1. Determining the Crystal-Melt Interfacial Free Energy via Molecular-Dynamics Simulation Brian B. Laird Department of Chemistry University of Kansas Lawrence, KS 66045, USA Ruslan L. Davidchack Department of Mathematics University of Leicester Leicester LE1 7RH, UK Funding: NSF, KCASC Prestissimo Workshop 2004

2. Observation #1 Not all important problems addressed with MD simulation are biological. In this work we describe application of molecular simulation to an important problem in materials science/metallurgy Material Properties Molecular Interactions

3. Problem: Can one calculate the free energy of a crystal-melt interface using MD simulation? Crystal-melt interfacial free energy,  cl –The work required to form a unit area of interface between a crystal and its own melt SolidLiquid

4. Why is the Interfacial Free Energy Important?  cl is a primary controlling parameter governing the kinetics and morphology of crystal growth from the melt Example: Dendritic Growth The nature of dendritic growth from the melt is highly sensitive to the orientation dependence (anisotropy) in  cl. Data from simulation can be used in continuum models of dendrite growth kinetics (Phase-field modeling) Experiment Model Growth of dendrites in succinonitrile (NASA microgravity program)

5. Why is the Interfacial Free Energy Important?  cl is a primary controlling parameter governing the kinetics and morphology of crystal growth from the melt Example: Crystal nucleation The rate of homogeneous crystal nucleation from under-cooled melts is highly dependent on the magnitude of  cl Nucleation often occurs not to the thermodynamically most stable bulk crystal phase, but to the one with the lowest kinetic barrier (i.e. lowest  cl ) - Ostwald step rule Observation of nucleation in colloidal crystals: Weitz group (Harvard)

6. Why are simulations necessary here? Direct experimental determination of  cl is difficult and few measurements exist Direct measurements: (contact angle) Only a handful of materials: water, cadmium, bismuth, pivalic acid, succinonitrile Indirect measurements: (nucleation) Primary source of data for  Accurate only to 10-20% Simulations needed to determine phenomenology

7. Indirect experimental measurement of  cl (Nucleation data and Turnbull’s rule)  cl can be estimated from nucleation rates: typically accurate to 10-20% Turnbull (1950) estimated  cl from nucleation rate data and found the following empirical rule:  cl    C T  fus H where the Turnbull Coefficient C T  is ~.45 for metals and ~ 0.32 for non- metals Can we understand the molecular origin of Turnbull’s Rule???

8. Calculating Free Energies via simulation: Why is Free Energy hard to measure? Unlike energy, entropy (& free Energy) is not the average of some mechanical variable, but is a function of the entire trajectory (or phase space) Free energy, F = E - TS, calculation generally require a series of simulations slowly transforming the system from a reference state to the state of interest Thermodynamic Integration Frenkel Analogy Energy  Depth of lake Entropy  Volume of Lake

9. Observation #2 In molecular simulation there are almost always two (or more) very different methods for calculating any given quantity Calculating a quantity of interest using more than one method is an important check on the efficacy of our methods Cleaving Method Fluctuation Method  cl

10. Fluctuation Method Method due to Hoyt, Asta & Karma, PRL 86 5530, (2000) h(x) = height of an interface in a quasi two- dimensional slab If  = angle between the average interface normal and its instantaneous value, then the stiffness of the interface can be defined The stiffness can be found from the fluctuations in h(x) Advantage: Anisotropy in  precisely measured Disadvantages: Large systems (N = 40,000 - 100,000) Low precision in  itself h(x)h(x) W b

11. Cleaving methods for calculating interfacial free energy crystal crystal liquid liquid + + cryuid cry Cleaving Potentials: Broughton & Gilmer (1986) - Lennard-Jones system Cleaving Walls: Davidchack & Laird (2000) - HS, LJ, Inverse-Power potentials Advantages: very precise for , relatively small sizes (N = 7000 - 10,000) Disadvantages: Anisotropy measurement not as precise as in fluctuation method uid Calculate  directly by cleaving and rearrang- ing bulk phases to give interface of interest THERMODYNAMIC INTEGRATION

12. How is the cleaving done? We employ special cleaving walls made of particles similar to those in the system Choose a dividing surface – particles to the left of the surface are labeled “1”, those to the right are labeled “2” Wall labeled “1” interacts only with particles of type “1”, same for “2” As walls “1” and “2” are moved in opposite directions toward one another the two halves of the system are separated If separation done slowly enough the cleaving is reversible Work/Area to cleave measured be integrating the pressure on the walls as a function of wall position.

13. Observation #3 Physical reality is overrated in molecular simulation In calculating free energies via simulation, we only care that the initial and final states are “physical”, we can do (almost) anything we want in between

14. Cleaving methods for calculating interfacial free energy SolidLiquid Start with separate solid and liquid systems equilibrated at coexistence conditions: T,  c,  f Steps 1, 2: Insert suitably chosen “cleaving” potentials into the solid and liquid systems Step 1Step 2 AABB Step 3 Step 3: Juxtapose the solid and liquid systems by rearranging the boundary conditions while maintaining the cleaving potentials BAAB Step 4 Step 4: Remove the cleaving potentials from the combined system  w 1 + w 4 + w 3 + w 4

15. Systematic error: hysteresis The main source of uncertainty in the obtained results is the presence of a hysteresis loop at the fluid ordering transition in Step 2 Reducing Hysteresis longer runs improve cleaving wall design cleave fluid at lower density (adds an extra step)

16. Our approach: a systematic study of the effect of inter-atomic potential on  cl Simplest potential - Hard spheres Effect of Attraction - Lennard Jones Effect of range of repulsive potential - inverse power potentials

17. First Study: The Simplest System Hard Spheres Why hard spheres? Hard Sphere Model Typical Simple Material The freezing transition of simple liquids can be well described using a hard-sphere model

18. The Hard-Sphere Crystal Face-Centered Cubic (FCC)

19. Simulation Details for Hard-Spheres Hard-sphere MD algorithmically exact: Chain of exactly resolved elastic collisions Rappaport’s cell method: dramatically speeds up collision detection (100), (110) and (111) interfaces studied N ~ 10,000 particles Phase coexistence independent of T:    crystal); 0.939 (fluid)    k B T/  

20. Results for hard-spheres  Davidchack & Laird, PRL 85 4751 (2000)]  kT/    kT/    kT/   How do these numbers fit in with other estimates? From Nucleation Experiments on silica microspheres 0.54 to 0.55 kT/   (Marr&Gast 94, Palberg 99) From Density-Functional Theory: predictions range from 0.28 - 2.00 kT/   (WDA of Curtin & Ashcroft gives 0.62 kT/    From Simulation: Frenkel nucleation simulations: 0.61 kT/  

21. Can Turnbull’s rule be explained with a hard- sphere model? For hard-spheres, Turnbull’s reduced interfacial free energy scales linearly with the melting temperature  = 0.57 (0.55) kT m /  2  s -2/3 = 0.55 (0.53)T m since  s  -3 If a hard-sphere model holds one would predict that  s -2/3 = C T m with C ~ 0.5-0.6

22. Hard-Sphere Model for FCC forming metals Turnbull’s rule follows since  fus S =  fus H/T m is nearly constant for these metals  s -2/3 ~ 0.5 kT m

23. Continuous Potentials Lennard-Jones Inverse-Power Potentials Differences with Hard-spheres  w 3 is non-zero  More care must be taken in construction of cleaving wall potential  need to use NVT simulation to maintain coexistence temperature throughout simulation (e.g., use Nose´-Hoover or Nose´-Poincare methods)

24. Observation #4 In precise simulation work it is important to always be aware of the damage done to statistical mechanical averages by the discretization Free energy simulations involving phase equilibrium require highly precise simulations and discretization error in averages can be important Need: a detailed statistical mechanics of numerical algorithms

25. Example of the effect of discretization error in Nose´-Poincare´ MD In Nose´NVT dynamics, constant T is maintained by adding two new variables to the Hamiltonian In Nose´-Poincare´ (Bond, Leimkuhler & Laird, 1999) the Nose´ Hamiltonian is time transformed to run in real time Can be integrated using the Generalized Leapfrog Algorithm (GLA) GLA is symplectic g = Number of degrees of freedom NVE dynamics generated by H N, after time transformation d  /dt=s, yields a canonical (NVT) distribution in the reduced phase space

26. Example of the effect of discretization error in Nose´-Poincare´ MD If canonical distribution is correctly obtained then the equipartition theorem holds The difference between T and T inst is a measure of the uncertainty in T due to the discretization For Nose´-Poincare´ with GLA this can be worked out (S. Bond thesis). Similar formulae for Nose´-Hoover integrators

27. The Lennard-Jones System Davidchack & Laird, J. Chem Phys. 118, 7651 (2003) LJ Potential: (Modified to go smoothly to zero at 2.5  Phase Diagram:

28. Results for the truncated Lennard-Jones system Results for the truncated Lennard-Jones system (Davidchack & Laird, J. Chem. Phys., 118, 7651 (2003) kT/  OrientationThis work  (   ) Broughton & Gilmer * Morris & Song 0.617 (tp)(100)0.371(3)0.34(2)0.369(8) (110)0.360(3)0.36(2)0.361(8) (111)0.347(3)0.35(2)0.355(8) 1.0(100)0.562(6)N/A- (110)0.543(6)N/A- (111)0.508(8)N/A- 1.5(100)0.84(1)N/A- (110)0.80(1)N/A- (111)0.75(1)N/A- * J.Chem.Phys. 84, 5759 (1986). Note that anisotropy in LJ differs from HS in that the order of (110) and (111) are switched.

29. Results for Truncated L-J System As predicted by HS model,  is roughly linear in T – with a slope averaging 0.53 for the three temperatures

30. ANISOTROPY in Interfacial Free Energy Lennard-Jones SystemHard Spheres T * = 0.6171.01.5  0 (    0.360(2)0.539(4)0.798(6) 0.573(6)kT/    0.093(17)0.13(3)0.16(4)0.09(4)  -0.011(4)-0.022(9)-0.019(6)-0.005(11) (   -    0 0.03(1)0.035(15)0.050(7)0.036(16) (   -    0 0.07(1)0.10(2)0.113(7)0.061(16) Expansion in cubic harmonics (Fehlner & Vosko)

31. Inverse Power Potentials Important reference system for studying the effect of potential range For n=∞, we have the hard-sphere system. Phase Diagram: For n>7, crystal structure is FCC, for 4<n<7 system freezes to BCC transforming to FCC at lower temperatures (higher densities) fccfluid density n > 7 n < 7 fccbccfluid

32. Inverse Power Potential Scaling Only one parameter  n Excess thermodynamic properties only depend upon  n    k  ) -3/n =     -3/n Phase diagram is one dimensional, only depends upon  n Along coexistence curve: P c = P 1 T (1+3/n) ;   T (1 + 2/n)

33. Inverse Power Potentials and Turnbull’s rule From the scaling laws So… And since  fus  T  S fus Turnbull’s rule is exact for inverse power potentials! as in hard spheres, we see scaling with T m

34. Results for the Inverse-Power Series Similar to Fe (Asta, et al.) the bcc interface has a lower free energy

35. Turnbull Coefficient Similar to Fe (Asta, et al.) the bcc interface has a lower free energy

36. Results for the Inverse-Power Series (Anisotropy) Similar to Fe (Asta, et al.) the bcc interface has a anisotropy

37. Summary We have measured the crystal/melt interfacial free energy, g, for hard-spheres, Lennard-Jones and inverse-power series. Our simulations can resolve the anisotropy in this quantity. Comparison of data from fluctuation method and cleaving method shows the two methods to be consistent and complementary We show the molecular origin of Turnbull’s rule and give an alternate formulation For the inverse-power series  bcc <  fcc consistent with fluctuation model calculations and nucleation experiments - also bcc is less anisotropic than fcc.