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A dynamic theory for contact angle hysteresis on chemically rough boundary

  • * Corresponding author: Xiao-Ping Wang

    * Corresponding author: Xiao-Ping Wang 
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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  • 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.

    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

    Figure 2.  Interface motion in a channel

    Figure 3.  The phase plane for the homogeneous boundary

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

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

    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.

    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

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

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    [3] A. DeSimoneN. Gruenewald and F. Otto, A new model for contact angle hysteresis, Networks and Heterogeneous Media, 2 (2007), 211-225.  doi: 10.3934/nhm.2007.2.211.
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    [6] T. QianX. Wang and P. Sheng, Power-law slip profile of the moving contact line in two-phase immiscible flows, Phys. Rev. Lett., 93 (2004), 094501.  doi: 10.1103/PhysRevLett.93.094501.
    [7] A. Turco, F. Alouges and A. DeSimone, Wetting on rough surfaces and contact angle hysteresis: Numerical experiments based on a phase field model, ESAIM: Mathematical Modelling and Numerical Analysis, 43 (2009), 1027-1044, http://www.esaim-m2an.org/article_S0764583X09000168.
    [8] X. WangT. Qian and P. Sheng, Moving contact line on chemically patterned surfaces, Journal of Fluid Mechanics, 605 (2008), 59-78.  doi: 10.1017/S0022112008001456.
    [9] X. Xu and X. P. Wang, Analysis of wetting and contact angle hysteresis on chemically patterned surfaces, SIAM J. Appl. Math., 71 (2011), 1753-1779.  doi: 10.1137/110829593.
    [10] T. Young, An essay on the cohesion of fluids, Philos. Trans. R. Soc. London, 95 (1805), 65-87. 
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