Molecular Modeling in Engineering: Methods and Applications
Transcript of Molecular Modeling in Engineering: Methods and Applications
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Molecular Modeling in Engineering:
Methods and Applications
Lecture: Molecular origin of surface tension
Nikolai V. Priezjev
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Pressure-stress 3x3 tensor: equilibrium properties
( )i i ij ij
i i j i
P V mv v r F r
Pxx Pxy Pxz
Pyx Pyy Pyz
Pzx Pzy Pzz z y, x, and zor y,x,
• At equilibrium (i.e. no shear flow or external forces) non-diagonal elements of
the pressure-stress tensor are zero (Pxy = Pxz = Pyz = 0 ).
• The diagonal elements are pressures in the x, y, or z directions (Pxx = Pyy = Pzz ).
• First term is just a sum of all kinetic energies in the x, y, or z directions (ideal gas law).
• Second term is the virial: sum of the product Fxrx of all pairs within the cutoff radius.
4
2
0
-2
2.52.01.51.0
Separation, r/s
Energy Force
s6/1 2crs5.2cr
N
i
iiB vmN
Tk
1
2
2
1
2
3
?)( crP
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Examples of surface tension force in nature (water striders)
D. Hu, B. Chan, J. W. M. Bush. 2003. The
Hydrodynamics of Water Strider Locomotion.
Nature 424, 663-666.
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D. Hu, B. Chan, J. W. M. Bush. 2003. The Hydrodynamics of Water Strider Locomotion. Nature 424, 663-666.
Examples of surface tension force in nature (water striders)
Maximum
Curvature
Force
Body weight
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http://www-math.mit.edu/~dhu/Climberweb/climberweb.html
Examples of surface tension force in nature (water striders)
The deformation of the water surface near the
head and tail of the larva is clearly visible.
In these images, it approaches an emerging
wetted leaf.
, MIT
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Net forces at the liquid-vapor interface
( )i i ij ij
i i j i
P V mv v r F r
=
Force per unit length
4
2
0
-2
2.52.01.51.0
Separation, r/s
Energy Force
attraction
repulsion
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Why is surface tension a force parallel to the interface?
The total pressure p:
The molecular mechanism of surface tension. M. V. Berry 1971 Phys. Educ. 6, 79
n=normal t=tangential
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Why is surface tension a force parallel to the interface?
Antonin Marchand, Joost H. Weijs, Jacco H. Snoeijer, Bruno Andreotti, Am. J. Phys. 79, 999 (2011).
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Pressure and density profiles at the liquid-vapor interface
The transition from the high
density liquid to the low density
gas takes place in a very narrow
region that is only a few
molecules wide.
dzzPzP TN )]()([
Surface tension (s2):
s
s
10
10
))]()((2
1)([ dzzPzPzP yyxxzz
Surface tension is very sensitive
to the cut-off radius of the LJ
potential. Rule of thumb:
to avoid dependency on rc.
N molecules > 1000
s6cr
Antonin Marchand, Joost H. Weijs, Jacco H. Snoeijer, Bruno Andreotti, Am. J. Phys. 79, 999 (2011).
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Temperature dependence of density profiles and surface tension
s
s
24
0
))]()((2
1)([ dzzPzPzP yyxxzz
and then divided by 2 interfaces
Ph.D. thesis by Shashank Sinha, 2004. UCLA. Molecular Dynamics
Simulation of Interfacial Tension and Contact Angle of Lennard-Jones Fluid
Ultra-thin Lennard-Jones film:
Etomica module:
Interfacial tension
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Laplace pressure or why do soap bubbles break?
Surface tension forces acting on a tiny (differential)
patch of surface. δθx and δθy indicate the amount of
bend over the dimensions of the patch. Balancing the
tension forces with pressure leads to the Young-Laplace
equation:
where Rx and Ry are radii of curvature in each of the
axes that are parallel to the surface, is the surface
tension, and is the pressure difference across the
interface.
For a spherical bubble R1 is being equal to R2
and the Laplace pressure difference becomes
Pressure inside the bubble is equal to the
atmospheric pressure plus the Laplace
pressure (note 2 interfaces!)
RP
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Surfactants lower surface
tension of water, increase
elasticity and reduce
evaporation. Bubbles last
longer on a rainy day:
reduced evaporation.
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The contact angle and surface energies (gas, liquid, solid)
The molecular mechanism of surface tension. M. V. Berry 1971 Phys. Educ. 6, 79
Young’s law: Thomas Young (1773-1829)
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Summary
( )i i ij ij
i i j i
P V mv v r F r
Pxx Pxy Pxz
Pyx Pyy Pyz
Pzx Pzy Pzz z y, x, and zor y,x,
(P = Microscopic Pressure-Stress Tensor)
dzzPzP TN )]()([
Surface tension:
s
s
10
10
))]()((2
1)([ dzzPzPzP yyxxzz
4
2
0
-2
2.52.01.51.0
Separation, r/s
Energy Force
attraction
repulsion
Surface tension a force per unit length parallel
to the liquid-vapor interface.
At equilibrium, only
diagonal elements are
non zero.