Determining the Crystal-Melt Interfacial Free Determining the Crystal-Melt Interfacial Free Energy via Molecular-Dynamics SimulationEnergy via Molecular-Dynamics Simulation

Brian B. Laird Department of Chemistry

University of KansasLawrence, KS 66045, USA

Ruslan L. DavidchackDepartment of MathematicsUniversity of LeicesterLeicester LE1 7RH, UK

Funding: NSF, KCASC

Prestissimo Workshop 2004

Observation #1Observation #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

Problem: Can one calculate the free energy of a Problem: Can one calculate the free energy of a crystal-melt interface using MD simulation?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

Solid Liquid

Why is the Interfacial Free Energy Important?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 GrowthThe 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)

Why is the Interfacial Free Energy Important?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 nucleationThe 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€

knucleation = Ae−Bγ cl / kT

Observation of nucleation in colloidal crystals: Weitz group (Harvard)

Why are simulations necessary here?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

Indirect experimental measurement of Indirect experimental measurement of clcl

(Nucleation data and Turnbull’s rule)(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 CT fusHwhere the Turnbull Coefficient CT is ~.45 for metals and ~ 0.32 for non-metals

Can we understand the molecular origin of Turnbull’s Rule???

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

€

E = E({r p ,

r q }); S = k ln δ(H({

r p ,

r q }

r ) dΓ

Γ∫[ ]

€

H( p,q) = H0(p,q) + λH1(p,q)

F( H) = F( H0) + H1 dλ0

1∫

Observation #2Observation #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

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

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

h(x)

€

ˆ γ = γ + d2γdθ 2

€

< h(k) 2 >= kBTbW ˆ γ k 2

Wb

Cleaving methods for calculating interfacial free energy

crystal cry stal

liquid liq uid

+

+

cry uid 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

How is the cleaving done?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.

Observation #3Observation #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

Cleaving methods for calculating interfacial free energy

Solid Liquid

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 1 Step 2

A ABB

Step 3 Step 3: Juxtapose the solid and liquid systems by rearranging the boundary conditions while maintaining the cleaving potentials

BAA B

Step 4 Step 4: Remove the cleaving potentials from the combined system w1 + w4 + w3 + w4

Systematic error: hysteresisSystematic error: hysteresisThe 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)

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

First Study: The Simplest SystemFirst Study: The Simplest SystemHard SpheresHard 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

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

Simulation Details for Hard-SpheresSimulation 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)

• kBT/

Results for hard-spheresResults for hard-spheres Davidchack & LairdDavidchack & Laird, , PRLPRL 85 85 4751 (20004751 (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/

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) kTm/2

s-2/3 = 0.55 (0.53)Tm since s -3

If a hard-sphere model holds one would predict that s

-2/3 = C Tm with C ~ 0.5-0.6

Hard-Sphere Model for FCC forming metals

Turnbull’s rule follows since

fusS = fusH/Tm

is nearly constant for these metals

s-2/3 ~ 0.5 kTm

Continuous Potentials

•Lennard-Jones

•Inverse-Power Potentials

Differences with Hard-spheresw3 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)

Observation #4Observation #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

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

€

HN =˜ p i

2

2ms2∑ + V (q) + π 2

2Q+ gkT ln s

• g = Number of degrees of freedom

•NVE dynamics generated by HN , after time transformation d/dt=s, yields a canonical (NVT) distribution in the reduced phase space

)]0([ − HHH NNNP

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 Tinst 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

€

T = Tinst ≡ 1g kB

pi2

mi∑

€

ANP

= AC

− h2

kTAε2 C

− AC

ε2 C{ }

ε2( p,q) = 112

pT M –1 ′ ′ V − 12Q

(pT M –1p − gkBT)2 − 12

′ V T M –1 ′ V + kTQ

(pT M –1p − gkBT) ⎡ ⎣ ⎢

⎤ ⎦ ⎥

Results for the truncated Lennard-Jones systemResults for the truncated Lennard-Jones system(Davidchack & Laird, J. Chem. Phys., 118, 7651 (2003)

kT/ Orientation This 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.

ANISOTROPY in Interfacial Free Energy

Lennard-Jones System Hard Spheres

T* = 0.617 1.0 1.50( 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)

€

( ˆ n ) = γ 0 1+ ε1 ni4 − 3

5i=1

3

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟+ ε2 ni

4 + 66n12n2

2n32 − 17

7i=1

3

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥

Expansion in cubic harmonics (Fehlner & Vosko)

Inverse Power Potentials

€

V (r) = ε σr ⎛ ⎝ ⎜

⎞ ⎠ ⎟n

•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

densityn > 7

n < 7fccbccfluid

Inverse Power Potential Scaling

€

V (r) = ε σr ⎛ ⎝ ⎜

⎞ ⎠ ⎟n

•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:

Pc = P1T(1+3/n); T(1 + 2/n)

Inverse Power Potentials and Turnbull’s ruleInverse Power Potentials and Turnbull’s rule

€

1 T (1+2 / n ) ; Γn (solid)=ρ s Tm−3 / n ⇒ ρ s = ΓnTm

3 / n

From the scaling laws

So…

€

˜ γ ≡γρs−2/3

=γ1Tm1+2/n

[Γn(s)Tm3/n

]−2/3

⇒˜ γ =[γ1Γn−2/3

(s)]Tm

And since Hfus TSfus

€

˜ γ =γ1 Γn,s

−2 / 3

ΔS fus

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ΔH fus = CT ,n ΔH fus

Turnbull’s rule is exact for inverse power potentials!

as in hard spheres, we see scaling with Tm

Results for the Inverse-Power SeriesResults for the Inverse-Power Series

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

Turnbull CoefficientTurnbull Coefficient

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

Results for the Inverse-Power SeriesResults for the Inverse-Power Series(Anisotropy)(Anisotropy)

Similar to Fe (Asta, et al.) the bcc interface has a anisotropy

SummarySummary• 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.