# 接觸角

## 热力学

${\displaystyle \gamma _{\mathrm {SG} }-\gamma _{\mathrm {SL} }-\gamma _{\mathrm {LG} }\cos \theta _{\mathrm {C} }=0\,}$

${\displaystyle \gamma _{\mathrm {LG} }(1+\cos \theta _{\mathrm {C} })=\Delta W_{\mathrm {SLG} }\,}$

### 修改的杨氏方程

${\displaystyle \cos(\theta )={\frac {\gamma _{SV}-\gamma _{SL}}{\gamma _{LV}}}+{\frac {\kappa }{\gamma _{LV}}}{\frac {1}{a}}}$

### 在考虑线张力和拉普拉斯压力的同时预测接触角

${\displaystyle 0={\frac {dA_{LV}}{dA_{SL}}}+{\frac {\gamma _{SL}-\gamma _{SV}}{\gamma _{LV}}}-{\frac {\kappa }{\gamma _{LV}}}{\frac {dL}{dA_{SL}}}-{\frac {V}{\gamma _{LV}}}{\frac {dP}{dA_{SL}}}}$

${\displaystyle \cos(\theta \mp \alpha )=A+B{\frac {\cos(\alpha )}{a}}\pm C\sin(\theta \mp \alpha )(\cos(\theta )+1)^{2}{\biggl (}{\frac {\sin(\alpha )(\cos(\alpha )+2)}{(\cos(\alpha )+1)^{2}}}\mp {\frac {\sin(\theta )(\cos(\theta )+2)}{(\cos(\theta )+1)^{2}}}{\biggr )}}$

${\displaystyle \cos(\theta )={\frac {\gamma _{SV}-\gamma _{SL}}{\gamma _{LV}}}+{\frac {\kappa }{\gamma _{LV}}}{\frac {1}{a}}-{\frac {\gamma }{3\gamma _{LV}}}(2+\cos(\theta )-2\cos ^{2}(\theta )-\cos ^{3}(\theta ))}$

### 接触角滞后

${\displaystyle \theta _{\mathrm {c} }=\arccos \left({\frac {r_{\mathrm {A} }\cos \theta _{\mathrm {A} }+r_{\mathrm {R} }\cos \theta _{\mathrm {R} }}{r_{\mathrm {A} }+r_{\mathrm {R} }}}\right)}$

${\displaystyle r_{\mathrm {A} }=\left({\frac {\sin ^{3}\theta _{\mathrm {A} }}{2-3\cos \theta _{\mathrm {A} }+\cos ^{3}\theta _{\mathrm {A} }}}\right)^{1/3}~;\qquad r_{\mathrm {R} }=\left({\frac {\sin ^{3}\theta _{\mathrm {R} }}{2-3\cos \theta _{\mathrm {R} }+\cos ^{3}\theta _{\mathrm {R} }}}\right)^{1/3}}$

### 粗糙度对接触角的影响

${\displaystyle \cos(\theta _{m})=r\cos(\theta _{Y})}$

## 接触角曲率

${\displaystyle \kappa _{m}={\frac {1}{2}}{\frac {(1+{f_{x}}^{2})f_{yy}-2f_{x}f_{y}f_{xy}+(1+{f_{y}}^{2})f_{xx}}{(1+{f_{x}}^{2}+{f_{y}}^{2})^{3/2}}}.}$

## 參閱

1. III. An essay on the cohesion of fluids. Philosophical Transactions of the Royal Society of London. January 1805, 95: 65–87. ISSN 0261-0523. doi:10.1098/rstl.1805.0005.
2. ^ Gibbs, J. Willard (Josiah Willard). Scientific papers.. Dover Publications. 1961. ISBN 978-0486607214. OCLC 964884.
3. Jasper, Warren J.; Anand, Nadish. A generalized variational approach for predicting contact angles of sessile nano-droplets on both flat and curved surfaces. Journal of Molecular Liquids. May 2019, 281: 196–203. ISSN 0167-7322. doi:10.1016/j.molliq.2019.02.039.
4. ^ Jasper, Warren J.; Rasipuram, Srinivasan. Relationship between contact angle and contact line radius for micro to atto [10−6 to 10−18] liter size oil droplets. Journal of Molecular Liquids. December 2017, 248: 920–926. ISSN 0167-7322. doi:10.1016/j.molliq.2017.10.134.
5. ^ Hattori, Tsuyoshi; Koshizuka, Seiichi. Numerical simulation of droplet behavior on an inclined plate using the Moving Particle Semi-implicit method. Mechanical Engineering Journal. 2019, 6 (5): 19–00204–19–00204. ISSN 2187-9745. .
6. ^ Tadmor, Rafael. Line energy and the relation between advancing, receding, and Young contact angles. Langmuir. 2004, 20 (18): 7659–64. PMID 15323516. doi:10.1021/la049410h.
7. ^ Chibowski, Emil. Surface free energy of sulfur—Revisited I. Yellow and orange samples solidified against glass surface. Journal of Colloid and Interface Science. 2008, 319 (2): 505–13. Bibcode:2008JCIS..319..505C. PMID 18177886. doi:10.1016/j.jcis.2007.10.059.
8. ^ de Gennes, P.G. Wetting: statics and dynamics. Reviews of Modern Physics. 1985, 57 (3): 827–863. Bibcode:1985RvMP...57..827D. doi:10.1103/RevModPhys.57.827.
9. ^ Influence of surface roughness on contact angle and wettability (PDF). [2021-06-08]. （原始内容存档 (PDF)于2017-11-07）.
10. ^ Wenzel, Robert N. Resistance of Solid Surfaces to Wetting by Water. Industrial & Engineering Chemistry. 1936-08-01, 28 (8): 988–994. ISSN 0019-7866. doi:10.1021/ie50320a024.
11. ^ Cassie, A. B. D.; Baxter, S. Wettability of porous surfaces. Transactions of the Faraday Society. 1944-01-01, 40: 546. ISSN 0014-7672. doi:10.1039/tf9444000546.
12. ^ Marmur, Abraham. Solid-Surface Characterization by Wetting. Annual Review of Materials Research. 2009-07-06, 39 (1): 473–489. Bibcode:2009AnRMS..39..473M. ISSN 1531-7331. doi:10.1146/annurev.matsci.38.060407.132425.
13. Shi, Z.; et al. Dynamic contact angle hysteresis in liquid bridges. Colloids and Surfaces A: Physicochemical and Engineering Aspects. 2018, 555: 365–371. . doi:10.1016/j.colsurfa.2018.07.004.
14. ^ Chen Y, He B, Lee J, Patankar NA. Anisotropy in the wetting of rough surfaces (PDF). Journal of Colloid and Interface Science. 2005, 281 (2): 458–464 [2017-03-31]. Bibcode:2005JCIS..281..458C. PMID 15571703. doi:10.1016/j.jcis.2004.07.038. （原始内容 (PDF)存档于2017-08-10）.
15. ^ Kobrin, B.; Zhang, T.; Chinn, J. Choice of precursors in Vapor-phase Surface Modification. 209th Electrochemical Society meeting, May 7–12, 2006, Denver, CO.