Relationship between enthalpy fluctuation and nonexponential relaxation in glass-forming liquids

Abstract

A relationship between the equilibrium fictive temperature fluctuation and the nonexponentiality β_KWW is derived by extending the Adam–Gibbs equation within Donth's fluctuation–dissipation theorem to quantitatively account for dynamic heterogeneity in supercooled liquids. The framework is validated against three structurally, chemically, and kinetically distinct glass-forming systems, namely B₂O₃, Pd₄₃Cu₂₇Ni₁₀P₂₀, and PVAc, without any adjustable fitting parameters. A condition for time-temperature superposition (TTS) is further derived as a balance between the fictive temperature fluctuation rate and a configurational term composed of the configurational entropy and heat capacity difference between the glass and liquid. The present framework reveals the interplay between thermodynamic quantities, enthalpy fluctuations, and dynamic heterogeneity in glass-forming liquids.

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I. Introduction

Dynamic heterogeneity is a fundamental characteristic of glass-forming liquids.^1,2 As they approach the glass transition, distinct spatial regions relax at different rates, giving rise to a broad spectrum of relaxation times and the characteristic nonexponential relaxation response.³ This nonexponential behavior is commonly described by the Kohlrausch–Williams–Watts (KWW) function,^4,5 whose stretching exponent β_KWW reflects the width of the underlying distribution of relaxation times, where β_KWW = 1 corresponds to Debye relaxation and decreasing values of β_KWW correspond to an increasingly broader spectrum of relaxation times. Each spatial region may relax exponentially or may have an intrinsic nonexponentiality β₀.⁶ The global relaxation is then understood to be a superposition of local relaxation processes weighted by a distribution of relaxation times X(ln τ):where the right-hand side is the KWW function and β_KWW is the macroscopic stretching exponent, and τ_KWW is related to the average relaxation time as , where Γ is the gamma function. Eqn (1) links β_KWW directly to the underlying dynamic heterogeneity: in the homogeneous limit where all subsystems share a single relaxation time, X(ln τ) → δ(ln τ − ln τ₀), where δ denotes Dirac's delta function, and the macroscopic exponent reduces to the intrinsic value, β_KWW → β₀. A key observation is that β_KWW is not a material constant but depends on the temperature.^7–10

According to Angell, glass-forming liquids can be classified as strong or fragile depending on the temperature dependence of their relaxation time or, equivalently, their viscosity, and can be characterized by the fragility index m, defined as the slope d log₁₀ τ/d(T_g/T) evaluated at the glass transition temperature T_g.¹¹ Fragility has been correlated with several dynamic and thermodynamic properties, including the configurational entropy S_c¹² the heat capacity difference between the liquid and glass,^13,14 the stretching exponent at T_g β_KWW (T_g),¹⁵ and the temperature dependence of β_KWW.^16,17 When β_KWW remains constant over a range of temperatures, the relaxation-time spectrum preserves its shape, a condition known as time–temperature superposition (TTS). Dielectric studies^17,18 and shear mechanical spectroscopy¹⁶ across a wide range of glass-forming liquids have shown a clear trend where more fragile liquids tend to maintain TTS over a broader range of relaxation times. However, no unified thermodynamic framework has been established that quantitatively predicts the conditions under which TTS holds or breaks down.

Understanding the temperature dependence of β_KWW requires examining how relaxation time fluctuations arise from thermodynamic fluctuations. As established by Donth's fluctuation-dissipation theorem (DFDT),^19,20 the distribution of relaxation times maps onto the fictive temperature fluctuation , which, since the equilibrium fictive temperature is proportional to the enthalpy, is fundamentally rooted in the distribution of local enthalpies.^21,22 A distribution of enthalpy in supercooled liquids has been observed in both experiments and simulations.^22–25 While the connection between enthalpy fluctuations, the distribution of relaxation times, and β_KWW is physically well motivated, a quantitative relationship linking the dynamic heterogeneity characterized by nonexponential relaxation to the fictive temperature fluctuation through rigorously defined thermodynamic quantities, such as the configurational entropy and heat capacity, has not yet been developed.

In this work, we expand on a phenomenological model, the Takeda–Lucas (TL) model,²⁶ to develop a thermodynamic framework that connects the Adam–Gibbs relation²⁷ to dynamic fluctuations. While the TL model accounts for both equilibrium and nonequilibrium states, the present generalization focuses on the equilibrium supercooled liquid, where the thermodynamic quantities are well defined. This yields a parameter-free prediction connecting and β_KWW, through S_c(T) and ΔC_p(T). We validate the framework against three chemically, structurally, and kinetically distinct glass-forming systems: B₂O₃ (network oxide, m = 36²⁸), Pd₄₃Cu₂₇Ni₁₀P₂₀ (metallic alloy, m = 69²⁹), and PVAc (polymer, m = 95¹⁵). Furthermore, we derive a condition for TTS in terms of a balance between the fictive temperature fluctuation rate and a configurational term, revealing the interplay between the thermodynamic quantities S_c(T), ΔC_p(T), the enthalpy fluctuation captured by , and the temperature dependence of nonexponentiality in glass-forming liquids.

This paper is organized as follows. In Section II, we briefly review the TL model of dynamic heterogeneity. In Section III, we present the heterogeneous extension of Adam–Gibbs (AG) theory with DFDT formalism and derive the key equations. In Section IV, we validate the framework against three glass-forming systems. In Section V, we verify the predicted condition for TTS, in Section VI we discuss the implications of the developed framework, and Section VII concludes the paper.

II. TL model of supercooled liquids and glasses

The TL model accounts for both enthalpy fluctuation and temperature-dependent β_KWW by introducing a distribution of equilibrium fictive temperatures that maps to a distribution of relaxation times via DFDT.^19,26,30,31 The enthalpy fluctuation is modeled by the fictive temperature fluctuation, using the consideration that fictive temperature T_f is proportional to enthalpy H, T_f ∝ H. The model accounts for the heterogeneity of the supercooled liquid by introducing a distribution of equilibrium fictive temperatures T_f^e and for the temperature dependence of nonexponentiality by introducing a temperature dependent distribution of relaxation times. This is achieved using Cangialosi et al.'s formulation of dynamic heterogeneity, which relates the distribution X of relaxation times to a distribution Y of Vogel temperatures³² and further relates it to a distribution W^e of equilibrium fictive temperatures T_f^e as:

Xd ln τ = YdT_v = W^edT_f^e, (2)

The detailed equations of the TL model (discretized domain weights, relaxation time expressions, and skewness parameters) are provided in ref. 26. The key result relevant to the present work is the connection between the fluctuation of fictive temperatures, i.e. the width of the distribution or the variance of fictive temperatures and the fluctuation of relaxation times through the DFDT formalism:^19,26,30,31

where is the fluctuation (variance) of equilibrium fictive temperatures and σ_ln τ is the fluctuation of the relaxation time in equilibrium (see the detailed derivation of eqn (3) in the supplementary information, SI). Furthermore, the relaxation time for each domain i of the distribution is determined by the local Vogel temperature T_v,i using the Adam–Gibbs equation modified by Hodge (AGH)^33,34 as

In equilibrium, 〈T_f〉 = T, and eqn (4) adopts the Vogel–Fulcher–Tammann (VFT) form:

where D is the strength parameter related to the fragility index m.¹⁵ T_v,i determines the amplitude of the activation energy for each domain and its distribution is temperature independent (Fig. 1(a)). Finally, an expression for the equilibrium fictive temperature of ith domain T_f,i^e that satisfies both DFDT and the VFT equation in eqn (5), is given by:²⁶

Fig. 1 Schematics of the temperature dependence of the discrete distribution for (a) the Vogel temperatures, (b) the equilibrium fictive temperatures, and (c) the relaxation time.

Eqn (6) effectively makes the distribution of T_f^e temperature-dependent.

Indeed, it is important to recall that the distribution of local fictive temperatures is determined by both the weights W_i^e and the relative values of T_f,i^e. In our framework, the weights W_i^e are temperature-independent, so the temperature dependence of the distribution width is determined entirely by the relative positions of T_f,i^e, which evolve with temperature at rates defined by their respective T_v,i per eqn (6). Hence, the width of the distribution is temperature-dependent (Fig. 1(b)). This accounts for the temperature dependence of the distribution of enthalpy fluctuations.

Finally, a discretized distribution of weight X_i for τ_i is obtained from the weights Y_i of the Vogel temperatures T_v,i according to:

Consequently, according to eqn (6) and (7), both the shape and the width of the distribution of relaxation times are temperature dependent, which accounts for the temperature dependence of nonexponentiality (Fig. 1(c)).

III. Heterogeneous Adam–Gibbs theory applied to the TL model

The TL model is based on the AGH equation (eqn (4)). This form of the AG equation approximates the configurational entropy S_c by assuming the temperature dependence of the difference in heat capacity between the glass C_p^g and the liquid C_p^l to be inversely proportional to temperature as:where ΔC_p is the heat capacity difference defined as C_p^l − C_p^g, and C is the proportionality constant, where this form was conveniently chosen so that when T_f = T, eqn (4) goes back to the ordinary VFT equation in eqn (5). Thus, the formulated heterogeneous model above is limited to a system in which eqn (8) holds. Here, we generalize the TL–DFDT formalism to arbitrary temperature dependences of ΔC_p(T) commencing from the original Adam–Gibbs equation.^27,35 While doing so, we establish a connection between the nonexponentiality β_KWW, the enthalpy fluctuation , the configurational entropy S_c, and ΔC_p.

Let us begin by restating the original AG equation as

where A is the energy barrier for cooperative rearrangement. We assume that the heterogeneity in relaxation time comes entirely from the heterogeneity in the energy barrier A following the treatment by Gupta and Mauro in ref. 35. We further assume that the effect of heterogeneity can be captured by a number of discrete sets of domains N where the relaxation time of the ith domain based on eqn (9) becomeswhere A_i is the activation energy for domain i. We treat the configurational entropy as a bulk property that remains uniform across all domains, while A_i captures the local dynamic variation. Thus, the average relaxation time becomeswhere the angle brackets denote averaged quantities.

A. Enthalpy fluctuation and nonexponentiality

The DFDT framework establishes that fluctuations in relaxation time map to fluctuations in fictive temperature through eqn (3).

From eqn (11), the denominator in eqn (3) becomes

where we use the identity dS_c/dT = ΔC_p(T)/T.

Thus, by inserting eqn (12) into eqn (3), σ_ln τ can be expressed in terms of as

Richert and Richert show that the fluctuation (variance) of relaxation time can be expressed in terms of nonexponential parameter β_KWW and intrinsic nonexponentiality β₀ as:³⁶

We can then obtain an expression for β_KWW by inserting eqn (13) into eqn (14), and solving for β_KWW gives:

where Λ(T) is defined as

Thus, eqn (15) relates the dynamic heterogeneity β_KWW, the fictive temperature fluctuation , and the configurational entropy S_c(T), all of which are experimentally accessible. In most cases, the absolute value of S_c is not experimentally available, thus, one may use the approximation S_c ≈ S_ex, where S_ex is the excess entropy obtained by integrating ΔC_p(T)/T.

B. A condition for time-temperature-superposition (TTS)

One of the hallmark questions in glass science is in which systems, and over what temperature ranges, does time–temperature superposition (TTS) apply in glasses and glass-forming liquids. Based on eqn (15), we can determine the conditions under which β_KWW increases, remains constant, or decreases with increasing temperature by taking the temperature derivative, as follows:if we exclude the limiting case of Λ = 0. We note that Λ(T) cannot be negative by definition. Thus, the condition for TTS can be obtained by setting d ln Λ/dT = 0, and solving for the equilibrium fluctuation term in eqn (16) as:

We introduce two dimensionless parameters: the fictive-temperature fluctuation rate R(T), and configurational term Ξ(T), defined as

andwhere all the variables are experimentally measurable. Both R and Ξ are dimensionless ratios. Finally, we obtain the conditions for the temperature dependence of β_KWW, as compiled in Table 1.

Table 1 Derived conditions of temperature dependence of β_KWW

Condition	Temperature dependence of βKWW
R(T) < Ξ(T)	βKWW increases upon heating
R(T) = Ξ(T)	βKWW is constant (TTS condition)
R(T) > Ξ(T)	βKWW decreases upon heating

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IV. Validation

In this section, we validate the derived heterogeneous AG model with the DFDT formalism using three systems: B₂O₃, Pd₄₃Cu₂₇Ni₁₀P₂₀, and PVAc. We do so by using experimentally determined values of β_KWW, , and S_c to confirm the validity of the relation expressed in eqn (15) and (16) with no adjustable fitting parameters. These systems were chosen based on the availability of accurate experimental heat capacity data, temperature-dependent nonexponentiality values, and equilibrium fictive temperature fluctuations. Additionally, they span a wide range of fragility, specifically 36,²⁸ 69,²⁹ and 95,¹⁵ for B₂O₃, Pd₄₃Cu₂₇Ni₁₀P₂₀, and PVAc, respectively, and are structurally and chemically distinct.

In all the analyses below, we first need to obtain ΔC_p(T) and S_ex(T) to compute S_c(T).

To estimate configurational entropy for B₂O₃, we calculate the excess entropy as

where ΔS_m is the entropy of fusion and ΔC_p(T) is the heat capacity difference between liquid C_p^l and glass C_p^g, assuming that the vibrational contribution to the heat capacity in the liquid is the same as that of glassy B₂O₃.

For Pd₄₃Cu₂₇Ni₁₀P₂₀, we follow the procedure of Gallino et al.⁴⁰ using the temperature at which the viscosity is 1 Pa s as the reference. Details of these calculations are provided in the SI. For PVAc, where no melting temperature exists, the configurational entropy is obtained by integrating the heat capacity difference from the Kauzmann temperature T_k, estimated using the Vogel temperature. Thus, we assume that ΔC_p(T) values in three glass-forming systems are purely configurational in origin in all analyses.⁴¹

The heat capacity data for B₂O₃, Pd₄₃Cu₂₇Ni₁₀P₂₀, and PVAc are taken from ref. 42–44, respectively. The heat capacity of crystalline Pd₄₃Cu₂₇Ni₁₀P₂₀ was used for the heat capacity of a solid following the procedure of Gallino et al.³⁴

The other necessary parameters for determining S_c(T) for each system are ΔS_m for B₂O₃, S_c(T_r) for Pd₄₃Cu₂₇Ni₁₀P₂₀, and T_k for PVAc, and the values and corresponding references are compiled in Table 2. Fig. 2 shows the heat capacity data and corresponding fits for B₂O₃ as an example. The heat capacity data and their fits for Pd₄₃Cu₂₇Ni₁₀P₂₀ and PVAc, as well as the details of the fitting procedures, are shown in the SI.

Table 2 Summary of the model parameters for the three glass-forming systems. Configurational entropy is anchored at T_m (fusion) for B₂O₃, at T_r for Pd₄₃Cu₂₇Ni₁₀P₂₀, and at T_K (Kauzmann) for PVAc. T_r is the temperature at which viscosity is 1Pa s defined by Gallino et al.³⁴ Units: τ₀ in seconds and ΔS_m and S_c(T_r) in J (mol)⁻¹ (K)⁻¹. The numbers in square brackets indicate the corresponding reference

	B2O3	Pd43Cu27Ni10P20	PVAc
Tg [K]	560^17	568^34	311^37
Tm[K]	723^17	Not used	Not used
[K]	Not used	Not used	267^37
Tr[K]	Not used	900^34	Not used
Sc anchor	ΔSm = 30.5^38	Sc(Tr) = 11.26^34	Sc(Tk) = 0
β0	1.00	0.76^39	0.67^37
m0	16^15	Not used	Not used
log10 τ0	Not used	Not used	−12.08^37
〈A〉	448 023	114 870^34	88 332

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Fig. 2 Heat capacities of B₂O₃. The data are taken from ref. 42. Cooling and heating in the legend indicate that the data were collected during constant cooling and heating. The semiempirical equation for the heat capacity of the solid introduced by Tsao et al.⁴⁵ was used for C_p^g, and the C_p^l was assumed to be constant. The details of the fitting are given in the SI.

The calculated S_c and the ratio ΔC_p/S_c as a function of the reduced temperature T/T_g for the three glass-forming systems are shown in Fig. 3(a) and (b).

Fig. 3 Configurational entropy S_c in (a) and the ratio ΔC_p/S_c in (b) as a function of the reduced temperature T/T_g for the three glass-forming systems. The black vertical dashed line marks T/T_g = 1. The characteristic reference temperatures used to anchor S_c for each system are indicated by vertical lines whose color and style match the corresponding curve: T_m for B₂O₃; T_r for Pd₄₃Cu₂₇Ni₁₀P₂₀; and T_k for PVAc.

Next, 〈A〉 and β₀ must be determined to establish the correlation between β_KWW and using eqn (15) and (16).

First, 〈A〉 is derived based on the AG equation in eqn (11) as

A is determined from eqn (22) with m₀ = 16 for B₂O₃ and with τ₀ obtained by dielectric spectroscopy for PVAc. The experimentally determined A from viscosity³⁴ is used for Pd₄₃Cu₂₇Ni₁₀P₂₀. Second, β₀ is set as 1 for B₂O₃, 0.76 for Pd₄₃Cu₂₇Ni₁₀P₂₀ taken from ref. 39, and 0.67 for PVAc taken from ref. 37. The resulting parameters are summarized in Table 2.

Using the experimentally determined equilibrium fictive temperature fluctuation and independently evaluated thermodynamic quantities ΔC_p and S_c, we now test the quantitative validity of eqn (15) and (16) by comparing the predicted temperature dependence of the nonexponentiality parameter β_KWW with reported experimental values. Importantly, this comparison involves no adjustable fitting parameters.

Panels (a)–(c) in Fig. 4 show the equilibrium fictive temperature fluctuation for B₂O₃, Pd₄₃Cu₂₇Ni₁₀P₂₀, and PVAc, respectively. The reported values for B₂O₃ are measured via DSC and modulated-DSC.⁴⁶

Fig. 4 Equilibrium fictive-temperature fluctuations in (a) B₂O₃, (b) Pd₄₃Cu₂₇Ni₁₀P₂₀, and (c) PVAc. for B₂O₃ is taken from ref. 46; for Pd₄₃Cu₂₇Ni₁₀P₂₀, it is taken from ref. 47; for PVAc, it is calculated from the reported ΔT_β in ref. 48 (see the SI). The solid line in panel (b) shows the fit used to obtain the smoothed data for calculating the fictive temperature fluctuation rate R(T) in eqn (19). Panels (d–f) show the corresponding β_KWW predicted from eqn (15) and (16) (red circles) compared with experimental values (black markers). For B₂O₃ in (d), data labeled a is taken from ref. 49 and (b)–(g) from ref. 50. For Pd₄₃Cu₂₇Ni₁₀P₂₀ in (e), the β_KWW values reported by Liu et al.²⁹ are shown as a solid line. For PVAc in (f), black circles represent data from Alegria et al.³⁷ and black squares represent data from Sasabe et al.⁵¹

For Pd₄₃Cu₂₇Ni₁₀P₂₀, we use reported in ref. 47 measured via Flash-DSC. For PVAc, is obtained from the reported glass transition temperature width ΔT_β measured via Flash-DSC in ref. 48 using Schawe's vitrification function κ³¹ as

For , the independently reported values by Tombari et al.⁴⁰ and Hempel et al.⁴⁶ are used. This was done because we found the deviation of the reported in ref. 48 from the values measured via M-DSC.

Using these values as inputs, eqn (15) and (16) yield predictions for β_KWW. The results are shown as red circles in Panels (d)–(f) of Fig. 4 and are compared with experimental β_KWW values obtained from various techniques such as dielectric spectroscopy, mechanical spectroscopy, and ultrasonic sound-velocity measurements compiled from the literature (see Fig. 4 caption for details and the values listed in the SI). Despite the distinct differences in chemistry, structure, and kinetics (fragility) among the three systems, the predicted β_KWW values show quantitative agreement with the experiment across the accessible temperature range.

V. Time–temperature superposition

The assumption of time–temperature superposition (TTS) for analyzing relaxation dynamics in supercooled liquids and glasses works remarkably well near the standard T_g,^17,18,52 where the relaxation time is on the order of 100 seconds. However, numerous studies have reported that this assumption breaks down at temperatures far from T_g,^32,53,54 leading to what is commonly referred to as thermorheologically complex behavior.

The analysis developed in Section III.B suggests that the validity or breakdown of TTS is not simply a consequence of the temperature range considered, but instead reflects a balance between the fictive temperature fluctuation rate R(T) and the configurational term Ξ(T), as defined in eqn (19) and eqn (20), respectively. To validate these derived conditions, smoothed data for Pd₄₃Cu₂₇Ni₁₀P₂₀, obtained by fitting the two linear regimes with a sigmoid function (solid line in Fig. 4(c)), are used (see the SI for the detailed method). This system is chosen because the available data exhibit comparatively small experimental uncertainty.

As summarized in Table 1, when R(T) − Ξ(T) = 0, the system exhibits TTS, when R(T) − Ξ(T) < 0, β_KWW increases upon heating, and when R(T) − Ξ(T) > 0, β_KWW decreases upon heating. In Fig. 5, the difference between the calculated R(T), evaluated using eqn (19), and Ξ(T) evaluated using eqn (20) is plotted as a function of the predicted β_KWW. When R(T) − Ξ(T) = 0, the system exhibits TTS and when R(T) − Ξ(T) < 0, β_KWW increases upon heating in agreement with the predictions in Table 1 established from eqn (18)–(20). The horizontal error bars represent 95% confidence from the linear fits to in Fig. 4(b), while the vertical error bars are from the experimental uncertainty in shown in Fig. 4(e). The figure shows that when β_KWW remains approximately constant near T_g, the quantity R(T) − Ξ(T) approaches zero, thereby confirming the internal consistency of the present derivation.

Fig. 5 Difference R(T) − Ξ(T) evaluated using eqn (19) and (20), plotted as a function of the predicted β_KWW for Pd₄₃Cu₂₇Ni₁₀P₂₀. Error bars represent 95% confidence intervals from linear fits’ in Fig. 4(b), and the uncertainty in β_KWW arising from the experimental . The color indicates temperature (see color bar). The dot-dashed line denotes the experimental β_KWW near T_g, and the vertical dashed line indicates the TTS condition R(T) = Ξ(T) summarized in Table 1.

Discussion

The successful application of the proposed framework was demonstrated for three chemically, structurally, and kinetically distinct glass-forming systems. The agreement between predictions and measurements, obtained without any adjustable fitting parameters, demonstrates that the correlations between the nonexponentiality parameter β_KWW and the fictive temperature fluctuation , established in eqn (15) and (16) are physically consistent and applicable to a broad range of glass-forming systems.

The current approach does not rely on a particular functional form for ΔC_p(T), S_c(T), β_KWW, and . Furthermore, extending the Adam–Gibbs equation to account for dynamic fluctuations (eqn (11)) through the DFDT relationship allows us to investigate not only the relationship between the configurational entropy S_c(T), and the average dynamics 〈τ〉 as in the standard AG equation (eqn (9)), but also the fluctuations of the dynamic and thermodynamic quantities, σ_ln τ and . Since the equilibrium fictive temperature is proportional to the enthalpy, directly reflects the magnitude of local enthalpy fluctuations in the supercooled liquid. The framework thus provides a quantitative link between the enthalpy fluctuation and the nonexponential relaxation spectrum through the configurational entropy and heat capacity.

The model further yields a thermodynamic condition for time–temperature superposition (TTS), requiring a balance between the temperature dependence of , defined as the fictive temperature fluctuation rate R(T), and the configurational term Ξ(T). Wang and Richert¹⁷ found that β_KWW remains constant from T_g up to a crossover temperature T_c (TTS regime), then rises steeply toward 1 at high T. They plot β_KWW against log₁₀ν_max, the peak frequency of the α-relaxation loss spectrum at different temperatures. The range of log₁₀ν_max over which β_KWW stays flat, expands systematically with fragility m: strong liquids show no TTS at all, while fragile liquids maintain TTS from T_g all the way to T_c, covering up to ∼8 decades. Above T_c, all systems converge to Debye relaxation (β = 1) by ν_max ∼ 10 GHz regardless of m. Similar behavior was observed in shear mechanical spectroscopy measurements of inorganic systems by Sen and Lovi.¹⁶ In the current framework, the TTS condition (eqn (18)) is:

It has been previously established that ΔC_p(T)/S_c(T) and the rate of change of S_c(T) are directly correlated to fragility.^12,13 Indeed, for the three systems investigated, both the ratio ΔC_p(T)/S_c(T) and the magnitude of its slope are larger for more fragile systems, as shown in Fig. 3(b). This implies that the right-hand side of eqn (18) (shown above) will be larger for more fragile systems. Consequently, for TTS to hold, the rate at which the equilibrium fictive temperature fluctuation increases must also be larger. The empirical observation that fragile systems maintain TTS over a wider range of relaxation times thus indicates that the temperature dependence of the fictive temperature fluctuation is stronger in fragile systems. However, the fictive temperature fluctuation rate is itself temperature-dependent, as seen in the analysis of Pd₄₃Cu₂₇Ni₁₀P₂₀ in Fig. 5, supporting the eventual breakdown of TTS even for fragile liquids at high temperature, as observed by Wang and Richert.¹⁷

Conclusion

This work developed a framework that connects thermodynamic fluctuations and dynamic heterogeneity in supercooled liquids by extending the Adam–Gibbs relation through DFDT formalism. This approach establishes a direct relationship between the equilibrium fictive temperature fluctuation , which reflects the underlying distribution of local enthalpies, and the nonexponential parameter β_KWW through thermodynamic quantities: the configurational entropy S_c(T) and the heat capacity difference between the glass and liquid ΔC_p(T). The successful application of the framework was demonstrated for three chemically, structurally, and kinetically distinct glass-forming systems. The agreement between predictions and measurements, obtained without any adjustable fitting parameters, indicates that the derived expression in eqn (15) is physically consistent and applicable to a broad range of glass-forming systems. The model further yields a thermodynamic condition for time–temperature superposition (TTS), which requires a balance between the temperature dependence of defined as the fictive temperature fluctuation rate R(T), and the configurational term Ξ(T), related to S_c(T), and ΔC_p(T), and their temperature dependence. These results provide a thermodynamic connection to dynamic heterogeneity and TTS in supercooled liquids. Further validation would require a more integrated examination of the relationships among the temperature dependence of fictive temperature fluctuations (enthalpy fluctuation), configurational entropy, and dynamic heterogeneity in glass-forming systems. Future work combining experiments and atomistic simulations may further clarify the structural origins of these relationships.

Conflicts of interest

There are no conflicts to declare.

Data availability

This study does not report any new experimental data. The data reported in this study are modeled from experimental data available in the cited literature. All modeled data are included in the manuscript and the supplementary information (SI). Supplementary information is available. See DOI: https://doi.org/10.1039/d6ma00442c.

Acknowledgements

The authors acknowledge financial support from NSF-DMR under grant#: 1832817.

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