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# Viscosity of amorphous materials

Viscous flow in amorphous materials (e.g. in glasses and melts) [1][2][3] is a thermally activated process:

$\eta = A \cdot e^{Q/RT}$

where Q is activation energy of viscosity, T is temperature, R is the molar gas constant and A is approximately a constant.

The viscous flow in amorphous materials is characterised by a deviation from the Arrhenius-type behaviour: Q changes from a high value QH at low temperatures (in the glassy state) to a low value QL at high temperatures (in the liquid state). Depending on this change, amorphous materials are classified as either

• strong when: QHQL < QL or
• fragile when: $Q_H - Q_L \ge Q_L$

The fragility of amorphous materials is numerically characterized by the Doremus’ fragility ratio:

RD = QH / QL

and strong material have $R_D < 2\;$ whereas fragile materials have $R_D \ge 2$

The viscosity of amorphous materials is quite exactly described by a two-exponential equation:

$\eta = A_1 \cdot T \cdot [1 + A_2 \cdot e^{B/RT}] \cdot [1 + C \cdot e^{D/RT}]$

with constants A1,A2,B,C and D related to thermodynamic parameters of joining bonds of an amorphous material.

Not very far from the glass transition temperature, Tg, this equation can be approximated by a Vogel-Tammann-Fulcher (VTF) equation or a Kohlrausch-type stretched-exponential law.

If the temperature is significantly lower than the glass transition temperature, $T\ll T_g\;$, then the two-exponential equation simplifies to an Arrhenius type equation:

$\eta = A \cdot e^{Q_H/RT}$

with:

QH = Hd + Hm

where Hd is the enthalpy of formation of broken bonds (termed configurons) and Hm is the enthalpy of their motion.

When the temperature is less than the glass transition temperature, T < Tg, the activation energy of viscosity is high because the amorphous materials are in the glassy state and most of their joining bonds are intact.

If the temperature is highly above the glass transition temperature, $T \gg Tg$, the two-exponential equation also simplifies to an Arrhenius type equation:

$\eta = A\cdot e^{Q_L/RT}$

with:

QL = Hm

When the temperature is higher than the glass transition temperature, T > Tg, the activation energy of viscosity is low because amorphous materials are melt and have most of their joining bonds broken which facilitates flow.

An example of glass viscosity is given in Calculation of glass properties, in which the viscosity is around 1012 Pa·s at 400°C.

## References

1. ^ R.H.Doremus (2002). "Viscosity of silica". J. Appl. Phys. 92 (12): 7619-7629. ISSN 0021-8979.
2. ^ M.I. Ojovan and W.E. Lee (2004). "Viscosity of network liquids within Doremus approach". J. Appl. Phys. 95 (7): 3803-3810. ISSN 0021-8979.
3. ^ M.I. Ojovan, K.P. Travis and R.J. Hand (2000). "Thermodynamic parameters of bonds in glassy materials from viscosity-temperature relationships". J. Phys.: Condensed matter 19 (41): 415107. ISSN 0953-8984.