Tribology Online, Vol. 4, No. 2 (2009) pp. 46-49

Tribology Online, 4, 2 (2009) 46-49. ISSN 1881-2198

DOI 10.2474/trol.4.46

Copyright © 2009 Japanese Society of Tribologists 46

Short Communication

Numerical Simulation for Meniscus Bridge Formation and Breakage

Kentaro Tanaka1)*, Fumihiko Asami2) and Katsumi Iwamoto1)

1)Department of Marine Electronics and Mechanical Engineering, Tokyo University of Marine Science and Technology 2-1-6 Etchujima, Koto-ku, Tokyo 135-8533, Japan

2)Graduate School of Marine Science and Technology, Tokyo University of Marine Science and Technology 2-1-6 Etchujima, Koto-ku, Tokyo 135-8533, Japan *Corresponding author: [email protected]

( Manuscript received 4 Nobember 2008; accepted 9 March 2009; published 16 April 2009 )

( Presented at JAST Tribology Conference Nagoya, September 2008 )

For mechanical devices in micro or nanometer scale, interactions between surfaces have become very important. When two surfaces approach each other, in the presence of meniscus bridges by liquid film on the surface or condensed water from humid air, strong attractive force arises. And, the force heavily influences the operation of micro/nano devices. In this study, particle based numerical simulation was carried out to investigate the dynamic behavior of the meniscus bridge. By using Moving Particle Semi-implicit (MPS) method, it is easy to identifying the large deformation of the liquid surface. The results obtained are compared with experiments. The dynamic process of the meniscus bridge formation and breakage with large deformation of the liquid surface is qualitatively simulated. Keywords: liquid bridge, meniscus, surface tension, wettability, numerical simulation, MPS method

1. Introduction

Recent advances in ultra-fine processing technologies are remarkable. Such as micro machine or magnetic storage device, devices become smaller and the performance is far advanced. It becomes important to understand physics of microscopic phenomena. For mechanical devices in micro or nanometer scale, forces acting on surface have become very important. For example, it is frictional force, capillary force and viscous force. In the presence of liquid film, droplet or condensed water from humid air on a surface, meniscus bridge can be formed between two bodies. The meniscus bridge causes a strong attractive force. And, the force heavily influences the operation of micro/nano devices1). Significant studies have been carried out to theoretically and experimentally investigate the meniscus behavior between two surfaces2-4). However several problems are still remained, such as the dynamics of the meniscus bridge. In this study, numerical simulation by using particle method, are carried out to investigate the dynamic process of meniscus bridge formation and breakage.

2. Simulation method

To treat the meniscus related phenomena, the

numerical method should take into account a large deformation of the liquid-gas interface, a wettability of the substrate surface. However conventional grid based numerical methods, e.g. finite element method, finite difference method, usually have some difficulties in dealing with phenomena with free surface, deformable boundary or moving interface. Particle methods are meshless methods in which motion of continua is simulated with a finite number of particles. By tracking the motion of particles, it is easy to identifying the large deformation of the liquid surface5-7). Therefore, we use one of the particle methods, moving particle semi-implicit (MPS) method.

2.1. MPS method Moving particle semi-implicit method was

developed by Koshizuka and Oka7). The MPS method tracks the motion of particles as a function to time. To treat the fluid dynamics, the governing equations of particles involve the differential operators, e.g. gradient and Laplacian. These operators are discretized by the following procedure.

A weight function, w(r) is introduced to represent the interaction between particles,

( )⎪⎩

⎪⎨⎧

<

≤−=rr

rrrr

rwe

ee

0

1 (1)


Japanese Society of Tribologists (http://www.tribology.jp/) Tribology Online, Vol. 4, No. 2 (2009) / 47

where r is the distance between particles, re is cutoff length of an interaction. The gradient operator ∇ is given as a weighted average of gradient vectors by using the weight function.

( )( ) ( )∑≠ ⎥

⎥

⎦

⎤

⎢⎢

⎣

⎡−−

−

−=∇

ijijj

ij

iji

rrwrrrrn

d20

φφφ

(2)

where d is the system dimensions, n0 is the particle number density, ri and rj is position of particle i and j. This means the gradient vector of a scalar quantities φ at the particle i. Laplacian, ∇2 operator, is modeled by distribution of a quantity from a particle to its adjacent particles by using the weight function.

( ) ( )[ ]∑≠

−−=∇ij

ijijirrw

nd φφ

λφ 0

2 2 (3)

To adjust the analytical solution of the statistical variance increase, parameter λ is adopted.

( ) ( )∑∑≠≠

−−−=ij

ijij

ijij rrwrrwrr2

λ

(4)

In MPS method, for simulating incompressible flows, the pressure term is implicitly calculated with the constant particle number density, while the other term are explicitly calculated. The detail of the model and algorithm can be seen from the references7).

2.2. Governing equation and contact angle model The governing equations for the meniscus bridge

formation is as follows,

0=DtDρ (5)

nguu σκδρηρ ++∇+−∇= 2PDtD (6)

where ρ is density, t is time, u is velocity, P is pressure, η is viscosity, g is gravitational acceleration, σ is surface tension coefficient, κ is curvature of the surface, δ is the delta function, n is the unit vector normal to the surface. First equation, (5) represents the mass conservation. And second, (6) represents the momentum conservation (Navier-Stokes equation).

We use the surface tension model proposed by Nomura et al8) to calculate forth term of (6). And, to characterize the surface of solid, the wettability model is included. Static contact angle model9) is adopted to our simulation. In this model, surface tension near the contact line are forced to change its direction to satisfy the Young’s equation.

3. Results and discussions

Figure 1 shows the two-dimensional simulation model for the meniscus bridge formation. At the

beginning, particles of liquid droplet are arranged in a rectangular shape on a base wall. And another wall is set above the droplet. Physical properties of liquid are set to propylene glycol(ρ = 1038 kg/m3, η/ρ = 3.89×10-5 m2/s, σ = 0.0355 N/m), and the contact angle is 42°. This value corresponds to the contact angle from the experimental snapshot of the droplet on the optically flat glass (BK7). Navier-Stokes equation, (6) is integrated using time step 1.0×10-6 sec.

A comparison of the meniscus shape obtained by present simulation and images obtained using a CCD camera is shown in Fig. 2. In the experiment, the base wall is an optically flat glass (BK7), and the upper wall is a hemispherical lens (BK7) with 103.8 mm in radius of curvature. The flame rate of the camera is 200 flame/s. Propylene glycol droplet is on the base wall with 25 μl. The cross sectional area of the droplet about equals to the area of the simulation model.

In the simulation, the upper wall approaches the base wall with a velocity of 0.03 m/s until the wall touches the droplet. And in the experiment, the upper wall approaches by a manually operated micrometer stage. After contact with the upper wall (0.000 sec), the top of the droplet starts to spread and form the liquid bridge. The formation process of bridge by the simulation agrees qualitatively with experimental snapshots. However the liquid bridge has an unnatural curve at the beginning of the formation. This result can be explained by the static contact angle model. This model keep the direction of surface tension near the contact line to satisfy a fixed contact angle. Consequently, the surface distant from the wall has to deform. But actually, in experiment, static contact angle from Young’s equation is valid in an equilibrium state. During formation process, the contact angle near the wall would be close to an advancing or receding contact angle. In addition, MPS method has some shortcomings such as non-conservation of momentum and spurious pressure fluctuation10). These seem to cause the fluctuation of liquid surface and the asymmetric shape of liquid bridge.

Figure 3 shows the breakage process of liquid bridge. The simulation result also shows qualitative agreement with experimental snapshots. The thickness of the neck gradually becomes thin with increasing the distance between walls (~ −0.010 sec). And then, the neck rapidly shrinks (−0.005 ~ 0.000 sec). Finally, the bridge breaks into two droplets.

Fig. 1 Simulation model of liquid droplet and solid walls

Kentaro Tanaka, Fumihiko Asami and Katsumi Iwamoto


4. Conclusions

Numerical simulations of the meniscus bridge formation and breakage were carried out with the static contact angle model. The dynamic process of the meniscus bridge formation with large deformation of the liquid surface is qualitatively simulated by MPS method. To get the quantitative agreement with experiment, some improvements of the contact angle model are necessary. It would be include dynamic effects of the contact angle.

5. References

[1] Binggeli, M. and Mate, C. M., “Influence of Capillary Condensation of Water on Nanotribology Studied by Force Microscopy,” Appl. Phys. Lett., 65, 4, 1994, 415-417.

[2] Squire, S. T., “Axisymmetric Meniscus Formation: a Viscous-Fluid Model for Cones,” J. Fluid Mech., 129, 1983, 91-108.

[3] Debregeas, G. and Brochard-Wyart, F., “Nucleation Radius and Growth of a Liquid Meniscus,” J. of Colloid and Interface Sci., 190, 1997, 134-141.

Fig. 2 Process of meniscus bridge formation

Fig. 3 Process of meniscus bridge breakage



[4] Matsuoka, H. , Fukui, S. and Kato, T., “A Study on Vibration Transfer Characteristics of Macroscopic Liquid Meniscus Bridge-Frequency Dependence of the Spring Constant and the Damping Coefficient,” Tribologist, 45, 10, 2000, 757-768 (in Japanese).

[5] Gingold, R. A. and Monaghan, J. J., “Smoothed Particle Hydrodynamics: Theory and Application to Non Spherical Stars,” Mon. Not. Roy. Astron. Soc., 181, 1977, 375-389.

[6] Liu, G. R. and Liu, M. B., “Smoothed Particle Hydrodynamics, a Meshfree Particle Method,” World Scientific, 2003.

[7] Koshizuka, S. and Oka, Y., “Moving-Particle Semi-Implicit Method for Fragmentation of Incompressible Fluid,” Nucl. Sci. Eng., 123, 1996, 421-434.

[8] Nomura, K., Koshizuka, S., Oka, Y. and Obata, H., “Numerical Analysis of Droplet Breakup Behavior using Particle Method,” J. Nucl. Sci. Technol., 38, 2001, 1057-1064.

[9] Xie, H., Koshizuka, S. and Oka, Y., “Modeling of Wetting Effects on the Liquid Droplet Impingement using Particle Method,” The53rd Nat. Cong. of Theoretical & Applied Mechanics, 2004, 367-368.

[10] Khayyer, A. and Gotoh, H., “Modified Moving Particle Semi-Implicit Methods for the Prediction of 2D Wave Impact Pressure,” Coastal Eng., (in press).

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Tribology Online, Vol. 4, No. 2 (2009) pp. 46-49