Natural Hazards and Earth System Sciences Additional studies on heterodiffusion coefficients in periclase

By using a thermodynamical model, the activation entropy is extracted from the experimental diffusion data of Al in MgO. Furthermore, we estimate the activation volume. We find a value for the activation entropy of (4.21 ± 0.05) k. As for the activation volume, we calculate the value (3 .65± 0.16) cm3 mol−1 which is comparable with the experimental values when errors are considered.


Introduction
Periclase (MgO) is of great geophysical interest since it is the second major and abundant mineral of the Earth's mantle.It has been extensively studied (Ita and Cohen, 1997;Yang and Flynn, 1994;Van Orman andal., 2003, 2009;Amman et al., 2012) for a better understanding of chemical exchange mechanism and diffusivity under pressure and temperature conditions prevailing in the lower mantle.A large number of experimental data (Dewaele and Fiquet, 2000;Jacobsen et al., 2008;Hirose et al., 2008) serve to validate various theoretical models (Oganov and Dorogokupets, 2003;Wu et al., 2008) on thermodynamic properties of MgO, but significant discrepancies have been reported between experimental and theoretical findings.
Here we show that a thermodynamical model which interrelates point defect parameters to bulk and elastic properties, allows to extract from experimental diffusion data, the activation entropy and also theoretically estimate the activation volume.As an application we consider the heterodiffusion of Al in MgO, and the resulting activation volume was found to be in agreement with reported ones.

Thermodynamical model
According to a thermodynamical model, known as cB , the Gibbs activation energy g act is interconnected to the isothermal bulk modulus B through the relation (Varotsos, 1976;Varotsos and Alexopoulos, 1977a;1986;Varotsos et al., 1978) where is the mean atomic volume per atom and c act is a constant considered as independent of pressure and temperature.The above equation also holds for different process mechanisms (e.g.formation and migration) and has been tested in various cases such as in alkali and silver halides (Varotsos andAlexopoulos, 1977b, 1982), in lead fluorides (Varotsos, 2008), in diamonds (Varotsos, 2007a) in metals (Alexopoulos and Varotsos, 1981;Varotsos and Ludwig, 1978) etc.Furthermore, it has been shown to account for the thermodynamic parameters for the defect activation in ionic solids (Varotsos et al., 1999) upon increasing the pressure.This is of profound importance in describing the emission of electric signals before fracture, which explains the electric signals before earthquakes (Varotsos and Alexopoulos, 1984a, b;Varotsos et al., 1986Varotsos et al., , 1993;;Sarlis et al., 2008).
The activation entropy s act is derived when Eq. ( 1) is introduced to the relation s act = −(dg act /dT ) P as with β the volume thermal expansion coefficient and T the temperature while the combination of the thermodynamic relation h act = g act − T (dg act /dT ) P and Eq. ( 1) gives the activation enthalpy h act : Published by Copernicus Publications on behalf of the European Geosciences Union.

E. Dologlou: Additional studies on heterodiffusion coefficients in periclase
Thus, from Eqs. ( 2) and (3) we can estimate the activation entropy (Varotsos, 2007b(Varotsos, , 2008) ) as This entropy should not be confused with the entropy defined (Varotsos et al., 2003a) in a new time domain termed natural time (Varotsos et al., 2002(Varotsos et al., , 2003b) ) which has been shown to be optimal (Abe et al., 2005) in extracting the maximum information possible from an evolving dynamic system.
On the other hand the activation volume v act can be calculated by introducing Eq. ( 1) to the relation v act = dg act /dP which results in

Results and discussion
In order to test the credibility of the above thermodynamical model, we proceed to the estimation of the activation entropy and activation volume, based on experimental diffusion data of Al in MgO.We use the equation reported by Van Orman et al. (2009) activation energy h act = 213 ± 32 kJ mol −1 = 2.21 eV which was extracted from experimental diffusion data of Al in MgO.We consider at T = 2273 and room pressure, β = 3.16 × 10 −5 K −1 (Fabrichnaya et al., 2004).The isothermal B = 102.6GPa and the temperature derivative (dB/ dT ) P = (0.0301 ± 0.0005) GPa K −1 are obtained from Table 6 of Sushil (2005): the first from interpolation to the data for P = 0 and the second from the slope of these data.By inserting all the above relevant values in Eq. ( 4), we get for the activation entropy s act = (4.21± 0.05) k.Concerning now the estimation of the activation volume, we use Eq. ( 5).The pressure derivative of B at T = 2273 K is obtained from Table 6 of Sushil (2005) as follows: first by interpolation to the B-values at 2000 K and 2500 K, we get the B-values at T = 2273 K for the pressure range (0-56.6)GPa (see Table 1); then from the slope of these interpolated values versus pressure (Fig. 1), we have dB/dP = 3.8 ± 0.1.At last, considering that g act = h act − T s act , we get through Eq. ( 5) v act , = (3.65 ± 0.16) cm 3 mol −1 .This value, taking into account all uncertainties, is comparable with the reported one (3.22 ± 0.25) cm 3 mol −1 at T = 2273 K by Van Orman et al. (2009) and matches successfully a previously published one of (3.5±0.5)cm 3 mol −1 (Van Orman et al., 2003), when errors are considered.

Conclusions
A thermodynamical model interrelating defect parameters to bulk and elastic expansivity data allows to extract from experimental diffusion data of Al in MgO the activation entropy   1.

Fig. 1 .
Fig. 1.The pressure dependence of isothermal bulk modulus B at T = 2273 for MgO with correlation coefficient R = 0.999, based on data listed in Table1.

Table 1 .
Values of isothermal bulk modulus B (GPa) for MgO at different temperatures and pressures.