Fig. 1 presents the X-ray diffraction (XRD) pattern of the as-prepared Se_87.5Te₁₀Pb_2.5 glass, characterized by the absence of sharp reflections and the presence of a broad diffuse halo centered at approximately 2θ ≈ 28–32°, accompanied by a weaker, broader feature at higher angles. Such diffraction features are typical of amorphous chalcogenide glasses and arise from short-range atomic ordering without long-range periodicity.

Fig. 2 shows DSC curves of the Se_87.5Te₁₀Pb_2.5 glass collected at a wide range of heating rates (5 to 99 K·min⁻¹), depicting a single well-defined crystallization peak along with glass transition, and melting. Crystallization occurs around 370 to 480 K and the peak broadens with heating rate, reflecting a wider temperature range for the process. This broadening arises from thermal lag and sample temperature gradients, leading to overlapping nucleation and growth. Similar behavior has been reported for Se-Te-Pb glasses [9,18], where higher heating rates produced broader, higher-temperature peaks. Crystallization is a thermally activated process governed by nucleation and crystal growth, both of which require finite time for atomic rearrangement. At low heating rates, sufficient time is available for structural relaxation and nucleation at lower temperatures, whereas at higher heating rates the material rapidly traverses these temperature regions, shifting the onset of crystallization to higher temperatures. This kinetic trend arising from the interplay between atomic relaxation and thermal diffusion is evident in the onset of Fig. 2 and is typical of nonisothermal crystallization [18].

The crystallization process under nonisothermal conditions is described by the following kinetic equation [21,22,23]: (1) d α d t = k T f α = A α exp − E R T f α where α is the extent of crystallization, T is temperature, k T is the temperature-dependent rate constant, A is the pre-exponential factor, E is the apparent activation energy, R is the universal gas constant, and f α is the model function describing the mechanism of transformation.

Fig. 3a presents the extent of crystallization, α , versus temperature for heating rates between 5 and 99 K·min⁻¹. The sigmoidal profiles, characteristic of glass-to-crystal transformation, reflect sequential nucleation and growth. As the heating rate increases, the curves shift to higher temperatures, indicating that greater thermal input is required to achieve the same conversion due to reduced time for atomic rearrangement. Crystallization may involve overlapping nucleation and growth processes rather than strictly sequential stages [18]. These observations form the basis for subsequent kinetic analysis of the Avrami exponent, n , activation energy, E, and growth dimensionality.

Fig. 3b displays the corresponding crystallization-rate curves obtained by numerically differentiating the α − T data shown in Fig. 3a. Each curve represents the instantaneous rate of transformation from the amorphous to the crystalline state. The peak crystallization rate shifts systematically to higher temperatures with increasing heating rate, confirming the reduced time available for nucleation and growth at faster heating. This time–temperature compensation follows Arrhenius kinetics and enables the extraction of conversion-dependent activation energies using isoconversional methods [18].

It should be emphasized that the following isothermal crystallization curves are predictive simulations derived from nonisothermal DSC data using AKTS software The AKTS software plays a powerful role in predicting thermal transformation behavior by employing kinetic parameters, namely the activation energy and pre-exponential factor derived from nonisothermal experimental data [26,30]. Once these parameters are established, the software enables accurate simulations of crystallization under arbitrary thermal programs T(t), including strictly isothermal conditions. Importantly, this predictive capability does not require the selection of a specific reaction model. Within the isoconversional framework, the product A α f α is assumed to be constant at a fixed degree of conversion and remains invariant regardless of the functional form of f α associated with any particular kinetic model. Consequently, the time required to reach a given conversion under a prescribed temperature history can be calculated from Eq. (1).

This model-free predictive approach offers several important advantages. It provides reliable insight into long-term crystallization behavior and allows the optimization of thermal processing conditions without the need for time-consuming isothermal experiments [31]. Fig. 6 shows the predicted evolution of the crystallized fraction as a function of time under predictive isothermal conditions at temperatures ranging from 373 K to 453 K. The curves display the characteristic sigmoidal shape typical of thermally activated nucleation and growth processes. At lower temperatures, the transformation proceeds slowly, featuring extended induction periods and gradual progression toward completion (low slope). As the temperature increases, the curves shift markedly to shorter timescales, indicating a substantial acceleration in the crystallization kinetics (high slope). This behavior arises from enhanced atomic mobility and reduced energy barriers at elevated temperatures, which promotes faster nucleation and crystal growth.

The predicted isothermal conversion curves (Fig. 6) are utilized to determine the most appropriate crystallization reaction model at various temperatures. By evaluating the time required to reach a specific conversion level (commonly α = 0.632 ), t α = 0.632 , it is possible to apply the reduced reaction model formalism based on the relation [16]:

This formulation provides a dimensionless framework for comparing experimental and theoretical conversion behavior under isothermal conditions. Here, g α is the integral form of the reaction model, and B is constant that depends on the specific kinetic mechanism. When the reduced activation energy E / R T is sufficiently high (e.g., above 20), the value of B approaches unity, allowing direct comparison between the normalized time-dependent conversion data and theoretical models. Table 1 lists examples of the common Avrami–Erofeev models, used to describe solid-state reactions. Increasing the Avrami exponent, n , values correspond to higher growth dimensionality and a more complex nucleation behavior.

By plotting the normalized isothermal conversion data and fitting them to the various theoretical functions listed in Table 1, the dominant crystallization mechanism can be identified from the model that best reproduces the experimental trend. Each model reflects a specific combination of nucleation behavior and growth dimensionality. Therefore, when the experimentally reduced (normalized) curve shows close agreement with a particular functional form, it provides strong evidence that the corresponding crystallization mechanism governs the process at that isothermal temperature. As shown in Fig. 7, the degree of agreement between experimental and theoretical curves varies systematically with temperature. At lower isothermal temperatures, the experimental data exhibit an excellent fit with the A3 model ( n  ≈ 3), suggesting that crystallization proceeds primarily through a three-dimensional growth with instantaneous or diminishing nucleation. The relatively high activation energies observed in this regime reflect the increased structural rigidity and nucleation complexity that dominate the early stages of transformation. With increasing temperature, the best-fit model gradually shifts toward functions with lower Avrami exponents, A2, A1.5, and even A1.25, indicating a transition toward a lower-dimensional growth or a reduced nucleation intensity. This behavior implies that crystallization becomes kinetically simpler at elevated temperatures, where higher atomic mobility facilitates growth processes with fewer nucleation constraints. The corresponding decline in activation energy at higher temperatures supports this interpretation, as it reflects a transition from nucleation-controlled to growth-controlled crystallization. Similar temperature-dependent reductions in the Avrami exponent have been reported for other Se–Te-based chalcogenide systems and related amorphous materials, where increased thermal energy or heating rate promotes surface or one-dimensional growth modes at the expense of complex bulk nucleation mechanisms [32]. Accordingly, these predicted isothermal curves are interpreted only as illustrative extensions of the nonisothermal kinetic framework and not as experimental verification of isothermal crystallization behavior.

In nonisothermal experiments, the master plot method, z α , is commonly applied. The theoretical expression of z α is obtained by combining the differential form, f α , and the integral form, g α , of the reaction model, and is given as:

Experimentally, z(α) is given as [33]:

The above two equations can be normalized into the following form:

Fig. 8 presents the normalized master plots in the form of Z α / Z 0.5 versus α across various heating rates (7, 20, 30, 40, 70, and 90 K·min⁻¹), corresponding to the solid-state reaction mechanisms listed in Table 1, where the experimental datasets obtained at different heating rates from nonisothermal experiments are superposed onto the theoretical master curves derived from solid-state reaction models (solid and dashed lines). The experimental points (symbols) corresponding to multiple heating rates collapse into a single bell-shaped profile, demonstrating the robustness of the z-plot approach for nonisothermal crystallization analysis. A common maximum is observed around α ≈ 0.6, indicating that the normalized rate function attains its peak at this crystallized fraction irrespective of the heating rate. In the early region ( α < 0.5), the experimental outcomes are well described by diffusion-controlled models (F1–F2), reflecting the role of the diffusive transport of structural units to the growth front during the onset of crystallization. Once the transformation advances beyond α > 0.5, the datasets follow Avrami-type models (A1–A3), indicating a shift toward interface-controlled crystallization. The peak of the z-function near α ≈ 0.6 marks the balance between rapid growth and the impingement/exhaustion effects that dominate the later stage.

Taken together, the two analyses provide complementary perspectives on the same crystallization pathway. Both the nonisothermal z-master plot and the isothermal model fitting highlight a two-stage mechanism: an early stage characterized by diffusion influence and a complex three-dimensional growth, followed by a later stage dominated by an Avrami-type interface-controlled crystallization with progressively simpler kinetics. The fact that the z-plot remains invariant across heating rates, while the isothermal fits reveal a temperature-dependent shift toward lower Avrami exponents, demonstrates a coherent mechanistic picture: the pathway is structurally robust, but the balance between diffusion, nucleation, and growth is modulated by thermal energy. These convergent findings reinforce the conclusion that the crystallization of Se–Te–Pb glasses follows a consistent two-regime mechanism, in agreement with earlier studies on related chalcogenide systems [34,35,36].

The thermal metrics and glass-forming ability (GFA) of glassy materials are commonly assessed using empirical parameters derived from characteristic DSC temperatures, namely the glass transition temperature T g , the onset crystallization temperature T c , and the melting temperature T m . These quantities are used to construct several dimensionless or semi-empirical indicators describing the resistance of glasses to crystallization under heating. Since these metrics are derived from non-equilibrium DSC measurements, they primarily reflect kinetically controlled responses under a given heating protocol rather than intrinsic thermodynamic stability. One of the most frequently used indicators is the width of the supercooled liquid region given by [37]: (11) Δ T = T c − T g which represents the temperature interval over which the glass remains metastable before crystallization. Although a larger Δ T  is often associated with improved GFA, it is well established that Δ T  exhibits strong heating-rate dependence due to kinetic delay of crystallization relative to structural relaxation [25]. Another commonly used parameter is the Saad–Poulin stability parameter s, given by [38]: (12) s = T c − T g T p − T c / T g which relates the separation between the glass transition and crystallization temperatures to the remaining interval up to melting. The reduced glass transition temperature, T g / T m , provides a normalized measure of the glass transition relative to the melting point and is frequently employed as an empirical descriptor of glass-forming ability. Similarly, the Hruby number is given by [39]: (13) H r = T c − T g T m − T c which incorporates both crystallization onset and melting behavior and is widely used to assess kinetic resistance to crystallization. It should be noted that the absolute values and trends of all these parameters are strongly influenced by heating rate, thermal inertia, and finite heat-transfer effects inherent to DSC measurements [40,41].

Fig. 9 shows the heating-rate dependence of these thermal metrics. In Fig. 9a, the glass transition temperature increases from approximately 326 K at the lowest heating rate to about 343 K at the highest. This behavior reflects the well-known kinetic dependence of T g on heating rate [42], commonly described by the logarithmic relationship proposed by Lasocka [43] and consistent with kinetic models such as the Kissinger relation [44]. In addition to intrinsic kinetic effects, thermal inertia and instrumental lag contribute to systematic upward shifts of characteristic temperatures at high heating rates, even in the absence of changes in material stability [40].

The supercooled liquid region Δ T  broadens from ~30 K to ~55 K with increasing heating rate. This widening represents an apparent extension of the metastable temperature window under rapid heating, arising primarily from delayed crystallization onset caused by kinetic suppression of nucleation and growth rather than enhanced thermodynamic stability [40,43]. Similar heating-rate-induced broadening of Δ T  has been reported in polymeric and inorganic glasses studied by conventional DSC, where it was attributed to delayed structural relaxation and metastable structural states. The Saad–Poulin s -parameter increases nearly linearly with heating rate, indicating a proportional increase in the delay between T g and T c . This trend reflects a predictable, rate-dependent kinetic delay of crystallization relative to the glass transition, consistent with previous observations in chalcogenide glasses [25].

Fig. 9b shows that the reduced glass transition temperature T r g increases with heating rate and tends toward saturation at higher β. This behavior arises primarily from the upward shift of T g under faster heating due to diminished structural relaxation, while the melting temperature remains comparatively insensitive [43,45].

The obtained T r g values (0.648–0.674) follow the empirical ‘two-thirds rule’, indicating reasonable glass-forming ability [45]. The Hruby number H r also increases steadily with heating rate, reflecting an apparent enhancement of resistance to crystallization under rapid heating conditions, which is widely recognized as a kinetic effect rather than an intrinsic improvement in thermodynamic stability [39]. Comparable trends have been reported for Se–Te-based glasses, including Pb-containing systems, and are consistently attributed to kinetic suppression of crystallization under nonisothermal conditions [9,19,40].

Taken together, the thermal metrics in Fig. 9 demonstrate a clear rate-dependent postponement of crystallization onset as the heating rate increases. These parameters describe when crystallization begins under a given thermal protocol, whereas the activation energy obtained from kinetic analysis characterizes how crystallization proceeds once initiated. Delaying crystallization onset to higher temperatures places the system in a regime of enhanced atomic mobility, resulting in lower effective activation barriers during subsequent growth, consistent with isoconversional analyses and the Fisher–Turnbull framework [19,40].

While undoped Se–Te glasses exhibit a monotonic increase in thermal metrics such as Δ T , s, T rg , and H r with heating rate due to intrinsically kinetically controlled effects arising from delayed structural relaxation and crystallization [9,25], the present Se–Te–Pb glass preserves this general rate dependence but with systematically higher absolute values. The persistence of the same heating-rate trend indicates that Pb addition does not alter the fundamental kinetic origin of these thermal parameters but instead produces a composition-dependent upward shift superimposed on the general kinetic response characteristic of Se–Te glasses. Independent DSC studies on Se–Te–Pb glasses measured at fixed heating rates show that Pb incorporation increases the glass transition temperature and modifies the glass network through configurational changes and altered bonding environments, consistent with Pb acting as a network modifier [42]. Such network modification can suppress nucleation and delay crystallization, leading to an enhanced apparent resistance to crystallization under nonisothermal heating conditions.

By contrast, the activation energy obtained from the kinetic analysis characterizes how crystallization proceeds once it has initiated, rather than when crystallization begins. The observed decrease in activation energy with increasing extent of crystallization, as revealed by the isoconversional analysis, is fully consistent with the thermal-metric trends discussed above. The rate-dependent postponement of crystallization onset to higher temperatures places the system in a regime of enhanced atomic mobility, so that once nucleation occurs, the subsequent transformation proceeds with a lower effective kinetic barrier.

Within the Fisher–Turnbull framework, initiation at elevated temperatures reduces the effective nucleation barrier and promotes diffusion-controlled growth. Consequently, crystallization evolves toward a growth-dominated regime in which the apparent activation energy decreases progressively with conversion. This behavior is characteristic of nonisothermal crystallization in chalcogenide glasses and has been widely reported for Se–Te–Pb systems, where isoconversional methods consistently show a monotonic decline of activation energy as crystallization advances, with only minor methodological variation between integral approaches [19].

In this context, the thermal metrics describe a rate-dependent delay of crystallization onset, whereas the activation-energy analysis captures the kinetic efficiency of the transformation after onset. The thermal and kinetic results therefore probe complementary stages of the same crystallization pathway and are mutually reinforcing rather than contradictory.