Similar to configurational heat capacity Cpconf, the results of configurational entropy Sconf and Be parameter have not been shown before in a controlled series of compositions.
Together with higher energy, the distribution of configurational states occurs, what is measured by configurational entropy Sconf (Richet & Bottinga, 1995). The configurational entropy of the melt can be determined e.g. using heat capacity measurements:
( ) ( )
g
T conf
conf conf p
g T
S T S T C dT
= +
∫
T , (Eq. 60)where Sconf( )T is constant between 0 K and Tg and Cconfp =constant. Sconf
( )
Tg can bedetermined from viscosity measurements using the Adam-Gibbs relationship:
( )
T 0 exp T SconfBe( )
Tη
=η
⎛⎜⎜⎝ ⎞⎟⎟⎠, (Eq. 61)where Be is a constant (Richet & Neuville, 1992; Toplis, 1998).
Changes in the viscosity and fragility in the investigated melts as a function of composition are connected with changes in configurational parameters. The link between configurational entropy and viscosity lies in Adam – Gibbs theory (Adam & Gibbs, 1965;
Richet, 1984; Bottinga & Richet, 1996; Mysen, 1998):
ln ( )
e
c conf
A B
T S T
η
= + (Eq. 62)(logarithmic form of Eq. 61), where Ac and Be are constants, independent of temperature, but dependent on composition; and Sconf(T) is configurational entropy. This configurational entropy model bonds viscous behaviour of aluminosilicates and their thermodynamic behaviour (Richet & Neuville, 1992; Bottinga, 1994; Mysen, 1995b; Mysen, 1997). Adam – Gibbs theory says that structural relaxation time is function of energy barrier needed to relocate a single silicate unit, and the number of units that must at the same time conquers their energetic barrier to reach a change in configuration (Toplis, 2001).
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The height of the peak in the Cp curve (Fig. 66) upon heating can be described as a function of heating and cooling rate in terms of activation energy H, which is found to be identical to that obtained from viscosity measurements (Scherer, 1984; Stevenson et al., 1996); with the structural relaxation time being given by Adam-Gibbs model (Crichton &
Moynihan, 1988):
The Narayanaswamy model partitions the B S/ conf
( )
T term into a structure dependent term(
1−x H)
and a temperature dependent term xH (Wilding et al., 1995;and references therein). Thus the determination of Cp as a function of heating rate can be used to determine changes in melt structure with changing composition as well as the
( )
free (black points) and Fe-bearing (red points) melts as a function of compositional parameterγ
. The trends are the guides to the eye. In the case of Fe-free melts in the plot Sconf(T) vs. composition there is not enough data to support a breaking trend.0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 7
How to connect calculated configurational entropy and the structure of the glasses?
The studies of Toplis (1998; 2001) and Russel et al. (2003) (provide an average value of
η
0 for silicate melts which allows the calculation of the ratio Be/Sconf(T) and ultimately a calculation of Sconf(Tg). Sconf(Tg) can be divided into two parts: (1) the topological contribution due to the different bonding within the glass; and (2) the chemical contribution – due to the mixing of different elements on the same structural site (Richet &Neuville, 1992). Both contributions change and new entropies arise.
Similarly, Cconfp is made up of a chemical and topological contribution and can be used to support the interpretation of the variations in Sconf(T) with composition. The calculation of Sconf(T) as a function of composition will therefore allow discussion of the structure in terms of bonding and site preference of the Al3+ and Fe3+ atoms. Sconf(T) is a measure of the range of structure.
In Figure 77 the Be parameter (Fig. 77a) and configurational entropy Sconf(T) (Fig. 77b) for NS2, Fe-free and Fe-bearing melts as a function of compositional parameter γ = (Na2O+FeO)/(Na2O+Al2O3+FeO+Fe2O3) has been presented. Be is:
e B
B h f
= k T , (Eq. 64)
where h is Planck’s constant (6.626068·10-34 m2 kg s-1), f is frequency, kB is Boltzmann’s constant and T is temperature. Be was determined analytically from fit to the viscosity curve of the sample which heat capacity is measured (Richet, 1984; Bottinga &
Richet, 1996).
Be is a potential energy barrier to viscous flow and depends on the composition but is assumed to be temperature independent (Toplis, 1998). Be parameter has been then taken to determine configurational entropy Sconf(T) using an Eq. 61 where
( )T 0 exp(Be/Sconf( )T T)
η
=η
.η
0 (marked also as A0 or AVFT in Eq. 26) is the constant value of viscosity at infinite temperature. In this study this value has been fixed as -2.6 (Toplis, 1998). Choosing another value produces the same trend, just at slightly different absolute values.The size of the region in the rearranging structure is expressed by the configurational entropy of this region (Adam & Gibbs, 1965). Sconf(T) is the sum of the configurational entropy of the smallest rearranging unit occurring in the structure – more precisely: the number of atoms taking part in the viscosity flow (Toplis, 1998; Adam &
Gibbs, 1965).
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Sconf(T) reveals also a temperature dependence. The number of small subsystems increases and as, a consequence, Sconf(T) also increases with temperature (Mysen, 1997;
Toplis et al., 1997b; Toplis, 1998).
With change of the number of the smallest rearranging unit at glass transition [z*(Tg)] or height of the average potential energy barrier to viscous flow (∆µ), then the ratio Be/Sconf(T) also changes. Be/Sconf(T) relation shows a compositional dependence (Toplis, 1998).
In Figure 77 both Be and Sconf(T) represent the breaking points at γ ~ 0.5, only in the case of Fe-free melts in the plot Sconf(T) vs. composition there is not enough data to support a breaking trend. Be increases with increasing Al2O3 content. In the peralkaline range of composition this increase is small, but in peraluminous melts Be parameter rises very fast. The breaking point indicates a change in a potential energy barrier to viscous flow, it means that with increasing Al2O3 content (close to γ ~ 0.5) new structure with higher potential energy arises and the flow mechanism is different too.
Sconf(T) as a function of composition shows similar behaviour. Together with increasing Sconf(T) decreases the size and increases the number of the units causing the rearrangement of the structure. Fe-bearing melts have slightly higher Sconf(T) indicating smaller structural units in the network.
Because in the Be parameter and Sconf(T) plots is observed large scatter to describe a structure of the melts is used a ratio Be/Sconf(T). Be/Sconf(T) ratio for the investigated samples as a function of their composition is shown in Figure 78. The trend is exactly the same like on viscosity or shear modulus plot and shows a structural change close to subaluminous point.
Fig. 78. Be/Sconf(T) as a function of compositional parameter γ = (Na2O+FeO)/(Na2O+Al2O3+FeO+
+Fe2O3). Blue point is a NS2 melt (G0); black points are for Fe-free melts (G1-G7); red points are for Fe-bearing melts (G8-G14).