A dynamic theory for contact angle hysteresis on chemically rough boundary

Xiao-Ping Wang; Xianmin Xu

doi:10.3934/dcds.2017044

Article Contents

2017, Volume 37, Issue 2: 1061-1073. Doi: 10.3934/dcds.2017044

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A dynamic theory for contact angle hysteresis on chemically rough boundary

Xiao-Ping Wang^1, , and
Xianmin Xu^2,

1.
Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China
2.
LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, NCMIS, Chinese Academy of Sciences, Beijing 100190, China

* Corresponding author: Xiao-Ping Wang
* Corresponding author: Xiao-Ping Wang

Received: July 31, 2015

Revised: March 31, 2016

Published: May 2017

This publication was supported in part by the Hong Kong RGC-GRF grants 605513 and 605311, Hong Kong RGC-CRF grant C6004-14G and NSFC grant 11571354.

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Abstract

We study the interface dynamics and contact angle hysteresis in a two dimensional, chemically patterned channel described by the Cahn-Hilliard equation with a relaxation boundary condition. A system for the dynamics of the contact angle and contact point is derived in the sharp interface limit. We then analyze the behavior of the solution using the phase plane analysis. We observe the stick-slip of the contact point and the contact angle hysteresis. As the size of the pattern decreases to zero, the stick-slip becomes weaker but the hysteresis becomes stronger in the sense that one observes either the advancing contact angle or the receding contact angle without any switching in between. Numerical examples are presented to verify our analysis.

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Mathematics Subject Classification: Primary:41A60, 34A36;Secondary:76T10.

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Figure 1. Contact angle formed by the liquid-air interface with the solid boundary

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Figure 2. Interface motion in a channel

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Figure 3. The phase plane for the homogeneous boundary

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Figure 4. The phase plane and the attracting region bounded by the nullclines.

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Figure 5. The advancing contact angle trajectory for large (left) and small (right) pattern period $T$.

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Figure 6. Advancing and receding contact angles for large ($T=2\pi/5$) and small ($T=2\pi/40$) pattern period. For large $T$, we observe stick-slip of the contact point (see Fig. 7). For small $T$, we observe contact angle hysteresis.

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Figure 7. The contact point stick-slip behaviour for pattern surface with period $T=2\pi/5$. Top: advancing trajectory of contact angle $\theta$. Middle: contact point velocity $x_t$. Below: $x_t$ in log scale

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Figure 8. Advancing and receding contact angles for inhomogeneous surfaces with smooth $\theta_Y$ with large and small patterned period.

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