Molecular Dynamics Simulation of nanobubbles in the bulk and on
the substrate
Shunsuke Kohno1 and Sadayoshi Toh1
11

Graduate School of Science, Kyoto University, Japan

Introduction

Nanobubbles are considered to be one origin of characteristic behaviors of fluids in microscopic systems, such as
boundary slip in channels and long-range attractive force
acting between hydrophobic surfaces. In many recent experiments, nanobubbles on the hydrophobic substrates areobserved with atomic force microscopy , infrared spectroscopy and other techniques. Observed nanobubbles
are extraordinarily stable; for instance, Zhang et al. reported that air nanobubbles on hydrophobic walls survive for more than 4 days[1]. However, according to theclassical diffusion model based on Young-Laplace equation, nanobubbles cannot be stable because the gas in thebubble diffuses rapidly into the surrounding liquid due tohigher pressure in the bubble. The lifetime of the spherical bubble whose radius is 10-100nm, is estimated to be
1-100µs[2], which is at least 9 orders of magnitude shorter
than the observed lifetimes.
Why nanobubbles are stable? This is one of the outstanding questions in fluid dynamics and has remained
puzzling for decades. Mainly two types of stabilizing mechanisms are proposed in the preceding studies: (1) the effects of impurities on the liquid-gas interface, (2) nonequilibrium re-entrance of gas into the bubble. The former
idea is that impurities on the liquid-gas interface reduce
the surface tension and prevent diffusion of the gas into
the surrounding liquid. The latter idea which is recently
proposed by Brenner et al.[3] and supported by Weijs et al.
[4] is that both the circulating flow of gas above the bubble created by a Knudsen gas behavior in the bubble[5],and an attraction of gas on the hydrophobic surfaces induce the influx of gas into the bubble near the contact line,
which balances the diffusive outflux. There are some numerical results suggesting the existence of such the influx,
but the mechanism of nonequilibrium is not clear. Notethat nanobubbles in the bulk liquid are also observed to
be stable[6]. To explain their astonishingly long lifetime,
the former idea is partially applicable, but only the effects
of impurities on the liquid-gas interface are not enough.
There must be another stabilizing mechanism remaining
unrevealed.
Molecular dynamics simulations are suitable to treat
molecular interactions and observe gas behavior in the liquid. A lot of numerical simulations on nanobubbles havebeen reported so far, most of them treat not the collapse ofbubbles and shrinking or sustained bubbles, but the nucleation of bubbles and growing or stable bubbles. Besides,
only a few simulations have been performed in binary (liquid and gas) or ternary (liquid, gas and solid) systems.

Fig. 1. Inhomogeneous pressure on the substrate.

2

Numerical Method

We perform molecular dynamics simulations of shrinking nanobubbles in binary or ternary Lennard-Jones systems using GROMACS software to investigate the dynamics of nanobubbles. Liquid and gas particles are LennardJones particles modeled Ar and Ne respectively. Wall particles are also Lennard-Jones particles. In simulations, twotypes of wall are employed; the fixed wall consists of particles fixed to the FCC lattice positions and the thermal
wall consists of thermally vibrating particles confined tothe lattice positions by a harmonic potential. The simulations are performed under controlled temperature anddivided into two steps: (I) creation of bubbles and (II) simulations of shrinking bubbles. In the step (I), the system
is in a stretched state and nanobubbles are created andsustained stably. In the step (II), after the temperature israised, nanobubbles start to shrink and collapse in a shorttime. Note that in all the figures, t = 0 is set to the time
when the controlled temperature is raised.
33.1

Results and Discussion
shrinking nanobubbles in the bulk liquid oron the substrate

We performed MD simulations of shrinking nanobubbles
in the bulk liquid (system (A)) or on the substrate (system(B)) and compared the lifetimes of nanobubbles. In thesystem (A), 50688 liquid atoms and 4608 gas atoms are inperiodic cubic box 14.161nm on each side and gas atoms
are initially placed at the center of the simulation box. In
the system (B), the fixed wall of the thickness of 4.902nm
is attached to the cubic box and gas atoms are initiallyplaced near the wall.
In the steps (I), sustaining nanobubbles are created atthe temperature of 85K. In the steps (II), after the controlling temperature is suddenly raised to 104K, althoughfixed wall
thermal wall
0.5
S

V/V0

1

Rayleigh-Plesset
104k
108k
112k
116k

1

0.1

0-0.5

0.01
00.2

0.4

0.6

0.8
t/τ

1

1.2

0

1.4

500

1000
1500
time [ps]

2000

2500

Fig. 2. Time evolution of the bubble volume. Bold line iscalculated by Rayleigh-Plesset equation.

Fig. 3. Calculated anisotropy of motion of gas atoms innanobubbles.

both nanobubbles in the bulk liquid and on the substrate
begin to shrink rapidly, the lifetime of the nanobubble on
the substrate is more than 20 times longer than that in
the bulk liquid.
In the system (B), the inhomogeneity of the particle
density perpendicular to the wall was observed as schematically illustrated in Fig1. The low pressure near the bubble
extends the lifetime of the bubble and this is one cause of
relatively long lifetime of the nanobubble on the substrate.

mean that all the gas atoms are moving in directions perpendicular to the wall, parallel to the wall and in random
directions respectively. As shown in fig3, no anisotropic
motion of the gas particles indicating the Knudsen gas behavior is observed in the thermal wall case, same as in thefixed wall case.

3.2

collapse of nanobubbles in the bulk liquid

We have investigated the details of collapse of nanobubbles in the bulk liquid using molecular dynamics simulation of binary systems. We place 405504 liquid atoms and36864 gas atoms in the simulation box 28.322nm on eachside. In the step (I), the temperature of the system is controlled to be 85K and raised to various temperatures atthe beginning of the step (II).
Using the Rayleigh-Plesset equation, the volume of therapidly shrinking nanobubble at time t is4

Concluding remarks

In the simulations described in sec3.1 and sec3.2, thermostat is applied directly to liquid and gas atoms, which
can affect the dynamics of the gas in the liquid. To remove
the effect of thermostat, indirect temperature control isneeded.
In the simulations described in sec3.3, the thermal flux
is not sustained but transient. We have to introduce somemechanisms that drive the sustaining thermal flux. We
also have to reduce the density in the bubble, because thegas density in the simulated nanobubble was a little higher
than the condition proposed by Lohse et al.
Acknowledgements

V (t) = V0 (1 − t/τ )6/5 ,(1)

where V0 is the initial volume and τ is the lifetime of thebubble. As illustrated in fig2, the time evolution of bubble volume is well described by the equation(1), except
the beginning of the shrinking process which is influenced
by a sudden increase of the temperature. This is qualitatively consistent with the preceding simulations performedin single component systems[7].
3.3Knudsen gas behavior

Shrinking nanobubbles on a fixed wall and on a thermal
wall are compared in order to investigate the Knudsen gas
behavior of gas atoms in the bubble on the substrate. In
the fixed wall case, thermostat is applied to the liquid andgas atoms directly. In the thermal wall case, on the otherhand, thermostat is applied only to the wall atoms andthe temperature of the liquid and gas atoms is controlled
indirectly. In the thermal wall case, transient heat flux
from the wall to the liquid and gas particles exists for morethan 2ns, which may induce Knudsen gas behavior in thebubble.
The calculated anisotropy S = (3hcos2 θi − 1)/2 is described in fig3, where θ is the angle between the directionin which the gas atom moves and the upward direction
perpendicular to the wall. S = 1 , S = −1/2 and S = 0

Numerical computation in this work was partially carried out on SR16000 at YITP in Kyoto University.
References[1] X.H.Xhang et al., ”Nanobubbles at the Interface between
Water and Hydrophobic Solid”, Langmuir, 24, 4756, 2008[2] S.Ljunggren and J.C.Eriksson, ”The lifetime of a colloidsized gas bubble in water and the cause of the hydrophobicattraction”, Colloid Surf. A, 129-130, 151, 1997
[3] M.P.Brenner and D.Lohse, ”Dynamic Equilibrium Mechanism for Surface Nanobubble Stabilization”, Phys. Rev.
Lett., 101, 214505,[4] J.H.Weijs et al., ”Formation of Surface Nanobubbles andthe Universality of Their Contact Angles: A MolecularDynamics Approach”, 108, 104501, 2012[5] J.R.T.Seddon et al., ”Knudsen Gas Provides Nanobubble
Stability”, Phys. Rev. Lett., 107, 116101, 2011[6] K.Ohgaki et al., ”Physicochemical Approach to Nanobubble Solutions”, Chem. Eng. Sci., 65, 1296, 2010[7] R.Holyst et al., ”Large-scale molecular dynamics verification of the Rayleigh-Plesset approximation for collapse ofnanobubbles”, Phys. Rev. E, 82, 066309, 2010