The generalized Vogel-Fulcher-Tamman equation for describing the dynamics of relaxor ferroelectrics

Relaxor ferroelectrics (RF) are outstanding materials owing to their extraordinary dielectric, electromechanical, and electro-optical properties. Although their massive applications, they remain to be one of the most puzzling solid-state materials because understanding their structural local order and relaxation dynamics is being a long-term challenge in materials science. The so-called Vogel-Fulcher-Tamman (VFT) relation has been extensively used to parameterize the relaxation dynamics in RF, although no microscopic description has been firmly established for such empirical relation. Here, we show that VFT equation is not always a proper approach for describing the dielectric relaxation in RF. Based on the Adam-Gibbs model and the Grüneisen temperature index, a more general equation to disentangle the relaxation kinetic is proposed. This approach allows to a new formulation for the configurational entropy leading to a local structural heterogeneity related order parameter for RF. A new pathway to disentangle relaxation phenomena in other relaxor ferroics could have opened.

, (S1.1) which can be easily obtained directly from the experimental data.
The VFT equation relates the relaxation times with the temperature as follow: such that: Thus, the Stickel function: turns out to be a linear temperature-dependent function.
Consequently, the Stickel plot became a key tool for estimating the so-called dynamic crossover temperature between two dynamic domains. The application of such analysis showed that at least two VFT equations are desired for describing "previtrificational" slowing down in a broader range of temperatures. Therefore, the Stickel plot represents a valuable tool for predicting the crossover temperature from a simple linearization of the classical VFT equation.
The computed ϕ(T) as a function of the temperature for the studied PLZT relaxor ferroelectric is showed in Fig. S2. Two dynamic domains are identified, which are in accordance with the two dynamic regions identified in the reciprocal of the Grunieesen temperature index plot (i.e.,

S2 Generalized entropy equation for relaxor ferroelectrics
The Adam-Gibbs theory relates the apparent activation energy and the configurational entropy such that: 4 Considering the validity of Adam-Gibbs model, we can write: Finding a derivative of equation (S2.1) allows to obtain: Considering the definition of the temperature index: 5 and substituting equation (S2.3) in equation (S2.4), we obtain: Integrating equation (S2.5) from 0 → ∞ to a finite temperature T, and adopting a linear behavior for the inverse of the index, we can obtain, substituting equation (S2.7) in equation (S2.6), the following equation: Calculating the primitive in equation (S2.8): and after some math transformations, we can write: After evaluating the integral in equation (S2.10): we can obtain: leading to: The equation (S2.13) is labeled as the generalized configurational entropy relation, which after some variable adjustments can be written in the following way: being the configurational entropy at high temperatures defined as (∞) = 0 , the divergent temperature as = − ⁄ (also namely the Kauzmann temperature), and the exponent = −(1⁄ ) as an order parameter.
Adopting a hyperbolic temperature dependence for the specific heat, a configurational entropy equation for relaxor ferroelectrics was obtained by Pirc et al. 6,7 as follow: where is the Vogel-Fulcher-Tamman temperature and 0 the configurational entropy at higher temperatures.
The configurational entropy given by the equation (S2.14) generalizes the equation (S2.15), which can be recovered for the particular case of = 1 taking into account that is conceptually equivalent to the divergent temperature .
Assuming the validity of AG theory for relaxor ferroelectrics, the activation energy can be written from equation (S2.14) in the following way: where ∆ 0 defines the activation energy at higher temperatures.
Substituting equation (S2.16) in the SA relationship: a more general equation for the relaxation time can be written as: which after some transformations leads to: Finally, a generalized relaxation time equation is obtained as follow: For the particular case of = 1, the classical VFT equation is recovering from equation (S2.20), writing the divergent temperature as : Table S1 shows the values of the fitting parameters obtained from the fitting of the relaxation data of the studied materials by using both classical and generalized VFT functions given by equations (S2.21) and (S2.20), respectively. The fitting curves are shown in Fig. 4 of the main manuscript.