Shear modulus of melts - Physical and thermodynamic properties of aluminosilicate melts as a fu

3.7.8. Shear modulus of melts

The shear modulus of the silicate melts was determined with a set: ~30mm long, 8 mm diameter glass cylinder glued to an upper and lower Al₂O₃ torsion rod. The lower rod is ~295mm long and the upper rod is ~155±20mm long (depending on the length of the sample). Preliminary experiments found it difficult to achieve a good physical connection between the melt cylinder and the alumina torsion rod. First attempts were done with a platinum tube (the method of Bagdassarov et al., 1993). This uses a Pt-foil wrapped around the melt cylinder and alumina rods to hold everything together at high temperature. The result was only bubble filled interfaces between the rods and the melt sample (see Fig. 44). Thus we reverted to the simpler method (Webb, 1992a) of wrapping the glass cylinder and alumina rods in paper to hold the assembly together at high temperature.

It was also necessary to use a glue to bond the rods to the melt. This glue is a Si-rich Fe-bearing melt (composition of obsidian from Little Glass Butte, Oregon), which has a viscosity slightly higher than the most peraluminous melts used. After initial attempts in which the alumina rod melted into the rhyolite melt; a recipe for temperature and

Channel 1

1000°C 159.38 145.83 ¹⁴⁵ ₀ ₁₀₀ ₂₀₀ ₃₀₀ ₄₀₀ ₅₀₀ ₆₀₀ ₇₀₀ ₈₀₀ ₉₀₀ ₁₀₀₀ ₁₁₀₀

150 155 160 165

CHANNEL 1

G = 162.67(14)-5.87(24)x10^-3 T

CHANNEL 2

G = 161.54(12)-11.84(21)x10^-3 T

CHANNEL1 CHANNEL2

Shear Modulus, G (GPa)

Temperature, T (°C)

Figure 43. Measured shear modulus of the alumina rod from room temperature to 1000°C. The GAl2O3 used in calculation of shear modulus of the melt was set to be 160 GPa.

Tab. 7. Table of measured shear modulus of the alumina rod from room temperature to 1000°C.

Fig. 44. Bubble filled sample produced by wrapping both glass and Al2O3

torsion rod in Pt-foil and going to high temperature.

Fig. 45. Rhyolite infiltration ~50 µm into the alumina torsion rod measured with Quanta 200F FE-SEM (Crystallography Department, Georg-August-University Göttingen).

A thin film of this rhyolite melt is allowed to react with the Al₂O₃ rod (Fig. 45). The sample is then melted onto the rhyolite at the ends of the two torsion rods to produce a good contact between the Na₂O-Al₂O₃-SiO₂ melt and the torque rod (Webb, 1991). The

interface alumina rod

rhyolite

__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

presence of this thin film of highly viscous melt will introduce an error of ~1 GPa into the calculated shear modulus of the melt. Thus the measured shear moduli have an error of

±1.5 GPa due to the use of a temperature independent shear modulus for the torsion rods, and the use of a glue between the sample and the torsion rods.

The temperature inside the furnace has been measured with a type S thermocouple, and it was found to be constant ±3°C across the 40 mm where the melt

sample stands (Fig. 46). Therefore, only 10 mm of the alumina rod is at temperature of the measurement, ~66 mm are at ~room temperature (lower part) and the remaining length is in a thermal gradient from room temperature to the temperature of interest. The upper part of the rod is partially at high temperature (up to the upper channel) and then is in a thermal gradient to the room temperature.

Fig. 46. Temperature profile in the torsion machine. Location of the furnace, sample and the channel 1 and base plate has been marked.

The deformation analysis for the compound torsion rod (Al₂O₃ + melt + Al₂O₃) as a function of frequency is made in the same manner as for the simple alumina rod but the calculation is slightly different. For the measurement of the frequency dependent deformation of the melt, the torque is created by the application of a sinusoidal signal to a

66.22 mm

temperature profile

two alumina rods plus the viscoelastic deformation of the melt. Voltage signals coming from the wing “S” and both channels were also calculated the same way as in previous case. As a result one obtains a sinusoidal angle of twist for channels as a function of time for given frequency (see Fig. 47).

Based upon the earlier calibrations, the voltage from the transducers at channel 1 gives the applied stress, and the voltage at channel 2 gives the deformation of the compound torsion rod. The time delay – ∆

δ

between the applied stress and the resulting strain is also calculated;

1 2

δ δ δ

∆ = − . (Eq. 45)

With known voltages of the channels V1 and V2 as well as time delay ∆

δ

it is possible to calculate the angle of twist φ for both channels:

1 1

1 ( )

V f R r

ϕ

+ and ₂ ²

2 ( )

V f R r

ϕ

+ , (Eq. 46 a-b)

where R is the length of the wings (in mm) and r – radius of the rod (in mm).

Fig. 47. Plot of the relationship voltage vs. time showed as a sinusoidal signal (measured from the channels 1 and 2), where V1 and V2 correspond with the voltage of these channels;

δ

₁ and

δ

₂

describe the shift of the sinusoids.

Here it was unnecessary to do a weight analysis because the torque

τ

is calculated using the angle of twist of the channel 1

ϕ

₁:

The angle of twist

ϕ

₂ of the compound rod must be separated into the components due to the twist of the lower alumina rod –

ϕ

rod , the melt -

ϕ

melt ^{and the}

upper alumina rod −

ϕ

_rod₂. This is shown graphically in Figure 48. Here, the measured parameters are δ – the delay between the applied stress and the resulting strain of the compound rod; and

ϕ

2 - the deformation measured at channel 2. This must now be separated into the contribution from the elastic twist of rod 1 and rod 2:

1 τ is calculated from Eq. 47. The complex shear modulus of the melt is:

( )

The complex shear modulus can be separated into the real and imaginary components by:

( )

_im

( )

_melt ^sin ^''

( )

ω

ω γ

ω

(Eq. 51)

( )

_real

( )

_melt ^cos ^'

( )

ω

ω γ

ω

, (Eq. 52) where

2 1

tan sin

cos _rod

ϕ δ

γ γ

ϕ δ ϕ

= =

− . (Eq. 53)

Finally, the shear modulus of the melt can be described by:

( )

_melt ^''

( )

² ^'

( )

ω

= G

ω

. (Eq. 54)

Fig. 48. Plot of trigonometrical relationships for the deformation of the compound torsion rod.

φ melt

φ rod1

φ rod2

δ γ G’(ω)

φ 2 sinδ

real component

imaginary component

G’’(ω) φ 2

Im Dokument Physical and thermodynamic properties of aluminosilicate melts as a function of composition (Seite 74-80)