1. Enthalpy Landscape Analysis for Calculating the Melting Temperature of a Material Alex M. Nieves, Vaclav Vitek, and Talid Sinno University of Pennsylvania AIChE 2010 Annual Meeting Salt Lake City, Utah

2. Importance of Melting steel.nic.in Process Conditions & Material Selection and Design sti.nasa.gov simplystone.com.au Theoretical Understanding trucknetuk.com cseg.ca HOW? WHY?

3. Thermodynamic definition of melting Thermodynamics defines Melting as a first order phase transition: Crystalline Phase long range ordered state Liquid Phase short range disordered state The Thermodynamic Melting Temperature is obtained by comparing the Free Energy of the two phases: Solid Phase Liquid Phase T Free Energy T M

4. Thermodynamic Melting “Heterogeneous Melting” Heterogeneous melting mechanism at T M when a Liquid Phase already exits

7. Theories for the Limit of Superheating Lu & Li, Phys Rev Lett 80 (1998) Entropy Catastrophe (1988) Rigidity Catastrophe (1989) Volume Catastrophe (1989) Fetch and Johnson, Nature 334 (1988) Molar Volume of the Crystalline Phase and Liquid Phase are equal. Rigidity of the Crystalline Phase at T S is equal to Rigidity of the Liquid Phase at T M Tallon, Nature 342 (1989) Tallon, Nature 342 (1989) Classical Nucleation Theory gives nucleation rate of a liquid sphere in a crystalline bulk material Classical Nucleation Theory (1998)

8. “Idealized” Energy Landscape

9. The “Real” Energy Landscape

10. Sampling the Energy Landscape Discretization of continuous energy landscape into energy minima basins known as Inherent Structures . Sample energy landscape with MD. Histogram the visited basin minima by quenching the system. The energy of each basin is defined here by the formation energy of the inherent structure.

11. Formation Energies Formation Volume of a Defect: Formation Enthalpy of a Defect: Formation Energy of a Defect:

12. Calculation of the Density of States Using MD Simulations Density of States: Number of basins with energy Δ H Boltzmann Factor: Probability of being at basin with energy Δ H at T

13. Effect of density of states on probability distribution Δ H If rate of increase of g( Δ H) is lower than rate of decrease of exp (- Δ H/kT) Δ H Negative Slope If rate of increase of g( Δ H) is higher than rate of decrease of exp (- Δ H/kT) Positive Slope Δ H If rate of increase of g( Δ H) is equal to rate of decrease of exp (- Δ H/kT) Zero Slope Δ H Number of states increases with enthalpy

14. Extracting an Effective Temperature from the Density of States Function Density of States Calculation: Density of States Growth Rate can be approximated by an exponential fit . An effective temperature, T eff , can be extracted.

15. Revisiting the Probability Distribution Curve Giving: Using this approximation for the density of states the probability becomes: Slope of the Probability Distribution is:

16. Determining Melting Temperature from T eff Assuming that the slope of g( Δ H) remains constant: At T sim > T eff , P( Δ H) becomes unbounded with increasing energy, giving access to liquid states. T sim > T eff T sim = T eff T sim < T eff

17. Melting Temperatures Calculated at Different System Sizes

18. Effect of Pressure on Melting Young, Phase Diagrams of the Elements. (1991) Δ V M > 0 for most materials Δ V M < 0 for the few materials Latent Heat of Melting Change in Volume during Melting Clausius-Clapeyron equation gives the slope of the melting curve :

19. Interpreting the Effect of Pressure on the Density of States Aluminum: P = 8GPa P = 5GPa P = 3GPa P = 0GPa

20. Interpreting the Effect of Pressure on the Density of States Silicon: P = 8GPa P = 5GPa P = 3GPa P = 0GPa

21. Effect of Δ V M on the Mechanism of Melting Initiated at a Void Silicon ( Δ V M < 0) Aluminum ( Δ V M > 0)

29. “ Pretty” Picture of Melting Crystal Phase Liquid Phase Real Picture of Melting Crystal Phase Liquid Phase

30. Using the Enthalpy Landscape Analysis to Study Melting Stages of a Void Teff Teff Teff homogeneous