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Michael I. Ojovan, [Anh Khoa Augustin Lu](https://orcid.org/0000-0003-4702-0933), Dmitri V. Louzguine-Luzgin

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[Massive Fluctuations in the Derivatives of Pair Distribution Function Minima and Maxima During the Glass Transition](https://mdr.nims.go.jp/datasets/788d282b-6176-4390-8cde-6405be9b4ff9)

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Massive Fluctuations in the Derivatives of Pair Distribution Function Minima and Maxima During the Glass TransitionAcademic Editor: Golden KumarReceived: 12 June 2025Revised: 20 July 2025Accepted: 30 July 2025Published: 2 August 2025Citation: Ojovan, M.I.; Lu, A.K.A.;Louzguine-Luzgin, D.V. MassiveFluctuations in the Derivatives of PairDistribution Function Minima andMaxima During the Glass Transition.Metals 2025, 15, 869. https://doi.org/10.3390/met15080869Copyright: © 2025 by the authors.Licensee MDPI, Basel, Switzerland.This article is an open access articledistributed under the terms andconditions of the Creative CommonsAttribution (CC BY) license(https://creativecommons.org/licenses/by/4.0/).ArticleMassive Fluctuations in the Derivatives of Pair DistributionFunction Minima and Maxima During the Glass TransitionMichael I. Ojovan 1,* , Anh Khoa Augustin Lu 2,3,4 and Dmitri V. Louzguine-Luzgin 4,51 School of Chemical, Materials and Biological Engineering, The University of Sheffield, Sheffield S1 3JD, UK2 Department of Materials Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-8656, Japan;lu.augustin@cello.t.u-tokyo.ac.jp3 Research Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science (NIMS),Tsukuba 305-0044, Japan4 MathAM-OIL, National Institute of Advanced Industrial Science and Technology (AIST),Sendai 980-8577, Japan; dml@wpi-aimr.tohoku.ac.jp5 Advanced Institute for Materials Research (WPI-AIMR), Tohoku University, Sendai 980-8577, Japan* Correspondence: m.ojovan@sheffield.ac.uk; Tel.: +447883891379AbstractParametric changes in the first coordination shell (FCS) of a vitreous metallic Pd42.5Cu30Ni7.5P20alloy are analysed, aiming to confirm the identification of the glass transition temperature(Tg) via processing of XRD patterns utilising radial and pair distribution functions (RDFs andPDFs) and their evolution with temperature. The Wendt–Abraham empirical criterion of glasstransition and its modifications are confirmed in line with previous works, which utilised thekink of the temperature dependences of the minima and maxima of both the PDF and themaxima of the structure factor S(q). Massive fluctuations are, however, identified near the Tgof the derivatives of the minima and maxima of the PDF and maxima of S(q), which addsvalue to understanding the glass transition in the system as a true second-order-like phasetransformation in the non-equilibrium system of atoms.Keywords: metallic glasses; glass transition; fluctuations; second-order phase transition1. IntroductionGlass transition is a generic phenomenon characteristic of amorphous materials rang-ing from pure elements and oxides to complex polymeric and biological molecules thatexhibit solid-like behaviour below the glass transition temperature (Tg, i.e., in the vitreousstate and liquid-like state above it) [1,2]. Most often, Tg is determined using differentialscanning calorimetry (DSC), which shows a well-seen kink in the temperature dependenceof the heat flow evolution of the system at Tg [3,4], although other techniques to determineit are in use that are similar to DSC, i.e., simple (thermal mechanical analysis, TMA) and dif-ferential thermal analysis (DTA [5]), which employs various heating and cooling (thermal)cycles. Furthermore, dynamic mechanical analysis is used, in which mechanical stress isapplied to the sample, and the resultant strain is measured (dynamic mechanical analysis,DMA, [6]), including specific heat measurements, thermomechanical analysis, thermal ex-pansion measurements [7,8], micro-heat transfer measurement, isothermal compressibility,and heat capacity determination [9]. Structural rearrangements in the non-equilibriumsystem of species (atoms and molecules) forming a glass are not yet well understood despitenumerous works related to glass-forming systems, including bulk metallic glasses [10].Wendt and Abraham were the first to observe that Tg can be detected by analysing theMetals 2025, 15, 869 https://doi.org/10.3390/met15080869https://doi.org/10.3390/met15080869https://doi.org/10.3390/met15080869https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://www.mdpi.com/journal/metalshttps://www.mdpi.comhttps://orcid.org/0000-0001-8928-4879https://orcid.org/0000-0003-4702-0933https://orcid.org/0000-0001-5716-4987https://doi.org/10.3390/met15080869https://www.mdpi.com/article/10.3390/met15080869?type=check_update&version=1Metals 2025, 15, 869 2 of 12temperature behaviour of the maxima and minima of radial and pair (normalised accordingto the average atomic number density variation with the interatomic distance) distributionfunctions (RDFs and PDFs, respectively), which result from the processing of X-ray orneutron-scattering spectra, as well as from computer experiments using molecular dynamic(MD) simulation [11,12]. They proposed an empirical criterion for finding Tg from theratio RWA = PDFmin/PDFmax = 0.14 [13]. We showed that the experimentally found criterionRWA = 0.14 is explained in terms of percolation in a system of broken bonds in amorphousmaterials termed configurons [14]. Indeed, it is known that the rigidity threshold of anelastic percolating network is identical to the percolation threshold [15]; therefore, we canfind the critical temperature Tg when the solid-like behaviour changes to a liquid-likebehaviour, assigning it to the temperature when percolation via broken bonds occurs [16].The fraction of broken bonds (configurons) ϕ(T) as a growing function of temperatureequals ϕ(T) = PDFmin/PDFmax; therefore, the temperature of glass transition in amor-phous materials is indeed in line with the Wendt–Abraham empirical rule provided by thefollowing equation: ϕ(T) = θc. For metallic systems, θc = 0.15± 0.01, which is equal to theuniversal Scher–Zallen critical density in 3-D space: θc = 0.15 ± 0.01 [17,18]. The presenceof kinks in the glass transition was later confirmed by directly analysing the temperaturedependences of structural factors S(q), where q is the scattering vector [19].The presence of kinks at Tg in the temperature dependences of radial and pair dis-tribution functions (RDFs and PDFs) and the structure factor (S(q)) means that their firstderivatives experience a jump, which may be useful to more easily detect the glass transi-tion. Moreover, the second derivatives of RDFs, PDFs, and S(q) have a diverging characterat Tg, which could facilitate identification. We aim to analyse these features, which, al-though being present in the temperature behaviours of RDF, PDF, and S(q), were found tonot be amenable for practical usage due to the character of the glass transition, which, beinga second-order phase transformation, is accompanied by large and increasing amplitudefluctuations near Tg.2. Theoretical ConsiderationsThe pair distribution function, denoted as g(r), provides the probability of findinga particle at the distance R from another particle, i.e., it is the probability of findingtwo particles i and j at a particular separation r = |ri − rj| in the system. In contrast, thestructure factor S(q) found from measurements in scattering experiments is essentiallythe Fourier transform of the PDF (also denoted as g(r)), being related to each other viathe following:g(r) = 1 +12π2rρ0∫ ∞0q[S(q)− 1]sin(qr)dq, (1)where ρ0 is the average density and q is the scattering vector. PDF(r) and S(q) provide struc-tural and thermodynamic information about the system [14,19–21]. If there is more thanone atom type present, such as in the case of metallic alloys or oxide glasses, then PDF(r) istypically split into several terms, one for each pair of atomic specie types: e.g., in the caseof two species α and β, the partial pair distribution function that characterises correlationsbetween atoms of type α and β is PDFαβ(r) = gαβ(r), provided by the following [22]:gαβ(r) = 1 +12π2rρ0∫ ∞0q[Sαβ(q)− 1]sin(qr)dq, (2)where Sαβ(q) is the Faber–Ziman partial structure factor, which, similarly to monoatomicsystems, follows the rule Sαβ(q)→1 for all α and β. The functions PDFαβ(r) serve as measuresof the probabilities of finding a β atom at a distance r from an α atom and calculating theMetals 2025, 15, 869 3 of 12partial coordination numbers nβα which determine the average number of β atoms in aspherical shell around an α atom by integrating partial radial distribution functions.Typical investigations analyse the behaviour of the first sharp diffraction peak (FSDP)of the structure factor S(q) [19–22], which reveal features in the reciprocal space, whilst themost prominent and intuitively straightforward features in the real space are providedby peaks of RDF and PDF [13,14,19,20,23–25]. The forms of PDF(r) are hence used tounderstand changes that occur in glasses and melts on temperature variations includingstructural modifications at the glass transition. The maximum of PDF(r) is positioned atthe most probable radii where atoms reside, whereas the minimum of PDF(r) is related tobond distances. The PDFmin(r) is positioned at the end of the first coordination shell (FCS)and corresponds to bonds connecting atoms which, e.g., was illustrated by data for H2Othat produces a negative peak at the OH bond distance [26]. The PDF(r) has a peak at amean inter-particle distance and converges with the increase in distance r oscillating to unitg(r→∞) = 1 (Figure 1). Figure 1. Pair distribution function PDF = g(r) of amorphous Ti2Ni (Ti67Ni33) alloy obtained viamolecular dynamic (MD) simulation at a cooling rate of ≈1012 K/s (Adapted with permissionfrom [14]. Copyright 2020 American Chemical Society).From experiments for both monoatomic [20] and multiatomic systems [14,23,24,26,27](as Figure 1 demonstrates), it is known that upon an increase in temperature, the maximumof PDF(r) of amorphous materials (PDFmax) decreases its amplitude while the first PDF(r)minimum (PDFmin) is(i) Increasing its amplitude;(ii) Shifting its position to larger values.The shift of the position of PDFmin reflects [14](iii) The formation of configurons (broken chemical bonds);(iv) Enlarging the size of the first coordination shell (FCS).The position of PDFmin is giving the radius of FCS, which increases with the increasein temperature, i.e., this is characterising the thermal expansion of materials due to theincrease in temperature.Metals 2025, 15, 869 4 of 12The experiment is explicitly showing the maximal amplitude of structure factor S(q1),where q1 is the position of the scattering vector where the maximum occurs, with changesin temperature exhibiting a kink as follows:S(q1) = S0 − sgT at T < Tg andS(q1) = S0 − sgTg − sl(T − Tg)at T > Tg(3)The difference in behaviour of the structure factor below and above the Tg is illustratedby Figure 2a.  (a) (b) Figure 2. Variation with temperature of structural factor S(q) and pair distribution function PDF(r)on crossing the glass transition temperature: (a) the first maximum of the structure factor S(q)maxand its shifting position q1 reflecting the thermal expansion of Pd40Cu30Ni10P20 bulk metallic glass(Reprinted with permission from Ref. [19], AIP Publishing); (b) the values of the pair distributionfunction first minimum PDFmin of Cu following the method proposed in (Adapted from Ref. [14]) andratios of PDFmin/PDFmax after the Wendt–Abraham criterion (Adapted from Ref. [13]) as a functionof temperature, where the inset shows the definitions of parameters used with PDF(r) given forthree temperatures, T = 2500, 1400, and 300 K, respectively. Reproduced with permission fromRef. [25], MDPI.The coefficient of proportionality (s) for liquids (melts) is always larger compared tosolids (glasses): sl > sg. Due to this, from (3) we see that the temperature derivative of thestructure factor exhibits a stepwise change at Tg:∂S(q1)/∂T = −sg at T < Tg and∂S(q1)/∂T = −sl at T > Tg(4)The second derivative of the structure factor thus has a singularity at the glass transition:∂2S(q1)/∂T2 = −δ(T − Tg)(5)Because steps and deltas can be readily detected from available data, it would beuseful to attempt to use (3) and (4) in detecting the glass transition.The experiment also explicitly shows that the PDFmin follows the same character oftemperature dependence near the glass transition exhibiting at Tg a kink:Metals 2025, 15, 869 5 of 12PDFmin = P0 + fgT at T < Tg andPDFmin = P0 + fgTg + fl(T − Tg)at T > Tg(6)The coefficient of proportionality fl in (6) obeys the rule fl > fg as seen explicitly fromFigure 2b. From (6), we see that the temperature differential of the pair distribution functionexhibits a stepwise change at Tg:∂PDFmin/∂T = fg at T < Tg and∂PDFmin/∂T = fl at T > Tg(7)The second derivative of the minimum of the FSDM thus also has a singularity at theglass transition:∂2PDFmin/∂T2 = δ(T − Tg)(8)The utilisation of (5) and (8) along with the Wendt–Abraham empirical criterionwould hence be a powerful tool in detecting the glass transition in amorphous materials,making the detection of Tg much easier and obvious in practice compared to analyses ofdependences of S(q) and PDF(r) with temperature. Based on these ideas, we attemptedto process data on vitrifying the metallic alloy Pd-Cu-Ni-P, for which data are availablefor a confident analysis. As it can be explicitly seen from Figure 2 both in the case ofpolyatomic (Figure 2a, Equation (1) case) and monoatomic (Figure 2b, Equation (1) case)systems, the linear dependencies occur, which change their slope at the inflection point.This is a generically known fact for all glass-forming systems and serves as the basis oftest protocols aiming to identify the calorimetric glass transition [3–6,9]. It means thatEquations (4), (5), (7) and (8) are mathematically valid expressions if the linear dependen-cies are firstly found and then processed by applying these equations. Our task in this workwas to check whether the same results have been obtained if we directly process experi-mental data without first determining the two linear dependencies below and above the Tg.Two outcomes are then possible. The first expected result is confirming the same results andis for the case when fluctuations (noises) are not growing when approaching the infectionpoint. The second one is a complete failure and is expected when the fluctuations becomemassive upon approaching the infection point; this is the situation that is typical for phasetransformations and hence would indicate a phase change on passing the Tg.3. Experimental ResultsThe glass formation process of the Pd42.5Cu30Ni7.5P20 alloy was studied in situ in theprevious works [28,29]. In the present work, these data were processed using Equations(4), (5), (7) and (8), aiming to check whether the temperature derivative obeys the expecteddependences following these equations. There was not much attention paid to the jumpsof derivatives of PDFmin in the literature apart from [14,23–25], e.g., the step (jump) atTg = 794 ± 10 K of Cu was found to be as high as (fl − fg) = 96 ppm (see Figure 4 of [25]). Wealso accounted that although following the laws of second-order phase transformations, theglass transition has a dual nature and is kinetically controlled due to relaxation phenomenathat occur in parallel to structural rearrangements [30,31].Figure 3 shows the following: (a) the temperature dependence of the RDF minima ofthe Pd42.5Cu30Ni7.5P20 alloy, confirming via kinks observed that the vitrification occurs atTg ≈ 300 ◦C via minima of PDFs; and (b) the Wendt–Abraham criterion, which used thePDF’s minima to the PDF’s maxima ratios.Metals 2025, 15, 869 6 of 120 100 200 300 400 500 6000.60.70.80.91.0 Min1 Min2 Min3 Min4PDF(R) minT, oCTg~300 oC  (a) (b) Figure 3. Variation with temperature of (a) minima of PDFs; (b) ratios of minima to maxima of PDFfollowing Wendt–Abraham (Adapted from Ref. [13]). The glass transition temperature is identifiedas Tg ≈ 300 ◦C.It is worth comparing data obtained with earlier works by Mattern et al. [19], who anal-ysed the temperature dependences of structure factor S(q1), which followed dependences(4) and (5), with Figure 4 demonstrating the temperature dependence of the structure factorof the Pd42.5Cu30Ni7.5P20 alloy and hence confirming its behaviour being fully in line withprevious findings by Mattern et al. in [19]. Figure 4. Variation in the first maximum and minimum of the structure factor S(q) with temperature.Figure 5 shows the temperature dependences of the first and second derivatives of themaximal value of the structure factor S(q1) in our case after processing our (D.V.L.-L. et al.)previous data taken from references [28,29].Metals 2025, 15, 869 7 of 12  (a) (b) Figure 5. Variation with temperature of (a) the temperature derivatives of the first maximum valueof the structure factor |∂S(q1)/∂T| and (b) the second temperature derivative |∂2S(q1)∂T2|. Theidealised stepwise (in (a)) and delta-function-wise (in (b)) behaviours expected without account offluctuations are shown by grey colour. The glass transition temperature was previously identified asTg ≈ 300 ◦C.While the expected behaviour of |∂S(q1)/∂T| as shown in grey colour in Figure 5ais a stepwise function (the Heaviside step function) at Tg, the experimental data of thefirst derivative of the structure factor show an extremely noisy behaviour at it. Then,instead of the expected delta function as follows from (5) and shown in grey colour inFigure 5b, the experimental data of the second derivative demonstrate a completely noisyand uncontrolled spread including both positive and negative values. Thus, instead ofthe expected Heavyside step function and delta function at Tg, we observed massivefluctuations for experimental data. Similarly to the above, strongly fluctuating data wereobtained for ∂PDF/∂T and ∂2PDF/∂T2 at Tg, which is now expectedly appropriate, as thePDF(r) and S(q) are interrelated via Equations (1) and (2).4. DiscussionThe results obtained conform to previous data on the Wendt–Abraham empiricalcriterion of the vitrification of melts while being cooled fast enough [13] and its modificationfor the first minimum of PDF(r), denoted as PDFmin [14], which through this provides proofsof configuron formation and the expansion of the FCS, e.g., see Table 1 of [23]. The glasstransition temperature found agrees with data previously known—see, e.g., Figure 2a—although there is some spread of experimental points near the inflection point. Figure 2bwith more recent results from [25] also shows high deviations from the idealised linearapproximation near the inflection point, which identifies the Tg. In the meantime, theattempts to use the stepwise temperature dependence of ∂S/∂T and the diverging characterof ∂2S/∂T2 at Tg have clearly failed. The reason behind this is in the characteristics ofthe glass transition and massive fluctuations of both ∂S/∂T and ∂2S/∂T2 instead of theidealised dependences via Equations (4), (5), (7) and (8), which are also shown in Figure 5by grey coloured lines.The glass transition expresses itself at the calorimetric glass transition tempera-ture [3,32–34] as a second-order phase transformation with all its attributes followingthe Ehrenfest classification of phase transformations [35–37]. Namely, it is a continuoustransformation with continuous thermodynamic functions such as Gibbs free energy G(T,P),entropy S(T,P), volume V(T,P), and discontinuities in response functions (susceptibilities)such as heat capacity, compressibility, and the thermal expansion coefficient, and all theseare always seen during the glass transition [3,32–34,38]. Due to this, the International Unionof Pure and Applied Chemistry (IUPAC) defines the glass transition as a second-ordertransition in which a supercooled melt yields, on cooling, a glassy structure, so that belowMetals 2025, 15, 869 8 of 12the glass transition temperature, the physical properties vary in a manner similar to thoseof the crystalline phase [39].The theoretical analysis of phase transitions is well known, e.g., in the first-orderphase, transitions such as crystallisation of the correlation length (e.g., the size of the newphase) remains finite, while for continuous phase transitions the correlation length divergeswhen approaching the phase transition. The configuron percolation theory (CPT) of glasstransition [16,22,40] gives for the correlation length a diverging dependence at temperaturesapproaching Tg:ξ(T) ∝ξ0∣∣T − Tg∣∣ν (9)where ξ0 has the order of interatomic distance, the critical exponent ν in three-dimensionalspace is ν = 0.88, and ξ(T) diverges at Tg, in contrast to the structural coherence lengththat characterises the exponential decay of atomic pair distribution function oscillationsbeyond the first peak, which increases with decreasing temperature and freezes at theglass transition [41]. The amorphous material near the glass transition is dynamicallyinhomogeneous on length scales smaller than ξ(T), while at temperatures far from theTg, the correlation length becomes small, and the amorphous material is homogeneous.Fluctuations in the system of disordered species (atoms or molecules) become correlatedover all distances, and that forces the whole system to be in a unique phase, which is criticalat the phase transition [36,37,42,43].The phenomenon of increasing fluctuations in the vicinity of a phase transition isbest demonstrated with the critical opalescence phenomena and is already known to bepresent for glass transition [44,45]. Therefore, the massive fluctuations in approaching theglass transition temperature evidently seen in Figure 5 should not be surprising whileinterpreting the glass transition as a true second-order phase transformation, although itoccurs in a non-equilibrium system of atoms and molecules constituting the amorphousmaterial. It is now recognised that in addition to equilibrium phase transitions [36,45],non-equilibrium phase transitions are rather common across a wide range of scientificdisciplines [46–49], manifesting in a rich variety of both static and dynamic patternsincluding ergodicity breaking; the Mpemba [50–53], Bokov [44], and Kovacs effects [54,55];and the asymmetry of heating–cooling [56,57]. It is worth noting that the classification ofglass transition as a second-order transition in Ehrenfest terms remains debated within thescientific community. Most physicists and materials scientists agree that it does not meet thecriteria for a second-order equilibrium transition, mainly due to its non-equilibrium natureand time dependence. In this respect, the extension of the analysis of glass transition to thatbelonging to phase transformations in non-equilibrium systems is assisting in unveiling itsnature and dual, both kinetic and thermodynamic, character [30,40].Thus, the fluctuations (inherent noises) encountered upon analysing the glasstransition in the Pd-Cu-Ni-P alloy (a very fragile metallic glass with a fragility indexm = 60 [58,59]) are well expected within the theory of second-order (or second-order-like)phase transformations and once observed can be considered as an additional argument infavour of the effects associated with a true phase transformation. Moreover, there are somereasons to believe that fragility can be related to the properties of the resulting glass [60].The proposed delta-like or step-wise behaviour in structural derivatives (7) and (8) cannotbe directly revealed from experiments, namely due the intrinsic noises/fluctuations thatare always associated with second-order transitions.While the glass transition is traditionally viewed as a kinetic freezing process, strongevidence from modern studies supports its interpretation as a true phase transition [38,61]or a topological phase transition [62]. Unveiling the structural mechanisms behind the glasstransition then enables practical utilisation in various applications [63–66]. The Wendt–Metals 2025, 15, 869 9 of 12Abraham empirical criterion [13] and its analogues [14,27] enable the identification of theglass transition temperature Tg. However, why is the glass transition interval quite wide?We know that glass is not uniform: it has densely packed and loosely packed regions [67].It can be assumed that the glass transition of these regions occurs at slightly differenttemperatures, which gives the glass transition interval. Moreover, two relaxation processescompeting with each other related to the different diffusion coefficients of the alloyingelements were observed [68].5. ConclusionsThe Wendt–Abraham empirical law of glass transition is confirmed as valid for thePd-Cu-Ni-P metallic glass-forming alloy, which conforms to previous works. Attempts toprocess derivatives of the pair distribution function and structure factor failed because ofmassive fluctuations on approaching the glass transition, which confirms the concept ofglass transition as a second-order-like phase transformation in the non-equilibrium systemof atoms following the Ehrenfest classification scheme. Consequently, the observed massivefluctuations have a generic character for amorphous systems at the glass transition.Author Contributions: Conceptualization, M.I.O. and D.V.L.-L.; methodology, M.I.O.; software,A.K.A.L.; validation, D.V.L.-L. and A.K.A.L.; writing—original draft preparation, M.I.O., D.V.L.-L.and A.K.A.L.; writing—review and editing, M.I.O., D.V.L.-L. and A.K.A.L.; funding acquisition,A.K.A.L. All authors have read and agreed to the published version of the manuscript.Funding: This study was partly supported by JSPS KAKENHI under Grant No. 24K01284.Data Availability Statement: The original contributions presented in this study are included in thearticle. Further inquiries can be directed to the corresponding author.Conflicts of Interest: The authors declare no conflicts of interest.AbbreviationsThe following abbreviations are used in this manuscript:CPT Configuron percolation theoryFCS First coordination shellFSDM First sharp diffraction minimumDMA Dynamic mechanical analysis,DTA Differential thermal analysisPDF Pair distribution functionRDF Radial distribution functionTMA Thermal mechanical analysisReferences1. Kauzmann, W. The nature of the glassy state and the behavior of liquids at low temperatures. Chem. Rev. 1948, 43, 219–256.[CrossRef]2. Martinez, L.-M.; Angell, C.A. A thermodynamic connection to the fragility of glass-forming liquids. Nature 2001, 410, 663–667.[CrossRef]3. Zheng, Q.; Zhang, Y.; Montazerian, M.; Gulbiten, O.; Mauro, J.C.; Zanotto, E.D.; Yue, Y. Understanding glass through differentialscanning calorimetry. Chem. Rev. 2019, 119, 7848–7939. [CrossRef]4. 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MDPI and/or the editor(s) disclaim responsibility for any injury topeople or property resulting from any ideas, methods, instructions or products referred to in the content.https://doi.org/10.1134/S0022476625050208https://doi.org/10.1016/j.pmatsci.2019.04.005https://doi.org/10.1103/PhysRevB.81.144202 Introduction  Theoretical Considerations  Experimental Results  Discussion  Conclusions  References