Surface property corrected modification of the classical nucleation theory

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1 CCP5 Annual Meeting Surface property corrected modification of the classical nucleation theory Sheffield Hallam University, September 15, 2010 Martin Horsch, Hans Hasse, and Jadran Vrabec

2 The critical droplet nergy in un nits of kt free e is defined by a stable or unstable equilibrium with the vapour phase 10 Free energy of formation (CNT) pressur re in units of -3 ethane at T = 280 K ( s = 1.88 mol/l) mol/l LJ truncated-shifted 0.04 G* 0.00 G* mol/l T = /k T = 0.85 /k T = 0.75 /k T = 0.65 /k n* n* nucleus size in number of molecules specific volume in units of 3-0.6

3 Equilibrium vapour pressure y in units of kt p / kpa fre ee energ 100 Argon at 117 K Equilibrium condition for a droplet containing n 0 atoms: NpT: N = 6000, p = 1220 kpa NVT: N = 5000, = 1.85 mol/l NVT: N = 2000, = mol/l -300 p p T, n G at constant p and T: 1 unstable equilibrium 2000 F at constant V and T: unstable equilibrium 1 stable equilibrium droplet size in number of atoms

4 Canonical MD simulation of curved interfaces density in units of capillarity radius P = 2 o / p equimolar radius Q T = 0.75 /k (LJTS) distance from the centre of mass in units of

5 Droplet properties in equilibrium Irving-Kirkwood tensor: surface tension Laplace radius normal pressure p N (R) tangential pressure p T (R) 3 2 γ p outside 3 2 r dpn inside R 2γ p Q δ To olman lengt th in units of 0.9 /k /k /k 0.7 /k 1 LJTS surface of tension curvature 1/R in units of -1 f -2 surfa ace tension n in units of 0,6 0,4 = 0 Tolman equation = 0 T = 0.8 /k 0 T = 0.7 /k 0,2 T = 0.95 /k = Korr. 0, droplet size (number of molecules) Mol. Phys. 104: 1509 (2006)

6 Droplet properties in equilibrium: Discussion l droplet siz ze (numbe er of molec cules) critical /k 0.95 /k 0.7 /k 09/k 0.9 CNT present results Napari et al /k LJTS chemical potential supersaturation / P ratio Q radius Q ] [ equimolar = 1 0 = 2 0 = T = 0.65 /k T = 0.85 /k T = 0.95 /k equimolar curvature 1/Q in LJTS T = 0.65 /k T = 0.85 /k capillarity radius P []

7 Nucleation: Direct simulation vs. experiment Integration time step typically between 1 and 5 fs Feasible simulation time: on the order of nanoseconds A saturated vapor with V = m 3 contains: molecules (saturated methane at 114 K) molecules (saturated CO 2 at 253 K) Minimal nucleation rate accessible by direct simulation: #nuclei / (volume V x time t) = nucleation rate J 10 / ( m 3 x 10-9 s ) = /m 3 s Direct MD simulation above / m 3 s Experiment up to / m 3 s

8 Innovative HPC Methods and Applications for Highly Scalable Molecular Simulation (IMEMO) Project associates: Industrial associates:

9 Direct MD simulation of nucleation Yasuoka-Matsumoto method: Canonical MD simulation Limited time interval for nucleation Conditions change over time pressure / kp Pa dro oplets per nm NVT with =146mol/l Argon at T =965K time in units of ns

10 Direct MD simulation of nucleation Yasuoka-Matsumoto method: Canonical MD simulation Limited time interval for nucleation Conditions change over time pressure / kp Pa dro oplets per nm NVT with =146mol/l Argon at T =965K time in units of ns GCMD, i.e. MD steps alternating with GCMC insertion/deletion steps whenever n > θ Thermodynamic conditions of the supersaturated state are maintained

11 Direct MD simulation of nucleation Yasuoka-Matsumoto method: Canonical MD simulation Limited time interval for nucleation Conditions change over time pressure / kp Pa plets per nm 3 dro ,002 0,001 0,000 NVT with =146mol/l 1.46 VT with S = Argon at T =965K 96.5 interventions of McDonald's daemon 0,2 0,4 0,6 0,8 1,0 1,2 1,4 time in units of ns GCMD, i.e. MD steps alternating with GCMC insertion/deletion steps McDonald whenever n > θ Thermodynamic conditions of the supersaturated state are maintained

12 Video: MCDONALD s dæmon

13 Nucleation rates from GCMD + McDonald s dæmon m) 1/2 ts of -4 (/m n rate in uni nucleation /k 0.65 /k 0.45 /k 0.95 /k 0.9 /k 0.85 /k LJTS m) 1/2 rate in unit ts of -4 (/m nucleation LJTS 0.45 /k classical nucleation theory (CNT) McDonald's daemon MC-FFS (van Meel et al.) ) chemical potential supersaturation exp(/ kt) ) pressure supersaturation p / p s (T) J. Chem. Phys. 131: (2009)

14 Nucleus size distribution LJTS fluid at T = 0.7 ε/k: µvt (S µ = 2.866) and NVT (ρ = σ -3 ) simulation nits of -3 sity in un num mber den t NVT = CNT t NVT = 1050 µvt i= droplet size (number of molecules) Good agreement with CNT for n* and the number of small nuclei

15 Surface property corrected CNT Equilibrium condition for critical droplets yields 2 dv = R da and hence da 2dV R 8πQ 2 Q δ dq δ is positive da is larger than according to capillarity a approximation at o A A Correlated surface area CNT n T 75 c T 1 1 A Phys. Rev. E 78: (2008) CNT steric fac ctor (A/A NT )1/2 CN 1.6 correlation based on, n*, and J eff. increase by, i.e. (P + ) : P effective increase by D vap () T =09/k 0.9 T = 0.75 /k droplet size (number of molecules)

16 Surface property corrected nucleation theory Effect: A larger surface area compensates the lower surface tension critica al droplet siz ze (number of atoms) K K 110 K 124 K 131 K Argon vapour pressure in units of kpa of cm -3 s -1 nu ucleation ra ate in units K 110 K K 124 K 131 K 138 K pressure in units of kpa steady-state simulation YM simulation SPC CNT

17 Conclusion MD simulation of equilibria allows sampling over an arbitrary time interval, eventually leading to the desired level of accuracy Single droplets can be stabilized in the canonical ensemble A supersaturated vapour near the spinodal line can be stabilized by grand canonical simulation with McDonald s dæmon The classical theory leads to acceptable results for the LJTS fluid. However, it does not take into account curvature effects on the surface tension A consistent description is given by postulating an increased surface of tension

Dependence of the surface tension on curvature

Dependence of the surface tension on curvature rigorously determined from the density profiles of nanodroplets Athens, st September M. T. Horsch,,, S. Eckelsbach, H. Hasse, G. Jackson, E. A. Müller, G.