Computer Simulations, Nucleation Rate Predictions and Scaling Barbara Hale Physics Department and Cloud and Aerosol Sciences Laboratory, University of Missouri – Rolla Rolla, MO USA

Computer simulation techniques and predictions of vapor to liquid nucleation rates Molecular Dynamics: time dependent simulation of nucleation process in a (usually dense) vapor. Monte Carlo: statistical mechanical calculations of small n-cluster free energies of formation from the vapor,  G n

Monte Carlo Simulations Randomly generate molecular equilibrium configurations  use “importance ”, sampling in statistical mechanical ensemble:  U = U final –U initial = change in potential energy Canonical (N,T,V)  exp -[  U ] /kT Gibbs (N,T,P)  exp –[  U +P  V] /kT Grand Can. ( ,T,V)  exp –[  U -   N ] /kT Metropolis method: exp[ –  U / kT ] > Random # ; 0 < R < 1  accept Smart Monte Carlo: umbrella sampling, …

Nucleation rate via Monte Carlo  Calculate: = = free energy of formation (schemes for doing this depend on n- cluster definition, interaction potential and simulation efficiency…. must calculate free energy differences.)  N n /N 1 = exp - / kT = n-cluster size distribution or  Simulate: P n = in “larger” supersaturated (constrained) system   G n = - kT ln

Nucleation rate via Monte Carlo From simulation: J n = monomer flux ·N 1 e -  G(n)/kT Predict Nucleation rate, J: J  [N 1 v 1 4  r n* 2 ]· N 1 e -  G(n*)/kT ( often a classical model form is used) J -1 = [  n J n' ] -1 (steady-state nucleation rate summation)

Computer simulation study of gas-liquid nucleation in a Lennard-Jones system. P. R. ten Wolde and D. Frenkel J. Chem. Phys. 109, 9901 (1998) We report a computer-simulation study of homogeneous gas- liquid nucleation in a Lennard-Jones system. Using umbrella sampling, we compute the free energy of a cluster as a function of its size. A thermodynamic integration scheme is employed to determine the height of the nucleation barrier as a function of supersaturation. Our simulations illustrate that the mechanical and the thermodynamical surfaces of tension and surface tension differ significantly. In particular, we show that the mechanical definition of the surface tension cannot be used to compute this barrier height. We find that the relations recently proposed by McGraw and Laaksonen J. Chem. Phys. 106, 5284 (1997) for the height of the barrier and for the size of the critical nucleus are obeyed.

Numerical calculation of the rate of homogeneousgas–liquid nucleation in a Lennard-Jones system P. R. ten Wolde and D. Frenkel J. Chem. Phys. 110, 1591 (1999) We report a computer-simulation study of the absolute rate of homogeneous gas–liquid nucleation in a Lennard- Jones system. The height of the barrier has been computed using umbrella sampling, whereas the kinetic prefactor is calculated using molecular dynamics simulations. The simulations show that the nucleation process is highly diffusive. We find that the kinetic prefactor is a factor of 10 larger than predicted by classical nucleation theory.

P. R. ten Wolde and D. Frenkel J. Chem. Phys. 110, 1591 (1999) (N,P,T) Monte Carlo, N = 864 particles S = 1.53;  G/kT = 59.4 T = 80.9K =  ; J MC/MD = 4.5 x 10 5 cm -3 sec -1 T c = 130K =  (LJ truncated 2.5 , shifted)

Molecular Dynamics Simulations  Solve Newton’s equations, m i d 2 r i /dt 2 = F i = -  i  j≠i U(r j -r i ), iteratively for all i=1,2… n atoms;  Quench the system to temperature, T, and monitor cluster formation in supersaturated system.  Measure J  rate at which clusters form

Molecular dynamics of homogeneous nucleation in the vapor phase. I. Lennard-Jones fluid K. Yasuoka and M. Matsumoto J. Chem. Phys. 109, 8451 (1998) Molecular dynamics computer simulation was carried out to investigate the dynamics of vapor phase homogeneous nucleation at the triple point temperature under supersaturation ratio 6.8 for a Lennard-Jones fluid. To control the system temperature, the 5000 target particles were mixed with 5000 soft-core carrier gas particles. The observed nucleation rate is seven orders of magnitude larger than prediction of a classical nucleation theory. The kinetically defined critical nucleus size, at which the growth and decay rates are balanced, is 30–40, as large as the thermodynamically defined value of 25.4 estimated with the classical theory. From the cluster size distribution in the steady state region, the free energy of cluster formation is estimated, which diminishes the difference between the theoretical prediction and the simulational result concerning the nucleation rate.

K. Yasuoka and M. Matsumoto, LJ MD J. Chem. Phys. 109, 8451 (1998)  =2.15ps

K. Yasuoka and M. Matsumoto J. Chem. Phys. 109, 8463 (1998) MD Lennard-Jones S = 6.8 (monomer depletion occurs) T = 80.3K = 0.67  J MD  cm -3 sec -1  10 7 J classical LJ T c LJ = 161.7K = 1.35  LJ surface tension:  LJ >  experiment LJ potentials truncated at 12  ; ( r cutoff < 5  can alter T c and  LJ )

Molecular dynamics of homogeneous nucleation in the vapor phase. II. Water. K. Yasuoka and M. Matsumoto J. Chem. Phys. 109, 8463 (1998) Homogeneous nucleation process in the vapor phase of water is investigated with a molecular dynamics computer simulation at 350 K under supersaturation ratio 7.3. Using a method similar to Lennard-Jones fluid (Part I), the nucleation rate is three orders of magnitude smaller than prediction of a classical nucleation theory. The kinetically defined critical nucleus size is 30–45, much larger than the thermodynamically defined value of 1.0 estimated with the classical theory. Free energy of cluster formation is estimated from the cluster size distribution in steady state time region. The predicted nucleation rate with this free energy agrees with the simulation result. It is concluded that considering the cluster size dependence of surface tension is very important.

K. Yasuoka and M. Matsumoto J. Chem. Phys. 109, 8463 (1998) MD TIP4P water S = 7.3 ( monomer depletion occurs ) T = 350K;  TIP4P =39.29 erg cm -2 J MD  cm -3 sec -1  J Classical TIP4P T c model = 563K ? TIP4P surface tension:  TIP4P  0.65  experiment

Difficulties in comparing with experimental data  Simulation results depend on potential model. The classical atom-atom interaction potentials can have different surface tension, coexistence vapor pressure, and critical temperature.  Results can also depend on the definition of a cluster and the simulation technique.  How can the simulation results be evaluated and compared with experiment?

Proposal: Use Scaled Supersaturation B. N. Hale, Phys. Rev. A 33, 4156 (1986) Scaled nucleation rate: ln[J scaled /10 26 ] cgs  -(16  /3)  3 [lnS/(T c /T -1) 3/2 ] -2 s caled supersaturation (similar to Binder’s): lnS scaled = lnS/(T c /T -1) 3/2   2 ( argon excess surface entropy/molecule)   1.5 (water excess surface entropy/molecule)

Toluene (C 7 H 8 ) nucleation data of Schmitt et al plotted vs. scaled supersaturation, C o = [T c /240-1] 3/2 ; T c = 591.8K

Nonane (C 9 H 20 ) nucleation data of Adams et al. plotted vs. scaled supersaturation ; C o = [T c /240-1] 3/2 ; T c = 594.6K

Water nucleation rate data of Wölk and Strey plotted vs. lnS / [T c /T-1] 3/2 ; C o = [T c /240-1] 3/2 ; T c = 647 K

Example 1: using scaled supersaturation to compare experimental data and computer simulation predictions for J water Plot -log[J/10 26 ] cgs units vs. C o' [lnS/(T c /T -1) 3/2 ] -2 C o' = 23.1 = -(16  /3)  3 / ln(10) ;  = 1.47 T c = 647 K ; S exp or S model

Comments on water data and predictions for J Predicted rates using TIP4P are about 4 orders of magnitude too large, but appear to have correct “scaled supersaturation” dependence. TIP4P critical temperature < 647K MD and MC show similar results. Significance (if any) to “shifted” scaled supersaturation for TIP4P?

Example 2: using the scaled supersaturation to compare experimental data and computer simulation predictions for J argon. Plot -log[J/10 26 ] cgs units vs. C o' [lnS/(T c /T -1) 3/2 ] -2 C o' = 24.6 = -(16  /3)  3 ;  = 1.5 (T c = 150 K, S exp ) for data; (T c model, S model ) for simulation results.

Argon experimental rates Assume “onset” rates: shock tube:   2 cm -3 s -1 nozzle:   2 cm -3 s -1 Fast expansion chamber:  10 7  2 cm -3 s -1

+ Fladerer

 Zahoransky

+ Fladerer  Zahoransky ○ Stein

+ Fladerer  Zahoransky ○ Stein  Matthew et al.

+ Fladerer  Zahoransky ○ Stein  Matthew et al. X Wu et al.

+ Fladerer  Zahoransky ○ Stein  Matthew et al. X Wu et al. Calculations: ● ten Wolde

+ Fladerer  Zahoransky ○ Stein  Matthew et al. X Wu et al. Calculations: ● ten Wolde ■ Yasuoka and Matsumoto

+ Fladerer  Zahoransky ○ Stein  Matthew et al. X Wu et al. Calculations: ● ten Wolde ■ Yasuoka and Matsumoto ▼ Senger, Reiss, et al.

+ Fladerer  Zahoransky ○ Stein  Matthew et al. X Wu et al. Calculations: ● ten Wolde ■ Yasuoka and Matsumoto ▼ Senger, Reiss, et al. ▲ Oh and Zeng

+ Fladerer  Zahoransky ○ Stein  Matthew et al. X Wu et al. Calculations: ● ten Wolde ■ Yasuoka and Matsumoto ▼ Senger, Reiss, et al. ▲ Oh and Zeng  Chen et al.

+ Fladerer  Zahoransky ○ Stein  Matthew et al. X Wu et al. Calculations: ● ten Wolde ■ Yasuoka and Matsumoto ▼ Senger, Reiss, et al. ▲ Oh and Zeng  Chen et al. □ Hale

+ Fladerer  Zahoransky ○ Stein  Matthew et al. X Wu et al. Calculations: ● ten Wolde ■ Yasuoka and Matsumoto ▼ Senger, Reiss, et al. ▲ Oh and Zeng  Chen et al. □ Hale, Kiefer □ CNT

+ Fladerer  Zahoransky  Stein  Matthew et al. X Wu et al. Calculations: ● ten Wolde ■ Yasuoka and Matsumoto ▼ Senger, Reiss, et al. ▲ Oh and Zeng  Chen et al. □ Hale, Kiefer □ CNT --- Scaled model,  = Scaled model,  = 1.5

Comments on argon data and predictions for J Limited experimental rate data for argon; no rate dependence on temperature; rates are estimated from “onset” assumptions. Lennard-Jones MC and MD simulations at small scaled supersaturations (high nucleation rates) appear to be consistent. Monte Carlo LJ simulation and CNT results at higher scaled supersaturations (lower nucleation rates) are about smaller than experimental “onset” rates. ten Wolde’s LJ MC/MD predicted rate (  10 5 ) is closer to Fladerer’s experimental rate (10 7  2 ) ; T c model  130K and surface tension is smaller than experimental value.