D EPARTMENT OF M ECHANICAL E NGINEERING BCNucleation-Aggregation Workshop Grand canonical molecular dynamics simulation Grand canonical molecular dynamics simulation of homogeneous nucleation Universitat de Barcelona, Departament de Física Fonamental, June 18, 2009 M. Horsch, H. Hasse, and J. Vrabec SFB 716 T HERMODYNAMICS AND E NERGY T ECHNOLOGY P ROF. D R.- I NG. HABIL. J ADRAN V RABEC
D EPARTMENT OF M ECHANICAL E NGINEERING Molecular simulation of nucleation Indirect simulation: T Transition path sampling iti th li Determination of the critical size … by observing single droplets in non-equilibrium … by observing single droplets in equilibrium by observing single droplets in equilibrium Direct simulation: … of a metastable state far from the spinodal line … of nucleation at a high supersaturation, decreasing over time … of a metastable state near the spinodal line … of nucleation at a constantly high supersaturation T HERMODYNAMICS AND E NERGY T ECHNOLOGY P ROF. D R.- I NG. HABIL. J ADRAN V RABEC
D EPARTMENT OF M ECHANICAL E NGINEERING The critical nucleus … is defined by a stable or unstable equilibrium with the vapor. ethane at T = 280 K ( ρ s = 1.88 mol/l) 280 K ( ρ s ethane at T 1.88 mol/l) Free energy of formation Free energy of formation - Δ μ Δ Ω * ts of kT Positive surface contribution: 10 2.55 mol/l 2.55 mol/l - Δ μ ergy in uni = d Ω dA γ Δ Ω * A j 0 Negative volume contribution: N ti l t ib ti free en 2.8 mol/l = − liq d Ω dj μ μ V j -10 j * j j * j = liq li lim μ j μ 250 500 750 s → ∞ j nucleus size in number of molecules It i It is essential to know the supersaturation in terms of Δ µ . ti l t k th t ti i t f Δ T HERMODYNAMICS AND E NERGY T ECHNOLOGY P ROF. D R.- I NG. HABIL. J ADRAN V RABEC
D EPARTMENT OF M ECHANICAL E NGINEERING Equilibrium vapor pressure Equilibrium condition for 100 methane at 149 K of kT a droplet containing j molecules: molecules: 0 0 gy in units ( ) = p p T , j -100 free energ Δ G at constant p and T : NpT : N = 6000, p = 1170 kPa -200 NVT : N = 5000, ρ = 1.40 mol/l 1 unstable equilibrium NVT : N = 2000 ρ = 2 16 mol/l NVT : N = 2000, ρ = 2.16 mol/l f -300 Δ F at constant V and T : 2000 kPa 1 unstable equilibrium 1 unstable equilibrium 1000 1000 p / 1 stable equilibrium 0 10 100 1000 droplet size in number of molecules T HERMODYNAMICS AND E NERGY T ECHNOLOGY P ROF. D R.- I NG. HABIL. J ADRAN V RABEC
D EPARTMENT OF M ECHANICAL E NGINEERING Systems containing a single droplet � Vapor and liquid are equilibrated separately. q p y � A small ( j < 10000) droplet is inserted into droplet is inserted into the vapor. � If t he droplet cannot � If t he droplet cannot evaporate completely, an equilibrium is established established within a few ithin a fe nanoseconds. t t-s-LJ fluid ( r c = 2.5 σ ) LJ fl id ( 2 5 ) T HERMODYNAMICS AND E NERGY T ECHNOLOGY P ROF. D R.- I NG. HABIL. J ADRAN V RABEC
D EPARTMENT OF M ECHANICAL E NGINEERING Surface tension Integration of the p N ( r ) profile: 2 nits of εσ -2 0.6 T = 0.65 ε / k − − 2 ⎡ ⎤ ( p p ) dp ( r ) ∞ ∫ ∫ − = 3 3 γ γ l r N dr ⎢ ⎢ ⎥ ⎥ 8 dr ⎣ ⎣ ⎦ ⎦ sion in un 0 0.4 T = 0.8 ε / k Size dependence (Tolman): p ( ) urface ten ( ) 0.2 γ 2 δ − = + + 2 ∞ 1 T O R γ γ R γ T = 0 95 ε / k T = 0.95 ε / k su 0.0 Correlation from simulation data 0 5000 10000 for T = 0.65, 0.70, … 0.95 ε / k : , , droplet size in number of molecules p ● simulation ⎛ ⎞ δ 0 . 7 ⎜ ⎟ = − − 1 3 T 0 . 9 j ⎜ ⎟ — planar interface − R 1 T T ⎝ ⎝ ⎠ ⎠ γ γ c c - - - new correlation T HERMODYNAMICS AND E NERGY T ECHNOLOGY P ROF. D R.- I NG. HABIL. J ADRAN V RABEC
D EPARTMENT OF M ECHANICAL E NGINEERING Tolman equation Vapor-liquid interface of the t-s-LJ fluid 0.8 ε / k 10 gth 0.9 ε / k man leng simulation us / Tolm 5 correlation for γ , γ 2 δ ∞ = = 1 + 1 + using using T radi γ R γ 0 0 5000 10000 droplet size in number of particles The higher order terms of the Tolman equation should not be neglected. e g e o de te s o t e o a equat o s ou d ot be eg ected T HERMODYNAMICS AND E NERGY T ECHNOLOGY P ROF. D R.- I NG. HABIL. J ADRAN V RABEC
D EPARTMENT OF M ECHANICAL E NGINEERING Direct MD simulation of nucleation � Integration time step typically between 1 and 5 fs; Feasible simulation time: on the order of nanoseconds. � A saturated vapor with V = 10 -20 m 3 contains: � A d i h V 10 20 3 i 800,000 molecules (methane at 114 K = 0.6 T c ) 7,000,000 molecules (CO 2 at 253 K = 0.83 T c ) 7,000,000 molecules (CO 2 at 253 K 0.83 T c ) � Minimal nucleation rate accessible by direct simulation: #nuclei / (volume V x time Δ t ) = nucleation rate J 10 30 / m 3 s ( 10 -20 m 3 10 -9 s ) = x 10 ( ) / / Direct MD simulation Experiment above 10 30 / m 3 s up to 10 23 / m 3 s above 10 / m s up to 10 / m s T HERMODYNAMICS AND E NERGY T ECHNOLOGY P ROF. D R.- I NG. HABIL. J ADRAN V RABEC
D EPARTMENT OF M ECHANICAL E NGINEERING Grand canonical molecular dynamics Algorithm according to Cielinski: • fixed values of μ , V und T fi d l f V d T • test insertion of a molecule at a random position ⎡ − ⎤ ⎛ ⎞ μ Δ U V = ⎜ ⎟ P max 1 , exp ins ⎢ ) ⎥ ( ins 3 + kT Λ N 1 ⎝ ⎠ ⎣ ⎦ • test deletion of a random molecule ⎡ ⎤ − − ⎛ μ Δ U ⎞ V = = P P max max 1 1 , exp exp ⎜ ⎜ ins ⎟ ⎟ ⎢ ⎢ ⎥ ⎥ d l del 3 ⎝ kT ⎠ Λ N ⎣ ⎦ • equal number of test insertions and deletions (10 -5 – 10 -3 / step) T HERMODYNAMICS AND E NERGY T ECHNOLOGY P ROF. D R.- I NG. HABIL. J ADRAN V RABEC
D EPARTMENT OF M ECHANICAL E NGINEERING Supersaturation from NVT and µVT simulation 3 3.0 ration µVT NVT NVT supersatur S ρ 0.7 ε / k 2.5 ation upersatura potential s S p 2 2.0 S μ 0.7 ε / k chemical p s 1.5 0.85 ε / k 0.8 ε / k c 1.0 1 0.004 0.008 0.012 0.02 0.04 0.06 excess pressure in units of εσ - 3 excess pressure in units of εσ density in units of σ -3 density in units of σ T HERMODYNAMICS AND E NERGY T ECHNOLOGY P ROF. D R.- I NG. HABIL. J ADRAN V RABEC
D EPARTMENT OF M ECHANICAL E NGINEERING Szilárd‘s demon SZILÁRD SZILÁRD T HERMODYNAMICS AND E NERGY T ECHNOLOGY P ROF. D R.- I NG. HABIL. J ADRAN V RABEC
D EPARTMENT OF M ECHANICAL E NGINEERING McDonald‘s demon McDONALD McDONALD T HERMODYNAMICS AND E NERGY T ECHNOLOGY P ROF. D R.- I NG. HABIL. J ADRAN V RABEC
D EPARTMENT OF M ECHANICAL E NGINEERING Interactive presentation: McDonald‘s demon T HERMODYNAMICS AND E NERGY T ECHNOLOGY P ROF. D R.- I NG. HABIL. J ADRAN V RABEC
D EPARTMENT OF M ECHANICAL E NGINEERING Comparison: NVT and µVT simulation t-s-LJ fluid at T = 0.7 ε / k 80 i NVT = 25 NVT i NVT = 50 6 σ 3 i μ VT = 50 60 x 10 i NVT = 150 i NVT 150 40 40 ρ j 20 j μ VT > 25 0 3 σ 3 ε -1 20 µVT : S = 2.866 μ VT 15 10 NVT : ρ = 0 004044 σ -3 NVT : ρ = 0.004044 σ -3 p x 1 NVT 10 0 250 500 750 1000 1250 simulation time in units of σ ( m / ε ) 1/2 / ) 1/2 i l ti ti i it f ( T HERMODYNAMICS AND E NERGY T ECHNOLOGY P ROF. D R.- I NG. HABIL. J ADRAN V RABEC
D EPARTMENT OF M ECHANICAL E NGINEERING Nucleus size distribution t-s-LJ fluid at T = 0.7 ε / k : µVT ( S = 2.866) and NVT ( ρ = 0.004044 σ -3 ) simulation 10 -3 of σ -3 10 -4 y in units 10 -5 t NVT = 400 10 -6 CNT CNT density 10 -7 t NVT = 1050 µVT i =50 10 -8 0 10 20 30 40 50 60 nucleus size in number of molecules Good agreement with CNT for j * and the number of small nuclei. G d t ith CNT f j * d th b f ll l i T HERMODYNAMICS AND E NERGY T ECHNOLOGY P ROF. D R.- I NG. HABIL. J ADRAN V RABEC
D EPARTMENT OF M ECHANICAL E NGINEERING Threshold dependence of the intervention rate t-s-LJ fluid, T = 0.7, S = 2.496 Statistical probability for a nucleus of growing from size j j* CNT -15 to infinite size: to infinite size: m e logarithm ∞ ⎛ Ω ⎞ ( ) ( ) 2 ∫ ∫ -20 ∞ = = − ω ω ⎜ ⎜ ⎟ ⎟ P P j j 1 1 exp exp dj dj , vention rat Z ⎝ ⎠ kT j -25 such that h h interv CNT (ln J = -26.4) ( ) ( ) ∞ j ∗ = -30 1 P P j 20 20 40 40 60 60 80 80 2 2 threshold size in number of particles CNT predicts an acceptable value for j * and underestimates J significantly CNT predicts an acceptable value for j * and underestimates J significantly. T HERMODYNAMICS AND E NERGY T ECHNOLOGY P ROF. D R.- I NG. HABIL. J ADRAN V RABEC
D EPARTMENT OF M ECHANICAL E NGINEERING GCMD simulation of nucleation: Results 0.5 10 -6 0.65 ε / k 0.85 ε / k ε -0.5 m s of σ -4 ε - 10 -9 e in units NVT (YM method) McDonald‘s demon 10 -12 10 classical theory l i l th ation rate 0.95 ε / k 0.7 ε / k critical scaling EMLD-DNT EMLD DNT 10 -15 nuclea 0.008 0.012 0.016 0.03 0.04 0.05 pressure in units of εσ -3 pressure in units of εσ T HERMODYNAMICS AND E NERGY T ECHNOLOGY P ROF. D R.- I NG. HABIL. J ADRAN V RABEC
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