Molecular Modeling in Engineering: Methods and Applications
Lecture: Molecular origin of surface tension
Nikolai V. Priezjev 2 Pressure-stress 3x3 tensor: equilibrium properties
i i ij ij P P P N P V mv v r F() r xx xy xz 3 1 k T m v2 i i j i P P P B i i yx yy yz 2 2N i1 x, y, or z and x, y, z Pzx Pzy Pzz • 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.
P(r ) ? 4 c Energy Force 1/6 2 rc 2s rc 2.5s
0
-2
1.0 1.5 2.0 2.5 Separation, r/s 3 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. 4 Examples of surface tension force in nature (water striders)
Maximum Curvature Force
Body weight
D. Hu, B. Chan, J. W. M. Bush. 2003. The Hydrodynamics of Water Strider Locomotion. Nature 424, 663-666. 5 Examples of surface tension force in nature (water striders)
, MIT
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. http://www-math.mit.edu/~dhu/Climberweb/climberweb.html 6
Net forces at the liquid-vapor interface =
Force per unit length
4 repulsion Energy Force
2
0 attraction -2 i i ij ij P V mv v r F () r 1.0 1.5 2.0 2.5 i i j i Separation, r/s 7 Why is surface tension a force parallel to the interface?
n=normal t=tangential
The total pressure p:
The molecular mechanism of surface tension. M. V. Berry 1971 Phys. Educ. 6, 79 8 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). 9 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.
Surface tension (s2):
[P (z) P (z)]dz N T 10s 1 [P (z) (P (z) P (z))]dz zz xx yy 10s 2
Surface tension is very sensitive to the cut-off radius of the LJ
potential. Rule of thumb: rc 6s to avoid dependency on rc. N molecules > 1000
Antonin Marchand, Joost H. Weijs, Jacco H. Snoeijer, Bruno Andreotti, Am. J. Phys. 79, 999 (2011). 10 Temperature dependence of density profiles and surface tension
Ultra-thin Lennard-Jones film:
24s 1 [P (z) (P (z) P (z))]dz zz xx yy 0s 2 and then divided by 2 interfaces
Etomica module: Ph.D. thesis by Shashank Sinha, 2004. UCLA. Molecular Dynamics Interfacial tension Simulation of Interfacial Tension and Contact Angle of Lennard-Jones Fluid 11 Laplace pressure or why do soap bubbles break?
For a spherical bubble R1 is being equal to R2 and the Laplace pressure difference becomes
4 P R
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: Pressure inside the bubble is equal to the atmospheric pressure plus the Laplace pressure (note 2 interfaces!) Surfactants lower surface tension of water, increase elasticity and reduce where Rx and Ry are radii of curvature in each of the axes that are parallel to the surface, is the surface evaporation. Bubbles last tension, and is the pressure difference across the longer on a rainy day: interface. reduced evaporation. 12 The contact angle and surface energies (gas, liquid, solid)
Young’s law: Thomas Young (1773-1829)
The molecular mechanism of surface tension. M. V. Berry 1971 Phys. Educ. 6, 79 13 Summary
i i ij ij Pxx Pxy Pxz P V mv v r F () r At equilibrium, only i i j i Pyx Pyy Pyz diagonal elements are x, y, or z and x, y, z Pzx Pzy Pzz non zero.
(P = Microscopic Pressure-Stress Tensor) 4 repulsion Energy Force
2
0 attraction -2
1.0 1.5 2.0 2.5 Separation, r/s
Surface tension:
[P (z) P (z)]dz N T Surface tension a force per unit length parallel 10s 1 [P (z) (P (z) P (z))]dz to the liquid-vapor interface. zz xx yy 10s 2