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Phase separation of glass system

 Introduction

From a macroscopic observation, the glass often appears homogeneous; while narrowing down the scale to few hundred atoms, the microstructure starts to become inhomogeneous and appears some separate phases.

For glass-forming oxide system, such as well-known silica or boric oxide substrate glass, exist liquid-liquid immiscibility under certain composition and temperature, the homogeneous glass acquires driving force to separate into several more stable phases, this phenomenon is named as “phase separation”. There are two mechanisms result in separate phase, one is nucleation and growth from a supercooled liquid, the other is spinodal decomposition, both of them will be mentioned and discussed later.

To form a glass, it is inevitable to cool the glass-forming system from liquid phase to a metastable region at lower temperature without visualizing by naked eyes, since the significant properties of glass are high-viscosity and low diffusion rates under the glass transformation temperature. A subtle phase separation structure forms and can only be observed under electron microscopy.

Starting from the thermodynamics of mixing is helpful to get the whole concept of phase separation.

Thermodynamic perspective

Phase diagram

The simplest phase diagram is binary phase diagram, composed of two different components. Take Fig (1,a) for example, it starts from single liquid phase at high temperature, when decreases the temperature through a critical consolute temperature T_c, the two-liquid immiscible region exist. There are two types of phase diagrams exhibit different immiscibility situation.

The first type shows both stable and metastable immiscibility, a horizontal line separates unmixed stable area from unmixed metastable area. The examples for this phase diagram are MgO-SiO₂, CaO-SiO₂ and PbO-B₂O₃. The second type of phase diagram shows entire sub-liquidus metastable immiscibility region, the example for this type of phase diagram are BaO-SiO₂, Li₂O-SiO₂, and Na₂O-SiO₂. 

Free energy vs. composition relationship

Fig 1(a) Phase diagram

The Gibbs free energy versus composition curves represents the energy distribution at certain temperature of the phase diagram. From the curves, the driving force of the changing from single liquid phase to several more stable phases can be calculated.

Assumed the solution is composed of two components, A and B, at a certain temperature T and pressure P. There are three situations might appear: perfect mixed $(\bigtriangleup G_{m}<0)$, perfect unmixed $(\bigtriangleup G_{m}>0)$ and partially mixed. $\bigtriangleup G_{m}$ is defined as the change of Gibbs free energy after mixing solution $\bigtriangleup G_{m}=G_{AB}-[(1-x)G_{A}^{\circ }+xG_{B}^{\circ }]$, where $G_{AB}$ is the Gibbs free energy of the mixed system, $x$ is the mole fraction, and $G_{A}^{\circ }$, $G_{B}^{\circ }$ are the energies of pure A and B components. The Gibbs free energy change after mixing can be given as equation $\bigtriangleup G_{m}=\bigtriangleup H_{m}-T\bigtriangleup S_{m}$, where $\bigtriangleup H_{m}$ and $\bigtriangleup S_{m}$ are the enthalpy and entropy of mixing.

For ideal solutions, $\bigtriangleup H_{m}=0$, therefore, $\bigtriangleup G_{m}$ is always negative.

For regular solutions, $\bigtriangleup H_{m}\neq 0$. The free energy can be calculated quantitatively by using simple statistic model which assumed A and B component are randomly mixed on regular lattice. The regular solution model was first brought out by Hildebrand, $\bigtriangleup G_{m}=\bigtriangleup H_{m}-T\bigtriangleup S_{m}=\alpha x(1-x)+RT[x\ln x+(1-x)\ln(1-x)]$ , where $x$ stands for the mole fraction of B component, and $\alpha =NZ[E_{AB}-{\frac {1}{2}}(E_{AA}+E_{BB})]$ the excess interaction energy, where $E_{AA}$, $E_{BB}$ and $E_{AB}$ represents the bond energy between A and B components, $Z$ is the number of nearest neighbors, and $N$ is Avogadro’s number.

Fig 1(b) Gibbs free energy vs. composition under Tc and T2

$\bigtriangleup H_{m}$ can be positive or negative, depends on the sign of $\alpha$. When $\bigtriangleup H_{m}$ is negative, the system is exothermic, $\bigtriangleup G_{m}$ is always negative at all temperatures, therefore, the solution is a perfectly mixed solution system. However, when $\bigtriangleup H_{m}$ is positive, the system is endothermic, $\bigtriangleup G_{m}$ can be positive or negative depends on the temperature. For example, Fig (1,a). When T₁ >T_c, at sufficiently high temperature situation, $-T\bigtriangleup S_{m}$ term dominates, so $\bigtriangleup G_{m}$ is negative, the solution mixes well and the only phase appears is liquid. However, when decrease the temperature below T_c, $\bigtriangleup H_{m}$ term dominates and $\bigtriangleup G_{m}$ becomes positive, the solution starts to crystallize and separate. When the temperature reaches T₂, the compositions between $a,b$ points, which have a higher Gibbs free energy will tend to separate into two more stable compositions phases where the Gibbs free energy are lower, the equilibrium states $a,b$, Fig (1,b). The point $a$ and $b$ are both on the common tangent line, the chemical potential of A component is same in a and b composition, vice versa for B component. The amounts of phase $a$ and phase $b$ are determined by lever rule.

The equilibrium locations of the composition at different temperature on the phase diagram are determined by “binodal curve”, where the first derivative of the Gibbs free energy curves equals to zero, ${d\bigtriangleup G \over dx}=0$ . The "spinodal curve" in phase diagram is defined by the second derivatives of the Gibbs free energy curve equals to zero, ${\partial ^{2}\bigtriangleup G \over \partial {x}^{2}}=0$ , which is also called the “inflexion point”. The spinodal decomposition mechanism has a closed-relationship to phase separation.

For the symmetric system, free energy curve, binodal curve and spinodal curve all show the center point at $X=0.5$. However, for silicate and other oxide system, the structures are more complex and the phase diagrams are seldom symmetrical, such as binary Li₂O- and Na₂O- silica system, the critical compositions in these alkali glass are located around x=0.1 rather 0.5. The free energy curves are highly asymmetrical in these systems. 

Kinetics

Fig 2 G per mole vs. composition

We first start with the Gibbs free energy vs. composition diagram under the critical consolute temperature T_c, where the solution begins to separate. 

As Fig 2 shows, the composition of $a$ and $b$ are the two equilibrium phases where the curve has the minimum point and has the common tangent line. For example, when considering the initial composition $x$, the driving force for $b$ phase to separate out from the liquid is given by the line DE, which is the initial free energy subtract the equilibrium free energy of composition $b$.

From the method mentioned above, we can calculate the driving force at every composition of the solution. If we assumed $d$ point is the inflexion point, the composition from $a$ to $d$ is the “metastable” region. In this region, the phase separation can only occur after going through a thermodynamic barrier, this area lies between the binodal curve and spinodal curve in the phase diagram (Fig.1), and it is also named as “nucleation and growth” region. While, the composition lies between two inflexion points (on the right side of $d$) are unstable, the driving force for phase separation is negative, which represents there is no thermodynamic barrier to overcome; this region lies inside the spinodal curve in the phase diagram, and this region is called “spinodal decomposition”.

Homogeneous nucleation

Assumed the nucleus is distributed uniformly and the boundaries between nucleus and matrix is clear with the interface energy $\sigma$. For the nuclei to grow, the nuclei needs to have a radius larger than a critical radius $r^{*}$ . The nuclei growth rate (number of nuclei formed per unit volume per second) is defined by $I=K_{v}\exp[-(\bigtriangleup G_{D}+W^{*})/kT]$ , where $W^{*}$ represents the energy to form critical nucleus, $\bigtriangleup G_{D}$ is the activation energy to diffuse through the boundaries, and $K_{v}$ is a constant.

The nucleation rate $I$ and nuclei growth rate are temperature dependent below the miscibility temperature T_m. The nucleation and growth rate are determined by the atom diffusion, both rates decreases rapidly during larger undercooling in the reasons of slower atom diffusion activity in lower temperature, the viscosity also increases at low temperature.

Spinodal decomposition

Spinodal decomposition is a phase separation mechanism without overcoming the thermodynamic barrier. Hillert and Cahn have analyzed and introduced the quantitative parameters for spinodal decomposition. They found out in the spinodal system, the system is unstable when the wavelength of the sinusoidal fluctuation is larger than a critical value $\lambda _{crit}$ .

The diffusion equation for spinodal equation is ${\frac {\partial c}{\partial t}}=Mf''\bigtriangledown ^{2}c-2M\kappa \bigtriangledown ^{4}c$ , where $M$ is the mobility, $\bigtriangledown ^{2}c$ is the derivatives of the concentration, $f''$ is the derivatives of the Gibbs free energy. The equation only applies for the isotropic system without strain energy at boundaries; therefore, viscous liquid and glass are both taken into account.

The general solution for the diffusion equation is $c(r,t)-c_{0}=A(\beta ,t)\exp(i\beta r)$ , where $c$ equals to the average concentration, $\beta$ is the wave factor at time t, and $A$ is the amplitude of the wave function, and can be defined by $A(\beta ,t)=A(\beta ,0)\exp(R(\beta )t)$. $R(\beta )$ is only positive when $\beta$ is smaller than $\beta _{crit}$ .

The difference between nucleation and growth and spinodal mechanisms when forming phase separation is the boundaries of the two phases. For nucleation and growth mechanism, the composition of the second phase does not change with time, so that the connectivity of the spherical particles is low; however, for the spinodal decomposition, the sinusoidal fluctuation provides a continuous variation of concentration between two phases until the equilibrium compositions formed, hence the connectivity of the spherical particles is higher.

Applications

(1)  “Vycor” type of glasses: This type of glasses is a well-known Na₂O-B₂O₃-SiO₂ system. One of the separated phases is washed out by adding acid, the remaining porous silica structure will be sintering to produce a higher silica glass.

Further reading

1. Fundamentals of Inorganic Glass, Arun K. Varshneya, Academic Press, 1994
2. Phase Transformations in Metal and Alloys, David A. Porter, Kenneth E. Easterling, Mohamed Y. Sherif, CRC Press, 3rd edition, 2009
3. Introduction to the Thermodynamics of Materials, David R. Gaskell, CRC Press, 5th edition, 2008