADDRESS>
1 Department of Nursing and Midwifery, Research Centre for Education and Professional Practice, Faculty of Education and Health Sciences, University of Limerick, Limerick, Ireland

2 School of Education, Research Centre for Education and Professional Practice, Faculty of Education and Health Sciences, University of Limerick, Limerick, Ireland


**Sergey Vyazovkin**

Department of Chemistry, University of Alabama at Birmingham, 901 S. 14th Street, Birmingham, AL 35294, USA<EMAIL_ADDRESS>
Academic Editor: Giuseppe Cirillo Received: 25 May 2020; Accepted: 16 June 2020; Published: 18 June 2020

**Abstract:** The Kissinger method is an overwhelmingly popular way of estimating the activation energy of thermally stimulated processes studied by differential scanning calorimetry (DSC), differential thermal analysis (DTA), and derivative thermogravimetry (DTG). The simplicity of its use is offset considerably by the number of problems that result from underlying assumptions. The assumption of a first-order reaction introduces a certain evaluation error that may become very large when applying temperature programs other than linear heating. The assumption of heating is embedded in the final equation that makes the method inapplicable to any data obtained on cooling. The method yields a single activation energy in agreement with the assumption of single-step kinetics that creates a problem with the majority of applications. This is illustrated by applying the Kissinger method to some chemical reactions, crystallization, glass transition, and melting. In the cases when the isoconversional activation energy varies significantly, the Kissinger plots tend to be almost perfectly linear that means the method fails to detect the inherent complexity of the processes. It is stressed that the Kissinger method is never the best choice when one is looking for insights into the processes kinetics. Comparably simple isoconversional methods offer an insightful alternative.

**Keywords:** crosslinking polymerization (curing); decomposition; degradation; liquid and solid state; phase transitions; thermal analysis

### **1. Introduction**

The thermal behavior of materials is explored broadly by the techniques of differential scanning calorimetry (DSC), differential thermal analysis (DTA), and thermogravimetry (TGA). Kinetic analysis of the data obtained by these techniques provides important insights into the fundamental issues of the reactivity and stability of materials. When it comes to kinetics, the Kissinger method is by far the most popular way of evaluating the activation energy of thermally stimulated processes. The method was introduced in two successive publications [1,2] that, according to the Scopus database, have been cited over 12,000 times. Yet, science is not a popularity contest, so that a routine followed by the majority is not guaranteed to yield the best or even simply a correct result. As a matter of fact, the method is not among the techniques recommended [3] for advanced kinetic studies. Most of the time, it is employed in materials characterization work that among other quantities reports a number that presumably characterizes an energy barrier to a thermal process under study. In this situation, the Kissinger method provides an unbeatably simple way of estimating the activation energy.

While a desired trait, simplicity may carry the risk of trivializing the problem. This applies fully to the Kissinger method as its formalism largely oversimplifies the kinetics of the processes it treats. Nowadays, as never before, the kinetics community concerned with thermally stimulated processes has come to realize that these processes are commonly multi-step [3]. As such, they have more than one energy barrier that controls them, so that the temperature dependence of their rates cannot be described by a single activation energy. In contrast, the Kissinger method yields a single value of

the activation energy regardless of the process complexity. In other words, this method is destined to generally miss the actual kinetic complexity. Clearly, this is an essential limitation of the method. Nonetheless, it is just as clear that exposing this limitation is unlikely to stop the enormous usage of the Kissinger method. Respectively, this is not an objective of the present work. Rather, it aims at helping those who use the method to do it in a conscientious manner. It means to use the method with the clear realization that its application does not usually provide adequate insights into the processes kinetics and typically yields the results of a very limited value.

To accomplish the task, we consider a number of the applications of the Kissinger method that include the most popular ones such as chemical reactions, crystallization and glass transition. For each of the processes considered we give a brief theoretical discussion to explain the origins of the complex temperature dependence of the respective rate. Then we apply the Kissinger method to experimental data to see whether the said complexity manifests itself in the application. In addition to that, we briefly discuss some general limitations that are associated with the underlying assumptions of the method.

#### **2. Basics of the Method**

The simplest form of the Kissinger equation for estimating the activation energy, *E*, is as follows:

$$E = -R\frac{d\ln\left(\frac{\beta}{T\_p^2}\right)}{dT\_p^{-1}}\tag{1}$$

where *R* is the gas constant, β is the heating rate and *Tp* is the temperature that corresponds to the position of the rate peak maximum. Most commonly *Tp* is determined as the temperature of the peak signal (maximum or minimum) measured by DSC, DTA or derivative thermogravimetry (DTG).

A curious fact about Equation (1) is that it had been proposed in an obscure paper by Bohun [4] a few years before the famous publications [1,2] by Kissinger. A more instructive form of the Kissinger equation is the integral one:

$$\ln\left(\frac{\beta}{T\_p^2}\right) = \ln\left(-\frac{AR}{E}f'(\alpha\_p)\right) - \frac{E}{RT\_p} \tag{2}$$

where *f* 0 (α) = *d f*(α) *d*α . Equation (2) originates from the basic rate equation of a single-step process:

$$\frac{d\alpha}{dt} = A \exp\left(\frac{-E}{RT}\right) f(\alpha) \tag{3}$$

where α is the extent of conversion of the reactant to products, *t* is the time, *A* is the preexponential factor and *f*(α) is the reaction model. A list of the models is available elsewhere [5].

Both Equations (1) and (2) suggest that the activation energy can be evaluated as the slope of the Kissinger plot of ln β *T* 2 *p* ! vs. *T* −1 *p* . However, Equation (2) indicates that for this plot to be linear the intercept should be a constant independent of the heating rate. This condition is not generally satisfied because α*p* is known [6] to depend on β. To satisfy it, *f* 0 (α) should be independent of α. This is the case of a first order reaction model, *f*(α) = 1 − α, for which *f* 0 (α) = −1. However, for the majority of the reaction models *f* 0 (α) depends on α and, thus, on β. This introduces some inaccuracy in estimating the value of *E*. As shown by Criado and Ortega [7] for a large variety of models the respective error does not exceed 5% as long as *E*/*RT* > 10.

It needs to be stressed that checking for potential systematic variation of α*p* with β should be taken as a general prerequisite for using the Kissinger method. The existence of such variation can be an indication of the process complexity as discussed by Muravyev et al. [8]. They have also demonstrated that even a moderate change of 0.06 in α*p*, caused by a change of β from 1 to 10 K min−1 , can result in a systematic error in *E* as large as 15%.

The error caused by the dependence of α*p* on β is eliminated in isoconversional methods [5,9,10]. One of them is the Kissinger-Akahira-Sunose method [11]. It employs the same equations as the Kissinger method (i.e., Equations (1) and (2)) but replaces *Tp* with *T*α. The latter is the temperature related to a given conversion at different heating rates. This is a more accurate way of estimating the activation energy. A critical advantage of an isoconversional method over the Kissinger one is that it affords determining the activation energy, *E*α, as a function of conversion. A significant systematic variation of *E*α with α reveals that the process under study involves more than one step. As shown later, this is an essential piece of kinetic information that the Kissinger method tends to miss.

Obviously, the method is not highly accurate but its numerical accuracy is rather tolerable for many practical purposes. Holba and Sestak [12] have additionally questioned the accuracy of the Kissinger method due to not accounting for the thermal inertia component of the heat flow as measured by heat flux DSC (or DTA). Indeed, it is typically assumed that the process rate is directly proportional to the heat flow, *dQ*/*dt*:

$$\frac{d\alpha}{dt} = \frac{1}{Q\_0} \frac{dQ}{dt} \tag{4}$$

where *Q*0 is the total heat released or absorbed during the process. This assumption is due to Borchardt and Daniels [13], whose analysis suggests that the thermal inertia term can be neglected. Not correcting for this term should unavoidably cause some systematic error in the value of the activation energy. This is because the raw (i.e., uncorrected) DSC peaks appear at somewhat higher temperature than they should, and the magnitude of this temperature shift increases with increasing the heating rate. In regard to the Kissinger method, it means that the *Tp* values determined from uncorrected DSC peaks are shifted to a higher temperature.

The corrected heat flow is obtained via a relatively simple adjustment:

$$\frac{d\mathbb{Q}}{dt} = RHF + \tau \frac{d(RHF)}{dt} \tag{5}$$

where *RHF* is the raw heat flow as measured by DSC and the second addend is the thermal inertia term. This adjustment, however, requires estimating the time constant τ. This is done by analyzing the back tail of a DSC peak measured for melting of a pure metal [14]. The value of τ is proportional to the total heat capacity of the sample. On the other hand, the temperature correction is proportional to βτ. It means that the effect of thermal inertia decreases with using smaller sample masses and slower heating rates. As a rule of thumb, the International Confederation of Thermal Anlaysis and Calrimetry (ICTAC) recommendations [15] suggest that for kinetic studies the product of the mass and heating rate should be kept under 100 mg K min−1 .

Our previous study has demonstrated [16] that ignoring thermal inertia in decomposition of 3 mg samples of polystyrene at the heating rates 2–20 ◦C min−1 causes statistically insignificant error when estimating the activation energy by an advanced isoconversional method [17]. Here we use the same data set [18] to compare the effect of thermal inertia on the activation energy estimated by the Kissinger method. The results are presented in Figure 1. As expected, the Kissinger plot for the corrected data is shifted to lower temperatures. The magnitude of the shift is barely detectable at slower heating rates but becomes larger at the faster ones. Upon accounting for thermal inertia, the *E* value has increased from 183 ± 6 to 191 ± 7 kJ mol−1 , that is, by only 4%. However, with account of the respective uncertainties, the *t*-test suggests that the difference is not statistically significant. This example does not mean that the effect of thermal inertia is negligible in general. Ultimately, the effect is determined by the magnitude of the temperature shift. As long as the latter does not rise above 2–3 ◦C, the effect should be negligible [16].

Another rarely considered issue related to the accuracy of the Kissinger method is its applicability to the temperature programs other than the one of linear heating. It is noteworthy that the linear heating rate is introduced in the Kissinger derivations by replacing *dT*/*dt* with β. However, the related equation is derived for the condition of the rate maximum, that is, the respective values have the meaning of

the instantaneous heating rate, βp. The latter can be defined for a nonlinear heating program that, in principle, affords extending the Kissinger method beyond the standard linear heating [19]. This is important in connection with the sample controlled thermal analysis [20], in which the temperature program is controlled by the response of the reaction rate to heating. This idea is implemented commercially in the techniques of high or maximum resolution TGA. The application of the Kissinger method in the case of nonlinear heating programs has been scrutinized by Sanchez-Jimenez et al. [21]. They have demonstrated that in the case of nonlinear heating the value of α*p* can vary significantly with βp, so that the respective Kissinger plot yields a completely erroneous value of the activation energy. Therefore, the major conclusion here is that before applying the method to the data obtained under a program other than simple linear heating, one needs to make sure that there is no significant variation of *Molecules* **2020**, α *25 p* , x FOR PEER REVIEW with βp. 4 of 17

**Figure 1.** Kissinger plots for thermal degradation of isotactic polystyrene. Data from Liavitskaya and Vyazovkin [18] and Vyazovkin [16]. Solid and open circles represent *Tp* values obtained respectively from original (raw) data and data corrected for thermal inertia. The difference is obvious only for faster heating rates, 16 and 20 °C min−1. Solid and dash lines are the least square fits to original and corrected data. **Figure 1.** Kissinger plots for thermal degradation of isotactic polystyrene. Data from Liavitskaya and Vyazovkin [18] and Vyazovkin [16]. Solid and open circles represent *Tp* values obtained respectively from original (raw) data and data corrected for thermal inertia. The difference is obvious only for faster heating rates, 16 and 20 ◦C min−1 . Solid and dash lines are the least square fits to original and corrected data.

Another rarely considered issue related to the accuracy of the Kissinger method is its applicability to the temperature programs other than the one of linear heating. It is noteworthy that the linear heating rate is introduced in the Kissinger derivations by replacing *dT*/*dt* with *β*. However, the related equation is derived for the condition of the rate maximum, that is, the respective values have the meaning of the instantaneous heating rate, *β*p. The latter can be defined for a nonlinear heating program that, in principle, affords extending the Kissinger method beyond the standard linear heating [19]. This is important in connection with the sample controlled thermal analysis [20], in which the temperature program is controlled by the response of the reaction rate to heating. This In addition, many types of runs are carried out under cooling temperature programs. Whether cooling is linear or not, the data obtained cannot be treated by the Kissinger method. A simple indication is that Equation (2) contains a logarithm of β. As already stated, the value of β replaces the value of *dT*/*dt*, so that on cooling β is necessarily negative. Obviously one cannot take a logarithm of a negative value, which means that Equation (2) cannot be used for cooling data. Forcing Equation (2) to treat cooling data by dropping the negative sign of β results in obtaining completely invalid values of the activation energy [22,23]. That is, the Kissinger method should never be applied to the data obtained on cooling.

#### idea is implemented commercially in the techniques of high or maximum resolution TGA. The **3. Chemical Reactions**

data obtained on cooling.

**3. Chemical Reactions**

application of the Kissinger method in the case of nonlinear heating programs has been scrutinized by Sanchez-Jimenez et al. [21]. They have demonstrated that in the case of nonlinear heating the value of *αp* can vary significantly with *β*p, so that the respective Kissinger plot yields a completely erroneous value of the activation energy. Therefore, the major conclusion here is that before applying the method to the data obtained under a program other than simple linear heating, one needs to make sure that there is no significant variation of *αp* with *β*p. In addition, many types of runs are carried out under cooling temperature programs. Whether As already mentioned, the Kissinger method is based on Equation (3), which is the rate equation of a single step reaction. The temperature dependence of the latter is determined by a single activation energy that is readily evaluated by the Kissinger method. The problem, though, is that the reactions in the condensed (i.e., solid or liquid) phase typically involve multiple steps and, therefore, face more than a single energy barrier. This has important implications for the temperature dependence of the reaction rate.

(2) to treat cooling data by dropping the negative sign of *β* results in obtaining completely invalid values of the activation energy [22,23]. That is, the Kissinger method should never be applied to the

As already mentioned, the Kissinger method is based on Equation (3), which is the rate equation of a single step reaction. The temperature dependence of the latter is determined by a single activation energy that is readily evaluated by the Kissinger method. The problem, though, is that the reactions in the condensed (i.e., solid or liquid) phase typically involve multiple steps and, therefore, face more

cooling is linear or not, the data obtained cannot be treated by the Kissinger method. A simple indication is that Equation (2) contains a logarithm of *β*. As already stated, the value of *β* replaces the

Let us consider a very simple reaction that involves two competing steps, each of which follows the same model. The rate of this reaction is:

$$\frac{da}{dt} = k\_1(T)f(a) + k\_2(T)f(a) \equiv [k\_1(T) + k\_2(T)]f(a) = k\_{\epsilon f}(T)f(a) \tag{6}$$

where the subscripts 1 and 2 designate the rate constants, *k*(*T*), respectively related to the individual steps. In its turn, the temperature dependence of the rate constant is defined via the Arrhenius equation:

$$k(T) = A \exp\left(\frac{-E}{RT}\right) \tag{7}$$

Experimentally, the activation energy is determined from the slope of the plot of a logarithm of the rate constant vs. reciprocal temperature, that is, as the following derivative:

$$E = -R\frac{d\ln k(T)}{dT^{-1}}\tag{8}$$

Plugging *kef*(*T*) from Equation (6) into Equation (8) gives:

$$E = \frac{E\_1 k\_1(T) + E\_2 k\_2(T)}{k\_1(T) + k\_2(T)}\tag{9}$$

Equation (9) suggests that the experimentally determined activation energy for the above reaction will be temperature dependent, which also means that the respective ln*k*(*T*) vs. *T* −1 plot will be nonlinear as illustrated elsewhere [24].

Comparing Equation (8) with Equation (1) suggests that the Kissinger plot for the above reaction would also be nonlinear and the respective activation energy temperature dependent. If this nonlinearity were easy to detect, the application area of the Kissinger method could be extended to multi-step kinetics. In reality, detecting such nonlinearity is not easy and contingent on several conditions. One of them is the number of heating rates used, that is, the number of points on the Kissinger plot. It is nearly impossible to detect the nonlinearity with less than 5 points. However, the typical application of the Kissinger method is limited to 3–4 heating rates. Another important factor, is the width of the temperature range of the *Tp* values. The wider the range, the better chances to detect the nonlinearity. In experimental terms, a wider range of *Tp* means a wider range of β. The ratio of the maximum to minimum heating rate should be no less than 5. Even if all these conditions are met, the nonlinearity may still escape detecting because the difference in the activation energies of the individual steps is not large enough.

A much more sensitive way of detecting the reaction complexity is to use an isoconversional method [5,9,10] that allows one to determine the activation energy as a function of conversion. As an example, Figure 2 provides a comparison of the Kissinger plot with a dependence of the isoconversional activation energy (*E*α) on conversion for the thermal decomposition (dehydration) of calcium oxalate monohydrate as measured by DSC [25]. As one can see, the activation energy estimated by an isoconversional method varies from about 105 to 75 kJ mol−1 . This means that the respective Kissinger plot should be nonlinear. In particular, its lower temperature part should have a steeper slope than the high temperature one. This actually is the case, if one looks very closely. Yet, the change in the angle of the slope is a little over 3◦ and, thus, is easy to miss. Furthermore, treating this plot as linear, that is, ignoring the small nonlinearity, yields a high value of the correlation coefficient (*r*), which means that statistically this nonlinearity insignificant. All this illustrates how insensitive the Kissinger method is in detecting the reaction complexity. This is especially alarming considering that this Kissinger plot includes 10 heating rates ranging from 0.75 to 20 ◦C min−1 and covering a 53 ◦C interval.

product as [18]:

**Figure 2.** Kissinger plot for thermal dehydration of CaC2O4 H2O. Dash line is least square fit to experimental data (squares). Solid line is a fit to the three lower temperature data points. It demonstrates that the angle of the slope changes for higher temperature data. Inset shows variation in isoconversional activation energy. Data from Liavitskaya and Vyazovkin [25]. **Figure 2.** Kissinger plot for thermal dehydration of CaC2O4 H2O. Dash line is least square fit to experimental data (squares). Solid line is a fit to the three lower temperature data points. It demonstrates that the angle of the slope changes for higher temperature data. Inset shows variation in isoconversional activation energy. Data from Liavitskaya and Vyazovkin [25].

Naturally, the question can be raised whether the statistically insignificant nonlinearity is important. An answer depends on the purpose of the kinetic study. If one simply needs to obtain a ballpark estimate for the activation energy, such nonlinearity can be ignored. However, it is critically important when one strives for a mechanistic understanding of the estimate. For instance, the decreasing dependence of *Eα* shown in Figure 2 is not just something encountered in a particular instance of dehydration of calcium oxalate monohydrate. It is a general phenomenon observed for a wide variety of reversible decompositions [26] that include dehydration of diverse crystal hydrates. For this type of processes, the activation energy depends on the equilibrium pressure, *P0*, of the gas Naturally, the question can be raised whether the statistically insignificant nonlinearity is important. An answer depends on the purpose of the kinetic study. If one simply needs to obtain a ballpark estimate for the activation energy, such nonlinearity can be ignored. However, it is critically important when one strives for a mechanistic understanding of the estimate. For instance, the decreasing dependence of *E*α shown in Figure 2 is not just something encountered in a particular instance of dehydration of calcium oxalate monohydrate. It is a general phenomenon observed for a wide variety of reversible decompositions [26] that include dehydration of diverse crystal hydrates. For this type of processes, the activation energy depends on the equilibrium pressure, *P0*, of the gas product as [18]:

$$E = E\_1 + \Delta H^0 \frac{P}{P\_0 - P} \tag{10}$$

 − (10) where *E*1 is the activation energy of the forward reaction, Δ*H*0 is the reaction enthalpy and *P* is the partial pressure of the gas product. According to Equation (10), *E* should decrease with increasing temperature because *P0* increases, making the second addend increasingly smaller. Clearly, none of that information can be gained from the single value 82 ± 1 kJ mol−1 estimated by the Kissinger method where *E*1 is the activation energy of the forward reaction, ∆*H*0 is the reaction enthalpy and *P* is the partial pressure of the gas product. According to Equation (10), *E* should decrease with increasing temperature because *P0* increases, making the second addend increasingly smaller. Clearly, none of that information can be gained from the single value 82 ± 1 kJ mol−1 estimated by the Kissinger method (Figure 2).

(Figure 2). It is worth noting that Agresti [27] has proposed a modification to the Kissinger method that accounts for the pressure dependence. As expected, the resulting Kissinger plots are nonlinear and It is worth noting that Agresti [27] has proposed a modification to the Kissinger method that accounts for the pressure dependence. As expected, the resulting Kissinger plots are nonlinear and their curvature increases dramatically in the vicinity of equilibrium temperatures.

their curvature increases dramatically in the vicinity of equilibrium temperatures. Concerning the reaction complexity, there is a common belief that it has to manifest itself via the rate peaks that reveal shoulders or other aberrations of the regular bell-shaped form. Definitely, discovering such features is a sign of the reaction complexity. However, the opposite is not true, meaning that the absence of such features in DSC, DTA or DTG peaks does not mean that the reaction is simple, that is, single step. Figure 3 illustrates such situation in the case of epoxy-anhydride crosslinking polymerization (sometimes termed as curing). It is seen that the respective DSC peaks are of regular bell-shaped form without any obvious aberrations. Yet, the isoconversional activation energy demonstrates a significant increase from about 20 to 70 kJ mol−1, which again results from a multi-step reaction mechanism [28] At the same time, the Kissinger plot is almost perfectly linear (*r*  = −0.9999) and yields a single value of the activation energy, 71 kJ mol−1 [29]. The latter obviously Concerning the reaction complexity, there is a common belief that it has to manifest itself via the rate peaks that reveal shoulders or other aberrations of the regular bell-shaped form. Definitely, discovering such features is a sign of the reaction complexity. However, the opposite is not true, meaning that the absence of such features in DSC, DTA or DTG peaks does not mean that the reaction is simple, that is, single step. Figure 3 illustrates such situation in the case of epoxy-anhydride crosslinking polymerization (sometimes termed as curing). It is seen that the respective DSC peaks are of regular bell-shaped form without any obvious aberrations. Yet, the isoconversional activation energy demonstrates a significant increase from about 20 to 70 kJ mol−1 , which again results from a multi-step reaction mechanism [28] At the same time, the Kissinger plot is almost perfectly linear (*r* = −0.9999) and yields a single value of the activation energy, 71 kJ mol−1 [29]. The latter obviously gives no hints regarding the reaction complexity.

gives no hints regarding the reaction complexity.

**Figure 3.** DSC curves for crosslinking polymerization of an epoxy-anhydride system. The numbers by the curves are heating rates in °C min−1. The inset shows the conversion dependence of the isoconversional activation energy. Adapted with permission from Vyazovkin and Sbirrazzuoli [29]. Copyright 1999 Wiley-VCH. **Figure 3.** DSC curves for crosslinking polymerization of an epoxy-anhydride system. The numbers by the curves are heating rates in ◦C min−1 . The inset shows the conversion dependence of the isoconversional activation energy. Adapted with permission from Vyazovkin and Sbirrazzuoli [29]. Copyright 1999 Wiley-VCH.

Of course, there are cases of single-step reactions. These ordinarily are multi-step reactions whose overall kinetics is limited or dominated by one step. In these cases, the Kissinger method could serve as a basis for an adequate kinetic analysis. The problem, though, is that the method does not typically possess the sufficient sensitivity to differentiate reliably between the single and multi-step kinetics. The occurrence of single-step kinetics can be easily detected by an isoconversional method as the absence of any significant dependence of *Eα* on *α*. However, employing an isoconversional Of course, there are cases of single-step reactions. These ordinarily are multi-step reactions whose overall kinetics is limited or dominated by one step. In these cases, the Kissinger method could serve as a basis for an adequate kinetic analysis. The problem, though, is that the method does not typically possess the sufficient sensitivity to differentiate reliably between the single and multi-step kinetics. The occurrence of single-step kinetics can be easily detected by an isoconversional method as the absence of any significant dependence of *E*α on α. However, employing an isoconversional method for such purpose immediately makes the use of the Kissinger method redundant.

method for such purpose immediately makes the use of the Kissinger method redundant. Last but not least, the Kissinger method is a part of the ASTM E698 technique for Arrhenius kinetic constants for thermally unstable materials [30]. This technique relies on using a single value of the activation energy estimated by the Kissinger method to make predictions of a material behavior under isothermal conditions. Unfortunately, in the case of the reaction complexity this technique produces rather poor predictions, that is, significantly less accurate than the ones produced via isoconversional methods that use variable activation energy [31,32]. Last but not least, the Kissinger method is a part of the ASTM E698 technique for Arrhenius kinetic constants for thermally unstable materials [30]. This technique relies on using a single value of the activation energy estimated by the Kissinger method to make predictions of a material behavior under isothermal conditions. Unfortunately, in the case of the reaction complexity this technique produces rather poor predictions, that is, significantly less accurate than the ones produced via isoconversional methods that use variable activation energy [31,32].

#### **4. Crystallization**

**4. Crystallization**  There are about as many publications on the application of the Kissinger method to crystallization as to chemical reactions. The applications are so common that sometimes one can see the claims that the method was proposed by Kissinger for estimating the activation energy of crystallization. This, of course, is absolutely false. In reality, neither of his seminal papers [1,2] even There are about as many publications on the application of the Kissinger method to crystallization as to chemical reactions. The applications are so common that sometimes one can see the claims that the method was proposed by Kissinger for estimating the activation energy of crystallization. This, of course, is absolutely false. In reality, neither of his seminal papers [1,2] even contains the word "crystallization."

contains the word "crystallization." To understand the problems with this particular application, one needs to recognize that the temperature dependence of the crystallization rate differs dramatically from that of the reaction rate. Decreasing temperature makes chemical reactions to proceed slower. Crystallization rate depends on supercooling, Δ*T* = *Tm* − *T* with respect to the equilibrium melting temperature, *Tm*. At small supercoolings, crystallization accelerates with decreasing temperature until reaching the maximum rate at some temperature *Tmax*. At large supercoolings, that is, at temperatures below *Tmax*, the To understand the problems with this particular application, one needs to recognize that the temperature dependence of the crystallization rate differs dramatically from that of the reaction rate. Decreasing temperature makes chemical reactions to proceed slower. Crystallization rate depends on supercooling, ∆*T* = *Tm* − *T* with respect to the equilibrium melting temperature, *Tm*. At small supercoolings, crystallization accelerates with decreasing temperature until reaching the maximum rate at some temperature *Tmax*. At large supercoolings, that is, at temperatures below *Tmax*, the crystallization rate decreases with decreasing temperature.

crystallization rate decreases with decreasing temperature. This complex temperature dependence cannot be described by a single Arrhenius equation, which is the basis of the Kissinger method. It is described well by the models of nucleation or nuclei growth that combine the Arrhenius kinetics with the underlying thermodynamics of crystallization. This complex temperature dependence cannot be described by a single Arrhenius equation, which is the basis of the Kissinger method. It is described well by the models of nucleation or nuclei growth that combine the Arrhenius kinetics with the underlying thermodynamics of crystallization. The

rate of crystallization can be limited by the formation of nuclei or by the growth of existing nuclei. The temperature dependence of the nucleation rate is adequately represented by the Turnbull and Fisher model [33]:

$$n = n\_0 \exp\left(\frac{-E\_D}{RT}\right) \exp\left(\frac{-\Delta G^\circ}{RT}\right) \tag{11}$$

where *n* is the nucleation rate constant, *n*0 the preexponential factor, *ED* is the activation energy of diffusion and ∆*G* \* is the free energy barrier to nucleation. The size of this barrier for a spherical nucleus is as follows:

$$
\Delta G^\* = \frac{16\pi\sigma^3 T\_m^2}{3(\Delta H\_m)^2 (\Delta T)^2} = \frac{\Omega}{(\Delta T)^2} \tag{12}
$$

where σ is the surface energy (surface tension), ∆*Hm* is the enthalpy of melting per unit volume and Ω is a constant that collects all parameters that do not practically depend on temperature. The derivations can be found in various sources, for example, References [9,34–38].

If the crystallization rate, *u*, is limited by the growth of existing nuclei, it depends on temperature as follows [38]:

$$u = u\_0 \exp\left(\frac{-E\_D}{RT}\right) \left[1 - \exp\left(\frac{\Delta G}{RT}\right)\right] \tag{13}$$

where *u*0 the preexponential factor and ∆*G* is the difference in the free energy of the final (crystalline) and initial (liquid) phase. Surprisingly, the monographic literature does not consider Equation (13) as commonly as Equation (11). Thus, it needs a few comments here. First, it is readily derived as the difference between the rates of the forward and reverse transition. For the forward rate, the energy barrier is *ED*, whereas for the reverse rate it is *ED* − ∆*G* (∆*G* < 0). The frequency (preexponential) factor is assumed the same for both rates.

Second, the free energy terms in Equations (11) and (13) differ entirely in their meaning. The ∆*G \** term in Equation (11) is the energy barrier that makes it a positive value. The ∆*G* term in Equation (13) is the free energy change for a spontaneous process and, thus, negative. Note that Equation (13) is sometimes written with a negative sign in front of ∆*G* when it is referred to as the driving force. The latter in the strict thermodynamics sense [39] should be a positive quantity. This, however, may create extra confusion of dealing with positive ∆*G* for a spontaneous process. One way or another, the argument of the exponential function in the bracketed term of Equation (13) must be negative.

Despite their differences, Equations (11) and (13) suggest the existence of the rate maximum. In both equations, the acceleration at small supercooling is due to the thermodynamic terms. In Equation (11), it occurs because the nucleation barrier ∆*G \** decreases with increasing the supercooling (Equation (12)), that is, with decreasing temperature. In Equation (13), the dependence on supercooling is introduced via an approximate equality [35,36,38]:

$$
\Delta G^{\circ} = \Delta H\_c \left(\frac{T\_m - T}{T\_m}\right) \equiv \Delta H\_m \left(\frac{T - T\_m}{T\_m}\right) \tag{14}
$$

where ∆*Hc* is the enthalpy of crystallization. Then, as temperature lowers and supercooling increases the value of ∆*G* becomes increasingly more negative. As a result, the exponential function in the bracketed term (Equation (13) decreases toward zero, whereas the term itself increases toward unity. Thus, the acceleration is associated with the bracketed term that becomes larger at larger supercoolings.

In both cases (Equations (11) and (13)), the thermodynamic acceleration is counteracted by the diffusional retardation that originates from continuously growing viscosity of the melt. This behavior is represented by the exponential term containing *ED*. At some point the diffusional retardation starts to outweigh the thermodynamic acceleration, so that the rate begins to drop with decreasing temperature. As a result, the rate passes through a maximum.

Figure 4 (inset) displays the temperature dependence of the rate derived by combining Equations (13) and (14). Such dependence for the nucleation rate (*n* in Equation (11)) can be seen elsewhere [40]. Either of these dependencies passes through a maximum. However, in general

that yields the following:

the growth process tends to demonstrate the maximum at *Tmax*, which is larger (closer to *Tm*) than that for the nucleation process. growth process tends to demonstrate the maximum at *Tmax*, which is larger (closer to *Tm*) than that for the nucleation process.

**Figure 4.** Temperature dependence of the activation energy according to Equation (15). Inset shows temperature dependence of the growth rate according to Equation (13). **Figure 4.** Temperature dependence of the activation energy according to Equation (15). Inset shows temperature dependence of the growth rate according to Equation (13).

Most relevant to the Kissinger analysis is that the existence of the rate maximum on the temperature dependence entails the inversion of the sign of the experimentally determined activation energy. This is unavoidable because the activation energy is determined by the sign of the temperature derivative of the rate. In the temperature region above *Tmax* the rate decreases with increasing temperature; the sign of the experimental activation energy is negative. However, below *Tmax* the rate increases as temperature rises so that the sign is positive. An analytic expression for the temperature dependence of the activation energy is arrived at by plugging *u* for *k*(*T*) in Equation (8) Most relevant to the Kissinger analysis is that the existence of the rate maximum on the temperature dependence entails the inversion of the sign of the experimentally determined activation energy. This is unavoidable because the activation energy is determined by the sign of the temperature derivative of the rate. In the temperature region above *Tmax* the rate decreases with increasing temperature; the sign of the experimental activation energy is negative. However, below *Tmax* the rate increases as temperature rises so that the sign is positive. An analytic expression for the temperature dependence of the activation energy is arrived at by plugging *u* for *k*(*T*) in Equation (8) that yields the following:

$$E = E\_D + \frac{\Delta H\_m \exp\left[\frac{\Delta H\_m (T - T\_m)}{RTT\_m}\right]}{\exp\left[\frac{\Delta H\_m (T - T\_m)}{RTT\_m}\right] - 1} \tag{15}$$

 The resulting dependence is depicted in Figure 4. Indeed, it is seen that *E* takes on large negative values in the vicinity of *Tm* but increases toward 0 as temperature drops toward *Tmax*. Yet, in the limit of large supercoolings (*T* < *Tmax*) *E* is positive and decreases from *ED* toward 0 as temperature rises. The nucleation model (Equation (11)) gives rise to a different analytic expression for the temperature dependence of the activation energy [9], viz.: The resulting dependence is depicted in Figure 4. Indeed, it is seen that *E* takes on large negative values in the vicinity of *Tm* but increases toward 0 as temperature drops toward *Tmax*. Yet, in the limit of large supercoolings (*T* < *Tmax*) *E* is positive and decreases from *ED* toward 0 as temperature rises. The nucleation model (Equation (11)) gives rise to a different analytic expression for the temperature dependence of the activation energy [9], viz.:

$$E = E\_D - \Omega \left[ \frac{2T}{\left(T\_m - T\right)^3} - \frac{1}{\left(T\_m - T\right)^2} \right] \tag{16}$$

Nevertheless, the dependence predicted by Equation (16) is very similar to the one predicted by Equation (15) and has exactly the same asymptotes, that is, *ED* and −∞ for the infinitely large and small supercooling respectively. Nevertheless, the dependence predicted by Equation (16) is very similar to the one predicted by Equation (15) and has exactly the same asymptotes, that is, *ED* and −∞ for the infinitely large and small supercooling respectively.

The above has direct relevance to the experimental studies of the crystallization kinetics. It needs be recognized that experimentally the temperature region above *Tmax* is accessed by cooling melts, whereas the one below *Tmax* by heating glasses. The applications of the Kissinger method to crystallization of melts are usually encountered in the field of polymers. As explained earlier, the Kissinger method cannot be applied to the data obtained on cooling and when forced to do that yields entirely erroneous values of the activation energy [22,23]. When the Kissinger method is applied to crystallization of glasses, it yields positive activation The above has direct relevance to the experimental studies of the crystallization kinetics. It needs be recognized that experimentally the temperature region above *Tmax* is accessed by cooling melts, whereas the one below *Tmax* by heating glasses. The applications of the Kissinger method to crystallization of melts are usually encountered in the field of polymers. As explained earlier, the Kissinger method cannot be applied to the data obtained on cooling and when forced to do that yields entirely erroneous values of the activation energy [22,23].

When the Kissinger method is applied to crystallization of glasses, it yields positive activation energies that are commonly reported as constant temperature independent values. On the other hand, the theory predicts the activation energy of the glass crystallization to decrease with increasing temperatures. It means that the corresponding Kissinger plots should be nonlinear, or, more precisely, concave down. As already noted, detecting the curvature requires using multiple heating rates spread over a broad range. For example, the effect is detectable (Figure 5) in the data obtained for crystallization of Ga7.5Se92.5 and Si12.5Te87.5 glasses [41,42]. Note that in both cases the range of the heating rates used is unusually broad, 5–90 ◦C min−1 . However, if one looks at these plots within a more typical range 5–25 ◦C min−1 (first five points corresponding to the lower temperatures) the curvature is practically unnoticeable. Expectedly, it is much easier to detect a variation in the activation energy by using an isoconversional method. For these two glasses the isoconversional activation energy decreases 1.5 times (from 85 to 55 kJ mol−1 for Ga7.5Se92.5 and from 200 to 130 kJ mol−1 for Si12.5Te87.5) in the temperature range of crystallization [41,42]. It should be emphasized that the curvature of the Kissinger plots for crystallization increases dramatically when using ultrafast scanning calorimetry [43]. This technique permits employing both much faster heating rates and much broader range of them. hand, the theory predicts the activation energy of the glass crystallization to decrease with increasing temperatures. It means that the corresponding Kissinger plots should be nonlinear, or, more precisely, concave down. As already noted, detecting the curvature requires using multiple heating rates spread over a broad range. For example, the effect is detectable (Figure 5) in the data obtained for crystallization of Ga7.5Se92.5 and Si12.5Te87.5 glasses [41,42]. Note that in both cases the range of the heating rates used is unusually broad, 5–90 °C min−1. However, if one looks at these plots within a more typical range 5–25 °C min−1 (first five points corresponding to the lower temperatures) the curvature is practically unnoticeable. Expectedly, it is much easier to detect a variation in the activation energy by using an isoconversional method. For these two glasses the isoconversional activation energy decreases 1.5 times (from 85 to 55 kJ mol−1 for Ga7.5Se92.5 and from 200 to 130 kJ mol−1 for Si12.5Te87.5) in the temperature range of crystallization [41,42]. It should be emphasized that the curvature of the Kissinger plots for crystallization increases dramatically when using ultrafast scanning calorimetry [43]. This technique permits employing both much faster heating rates and much broader range of them.

*Molecules* **2020**, *25*, x FOR PEER REVIEW 10 of 17

**Figure 5.** Kissinger plots for crystallization of Ga7.5Se92.5 (squares) and Si12.5Te87.5 (circles) glasses. Dash lines connecting three lowest temperature points are a guide for the eye to better visualize the nonlinearity. The heating rates range from 5 to 90 °C min−1. Data from Abu El-Oyoun [41,42]. **Figure 5.** Kissinger plots for crystallization of Ga7.5Se92.5 (squares) and Si12.5Te87.5 (circles) glasses. Dash lines connecting three lowest temperature points are a guide for the eye to better visualize the nonlinearity. The heating rates range from 5 to 90 ◦C min−1 . Data from Abu El-Oyoun [41,42].

#### **5. Glass Transition 5. Glass Transition**

The glass transition appears to be the third most important application area of the Kissinger method. One of popular models used for describing the glass transition kinetics is the Tool-Narayanaswami-Moynihan model (TNM) [44–46]. It can be presented as follows: The glass transition appears to be the third most important application area of the Kissinger method. One of popular models used for describing the glass transition kinetics is the Tool-Narayanaswami-Moynihan model (TNM) [44–46]. It can be presented as follows:

$$
\ln \pi = \ln \pi\_0 + \frac{\mathbf{x}E}{RT} + \frac{(1-\mathbf{x})E}{RT\_f} \tag{17}
$$

where *τ* is the relaxation time, *τ*0 is the preexponential factor, *x* is the nonlinearity parameter and *Tf* is the fictive temperature. The relaxation time in Equation (17) is an analog of the reciprocal rate constant in the Arrhenius equation. The model indicates that the whole process of the glass transition is driven by a single constant activation energy. This is a simplification also known as thermorheological simplicity, which, by no means, is the general rule for the relaxation behavior [47]. An important limitation of the model is that it, in particular, predicts the Arrhenius temperature dependence for viscosity, whereas this dependence is generally of the Williams-Lander-Ferry (WLF) [48] or Vogel-Tammann-Fulcher (VTF) [37,49,50] type. where τ is the relaxation time, τ0 is the preexponential factor, *x* is the nonlinearity parameter and *Tf* is the fictive temperature. The relaxation time in Equation (17) is an analog of the reciprocal rate constant in the Arrhenius equation. The model indicates that the whole process of the glass transition is driven by a single constant activation energy. This is a simplification also known as thermorheological simplicity, which, by no means, is the general rule for the relaxation behavior [47]. An important limitation of the model is that it, in particular, predicts the Arrhenius temperature dependence for viscosity, whereas this dependence is generally of the Williams-Lander-Ferry (WLF) [48] or Vogel-Tammann-Fulcher (VTF) [37,49,50] type.

Based on Equation (17), Moynihan et al. [51] have proposed methods of estimating the activation energy of the glass transition from either cooling or heating data. For heating, *E* is evaluated from the

Based on Equation (17), Moynihan et al. [51] have proposed methods of estimating the activation energy of the glass transition from either cooling or heating data. For heating, *E* is evaluated from the heating rate dependence of the glass transition temperature:

$$E = -\mathcal{R}\frac{d\ln\beta}{dT\_{\mathcal{R}}^{-1}}.\tag{18}$$

In their paper, Moynihan et al. point out that *Tg* can be defined from calorimetric measurements as temperature of the extrapolated onset or inflection point or the heat capacity maximum. The latter corresponds to the endothermic peak that appears on heating in DSC at the end of the glass transition event.

The appearance of this peak, sometimes referred to as the glass transition peak, has inspired numerous applications of the Kissinger method for determination of the activation energy of the glass transition. Unfortunately, most of these application have been wrong. Apparently, many believe that simply observing a DSC peak that shifts with the heating rate justifies the application of the Kissinger method. This is certainly not true in the case of the glass transition. The problem is not specific to the Kissinger method itself, although it may appear as such [52]. Rather, it arises from heating the glasses not having a proper thermal history. As stressed by Moynihan et al. [51,53,54], obtaining a correct value of *E* from Equation (18) requires creating a glass of a specific thermal history. Namely, immediately before heating the glass has to be cooled at the rate, which is equal (or proportional) to the rate of heating. Also, cooling must occur from the equilibrium state, that is, from well above *Tg* down to well below *Tg*. Using other thermal histories gives rise to the *E* values that can deviate dramatically from the correct one [54].

While demonstrated [54] in the case of Equation (18), the importance of using the proper thermal history applies fully to the application of the Kissinger method. As a matter of fact, both the Moynihan (18) and Kissinger (1) equations yield nearly identical values of *E* when *Tg* is estimated as the peak temperature of the glass transition and they both produce equally wrong values in the case of not using the proper thermal history [55]. Nevertheless, the application of Equation (18) to *Tg* = *Tp* when using the proper thermal history does give rise to correct values of *E* [56]. Putting all these results together, we can conclude that one can use the Kissinger method to obtain the correct values of *E* as long as the glass transition measurements are performed on a sample exposed to the proper thermal history, for example, when using heating at β immediately preceded by cooling at −β, as mentioned before. More importantly, it makes no sense trying to determine the *E* values by the Kissinger or Moynihan method by heating the as-is glass samples. The resulting values would be largely the fortuitous ones that are impossible to interpret in a meaningful way.

Assuming that the glass transition measurements are performed under a proper thermal history, we can now return to aforementioned limitation of the TNM model associated with the oversimplified (i.e., Arrhenius) treatment of the temperature dependence of the relaxation time or viscosity. As stated, the more general is the WLF or the VTF dependence. The VTF equation for the relaxation time is:

$$
\ln \tau = \ln \tau\_0 + \frac{B}{T - T\_0} \tag{19}
$$

where *B* is a constant and *T*0 is a reference temperature. The respective Arrhenius plot, lnτ vs. *T* −1 is nonlinear and gives rise to the activation energy that decreases with increasing temperature as follows:

$$E = R \frac{BT^2}{\left(T - T\_0\right)^2} \tag{20}$$

A similar dependence derives [48] from the WLF equation.

This brings about an important question as to whether the activation energy of the glass transition should be a constant or temperature dependent (i.e., variable) value. Based on the Adam-Gibbs

theory [57], the activation energy in the Arrhenius equation is proportional to the size of the region that rearranges cooperatively during the transition. This size is inversely proportional to the configurational entropy that increases with *T* so that the experimental activation energy is expected to decrease with *T*. In practice, one may or may not detect this variation depending on the dynamic fragility [58] of the systems studied. According to Angell [58], there are strong and fragile glassformers that demonstrate distinctly different lnτ vs. *T* −1 plots. For the strong glass formers, these plots are nearly linear, that is, Arrhenian. For the fragile ones, they are nonlinear, that is, of the WLF/VTF type. Typical examples of the strong and fragile glassformers respectively are inorganics and polymers. On the other hand, organics and metals tend to fall between those two limits. *Molecules* **2020**, *25*, x FOR PEER REVIEW 12 of 17 plots are nearly linear, that is, Arrhenian. For the fragile ones, they are nonlinear, that is, of the WLF/VTF type. Typical examples of the strong and fragile glassformers respectively are inorganics and polymers. On the other hand, organics and metals tend to fall between those two limits.

In any event, when it comes to the Kissinger analysis of the glass transition one may obtain either linear or nonlinear Kissinger plots depending on the fragility of the systems studied. Examples of such plots are shown in Figure 6 for two glasses: boron oxide (B2O3) and polystyrene (PS). The *Tp* values have been extracted from the previously published DSC data [59,60] As seen from the figure, the Kissinger plot is practically linear for the strong glassformer, boron oxide, whereas it is nonlinear for the fragile one, polystyrene. That is, the activation energy of the glass transition is expected to be practically constant in the former case but temperature dependent in the later one. Note that detecting nonlinearity of the Kissinger plots requires using multiple heating rates in relatively broad range. Much more sensitive way of detecting a variation in the activation energy of the glass transition is to employ an isoconversional method that demonstrates clearly that the variability of the activation energy is proportional to the fragility [61]. Remarkably, the method has been capable of detecting a minor variation in the activation energy even in the case of boron oxide [60]. In any event, when it comes to the Kissinger analysis of the glass transition one may obtain either linear or nonlinear Kissinger plots depending on the fragility of the systems studied. Examples of such plots are shown in Figure 6 for two glasses: boron oxide (B2O3) and polystyrene (PS). The *Tp* values have been extracted from the previously published DSC data [59,60] As seen from the figure, the Kissinger plot is practically linear for the strong glassformer, boron oxide, whereas it is nonlinear for the fragile one, polystyrene. That is, the activation energy of the glass transition is expected to be practically constant in the former case but temperature dependent in the later one. Note that detecting nonlinearity of the Kissinger plots requires using multiple heating rates in relatively broad range. Much more sensitive way of detecting a variation in the activation energy of the glass transition is to employ an isoconversional method that demonstrates clearly that the variability of the activation energy is proportional to the fragility [61]. Remarkably, the method has been capable of detecting a minor variation in the activation energy even in the case of boron oxide [60].

**Figure 6.** Kissinger plots for the glass transition in polystyrene (PS, circles) and boron oxide (B2O3, squares). For PS, the plot is visibly nonlinear, the activation energy decreases with temperature, the heating rate range 2.5–25 °C min−1. For B2O3, the plot is practically linear, the activation energy is nearly constant, 203 ± 11 kJ mol−1, the heating rate range 5–25 **°**C min−1. Data from Vyazovkin et al. **Figure 6.** Kissinger plots for the glass transition in polystyrene (PS, circles) and boron oxide (B2O3 , squares). For PS, the plot is visibly nonlinear, the activation energy decreases with temperature, the heating rate range 2.5–25 ◦C min−1 . For B2O3 , the plot is practically linear, the activation energy is nearly constant, 203 ± 11 kJ mol−1 , the heating rate range 5–25 ◦C min−1 . Data from Vyazovkin et al. [59,60].

#### [59,60]. **6. Melting and Other Processes**

**6. Melting and Other Processes**  In addition to the three major application areas discussed above, the Kissinger method has been used for treating some other processes. Although these applications are relatively scarce, they are still of research interest. The most common and, perhaps, most confusing is melting. A detailed discussion In addition to the three major application areas discussed above, the Kissinger method has been used for treating some other processes. Although these applications are relatively scarce, they are still of research interest. The most common and, perhaps, most confusing is melting. A detailed discussion regarding the theory and practice of the melting kinetics is given elsewhere [40]. Here, we only reiterate a few important points directly relevant to the Kissinger method.

regarding the theory and practice of the melting kinetics is given elsewhere [40]. Here, we only reiterate a few important points directly relevant to the Kissinger method. The basic thermodynamics suggests that on heating of a substance its temperature remains The basic thermodynamics suggests that on heating of a substance its temperature remains constant throughout the melting process. This temperature is the equilibrium melting temperature,

heating rate. This effect is observed because the DSC peaks are typically presented not as a function of the sample temperature but as a function of the reference (furnace) temperature. The latter obviously increases during continuous heating at the rate *β*. In this situation, the value of *Tp* represents the reference temperature, at which the substance has finished melting. This temperature

shifts with the heating rate according to the following equation [62]:

which is independent of the heating rate. The confusion may arise from the fact that the position of the DSC peak (i.e., *Tp*) measured for melting usually demonstrates a noticeable increase with the heating rate. This effect is observed because the DSC peaks are typically presented not as a function of the sample temperature but as a function of the reference (furnace) temperature. The latter obviously increases during continuous heating at the rate β. In this situation, the value of *Tp* represents the reference temperature, at which the substance has finished melting. This temperature shifts with the heating rate according to the following equation [62]:

$$T\_p - T\_m = \sqrt{2R\_{sf} \Delta H\_m m \beta} \tag{21}$$

where *Rsf* is the thermal resistance and *m* is the mass.

As seen from Equation (21), for melting the shift of *Tp* with β is determined by physical parameters other than the activation energy. It means that, as a rule, the application of the Kissinger method to the *Tp* vs. β data would yield a number that does not represent the activation energy of melting. An exception to this rule are the compounds that undergo real superheating, that is, the compounds, whose temperature does rise during melting. For certain reasons [40], melting with superheating cannot be identified by plotting DSC signal against the sample temperature. A simple yet informative test [63] is checking whether the melting peak width increases with increasing either the heating rate or the sample mass. If it does, the melting occurs without superheating and the Kissinger method cannot be applied.

The absence of the aforementioned DSC peak broadening is indicative of melting with superheating. A more definitive (quantitative) criterion is based on determining the value of the exponent *z* in Equation (22):

$$T\_p = T\_m + \mathcal{B}\beta^z \tag{22}$$

where *B* and *z* are fit parameters. According to Toda [63], the *z* values markedly smaller than 0.5 indicate that melting occurs with superheating. In this circumstance, the Kissinger method can be applied to obtain an estimate for the activation energy of melting.

As an example of melting with superheating we consider the case of glucose. Superheating of this substance was reported by Tammann [64] and then reconfirmed by Hellmuth and Wunderlich [65]. For this compound, the *z* value (Equation (22)) has been found [66] to be 0.2 that confirms the occurrence of superheating. Applying the Kissinger method to the DSC data on glucose melting [67] gives rise to a plot (Figure 7) that is practically linear with a significant correlation coefficient (*r* = −0.9986). Correspondingly, the method yields a single constant activation energy 294 kJ mol−1 .Nonetheless, an isoconversional method yields activation energy that decreases from about 350 to 230 kJ mol−1 .

Important is that a decrease of the experimental activation energy of melting with increasing temperature is justified theoretically [40]. Theoretically, the kinetics of melting is treated by the nucleation and nuclei growth models (Equations (11) and (13)), that is, just as the kinetics of crystallization. Both models can be used to derive the temperature dependence of the activation energy. The growth model (Equation (13)) gives rise [67] to:

$$E\_{\perp} = E\_D - \frac{\Delta H\_m \exp\left[\frac{\Delta H\_m (T\_m - T)}{RTT\_m}\right]}{\exp\left[\frac{\Delta H\_m (T\_m - T)}{RTT\_m}\right] - 1} \tag{23}$$

and the nucleation model (Equation (11)) to:

$$E = E\_D + \Omega \left[ \frac{2T}{\left(T - T\_m\right)^3} + \frac{1}{\left(T - T\_m\right)^2} \right] \tag{24}$$

Note that Equation (24) is mathematically identical with Equation (16) as can be found by replacing (*T* − *Tm*) with − (*T* − *Tm*). Either Equation (23) or Equation (24) predicts E to decrease with increasing *T* and they both have the same asymptotes. When *T* is close to *Tm* (i.e., at small superheatings) *E* tends to +∞, whereas at large superheatings *E* tends to *ED*. gives rise to a plot (Figure 7) that is practically linear with a significant correlation coefficient (*r* = −0.9986). Correspondingly, the method yields a single constant activation energy 294 kJ mol−1.Nonetheless, an isoconversional method yields activation energy that decreases from about 350 to 230 kJ mol−1.

occurrence of superheating. Applying the Kissinger method to the DSC data on glucose melting [67]

*Molecules* **2020**, *25*, x FOR PEER REVIEW 13 of 17

As seen from Equation (21), for melting the shift of *Tp* with *β* is determined by physical parameters other than the activation energy. It means that, as a rule, the application of the Kissinger method to the *Tp* vs. *β* data would yield a number that does not represent the activation energy of melting. An exception to this rule are the compounds that undergo real superheating, that is, the compounds, whose temperature does rise during melting. For certain reasons [40], melting with superheating cannot be identified by plotting DSC signal against the sample temperature. A simple yet informative test [63] is checking whether the melting peak width increases with increasing either the heating rate or the sample mass. If it does, the melting occurs without superheating and the

The absence of the aforementioned DSC peak broadening is indicative of melting with superheating. A more definitive (quantitative) criterion is based on determining the value of the

where *B* and *z* are fit parameters. According to Toda [63], the *z* values markedly smaller than 0.5 indicate that melting occurs with superheating. In this circumstance, the Kissinger method can be

As an example of melting with superheating we consider the case of glucose. Superheating of

where *Rsf* is the thermal resistance and *m* is the mass.

applied to obtain an estimate for the activation energy of melting.

Kissinger method cannot be applied.

exponent *z* in Equation (22):

− = ට2௦Δ (21)

= + ௭ (22)

**Figure 7.** Kissinger plot for melting of glucose. Dash line is least square fit to experimental data (squares). Inset shows variation in isoconversional activation energy (circles). The heating rate range 1–11 °C min−1. Data from Liavitskaya et al. [67]. Reproduced from Ref. [67] with permission from the PCCP Owner Societies. **Figure 7.** Kissinger plot for melting of glucose. Dash line is least square fit to experimental data (squares). Inset shows variation in isoconversional activation energy (circles). The heating rate range 1–11 ◦C min−1 . Data from Liavitskaya et al. [67]. Reproduced from Ref. [67] with permission from the PCCP Owner Societies.

Important is that a decrease of the experimental activation energy of melting with increasing temperature is justified theoretically [40]. Theoretically, the kinetics of melting is treated by the nucleation and nuclei growth models (Equations (11) and (13)), that is, just as the kinetics of It should be emphasized that the curvature of the Kissinger plots can be more prominent than the one seen in Figure 7. This has been found in the melting kinetic studies on poly(ethylene terephthalate) [68] and poly(ε-caprolactone) [69]. The curvature is larger when melting occurs closer to *Tm*. In such a case, it can be sufficient to evaluate the actual dependence of *E* vs. *T* that can further be used for estimating the parameters of the nucleation (Equation (24)) and nuclei growth (Equation (23)) models via fitting [67–69].

Lastly, one should keep in mind that the models of nucleation and nuclei growth are potentially applicable to a wide variety of phase transitions taking place on heating and cooling. Since it has been already explained that the Kissinger method cannot be applied to the processes taking place on cooling, we only mention examples of phase transitions occurring on heating. These include gelation of aqueous solutions of methylcellulose [70], the coil-to-globule transition in aqueous solutions of poly(*N-*isopropylacrylamide) [71], and solid-solid transition in some salts [72]. All these transitions demonstrate the *E* values that decrease with temperature. In addition, the curvature of the Kissinger plots has been large enough to determine the *E* vs. *T* dependence [71,72]. Fitting Equation (24) to this dependence has afforded estimating the magnitude of the free energy barrier (Equation (12)) and its dependence on temperature.

#### **7. Conclusions**

An overview of the problems associated with the Kissinger method has been presented. The method is not very accurate. There are different kinds of inaccuracies associated with the assumptions made to derive its basic equations. The assumption of the first-order reaction model normally does not give rise to a large error in the activation energy when the process follows another reaction model. However, this error may become exceedingly large when the process is studied under a temperature program other than the linear heating. The assumption that the kinetics is measured on heating is embedded in the final equation of the method. This makes the Kissinger method unsuitable for any data obtained on cooling.

The method yields a single activation energy that is consistent with the assumption of single-step kinetics. This feature of the method is seen as its most essential problem because the majority of the processes it is applied to are not single-step kinetics. The problem has been illustrated by using common applications of the method. The Kissinger plots tend to be almost perfectly linear in the case of the multi-step kinetics, readily revealed by isoconversional methods. It means that the method tends to yield a single activation energy when a process is controlled by more than one energy barrier. For this reason, the Kissinger method is usually incapable of providing adequate insights into the processes kinetics. As shown by comparison, isoconversional methods provide a significantly more insightful alternative.

To finalize we should note that over the years there have been some developments of the Kissinger method. In particular, efforts have been made to obtain an analytical [73] and exact [74] solution to the Kissinger equation, to adjust it to melt crystallization [12] and reversible decompositions [27], and to exploit the nonlinearity of the Kissinger plots [10]. So far, these developments have had little impact on the mainstream usage of the method that boils down to the straightforward application of the original equation and reporting a single value of the activation energy.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The author declares no conflict of interest.

#### **References**

© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).


**5. Conclusions**

We presented an appropriate DR customer selection methodology for a Korean residential DR program to maximize the DR effect with lower customer enrollment. The proposed method showed better performance than other methods. Our method is divided into two parts. The first is customer segmentation according to load profile and consumption, and the second is targeted group selection based on two standards for DR participation. When we conducted customer segmentation, a two-stage clustering method was introduced. Customers were clustered by demand characteristics as variables in the first stage, and then segmented based on load patterns in the second stage. It can reflect more features of residential demand data than existing clustering methods, that makes better result in customer segmentation. Customer groups were classified as having higher DR potential by peak time and consumption patterns to select adequate groups having large potential in PTR program. As a result, the targeted groups were 1, 2, 3, 10, and 11 in our sample of residential customers in Korea, and their average demand reduction was 0.3496 (kWh), for an improvement of approximately 0.0876 (kWh), which increased savings by 33.44% compared to demand reduction due to opt-in enrollment. The proposed method allowed identifying enhanced DR effects. After the DR targeting demand reduction, we also conducted the cost-effective analysis of the PTR program from the perspective of the DR operator.

As a result, we observed that targeted DR capacity may be achieved with a small number of customers if targeted enrollment is implemented, which can use infrastructure and operation costs effectively. These results provide insights into the efficient use of DR in Korea. The number of customers and total DR capacity of targeted enrollment decreased compared with opt-in enrollment. However, if the number of customers who would like to participate in the DR program is high enough when the official full-scale program starts, selecting optimal customers among them would be more highly important. Therefore, the proposed method would be of great help in ensuring an efficient and economically sensible DR program in Korea.

We considered the residential customer targeting based on customer segmentation in demand response in this paper. Customer segmentation focus on the model structure to reflect features affecting demand response well. Some researches consider clustering model with heuristic algorithm in other areas, so we will apply this concept in further study.

**Author Contributions:** Conceptualization, E.L. and J.K.; data analysis, simulation, and methodology framework development, E.L.; writing, review, and editing, J.K.; supporting data collection and comments for improving the article, D.J. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by GIST Research Institute(GRI) grant funded by the GIST in 2020, and this work was supported by the Korea Institute of Energy Technology Evaluation and Planning(KETEP) and the Ministry of Trade, Industry & Energy(MOTIE) of the Republic of Korea (No.<PHONE_NUMBER>1930).

**Conflicts of Interest:** The authors declare no conflict of interest.


• Stage 1: driven by household demand, from 2000 to 2013

Since 2000, China has undergone rapid urbanization, along with a rapid increase in the application of domestic solar water systems. Strong demand from the consumption side drove the significant growth of the installation of solar water heating systems, which achieved 323.1 million m2 by the end of 2012 [4]. Incentive measures, i.e., subsidies, tax credit, and compulsory standards greatly promoted the deployment of solar heating. The solar heating market expanded rapidly with strong responses, since solar heating was integrated into the energy saving livelihood program (China started to implement the energy saving livelihood program in 2012, by providing subsidies for energy-saving products, including solar water products). This required direct delivery of subsidies for households that were equipped with solar heating systems. However, the year 2014 witnessed the beginning of a decline in the growth rate due to the expiration of the program. The market progress of solar heating is described in Figure 2.

In this stage, solar heating dominated the renewable heating market, with other renewable heating technologies accounting for little market share. This stage is marked by an initial stage of renewable heating, with household demand as the key driving factor.

**Figure 2.** The development of the solar heating market in China.

• Stage 2: driven by air pollution issues, from 2013 to 2017

In 2013, extreme haze weather hit northern areas of China with a high frequency. The increasing environmental awareness of the public and concerns over air pollution instigated heating reform focusing on clean energy transition [31]. In 2013, the state council (SC) issued the Air Pollution Prevention and Control Action Plan [32]. Following this, regulations on the substitution of coal for residential heating became a vital part of national air pollution control and environmental supervision. Moreover, renewable heat policies were introduced accordingly in this stage, focusing on specific technologies, such as biomass and geothermal energy. In 2013, a special policy titled "Guidance opinions on promoting utilization of geothermal energy" [24] was released, clarifying the main objectives, key tasks, and measures required to foster the geothermal heating industry. It is also worth noting that the first renewable heating target for the whole country was confirmed at this stage. In the 12th Five-Year Renewable Energy Development Plan [33], a quantitative indicator for renewable heating, namely the amount of substitution of fossil energy, was established. Subsequently, in the 13th Five-Year Renewable Energy Development Plan, the targets were further enhanced, from 10 billion tce by 2015 to 15 billion tce by 2020 [34].

At this stage, policies on renewable heating began to be more vigorously implemented, and specific quantitative targets for renewable heating development were determined for the first time. The primary policy measures in this stage were aimed at providing resolutions to air pollution issues, and renewable heating was deemed as a supplementary solution to replacing coal.

• Stage 3: driven by demand of higher air quality

In 2017, 10 ministries and commissions jointly promulgated the Plan for Clean Heating in Northern China (2017–2021) [23], which formally defined the energy sources of clean heating, including geothermal energy, biomass, solar energy, natural gas, electricity, industrial waste heat, clean coal combustion, nuclear energy, etc. In the policy, an ambitious target was proposed, striving to achieve a 50% clean heating rate by 2019 and 70% by 2021. Furthermore, renewable heating was for the first time identified as the main source of clean heating energy rather than playing a supplementary role in the previous stage. Specific goals were given for each renewable heating technology, i.e., one billion square meters for geothermal energy, 2.1 billion square meters for biomass, and 50 million square meters for solar energy by the end of 2021.

To ensure the implementation of the plan, regulations on subsidies and price mechanisms were also released at this stage. In 2017, the Ministry of Finance provided subsidies to 43 pilot cities, with an annual subsidy of one billion RMB for municipalities, 700 million RMB for provincial capitals, as well as 500 million RMB for other cities [35]. Following

this, the National Development and Reform Commission (NDRC) issued a policy titled "Opinions on Clean Heating Price in North China" to overcome the issue of price distortion, clarifying that renewable energy should be given high priority in heating energy supply and linkage of electric heating, wind, and solar power generation curtailment should be established [36]. Thereafter, the 13th Five-Year Development Plan of Geothermal Energy [37] specified the goals and strengthened the measures, in which specific geothermal modes, shallow geothermal systems, and hot rock geothermal systems were introduced to solve the problem of winter heating where applicable. Meanwhile, in the 13th Five-Year Development Plan of Biomass Energy [26], biomass heating, especially biomass co-generation, was highlighted to supplement the energy supply with regard to the implementation of "coal to gas" or "coal to electricity" policies.

At this stage, renewable heating was formally integrated into the clean heating system as a main source of energy supply, rather than a supplement. Applying more diverse policy instruments could contribute to ensuring the implementation of the clean heating plan.

• Stage 4: driven by ambitious climate change goals, 2021–

In 2021, a comprehensive policy titled "Notice on advancing the development of renewable heating according to local conditions" was launched [38], consisting of six aspects: (1) release special plans for renewable heating, including specific renewable heating goals aligned with the requirements of climate change goals; (2) promote various types of renewable heating technologies according to local conditions; (3) continue promoting pilot demonstration work and major project construction; (4) guarantee policy support of renewable heat; (5) strengthen R&D support for key technical equipment; and (6) improve the government management system for renewable heat. The policy symbolized the significant achievements of clean heating policies in China and initiated a new era of renewable heating development. The 30/60 targets served as the dominant driver for this stage. Subsequently, a series of policies concerning renewable heating were released. The most recent policy is "Several opinions on promoting the development and utilization of geothermal energy" [25], jointly issued by eight ministries, which highlights the importance of geothermal utilization.

In summary, the year 2021 witnessed the transformation of renewable heat policy with the release of a comprehensive renewable heating program. This indicates that renewable heating in China has been formally integrated into the energy transition system as a key measure to address climate change.

#### **4. Evaluation of Renewable Heat Policy**

In China, policies have varying impacts according to the authority level of the issuer. Regulations issued by the state council generally have the highest authority, followed by regulations or policies jointly issued by multiple ministries. Policies released by the signal ministry or province (autonomous region, municipality) have the lowest authority. Furthermore, the efficacy of a policy is also greatly related to the document type, i.e., law has the highest efficacy, followed by strategy.

A policy instrument provides the link between policy formulation and policy implementation. The instrument used in a policy has a significant impact on how well it is implemented [39]. Policies can be divided into four categories using the instrument applied: regulation-based policy, fiscal instrument-based policy, price mechanism-based policy, and financial instrument-based policy. Regulation-based instruments include targets, planning, compulsory standards, etc. Fiscal instruments commonly include subsidies, grants, tax credits, etc. Price-based instruments include heat-trading mechanisms, carbon markets, etc. Typical financial instruments are bonds, loans, and direct equity investments [4].

In this section, renewable heat policies are evaluated from two perspectives. First, we focus on the instrument in the policies to find out how the policy is implemented. Second, PII is used to further understand the intensity of the existing policies and their efficacy.

#### *4.1. The Instruments of Renewable Heat Policies*

In this paper, the cumulative number is used to quantify the instrument applied in policies. To be specific, policies are classified according to the type of instrument and a specific instrument that appeared in a specific year was counted in the subsequent years. Policies that use several instruments are counted into primary instrument types to avoid double counting. The cumulative numbers of the four types of policies according to instruments applied from 2000 to 2021 are illustrated in Figure 3.

**Figure 3.** Development of policies according to instruments.

Overall, it demonstrated an upward tendency of all four types of renewable policies since stage 2, particularly for regulation-based policies, which increased from 20 to more than 100 over seven years. Comparatively, it can be seen that the other three types all showed some degree of delay. The amount of fiscal incentive policies has grown moderately since 2015, while the price and financial instrument-based policies began to increase steadily only after 2017.

In terms of the amount of the four types of policies, regulation-based policies are largest with a share of approximately 70% by 2021. Fiscal incentives began to increase in 2017, leading to a share of 11% by the end of 2021, with the remaining two types at less than 20%. Regulation-based policies are preferred, with an amount much higher than the other three instruments, showing a structured imbalance in the application of different instruments.

#### *4.2. Evaluation of Policy Intensity*

To further illustrate the dynamic changes in renewable heat policies over the last 20 years, the policy intensity index from 2012–2021 was gained using the model introduced in Section 2. The results are shown in a bubbling figure (Figure 4), where the size of the bubble represents the intensity of the policies of a specific year.

It can be seen that the intensity of the four types of policies showed different patterns, among which regulation-based policies showed the strongest intensity of all four stages. The intensity of regulation-based policies reached the highest level in the year 2017, which can be attributed to a series of policies issued by high-level authorities, including the SC. Fiscal relevant policies followed after and financial instrument-based policies had the lowest intensity. The years following 2017 witnessed an increase in the intensity of the four types of policies, indicating the increasing political will for renewable heating from the policy maker.

**Figure 4.** Policy intensity index of the four types of policies.

For the four types of policy instruments, the following can be found if Figures 3 and 4 are combined:

Regulation-based policies dominated in the policy system, which were both largest in amount and strongest in stringency. The high-frequency application of regulationbased instruments in renewable heat policies reflects that the renewable heating market is still primarily driven by governments and is still at an initial stage, where mandatory regulations are greatly needed. However, it is not recommended that regulation-based policies are applied long term, due to the lack of flexibility and cost effectiveness.

Fiscal incentives have been implemented since 2017 when the clean heating program was initiated, marking a major step. However, the incentive options are limited, and subsidy is the most common approach. Fiscal instruments can play a great role in the initial stages by supplementing the capital cost, but put great pressure on the government. More diverse instruments can be deployed when providing fiscal support. One example is the renewable heating incentive (RHI) from the United Kingdom, which is similar to the tariff mechanism in the generation market. The tariff mechanism can be more result-orientated and flexible compared to subsidy.

Market-based policies, including price mechanism and financial incentives, are least used. Market-based policies are seen as the most cost-efficient way to foster a renewable market by internalizing the environmental and economic cost. The application of marketbased policies should be greatly accelerated.

#### **5. Conclusions and Suggestions**

#### *5.1. Conclusions*

China's renewable heat policies have advanced quickly in the last two decades. Numerous policies have been implemented. However, few studies have examined the evolution of these policies. We used the PII model to present the dynamic changes in the renewable heat policies and three conclusions were summarized as follows.

Firstly, we systematically reviewed and illustrated the overall development of renewable heat policies from 2000 to 2021. China has established a renewable heat policy system, which consists of five levels of policy, each with different power and authority. The top levels are law, strategies, and plans, which provide macro guidance for long-term development. The middle levels are comprehensive policies and regulations in the form of five-year plans, special plans, etc., to further specify targets and measures. The policies at the bottom levels are commonly designed for guiding implementation, including supporting policies with incentives, special plans, or regulations, targeting specific renewable heating technology models and local policies.

Secondly, the 20 years were divided into four stages. In stage 1, the main driver was the increasing heating demand from households, when the market for solar water heating soared quickly. In this stage, regulation-based policies, such as compulsory standards, pushed the expansion of the market. Furthermore, incentives also played an important role. The second stage began in 2013. In this stage, issues with air pollution pushed the development of renewable heating, with more political attention paid to renewable heating to supplement the coal-to-gas program in fighting against air pollution. In 2017, China initiated its clean heating plan, when renewable heating began to formally integrate into the clean heating system. The fourth stage was driven by new climate change goals. In this stage, the development of renewable heating became a certainty. The renewable heating development of the past 20 years has a strong link with the socio-economic context, showing various features in different stages. Thirdly, to further dive into the renewable heat policy, we classified the existing renewable heat policies into four categories according to the instruments applied. The results showed that the structure of the four instruments were in disproportion, with regulation-based instruments the most preferred. While the government began to apply other instruments to overcome price distortion and other issues, the impact is still limited. The number of policies with all four instruments increased dramatically since stage 3, when China began to implement its clean heating program. To enhance the implementation of the 30/60 targets, the application of market-based policies are greatly needed.

#### *5.2. Policy Suggestions*

To further accelerate the deployment of renewable energy in the heating sector, the following suggestions are proposed:

Firstly, multiple barriers in the renewable heating sector call for a range of policy instruments, often in combination. Regulation-based instruments, which generally lack flexibility and cost-effectiveness, should be supplemented by other policies, especially market-orientated policies. The application of regulation instruments can be strengthened by increasing the application of other types of policy instruments. Instrument diversification is required to strike a balance between cost, adaptability, and efficacy.

Secondly, the incentive system for renewable heating needs to be strengthened. The externality of renewable heating can be partially compensated by incentives. Instead of relying solely on subsidies, the government can use a variety of incentives, such as tax breaks, tariffs for renewable heating sources, etc., to alleviate the financial load. Additionally, innovative measures, such as result-based incentives and loan guarantees, can also be applied when applicable.

Thirdly, market-orientated instruments should be further enhanced in policies. Carbon market and green finance instruments have great potential in further enhancing the development of the renewable heating market. Financial instruments should be more embodied in future policies, such as green loans and green bonds.

Finally, policy coordination should be further strengthened. Coordination can improve the efficiency of resource allocation, as well as avoid unnecessary conflict in implementation.

The present work provides a first and comprehensive review of China's renewable heating policy at the national level. Further research regarding the role of the policy's executor, the local government, would be of great help in understanding the effectiveness of renewable heating. In addition, a considerable amount of work is needed to evaluate the impact of renewable heating on energy transition and carbon emission, to examine the evolution of household heating patterns, and to explore the possible demand-side management option using an experimental approach.

**Author Contributions:** Writing—original draft preparation, C.X.; supervision, M.S.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.


**2. The Sahel: geography, livelihood and overview of climate change**

The Sahel is the semiarid strip of land located between the tropical rainforest in the south and the arid north of Africa and covers an area of about 3.053 × 103 km2 and has about 60 million inhabitants [43]. The Sahel is located between latitude 10° and 20° north and extends from about 5000 km around northern Senegal around the west towards southern Mauritania, central Mali, northern Burkina Faso, south-western Niger, northern Nigeria, central Chad,

*Recent Climate Change Adaptation Strategies in the Sahel: A Critical Review DOI: http://dx.doi.org/10.5772/intechopen.100110*

#### **Figure 1.**

*Location of the various (a) countries and (b) regions of the Sahel.*

north of Cameroon, Central African Republic, central Sudan and southern Sudan, northern Eritrea, extreme north of Ethiopia, to Somalia in the east and south east of the Sahel into Kenya (**Figure 1**).

In the Sahel, the vegetal landscape is covered by open Acacia shrubs and grassland. The Sahel represents a transition between the humid savanna in the south and the desert in the north [44, 45]. In terms of rainfall, the Sahel experiences declining rainfall with increasing latitude. At the southern border of the Sahel, about 450– 500 mm of rainfall are recorded yearly while towards the higher latitudes less than 200 mm of rainfall are recorded yearly [46, 47]. Between 1930 and 1965 and 1966 and 2000, the Sahel recorded about 100 mm of rainfall per year [48]. The rainfall pattern in the Sahel is tied to the migration of the Inter-tropical Convergence Zone (ITCZ) [44, 46, 47, 49]. In the Sahel, a rainfall gradient of between 250 and

300 mm between the southern and northern spheres of the region is recorded. At 17° latitude north less than 200 mm of rainfall are recorded annually while southwards at about 15° latitude north (southern boundary of the Sahel), more rainfall of about 450–500 mm is recorded annually [44, 46, 47].