Non-isothermal free-models kinetic analysis on crystallization of europium-doped phosphate glasses

Abstract

A kinetic analysis under non-isothermal conditions has been performed on europium-doped phosphate glasses with molar composition 20.42 Li₂O–10.25 Al₂O₃–58.49 P₂O₅–7.23 BaO–1.44 La₂O₃–2.16 Eu₂O₃ and particle size <30 µm, >50 µm. Differential scanning calorimetry measurements at four heating rates, from room temperature up to 900 °C using synthetic air have revealed that no significant mass change occurs during the heat treatment, as normally expected for glass samples. The step changes in heat flow signals are associated with the presence of three important effects: the first one, that occurs as a slope corresponds to the glass transition effect (T_g) is followed by two exothermic peaks, first of them more pronounced, associated with the first crystallization process (T_p1), while, the second one, lower in intensity corresponds to the second crystallization (T_p2) process. As expected, all peaks (T_g, T_p1 and T_p2) increase with the increasing of the heating rate. Both the activation energy of crystallization process and the crystallization mechanism were comparatively analyzed by two free-model estimations and using the formal theory of transformations for heterogeneous nucleation.

Introduction

Rare-earth-doped phosphate glasses are of scientific and technological interest due to their structural and optical behavior which gives them a wide range of applications [1], [2], [3], [4]. An essential condition to obtain desired properties refers to the rigorous control of processing parameters, and this is done only by knowing and understanding the formation mechanism of these materials. Detailed kinetic understanding on the nucleation and crystallization process is therefore fundamental to design condition for the controlled crystallization and to regulate the morphology and the microstructure of the material [5], [6], [7], [8], [9]. The thermokinetic analysis produces an adequate description of the processes in terms of the reaction model and Arrhenius parameters. All kinetic studies are developed by assuming that the isothermal conversion rate (dx/dt), with

$$x=\frac{{\int }_{t_f}^t[S(t)-B(t)]dt}{{\int }_{t_s}^{t_f}[S(t)-B(t)]dt},$$

where S(t) is the signal at time t, and B(t) is the baseline at time t, is a linear function of a temperature-dependent rate constant, k. This means that:

$$\frac{dx}{dt}=kf(x),$$

expressing conversion rate, dx/dt, at a constant temperature (T) as a function of the reactant concentration loss [10]. k(T) is always described by the Arrhenius equation [11], [12]:

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

where A represents the pre-exponential factor, E, the activation energy and R, the general constant for gases. Since thermal analysis in non-isothermal conditions is carried out at a constant heating rate (β=dT/dt), the substitution dt=dT/β can be made leading to the following differential equation by combining (2) and (3) formulae:

$$\frac{dx}{dT}=\frac{A}{\beta }\exp \left(-\frac{E}{RT}\right)f(x)$$

or

$$\frac{dx}{f(x)}=\frac{A}{\beta }\exp \left(-\frac{E}{RT}\right)dT$$

By integrating this equation in the temperature range T₀ (initial temperature, corresponding to a degree of conversion of x₀) up to T_p (inflection temperature, where x=x_p), the following formula can be obtained:

$${\int }_{x_0}^{x_p}\frac{dx}{f(x)}=\frac{A}{\beta }{\int }_{T_0}^{T_p}{e}^{-E/RT}dT$$

If T₀ value is low, is reasonable to assume that x₀=0 and considering that there is no reaction between 0 °C and T₀ [13], it can be calculated:

$$g(x)={\int }_0^{x_p}\frac{dx}{f(x)}=\frac{A}{\beta }{\int }_o^{T_p}{e}^{-E/RT}dT,$$

where g(x) represents the integral function of conversion.

It is quite obvious to consider that the conversion rate is proportional with the material concentration that has to react, according to the following formulae f(x)=(1−x)ⁿ, where n is the reaction order. This kind of approach is used by several models, the best known being Friedman and Ozawa–Flynn–Wall [13], [14], [15], [16]. When Ozawa–Flynn–Wall model is chosen, Eq. (6) is integrated using Doyle approximation [17]. The results of integration after taking logarithms can be calculated by:

$$\log \beta =\log \frac{AE}{g(x)R}-2.315-\frac{0.457E}{RT}$$

Ozawa–Flynn–Wall model is used to calculate the activation energy for certain given values of the conversion. The activation energy for different conversion values can be determined by applying a log β versus 1000/T plot. The pre-exponential factor is calculated as an average value over all dynamic heating rates. Another intensively used model is that of Friedman, which represents the most general of the derivative techniques and is based on the inter-comparison of dx/dt for a given degree of the conversion x determined when using different heating rates β [18]. This method uses the following logarithmic differential equation:

$$\log \frac{dx}{dt}=\log x\frac{dx}{dT}-\log Af(x)-\frac{E}{4.575T}$$

The equation offers the possibility to obtain E_a values for a wide conversion range by plotting log (dx/dt) versus 1/T for a constant x value [19], [20].

In this paper, the crystallization kinetics of Eu-doped phosphate glass has been studied by differential thermal analysis (DSC) under non-isothermal conditions. The kinetic parameters of the crystallization process were comparatively determined using Friedman and Ozawa–Flynn–Wall analysis. Crystalline phases formed during the thermal treatment of the glass were identified by X-ray diffraction.