# Analysis of Nucleation and Glass Formation by Chip Calorimetry
## 3 Results and Discussions
### Nucleation and Primary Crystallization
#### 3.1.2 Delay Time Determination in Various Amorphous Materials  
An important application of amorphous materials is to serve as a precursor for the nucleation of crystalline phases that often develop with nanocrystalline sizes. The typical crystallization reaction is illustrated in Figure 2a. Upon isothermal annealing, initially no crystals are present. During this delay time, \(\tau\) period crystalline embryos evolve with increasing size towards a critical size range where they can then proceed to grow as crystals. Within this delay time, there is a transient nucleation as the equilibrium cluster size distribution is established to develop into a steady state nucleation rate. The steady state rate continues until a saturation of nucleation sites is reached and is then followed by a size coarsening regime. The transient nucleation rate, \(J(t)\), is represented by [13; 14]. \[J(t)=J_{ss}exp[-\tau/t] \tag{2}\]  
where \(t\) is the time and \(J_{ss}\) is the homogeneous nucleation that is given by  
\[J_{ss}=\Sigma ZC_{0}exp[-\Delta G^{*}/kT] \tag{3}\]  
where \(\Sigma\) is the atom attachment frequency to the evolving nucleation clusters, \(Z\) is the Zeldovich factor that has a typical value of 0.1, \(C_{0}\) is the nucleation site density, \(\Delta G^{*}\) is the work for nucleus formation, \(k\) is the Boltzmann constant, and \(T\) is the temperature. Often \(\Sigma\) is taken as \(D/a^{2}\), where \(D\) is the bulk diffusivity and a is the atom spacing. Furthermore, since \(D\) is not always available it is estimated from the Stokes-Einstein relation. \[D=kT/[\theta\tau\eta] \tag{4}\]  
where \(\eta\) is the viscosity and \(r\) is the atom size. However, this approach has problems since the bulk diffusivity does not necessarily represent the interface attachment process governing cluster growth, and it has been demonstrated that the Stokes-Einstein relation breaks down near \(T_{g}\)[15]. However, from nucleation theory there is a direct relation between \(\Sigma\) and \(\tau\) given by [14]. \[\Sigma=1/2\tau Z^{2} \tag{5}\]Thus, a measurement of \(\tau\) offers an accurate approach to evaluating \(\beta\) and the nucleation rate. Moreover, the magnitude of \(\tau\) also provides an effective evaluation of the glass formation ability as large \(\tau\) values are necessary for bulk glass formation. For example, a longer delay time at one temperature endows the higher thermal stability and the better glass formation ability, which is critical for developing bulk MGs [15]. Thus, it is of great importance to experimentally and precisely determine the delay time corresponding to different temperatures. There are two common methods to experimentally measure the delay time for primary crystallization in amorphous materials: I. TEM method by plotting the measured crystal number density and the annealing time at one temperature and then extrapolating the steady state nucleation slope line to the time axis, the delay time can be established (see Figure 2a) [16; 17]; II. isothermal test method by holding the amorphous sample at one temperature and then observing the onset time for the crystallization reaction (see Figure 2b) [18; 19]. For method I, a great deal of TEM sample preparation and testing is required, and thus it is very time-consuming and is not a preferential method. For method II, the crystallization signal should be strong enough to show up in the isothermal annealing DSC trace, which makes it possible to directly measure the delay time for crystallization by isothermal tests. Thus, the amorphous system should have good glass formation ability and thermal stability, such as the Zr-, Pd-, and Ce-based bulk MG systems [18; 19]. However, for some marginal amorphous materials, such as the typica marginal Al-based MGs with poor glass formation ability, there usually appears a primary nucleation peak for Al nanocrystals followed by secondary or tertiary crystallization reactions involving intermetallic phases [20].