Theory and simulation of angular hysteresis in sessile drops

Theory and simulation of angular hysteresis in sessile

dropsM.J. Santos and J.A. WhiteDepartamento de Fısica Aplicada,

Facultad de Ciencias. Universidad de Salamanca, Spainhttp://campus.usal.es/gtfe

smjesus@usal.es, white@usal.es

1. Abstract

This work tries to reproduce the experimental results obtained by C.N.N. Lam et al.[1] in a hysteresis cycle of a water droplet deposited on poly(latic acid)-coated silicon wafer. Two methods are used: onthe one hand the solution of the Young-Laplace equation [2] in cylindrical coordinates and on the other hand a simulation with the Surface Evolver[3] software. In both cases, a friction term is introducedthat can adequately describe the contact angle hysteresis.

2. Introduction

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3. Method

3.1 Axisymmetric geometryIn cylindrical symmetry the Young-Laplace equation [2] can berewritten in the simple form:

1ρ(z)

√1+ρ ′(z)2

− ρ ′′(z)

(1+ρ ′(z)2)3/2 =2r0+

zl20, (1)

where (ρ,ϕ,z) are the cylindrical coordinates, being ρ =√x2+ y2 the radial coordinate, r0 is the curvature radius at the

apex 1r0≡ ∆P0

2σ, l0 =

√γ

ρg the liquid capillary length, γ the surfacetension, ρ the density and g the gravitational constant.Equation (1) can be straightforwardly solved for a given dropvolume V allowing to determine the contact angle θ as a func-tion of the contact radius R and viceversa. Recall that this isthe method used in the Axisymmetric Drop Size Analysis of thecontact Diameter (ADSA-D).In order to obtain a hysteresis cycle, the drop volume is in-creased in small increments from Vi to Vf and then decreasedat the same rate.

Determine contact angle θ

Initial situation

Keep contact radius R fixed θ r< θ < θa ?

Maintain contact angle θ=θa

Calculate R

Yes

No

No

Change volume V

θ=θa?

Maintain contact angle θ=θr

Yes

Calculate θ

Figure 1: Flowchart of the algorithm used with Young-Laplace.

Figure 1 shows a flowchart of the algorithm used to obtain ahysteresis cycle of a liquid drop, solving the Young-Laplaceequation with the help of MathematicaT M.

3.2 Nonaxisymmetric geometrySurface Evolver [3] is a software designed to model surfacessubject to different forces or constraints. Due to its character-istics Surface Evolver is an ideal tool for the analysis of ses-sile drops from its interfaces. These are described througha triangulation (fig. 2) that can be modified (refined) and ad-justed during the minimization of the energy of the system.

Figure 2: Shape of a drop during the process of change in volume.

Figure 3 shows a flowchart of the algorithm used to obtain ahysteresis cycle of a liquid drop, introducing a friction term inSurface Evolver.

Trial move of the drop: = ,

Store current values of the triple-line vertices coordinates

, =

Calculate forces acting on triple-line vertices

Advancing vertex ? Calculate

max , = ( )

Calculate max , = ( )

< max ?

Yes

No

No

Yes Restore old vertex position = ,

Allow vertex to move

Iterate

{

Figure 3: Flowchart of the algorithm used in Surface Evolver.

This method is valid both for axisymmetric and nonaxisymmet-ric drops.

4. Results

Figure 4 shows the results for the contact radius, ρ ,and the contact angle θ , as a function of time. Italso shows the variation of volume V with time.

0 100 200 300 400 500

60

65

70

75

80

θ(de

gree

s)

t

0 100 200 300 400 500

0.30

0.35

0.40

0.45

t

ρ(cm

)

0 100 200 300 400 500

0.04

0.06

0.08

0.10

0.12

t

V(c

m³)

Figure 4: Hysteresis cycle of a water droplet deposited on poly(laticacid)-coated silicon wafer. Comparison of the experimental results of Lamet al. (blue dots) with those obtained from the Young-Laplace equation (redcontinuous line) and simulation with Surface Evolver (black dashed line).

Comparison of our theory and simulation results with experi-mental data of Lam et al. yields excellent agreement.

Figure 5: Shape of a drop on a tilted plate.

Figure (5) shows a drop on a titled plate where one can appre-ciate the absence of cylindrical symmetry and the appearanceof a distribution of contact angles.

5. Summary

1. For axisymmetric drops the Young-Laplace equation hasbeen solved in cylindrical coordinates to reproduce a hys-teresis cycle of a liquid drop on a surface.

2. For nonaxisymmetric drops the Young-Laplace differentialequations becomes more complex. Therefore we have usedthe Surface Evolver software.

3. Contact angle hysteresis has been implemented in SurfaceEvolver through the algorithm of figure (3).

References

1. C.N.C. Lam et al. Advances in Colloid and Interference Science 96, 169(2002).

2. P. S. Laplace, Traite de mecanique celeste, Gauthier-Villars, Paris. 4,Supplement to Book 10, (1806)

3. K.A. Brakke, Exp. Math. 1(2), 141 (1992).

4. T. Young, Philos. Trans. R. Soc. London, 95, 65 (1805).

We thank financial support by Ministerio de Educacion y Ciencia of Spain

under Grant FIS2009-07557.