## 2 Modeling  
According to [PERSON]'s absolute rate theory, the viscosity of a liquid is given by [27; 28]:  
\[\eta=\frac{hN_{A}}{V_{m}}\exp\left(\frac{E_{0}}{RT}\right) \tag{1}\]  
where \(\eta\), \(h\), \(N_{A}\), \(V_{m}\), \(E_{0}\), and \(R\) are the dynamic viscosity, Planck's constant, Avogadro's number, molar volume of the liquid, activation free energy for viscous flow, and the ideal gas constant, respectively.  
To describe the pressure dependence of viscosity, the volume of activation was introduced as follows [29; 30]:  
\[\eta=\mathrm{A}\exp\left(\frac{pV_{0}}{RT}\right). \tag{2}\]where \(A\) is a prefactor, which is dependent on temperature, \(p\) is pressure, and \(V_{0}\) is the volume of activation. Parameter \(A\) should be expressed as a function of temperature to describe the temperature dependence of glass viscosity. Combining Equations (1) and (2), the Eyring model can be expressed as follows, in which, the similar form can be found in the literature [31]:  
\[\eta=\frac{B}{V_{m}}\exp\left(\frac{pV_{0}}{RT}\right) \tag{3}\]  
where \(B\) is a prefactor that is independent of temperature. The volume of activation is often considered as a constant [24]. However, it is dependent on pressure and temperature. Due to having little practical consequence for glass-forming or glass-processing, pressure is not taken into account [32], so \(V_{0}\) can be described as [22]:  
\[V_{0}=k(T)V_{m} \tag{4}\]  
where \(k\) is the ratio of the activation volume to molar volume and the function of temperature. This ratio should be equal to that ratio of the number of activated particles to the total number of particles--that is, the probability of the occurrence of an activated particle. According to the Boltzmann distribution law, \(k\) can be given by:  
\[k(T)=\alpha e^{-\frac{\beta}{T}} \tag{5}\]  
where \(\alpha\) and \(\beta\) are composition-dependent constants. The change in volume during glass forming is shown in Figure 1. Above \(T_{g}\), the molar volume (\(V_{m}\)) changes almost linearly with temperature. The crossover of the nonequilibrium state (at low temperatures) and the linear expansion of the equilibrium state (at high temperatures) allow the \(T_{g}\) values to be determined, which has been confirmed by both experiments and simulations [33; 34; 35; 36]. Despite a sudden change in the slope of \(V_{m}\) versus \(T\) in the glass transition region, it does not often have a significant effect on the relationship between viscosity and temperature, because the casting temperature is above \(T_{g}\). If the mass of glass remains constant, we can assume when \(T>T_{g}\), molar volume is an appropriate linear function of temperature:  
\[V_{m}=aT+b \tag{6}\]  
where \(a\) and \(b\) are related to glass compositions.  
Figure 1: Schematic representation of molar volume as a function of the temperature of a glass liquid [37].  
Putting Equations (4)-(6) into (3), the following Equation (7) is obtained:  
\[\eta=\frac{B}{aT+b}\exp\left(\frac{p\alpha e-\frac{\beta}{T}}{RT}(aT+b)\right) \tag{7}\]  
Equation (7) becomes:  
\[\eta=\frac{1}{CT+D}\exp\left(\frac{He-\frac{\beta}{T}}{RT}(CT+D)\right)\ln\eta= \ln\frac{1}{CT+D}+\frac{He-\frac{\beta}{T}}{RT}(CT+D) \tag{8}\]  
where \(C=a/B\), \(D=b/B\), and \(H=Bp\alpha\). \(C\), \(D\), \(H\), and \(\beta\) are regressed by experimental data.