Real and imaginary shear modulus

4.5. Heat capacity

4.6.1. Real and imaginary shear modulus

The frequency dependent deformation data were obtained from 300-1000°C. The maximum temperature is that at which it is no longer possible to measure the elastic (instantaneous) component of deformation; this occurs at viscosity of ~10⁹ Pa s. The minimum temperature is defined by the point at which the melt breaks away from the alumina torsion rod due to differences in thermal expansion of the two materials.

Figure 67 shows the variation in real and imaginary shear modulus as a function of angular frequency (

ω

π

f ) times Maxwell relaxation time (

τ

_M). Data of real and imaginary shear modulus are presented in Appendix 2. The relaxation time is calculated from the Maxwell equation

τ

_M( )T =

η

_N( ) /T G_∞ for Newtonian viscosity

η

_N - determined from the micropenetration measurements, and elastic (infinite frequency) shear modulus G_∞, which is taken to be the shear modulus at 1 Hz at the lowest temperature of measurement (see Tab. 16). The relaxation times for all of the melts are presented in Appendix 3. The studies of Rivers and Carmichael (1987), Webb (1992b) and Farnan and Stebbins (1994) (as well as many others) show that the Maxwell relationship successfully calculates the structural relaxation time of silicate melts. Plotting data from different temperatures on the same curve uses the principle of “thermorheological simplicity” which assumes that the structure and mechanism of flow in these melts does not change over the temperature interval of the measurements (e.g. Narayanaswamy, 1988).

Herzfeld & Litovitz (1959) developed a general equation to describe the frequency dependence of the shear modulus of a material independent of its structure;

( )

^{2 2}

2 2 2 2

1 1

G G

ω ω τ

ωτ

ω τ ω τ

∞ ∞

= +

+ + (Eq. 57)

where G_∞ is infinite frequency elastic shear modulus, τ is the structural relaxation time and ω is angular frequency (see also Nowick & Berry, 1972; Webb, 1991). The dotted lines in Figure 67 are the real and imaginary shear modulus expected from the theory of Herzfeld & Litovitz (1959). As can be seen in Figure 67, none of the present data is adequately described by this equation. The solid lines are fits to the data (see also Figures 68 and 69) based on the assumption that there is a distribution of relaxation times. Data of the lines are given in Appendix 4a,b (for samples G0 and G1-G7) and in Appendix 4c,d (for samples G8-G14).

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Fig. 67. Real and imaginary components of the shear modulus of the present melts as a function of log10 ωτ_M. Open circles are for real shear modulus; solid circles are for imaginary shear modulus;

the dotted line are the model of Herzfeld & Litovitz (1959) for a single structure melt; solid line of the fitting of the data points; grey line is fitting of the imaginary parameters to the real shear modulus of the sample. Not all the data has been shown; for some of the samples the plot has been cut at log10 ωτM = 10.

-2 -1 0 1 2 3 4 5 6 7 8 9 10

0 5 10 15 20 25 30 35 40

G1 real G1 imaginary FIT to G1_im

Shear Modulus, G (GPa)

log ₁₀ωτ_M

0 5 10 15 20 25 30 35 40

G2 real G2 imaginary FIT to G2_im Shear Modulus, G (GPa)

Fig. 67. continuation…

-2 -1 0 1 2 3 4 5 6 7 8 9 10

0 5 10 15 20 25 30 35 40

G3 real G3 imaginary FIT to G3_im

log ₁₀ωτ_M

Shear Modulus, G (GPa)

-2 -1 0 1 2 3 4 5 6 7 8 9 10

0 5 10 15 20 25 30 35 40

G4 real G4 imaginary FIT to G4_im

Shear Modulus, G (GPa)

log ₁₀ωτ_M

Fig. 67. continuation…

-2 -1 0 1 2 3 4 5 6 7 8 9 10

0 5 10 15 20 25 30 35 40

G5 real G5 imaginary FIT to G5_im

Shear Modulus, G (GPa)

log ₁₀ωτ_M

-2 -1 0 1 2 3 4 5 6 7 8 9 10

0 5 10 15 20 25 30 35 40

G6 real G6 imaginary FIT to G6_im

Shear Modulus, G (GPa)

log ₁₀ωτ_M

Fig. 67. continuation…

-2 -1 0 1 2 3 4 5 6 7 8 9 10

0 5 10 15 20 25 30 35 40

G7 real G7 imaginary FIT to G7_im

Shear Modulus, G (GPa)

log ₁₀ωτ_M

-2 -1 0 1 2 3 4 5 6 7 8 9 10

0 5 10 15 20 25 30 35 40

G8 real G8 imaginary FIT to G8_im

Shear Modulus, G (GPa)

log ₁₀ωτ_M

Fig. 67. continuation…

-2 -1 0 1 2 3 4 5 6 7 8 9 10

0 5 10 15 20 25 30 35 40

G9 real G9 imaginary FIT to G9_im

Shear Modulus, G (GPa)

log ₁₀ωτ_M

-2 -1 0 1 2 3 4 5 6 7 8 9 10

0 5 10 15 20 25 30 35 40

G10 real G10 imaginary FIT to G10_im

Shear Modulus, G (GPa)

log ₁₀ωτ_M

Fig. 67. continuation…

-2 -1 0 1 2 3 4 5 6 7 8 9 10

0 5 10 15 20 25 30 35 40

G11 real G11 imaginary FIT to G11_im

Shear Modulus, G (GPa)

log ₁₀ωτ_M

-2 -1 0 1 2 3 4 5 6 7 8 9 10

0 5 10 15 20 25 30 35 40

G12 real G12 imaginary FIT to G12_im

Shear Modulus, G (GPa)

log ₁₀ωτ_M

Fig. 67. continuation…

-2 -1 0 1 2 3 4 5 6 7 8 9 10

0 5 10 15 20 25 30 35 40

G13 real G13 imaginary FIT to G13_im

Shear Modulus, G (GPa)

log ₁₀ωτ_M

-2 -1 0 1 2 3 4 5 6 7 8 9 10

0 5 10 15 20 25 30 35 40

G14 real G14 imaginary FIT to G14_im

Shear Modulus, G (GPa)

log ₁₀ωτ_M

Fig. 68. The fits to the shear modulus data for NS2 (G0) and Fe-free melts (G1-G7), using Eq. 57.

a) fitted lines for real shear modulus data; b) fitted lines for imaginary shear modulus data. See also Appendix 4a,b.

Real Shear Modulus, G real (GPa)

∆ log₁₀τ_M (s)

Imaginary Shear Modulus, G im (GPa)

Fig. 69. The fits to the shear modulus data for Fe-bearing melts (G8-G14), using Eq. 57.

a) fitted lines for real shear modulus data; b) fitted lines for imaginary shear modulus data. See also

-2 -1 0 1 2 3 4 5 6 7 8 9 10

Real Shear Modulus, G real (GPa)

∆ log

Imaginary Shear Modulus, G im (GPa)

Instead of assuming the form of the distribution of relaxation time, the data have been fit by a summation of Eq. 57:

( )

^1.5

* *

8.5

( 10 )^x

x x

ωτ

=−

∑

⋅ (Eq. 58)

The real and imaginary parts of the data have been fit separately. The resulting calculated distribution of relaxation time is shown in Figures 70 and 71 and the parameters of the fit are given in Appendix 5a,b.

Table 16 compares the G_∞ values determined at 1 Hz, with the average of the frequency independent G_av values determined from the forced oscillation measurements, and the room temperature elastic shear moduli determined on the glasses by ultrasonic methods - Gultra. The Gav moduli were determined over the temperature range between lowest measured temperatures up to 500-600°C (depending on the sample) and appear to be frequency independent and shows very little temperature dependence over the 50-200°C (depending on the sample) range of calculation. The values of G_av and G_ultra are shown in Figure 57a,b as a function of the compositional parameter γ.

Tab. 16. Shear moduli Gav and G∞ from the torsion data together with that determined by ultrasonic techniques at room temperature Gultra.

Torsion Measurements Pulse Echo Overlap Technique Gav (GPa) G∞ (GPa) Gultra (GPa)

G0 27.730 ± 0.278 27.45 29.887 ± 0.232 G1 34.431 ± 0.479 34.26 37.389 ± 0.754 G2 34.663 ± 0.265 34.78 37.842 ± 1.091 G3 34.358 ± 0.402 34.43 38.036 ± 0.147 G4 33.984 ± 0.581 34.23 38.074 ± 0.625 G5 33.863 ± 0.219 34.01 37.122 ± 0.155 G6 33.485 ± 0.349 33.47 36.650 ± 0.191 G7 31.034 ± 0.376 31.68 34.284 ± 0.198 G8 31.998 ± 0.746 32.45 34.228 ± 0.292 G9 32.108 ± 0.636 32.94 34.090 ± 0.129 G10 31.825 ± 0.373 32.20 33.555 ± 0.514 G11 31.775 ± 0.472 32.09 33.540 ± 0.362 G12 31.078 ± 0.292 31.42 32.230 ± 0.274 G13 29.810 ± 0.489 30.89 31.256 ± 0.247 G14 29.261 ± 0.378 29.96 30.938 ± 0.189

Fig. 70. The distribution of relaxation times for Fe-free samples (G0 and G1-G7) calculated from Eq. 58. The lines on the plot a) are calculated from the real component of the shear modulus, the lines from plot b) show the distributions calculated from the imaginary part of the modulus. The structural relaxation of G7 and G0 is centred on ∆ log10 τ ~ 0, while that of G2 is centred on a timescale 1.5 order of magnitude faster than the Maxwell relaxation time. In all cases, a second

-2 -1 0 1 2 3 4 5 6 7 8 9

Real Shear Modulus, G real (GPa)

∆ log τ (s)

Imaginary Shear Modulus, G im (GPa)

Fig. 71. The distribution of relaxation times for Fe-bearing samples (G8-G14) calculated from Eq. 58. The lines on the plot a) are calculated from the real component of the shear modulus, the lines from plot b) show the distributions calculated from the imaginary part of the modulus. There is no sample with structural relaxation centred on ∆ log10 τ = 0. The other peaks are centred on a timescale 0.5-1.5 order of magnitude faster than the Maxwell relaxation time. Location of the second relaxation is not clear. See also Appendix 5b.

-2 -1 0 1 2 3 4 5 6 7 8 9

0 2 4 6 8 10 12 14 16

Fe-bearing melts G8 G9 G10 G11 G12 G13 G14

Real Shear Modulus, G real (GPa)

∆ log τ (s)

-2 -1 0 1 2 3 4 5 6 7 8 9

0 2 4 6 8 10 12

Fe-bearing melts

∆ log τ (s)

Imaginary Shear Modulus, G im (GPa)

G8 G9 G10 G11 G12 G13 G14 a)

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 27

28 29 30 31 32 33 34 35 36

G13 G14 G12 G11 G10 G8 G9

G0 G7

G6 G5 G4 G2G3

(Na₂O + FeO) / (Na₂O + Al₂O₃ + FeO + Fe₂O₃)

NS Fefreemelts Febearingme

Average Shear Modulus, G av (GPa)

Fig. 72. The average elastic shear moduli as a function of composition – data obtained from torsion apparatus; blue point is for sodium silicate melt NS2 (G0), black points are for data of G_av for Fe-free samples (G1-G7), red points – for Fe-bearing melts (G8-G14). The dashed lines are the suggested trends in the data.

Two peaks are observed in the imaginary component of the shear modulus. Both peaks are accompanied by an increase in the real part of the shear modulus. Each peak in the imaginary modulus must be accompanied by an increase in shear modulus. The existence of this second fast relaxation peak is confirmed by the fitting of Eq. 58 independently to the real and imaginary modulus data.

The peak which occurs at the lowest frequencies is the α-relaxation and therefore caused by the loss of energy to the motion of Si and O ions in the melt – the glass transition - which results in the change in shear modulus from 0 to ~35 GPa. The second peak occurs ~5.5-7.5 orders of magnitude faster than the α-peak (depending on the location of α-peak) and is accompanied by a 2-3 GPa increase in the real part of the modulus. There are two energy loss processes (other than that associated with the life-time of Si-O bonds) expected in these melts – the motion of Al³⁺ ions and the motion of Na⁺ ions – see Figure 73.

Fig. 73. The known change in modulus with ωτ_M for the motion of Si and O atoms in silicate melts together with theoretically expected loss modulus associated with the motion of Al³⁺ and Na⁺ atoms in the present melts.

The width of the α-imaginary peak is the same for all the melt compositions with the FWHM (full width at half maximum) for NS2 melt equals 1.13 and the range 1.40-2.68 for G1-G14 (see Table 17) with no clear compositional dependence. However, the position of the peak moves from being centred on τM – the Na-rich compositions, to being centred on a timescale 1.5 order of magnitude smaller – the Al-rich compositions.

Such fast relaxation has never been seen before in a silicate melt. The data of Mills (1974), Bagdassarov et al. (1993) and Webb (1992a) all show that the α-peak in silicate melts determined by mechanical spectroscopy is centred on log₁₀ωτ = 0 ±0.5.

Melt number FWHM log10 ωτM

G0 1.13

G1 1.90

G2 2.68

G3 1.80

G4 1.40

G5 2.01

G6 1.82

G7 1.52

G8 1.83

G9 1.57

G10 1.67 G11 1.81 G12 1.67 G13 1.69 G14 1.80

Shear Modulus, G (GPa)

log 10 ω τ_Μ

REAL COMPONENT

IMAGINARY COMPONENT

Tab. 17. FWHM (full width at half maximum) for α-peak in investigated silicate melts: NS2 (G0), Fe-free (G1-G7) and Fe-bearing (G8-G14)

Im Dokument Physical and thermodynamic properties of aluminosilicate melts as a function of composition (Seite 101-116)