4.1 Liquid Vapor Interface Surface Tension Young-Laplace Equation Condensation Coefficient 4.2 Liquid-Solid-Vapor Contact
Liquid Droplet on a Solid Surface Vapor Bubble on a Solid Surface Contact Angle and Young’s Equation Thermal Boundary Resistance
between Liquid and Solid
4. Molecular Dynamics of Phase-Interface 4. Molecular Dynamics of Phase-Interface
Liquid-Vapor Interface Liquid-Vapor Interface
Liquid Droplet
Flat Interface
Surface Tension Surface Tension
[ ]
³ −
=
GL
) ( )
(
TN LG
z
z
P z P z dz
γ
Densityρ [kg/m3]
Position z[nm]
1000 0–4 –2 0 2 4
Surface Tension of Droplet Surface Tension of Droplet
2 ) ( L G
LG
R P P− γ =
¦
− 2
=
k
fk
r m T r k r
P ρ π
4 1 ) ) (
( B
N
¦
⋅−
=
k ij
ij ij
ij r
r r r m T r k r
P d
) ( 1d 4
1 ) ) (
( B 3
N
φ π
ρ r r
G 3 e 3 L 3
e 3
4 3
4π ρ π ρ
¿¾
½
¯®
− +
= R L R
mN
( )
e2e LG LG
1 2 −
∞ +
¸¸¹·
¨¨©§ −
= OR
R γ δ γ
Young-Laplace Equation
Equilmolar Dividing Radius
Tolman Length δ
Condensation Coefficient Condensation Coefficient
Matsumoto et al.
Condensation coefficient α
Flux Molecule Incident
Flux Molecule on Condensati
= α
Molecular Exchange = Boiling off other molecules Tsuruta et al.
Connection to DSMC
Liquid Droplet on Solid Surface
Liquid Droplet on Solid Surface
Effect of Potential of Solid-Liquid Molecules
Effect of Potential of Solid-Liquid Molecules
E1: Weaker Interaction E4: Stronger Interaction
Effect of Temperature Effect of Temperature
1536 Molecules
100K 130K
10 20
10 20 30
height (Å)
Density Profile
50
40
30 400 radius (Å)
0
Layered Liquid Structure
0 2 4 6 8
-4 -2 0 2
0 0.05
Vibration Range K. E. = kBT [X10-21]
Density Prof.
σINT
Φ(z)
ε SURF
Distance from Surface z [Å]
Potential Energy [J] Number Density [1/Å]
Layered Liquid Structure
10 0
0.1 0.2
WFE0 WFE1 WFE2 WFE3 WFE5WFE4
σINT σAR
5
Distance from Surface [Å] Number Density [1/Å3]
Two-dimensional density distributions for droplet
0.025
0.000 h [Å]
10 20 30
r [Å]
0
10 20 30 40 0
ε*SURF E1=1.29 θ=135.4°
10 20 30
r [Å]
0
10 20 30 40 0
ε*SURF E2=1.86 θ=105.8°
10 20 30
r [Å]
0
10 20 30 40 0
ε*SURF E3=2.42 θ=87.0°
10 20 30
r [Å]
0
10 20 30 40 0
ε*SURF E4=2.99 θ=55.2°
wettable
Density and Potential Profiles
Nliq= 130 T= 92 K θ= 71o
Nliq= 360 T= 99 K θ= 85o
Nliq= 330 T= 85 K θ= 90o Nliq= 340
T= 113 K θ= 86o
Nliq= 1600 T= 92 K θ= 90o
Departure of Droplet for ε*
SURF< 1.0
t = 0 ps t = 30 ps t = 60 ps
Smaller System ε *
SURF=1.86
N = 200, NLiquid= 114 N = 144, NLiquid= 66
Density and Potential Profiles
1 2 3
0 1 2 3
θ
zc
10 nm-3
r[nm]
z[nm]
Vapor Bubble on Solid Surface
Snapshots of bubble formation for E3
Sliced view (central 10Å) All molecules
Two-dimensional density distributions
h [Å]
r [Å]
0 10 20 30 40 50
10 20 30 40 0
ε*SURF E2=1.86
r [Å]
0 10 20 30 40 50
10 20 30 40 0
ε*SURF E3=2.42
r [Å]
0 10 20 30 40 50
10 20 30 40 0
ε*SURFE4=2.99
r [Å]
0 10 20 30 40 50
10 20 30 40 0
ε*SURFE5=3.56
0.025
0.000
wettable
Contact angle correlated with ε*SURF
Contact angle correlated with ε*SURF
1 2 3 4
–1 0 1
ε*SURF=εSURF/εAR Contact angle Hc/R1/2(=cosθ)
Bubble(100K)
Solid: Density Open: Potential Droplet
Bubble(110K)
2 / 1
cos R
HC
= θ
2 /
R1
HC
HC<R1/2 HC>R1/2
θ R1/2 HC Solid surface
Liquid
Vapor bubble
HC R1/2
Solid surface Liquid
Vapor bubble
Two-dimensional density distribution for P5
Definition of contact angle cosθ for E5 and P5
Young’s Equation (Macroscopic) Young’s Equation (Macroscopic)
σ
LGσ
SGσ
SLSG SL
LG
θ + σ = σ
σ cos
LG SL SG
σ σ
−
= σ θ cos
θ
Solid Liquid
Gas
Thermal Boundary Resistance over Liquid-Solid Interface: 10-7∼10-6m2K/W
Therm. Sci. Eng., 1999, vol. 7, no. 1, pp. 63-68.
Thermal Boundary Resistance over Liquid-Solid Interface: 10-7∼10-6m2K/W
Therm. Sci. Eng., 1999, vol. 7, no. 1, pp. 63-68.
213.570 A
5 5.400 A 52.776 A }
} L iquid
Liq uid Vapor
3 solid layers 3 solid layers
Heating Cooling
0 0.01 0.02 0.03
100 110 120
0 100 200
0 4 8 12
Positon [Å]
Temperature [K]Density [1/Å3]
Temperature jump: 6.4 K
5.3 K 2000–5000 ps
2000 ps 3500 ps
5000 ps
Velocity [m/s]
<vz>
Thermal Resistance over Liquid-Solid Interface Thermal Resistance over
Liquid-Solid Interface
0 2500 5000
0 1000 2000 3000
0 20 40 100
120
Time [ps]
Number of MoleculesTemperature [K] Heat Flux [MW/m2]
qV NL
cond
NLevap NV TScond
TSevap
TL cond TL TV evap
Thermal Resistance over Liquid-Solid Interface Thermal Resistance over
Liquid-Solid Interface
0 0.01 0.02 0.03
100 110 120
0 100 200
0 4 8 12
Positon [Å]
Temperature [K]Density [1/Å3]
Temperature jump: 6.4 K
5.3 K 2000–5000 ps
2000 ps 3500 ps
5000 ps
Velocity [m/s]
<vz>
) / /( T z qW
L= ∂ ∂
λ
0.082 W/m⋅K
Handbook value 0.097 W/m⋅K
at the saturated temperature of 110 K Thermal Conductivity
Thermal Resistance over Liquid-Solid Interface Thermal Resistance over
Liquid-Solid Interface qWcond
qWevap qV
TJUMPevap
TJUMPco nd LR
LR qLevap qLcond
M Heat
M ass
S olid Liquid
Solid Liquid Vapor
z
Thermal Resistance over Liquid-Solid Interface Thermal Resistance over
Liquid-Solid Interface
1 2 3 4
0 10 20
–1 0 1
ε*SURF=εSURF/εAR
Contact anglecos
θ
LR[nm]
LR
cos θ E2
E3 E4
