1.  Sketch a TTT curve for a system containing 1 ppm of crystalline material. Add a curve for 1% crystallinity. Label each axis. Draw curves for constant cooling rates which are greater than, exactly equal to, and less than the critical cooling rate for the 1 ppm crystalline sample. Discuss HOW one might determine the critical cooling rate for a given glass.
 

See Shelby, pp. 16-20

Constant cooling rate < critical cooling rate: line a

Constant cooling rate = critical cooling rate: line b

Constant cooling rate > critical cooling rate: line c

Experiments to determine critical cooling rate? (See Shelby, pp. 20-24)

• Cool block of glass from melt at different controlled rates and examine for evidence of crystallinity (e.g., visual, x-ray diffraction, light scattering, electron microscopy, etc.)
• Perform DTA (Differential Thermal Analysis) or DSC (Differential Scanning Calorimetry) experiments at different cooling rates and determine minimum rate for which no crystallization peak can be detected.

 2.  Sketch the curves of nucleation and crystal growth rate on the figure below. (Note that this is a different axis-convention than we discussed in class). Label all important features. Explain how the nucleation rate curve represents a competition between kinetic and thermodynamic factors. Explain why the nucleation curve will always occur at temperatures below the crystal growth curve. Discuss the concept of a critical radius for a nucleus. (See Shelby, pp. 10-16).
 

 

• Consider Shelby’s eq. 2.16; the nucleation rate (I) depends on the inverse of viscosity (kinetic barrier) and the magnitude of the thermodynamic driving force. The thermodynamic driving force is a balance between the volume free energy of the stable nuclei (proportional to r³, where r is the nucleus radius) and the surface energy associated with the creation of new solid-liquid interfaces (proportional to r²). The critical nucleus radius is that where an increase in size means a decrease in energy, where the volume energy gained during nucleus formation overcomes the surface energy expended. The nucleation rate curve is at a lower temperature than the crystal rate curve because of this surface energy barrier. Undercooling the melt below the melting point (T_m), increases the thermodynamic contribution to the rate equation, increasing I. At the same time, viscosity is increasing, slowing material transport. When cooled to low enough temperature, approaching the glass transition temperature (T_g) the nucleation and crystal growth rates go to zero.

• Crit. Cooling rate = (T_m-T_n)/t_n = 100 K/1000 s = 0.1 K/s

• Crit. Cooling rate = (T_m-T_n)/t_n = 100 K/10000 s = 0.01 K/s

• High initial viscosity acts as kinetic barrier to both nucleation and subsequent crystallization; low diffusion coefficients in high viscosity melts, slow material transport to growing nuclei/crystals.

• Kinetic barrier to nucleation/crystallization is initially low at T_m- but the rapid increase in viscosity (large dh/dT) with decreasing temperature will cause the kinetic barrier to increase, counteracting the increasing thermodynamic driving force due to undercooling.