Skip to main content

Advertisement

SpringerLink
Log in

On the applicability range of Cassie–Baxter and Wenzel equation: a numerical study

  • Technical Paper
  • Published:
Journal of the Brazilian Society of Mechanical Sciences and Engineering Aims and scope Submit manuscript

Abstract

In this study, the range of applicability for Cassie–Baxter and Wenzel equations for estimating apparent contact angle on rough surfaces is numerically discussed. To do this, circular drops with different sizes are simulated on rough surfaces with a square pillar pattern and randomly distributed cylindrical pillar. With the aid of numerical method, the local surface fraction, local length fraction and local roughness factor for drops with different sizes on the surface are computed. Then, the global surface fraction and global roughness factor have been compared with the local surface fraction and local roughness factor, respectively. Local surface and local length fractions, as well as local roughness factor, behave oscillatory. It has been found that when drop radius is nearly 3 times greater than summation of pillar width and pillar edge-to-edge separation distance, the contact angles calculated through involving local and global surface fractions in Cassie–Baxter equation for the case of square pillar pattern, provided differences lower than ± 1° which is in agreement with the results of Marmur and Bittoun (Langmuir 25(3):1277–1281, 2009). The similar differences limit for contact angles calculated from local and global roughness factors are observed for square pillar pattern at Wenzel state when drop radius is nearly two times greater than the summation of pillar width, pillar edge-to-edge separation distance, and pillar height. Also, when drop size is almost 1000 times greater than the summation of pillar width and pillar edge-to-edge separation distance, the contact angle calculated through involving local surface and length fractions in Cassie–Baxter equation are different by ± 1°. Results of this work determine the minimum drop radius after which, different forms of Cassie–Baxter, as well as Wenzel equations, tend to the answer of their general equations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

Abbreviations

LLF:

Local length fraction

LSF:

Local surface fraction

GSF:

Global surface fraction

CA:

Contact angle

RF:

Roughness factor

LRF:

Local roughness factor

GRF:

Global roughness factor

A :

Pillar width (µm)

B :

Pillar edge-to-edge separation distance (µm)

C :

Indicator function of drop base area (–)

C′:

Indicator function of drop apparent surface area (–)

E A :

Error function of area (–)

E P :

Error function of perimeter (–)

F :

Local surface fraction (–)

f g :

Global surface fraction (–)

\(f^{*}\) :

Length fraction (–)

K :

Maximum cylindrical pillar diameter (µm)

L :

Length of imaginary cell (µm)

m :

Number of grids in x direction (–)

n :

Square root of pillar number (–)

P :

The pillar indicator function (–)

Q :

The number of faces of each block in the vicinity of the drop (–)

R :

Drop radius (µm)

R′:

The ratio of actual surface area to apparent surface area (–)

R max :

The minimum drop radius after which |θsf − θlf| < 1 (µm)

R :

Roughness factor (–)

r l :

Local roughness factor (–)

r g :

Global roughness factor (–)

S C :

The pillar indicator function inside the drop (–)

\(S_{C}^{\prime }\) :

The indicator function of actual surface area (–)

S p :

Indicator function of pillar under drop perimeter (–)

ω :

The maximum number of faces in access to drop (–)

x c, y c :

Drop center (µm)

x, y :

Cartesian coordinates (µm)

(x p, y p):

The pillar center (µm)

Z :

Phase function of surface in Cassie–Baxter model (–)

Z′:

Phase function of surface in Wenzel model (–)

θ :

Contact angle (°)

θ 0 :

Contact angle of smooth surface in the Wenzel model (°)

θ h :

Apparent contact angle (°)

θ s :

Apparent contact angle of solid fraction of surface in Cassie–Baxter equation (°)

θ sf :

Apparent contact angle through involving local surface fraction in Cassie–Baxter equation (°)

θ lf :

Apparent contact angle through involving local length fraction in Cassie–Baxter equation (°)

θ g :

Apparent contact angle through involving global surface fraction or global roughness factor in Cassie–Baxter or Wenzel equation, respectively (°)

θ l :

Apparent contact angle through involving local roughness factor in Wenzel equation (°)

References

  1. Amirfazli A, Kwok D, Gaydos J, Neumann A (1998) Line tension measurements through drop size dependence of contact angle. J Colloid Interface Sci 205(1):1–11

    Article  Google Scholar 

  2. Azadi Tabar M, Ghazanfari MH (2019) Experimental study of surfapore nanofluid effect on surface free energy of calcite rock. Modares Mech Eng 19(3):709–718

    Google Scholar 

  3. Azadi Tabar M, Ghazanfari MH, Monfared AD (2019) On the size-dependent behavior of drop contact angle in wettability alteration of reservoir rocks to preferentially gas wetting using nanofluid. J Petrol Sci Eng 178:1143–1154

    Article  Google Scholar 

  4. Barbosa TP, Naves MM, Menezes HHM, Pinto PHC, de Mello JDB, Costa HL (2017) Topography and surface energy of dental implants: a methodological approach. J Braz Soc Mech Sci Eng 39(6):1895–1907

    Article  Google Scholar 

  5. Bartell F, Shepard J (1953) The effect of surface roughness on apparent contact angles and on contact angle hysteresis. I. The system paraffin–water–air. J Phys Chem 57(2):211–215

    Article  Google Scholar 

  6. Bartell F, Shepard J (1953) Surface roughness as related to hysteresis of contact angles. II. The systems paraffin–3 molar calcium chloride solution–air and paraffin–glycerol–air. J Phys Chem 57(4):455–458

    Article  Google Scholar 

  7. Bormashenko E (2008) Why does the Cassie–Baxter equation apply? Colloids Surf A Physiochem Eng Aspects 324(1–3):47–50

    Article  Google Scholar 

  8. Brandon S, Wachs A, Marmur A (1997) Simulated contact angle hysteresis of a three-dimensional drop on a chemically heterogeneous surface: a numerical example. J Colloid Interface Sci 191(1):110–116

    Article  Google Scholar 

  9. Cansoy CE (2014) The effect of drop size on contact angle measurements of superhydrophobic surfaces. RSC Adv 4(3):1197–1203

    Article  Google Scholar 

  10. Chen S-Y, Kaufman Y, Schrader AM, Seo D, Lee DW, Page SH, Israelachvili JN (2017) Contact angle and adhesion dynamics and hysteresis on molecularly smooth chemically homogeneous surfaces. Langmuir 33(38):10041–10050

    Article  Google Scholar 

  11. Dehghan Monfared A, Ghazanfari MH, Jamialahmadi M, Helalizadeh A (2016) Potential application of silica nanoparticles for wettability alteration of oil-wet calcite: a mechanistic study. Energy Fuels 30(5):3947–3961

    Article  Google Scholar 

  12. Drelich J, Miller JD (1993) Modification of the Cassie equation. Langmuir 9(2):619–621

    Article  Google Scholar 

  13. Drelich JW (2019) Contact angles: From past mistakes to new developments through liquid-solid adhesion measurements. Adv Colloid Interface Sci. https://doi.org/10.1016/j.cis.2019.02.002

    Article  Google Scholar 

  14. Erbil HY (2014) The debate on the dependence of apparent contact angles on drop contact area or three-phase contact line: a review. Surf Sci Rep 69(4):325–365

    Article  Google Scholar 

  15. Erbil HY, Cansoy CE (2009) Range of applicability of the Wenzel and Cassie − Baxter equations for superhydrophobic surfaces. Langmuir 25(24):14135–14145

    Article  Google Scholar 

  16. Erfani Gahrooei HR, Ghazanfari MH, Karimi Malekabadi F (2017) Wettability alteration of reservoir rocks to gas wetting condition: a comparative study. Can J Chem Eng 96(4):997–1004

    Article  Google Scholar 

  17. Extrand C (2003) Contact angles and hysteresis on surfaces with chemically heterogeneous islands. Langmuir 19(9):3793–3796

    Article  Google Scholar 

  18. Gahrooei HRE, Ghazanfari MH (2017) Application of a water based nanofluid for wettability alteration of sandstone reservoir rocks to preferentially gas wetting condition. J Mol Liq 232:351–360

    Article  Google Scholar 

  19. Gahrooei HRE, Ghazanfari MH (2017) Toward a hydrocarbon-based chemical for wettability alteration of reservoir rocks to gas wetting condition: implications to gas condensate reservoirs. J Mol Liq 248:100–111

    Article  Google Scholar 

  20. Gao L, McCarthy TJ (2007) How Wenzel and Cassie were wrong. Langmuir 23(7):3762–3765

    Article  Google Scholar 

  21. Heitich L, Passos J, Cardoso E, da Silva M, Klein A (2014) Nucleate boiling of water using nanostructured surfaces. J Braz Soc Mech Sci Eng 36(1):181–192

    Article  Google Scholar 

  22. Li W, Amirfazli A (2005) A thermodynamic approach for determining the contact angle hysteresis for superhydrophobic surfaces. J Colloid Interface Sci 292(1):195–201. https://doi.org/10.1016/j.jcis.2005.05.062

    Article  Google Scholar 

  23. Li W, Amirfazli A (2007) Microtextured superhydrophobic surfaces: a thermodynamic analysis. Adv Colloid Interface Sci 132(2):51–68. https://doi.org/10.1016/j.cis.2007.01.001

    Article  Google Scholar 

  24. Li X-M, Reinhoudt D, Crego-Calama M (2007) What do we need for a superhydrophobic surface? A review on the recent progress in the preparation of superhydrophobic surfaces. Chem Soc Rev 36(8):1350–1368

    Article  Google Scholar 

  25. Marmur A (1994) Contact angle hysteresis on heterogeneous smooth surfaces. J Colloid Interface Sci 168(1):40–46

    Article  Google Scholar 

  26. Marmur A (2003) Wetting on hydrophobic rough surfaces: to be heterogeneous or not to be? Langmuir 19(20):8343–8348

    Article  Google Scholar 

  27. Marmur A, Bittoun E (2009) When Wenzel and Cassie are right: reconciling local and global considerations. Langmuir 25(3):1277–1281

    Article  Google Scholar 

  28. McHale G (2007) Cassie and Wenzel: were they really so wrong? Langmuir 23(15):8200–8205

    Article  Google Scholar 

  29. Milne A, Amirfazli A (2012) The Cassie equation: how it is meant to be used. Adv Coll Interface Sci 170(1):48–55

    Article  Google Scholar 

  30. Nosonovsky M (2007) Multiscale roughness and stability of superhydrophobic biomimetic interfaces. Langmuir 23(6):3157–3161

    Article  Google Scholar 

  31. Nosonovsky M (2007) On the range of applicability of the Wenzel and Cassie equations. Langmuir 23(19):9919–9920

    Article  Google Scholar 

  32. Nosonovsky M, Bhushan B (2008) Patterned nonadhesive surfaces: superhydrophobicity and wetting regime transitions. Langmuir 24(4):1525–1533

    Article  Google Scholar 

  33. Sun Y, Jiang Y, Choi C-H, Xie G, Liu Q, Drelich JW (2018) The most stable state of a droplet on anisotropic patterns: support for a missing link. Surf Innov 6(3):133–140

    Article  Google Scholar 

  34. Taghipour-Gorjikolaie M, Motlagh NV (2018) Predicting wettability behavior of fluorosilica coated metal surface using optimum neural network. Surf Sci 668:47–53

    Article  Google Scholar 

  35. Wenzel RN (1936) Resistance of solid surfaces to wetting by water. Ind Eng Chem 28(8):988–994

    Article  Google Scholar 

  36. Wolansky G, Marmur A (1999) Apparent contact angles on rough surfaces: the Wenzel equation revisited. Colloids Surf A Physiochem Eng Aspects 156(1–3):381–388

    Article  Google Scholar 

  37. Wu J, Xia J, Lei W, Wang B-P (2013) Advanced understanding of stickiness on superhydrophobic surfaces. Sci Rep 3:3268

    Article  Google Scholar 

  38. Yarom M, Marmur A (2015) Condensation enhancement by surface porosity: three-stage mechanism. Langmuir 31(32):8852–8855

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Chemical and Petroleum Engineering, Sharif University of Technology, Tehran, Iran

    Mohammad Azadi Tabar, Farzad Barzegar, Mohammad Hossein Ghazanfari & Mohammad Mohammadi

Authors
  1. Mohammad Azadi Tabar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Farzad Barzegar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mohammad Hossein Ghazanfari
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Mohammad Mohammadi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohammad Azadi Tabar.

Additional information

Technical Editor: Jader Barbosa Jr.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Azadi Tabar, M., Barzegar, F., Ghazanfari, M.H. et al. On the applicability range of Cassie–Baxter and Wenzel equation: a numerical study. J Braz. Soc. Mech. Sci. Eng. 41, 399 (2019). https://doi.org/10.1007/s40430-019-1908-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40430-019-1908-3

Keywords

Search

Navigation