Constrained LDFT To stabilize a nucleus,we introduced a suitable constraint in the framework of the LDFT Volume constraint E=2+2'=kT>[e,In p:+(1-p.)In(1-p.)] 0, P,∈vapor -号∑∑Pp,+∑AA-+g-Σx1-们 p,∈liquid Surface constraint E=2+2'=kT∑[p,lnp,+(I-p)ln(l-p】 [0. P,∈vapO -受∑∑P.P+∑A(-)+N-∑】= P,∈surface 0, p,∈liquid Men,Zhang and Wang,Phys.Rev.E 2009,79,051602
Constrained LDFT • To stabilize a nucleus, we introduced a suitable constraint in the framework of the LDFT – Volume constraint – Surface constraint ( ) [ ] 2 [ ln (1 )ln(1 )] i i 0 i i i i a i i a ff i i i i i ∑ ∑ ∑ ∑ ∑ − + − + − = Ω + Ω′ = + − − ρ ρ + ρ φ µ κ χ ε ρ ρ ρ ρ N L E kT ( ) [ ] 2 [ ln (1 )ln(1 )] i Si 0 i i i i a i i a ff i i i i i ∑ ∑ ∑ ∑ ∑ − + − + − = Ω + Ω′ = + − − ρ ρ + ρ φ µ κ χ ε ρ ρ ρ ρ Ns E kT liquid vapor i i i ∈ ∈ ⎩ ⎨ ⎧ = ρ ρ χ , , 1 0 liquid surface vapor i i i Si ∈ ∈ ∈ ⎪ ⎩ ⎪ ⎨ ⎧ = ρ ρ ρ χ 0, 1, 0, Men, Zhang and Wang, Phys. Rev. E 2009, 79, 051602
Nucleation in a infinitely long pore using the constrained LDFT,one can obtain the nucleation barrier and the morphology of critical nuclei 400 a =2.5 -●-adsor. 300 0- desor bulk -■-adsor. 200 -0- desor. 100 a000 033 3.2 3.1 3.0 -2.9 2.8 Chemcial potentialμ Men,Zhang,and Wang,Phys.Rev.E.79,051602(2009)
Nucleation in a infinitely long pore using the constrained LDFT，one can obtain the nucleation barrier and the morphology of critical nuclei -3.3 -3.2 -3.1 -3.0 -2.9 -2.8 -2.7 0 100 200 300 400 Nucleation barrier ∆Ω Chemcial potential µ w=2.5 adsor. desor. bulk adsor. desor. a Men, Zhang, and Wang, Phys. Rev. E. 79, 051602 (2009)
Capillary liquid bridges in AFM:Formation, rupture,and hysteresis With constrained LDFT,we can get the stable, metastable,and transition state in the formation and rupture process of capillary bridge -1600 a-o-h=9-●-h=8 -0-h=7-■一h=6 -2000 -a-h=5--h=4 -a-h=3-*-h=2 -女一h=1-+-h=0 0-0 -2400 00 -2800 (a)Constrained grand potential 线年4 as a function of liquid bridge size -3200 RH=90% w=0.950 女省-的草 -36001 0 5101520 2530 Radius of water bridge Men,Zhang,and Wang,J.Chem.Phys.131,184702(2009)
Capillary liquid bridges in AFM: Formation, rupture, and hysteresis • With constrained LDFT, we can get the stable, metastable, and transition state in the formation and rupture process of capillary bridge 0 5 10 15 20 25 30 -3600 -3200 -2800 -2400 -2000 -1600 Grand potential Radius of water bridge h=9 h=8 h=7 h=6 h=5 h=4 h=3 h=2 h=1 h=0 RH=90% wt=0.950 a (a) Constrained grand potential as a function of liquid bridge size Men, Zhang, and Wang, J. Chem. Phys. 131, 184702 (2009)
Capillary liquid bridges in AFM:Formation, rupture,and hysteresis 400 80 RH=90% 350 b 9200 40 W=0.950 -o-formation 300 100 0 -0-01 -e-rupture 250 200 40 RH=90% 1.21.62.0242.8 0-0-000-0 150 w=0.950 Tip-substrate distance/nm -80 a 100 -o-formation 23 一。-rupture -120 50 0 ● 160 01234 0 0 Tip-substrate distance/nm 200 5 6 0 2 4 6 8 10 12 1416 Tip-substrate distance Tip-substrate distance (b)and(c)show the corresponding energy barrier and capillary force as a function of tip-substrate distance. Men,Zhang,and Wang,J.Chem.Phys.131,184702(2009)
Capillary liquid bridges in AFM: Formation, rupture, and hysteresis 345678 0 50 100 150 200 250 300 350 400 1.2 1.6 2.0 2.4 2.8 0 100 200 Energy barrier/X10 -20J Tip-substrate distance/nm Energy barrier Tip-substrate distance formation rupture RH=90% wt=0.950 b (b) and (c) show the corresponding energy barrier and capillary force as a function of tip-substrate distance. 0 2 4 6 8 10 12 14 16 -200 -160 -120 -80 -40 0 40 80 01234 -4 -3 -2 -1 0 1 Force/nN Tip-substrate distance/nm Force Tip-substrate distance formation rupture RH=90% wt=0.950 c Men, Zhang, and Wang, J. Chem. Phys. 131, 184702 (2009)
How nanoscale seed particles affect vapor- liquid nucleation With constrained LDFT,the shape of critical nucleus and height of the nucleation barrier can be determined without using a predefined nucleus as input. 140 120 100 80 60 -2.8电m 40 u=.2.88 u=-2.89 u=-2.91 △ 4=-2.89 u=-2.87 20 μ=-2.90 ◇μ=-2.91 0 1500 3000 4500 nucleus volume,Vd homogeneous nucleation at different chemical potentials Liu and Zhang,J.Chem.Phys.2011,(accepted)
• With constrained LDFT, the shape of critical nucleus and height of the nucleation barrier can be determined without using a predefined nucleus as input. 0 1500 3000 4500 0 20 40 60 80 100 120 140 energy barrier, ∆Ω nucleus volume, Vd µ=-2.87 µ=-2.88 µ=-2.89 µ=-2.90 µ=-2.91 homogeneous nucleation at different chemical potentials Liu and Zhang, J. Chem. Phys. 2011, (accepted) How nanoscale seed particles affect vaporliquid nucleation