Measurement of the liquid/solid contact angle provides useful information on the wetting properties of fluids. In 1870, the German physicist Georg Hermann Quincke (1834–1924) published the functional relation between the maximum height of an enlarged drop and its contact angle. Quincke’s relation offered an alternative to the direct measurement of contact angle, which in practice suffers from several experimental uncertainties. In this paper, we review Quincke’s original derivation and show that it is based on a hidden assumption. We then present a new derivation that exposes this assumption and clarifies the conditions under which Quincke’s relation is valid. To explore Quincke’s relation experimentally, we measure the maximum height of enlarged water drops on several substrates and calculate the contact angle in each case. Our results are in good agreement with contact angles measured directly from droplet images.
REFERENCES
1.
F.
Behroozi
, K.
Macomber
, J. A.
Dostal
, C. H.
Behroozi
, and B.
Lambert
, “The Profile of a Dew Drop
,” Am. J. Phys.
64
, 1120
–1125
(1996
). 2.
E. L.
Decker
, B.
Frank
, Y.
Suo
, and S.
Garoff
, “Physics of contact angle measurement
,” Colloids Surf., A
156
, 177
–189
(1999
). 3.
D. E.
Sullivan
, “Surface tension and contact angle of a liquid-solid interface
,” J. Chem. Phys.
74
, 2604
–2615
(1981
). 4.
R. J.
Good
, “Contact angle, wetting, and adhesion: a critical review
,” J. Adhes. Sci. Technol.
6
, 1269
–1302
(1992
). 5.
B. J.
Ryan
and K. M.
Poduska
, “Roughness effects on contact angle measurements
,” Am. J. Phys.
76
, 1074
–1077
(2008
). 6.
J.
Drelich
and J. D.
Miller
, “The effect of solid surface heterogeneity and roughness on the contact angle/drop (bubble) size relationship
,” J. Colloid Interface Sci.
164
, 252
–259
(1994
). 7.
For a short review of the experimental techniques for measuring contact angle see
P. G.
de Gennes
, F.
Brochard-Wyart
, and D.
Quere
, Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves
(Springer
, New York
, 2004
), pp. 38
–43
. 8.
Y.
Rotenberg
, L.
Boruvka
, and A. W.
Neumann
, “Determination of surface tension and contact angle from the shapes of axisymmetric fluid interfaces
,” J. Colloid Interface Sci.
93
, 169
–183
(1983
). 9.
D. Y.
Kwok
, T.
Gietzelt
, K.
Grundke
, H. J.
Jacobasch
, and A. W.
Neumann
, “Contact angle measurements and contact angle interpretation. 1. contact angle measurements by axisymmetric drop shape analysis and a goniometer sessile drop technique
,” Langmuir
13
, 2880
–2894
(1997
). 10.
D. Y.
Kwok
, C. N. C.
Lam
, A.
Li
, A.
Leung
, R.
Wu
, E.
Mok
, and A. W.
Neumann
, “Measuring and interpreting contact angles: a complex issue
,” Colloids Surf., A
142
, 219
–235
(1998
). 11.
D. Y.
Kwok
and A. W.
Neumann
, “Contact angle measurement and contact angle interpretation
,” Adv. Colloid Interface Sci.
81
, 167
–249
(1999
). 12.
S.
Srinivasan
, G. H.
McKinley
, and R. E.
Cohen
, “Assessing the accuracy of contact angle measurements for sessile drops on liquid-repellent surfaces
,” Langmuir
27
, 13582
–13589
(2011
). 13.
W. A.
Zisman
, “Relation of the equilibrium contact angle to liquid and solid constitution
,” Adv. Chem.
43
, 1
–51
(1964
). 14.
G. H.
Quincke
, “Ueber Capillaritats-Erscheinungen an der gemeinschaftlichen Oberflche von Flssigkeiten
,” Ann. Phys. Chem.
139
, 1
–89
(1870
). 15.
W. F.
Magie
, “The contact angle of liquids and solids
,” Philos. Mag.
26
, 162
–183
(1888
). 16.
J. H.
Poynting
and J. J.
Thomson
, “A University Text-book of Physics
,” in Properties of Matter
, 14th ed. (Revised by G.W.
Todd
) (Griffin
, London
, 1947
), Vol. 1
, pp. 191
–193
. 17.
P. G.
de Gennes
, F.
Brochard-Wyart
, and D.
Quere
, Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves
(Springer
, New York
, 2004
), pp. 35
–37
. 18.
P. G.
de Gennes
, F.
Brochard-Wyart
, and D.
Quere
, Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves
(Springer
, New York
, 2004
), p. 17
. 19.
H. S. M.
Coxeter
, Introduction to Geometry
, 2nd ed. (John Wiley & Sons
, New York
, 1969
), pp. 352
–353
. 20.
F.
Behroozi
and P. S.
Behroozi
, “Determination of surface tension from the measurement of internal pressure of mini soap bubbles
,” Am. J. Phys.
79
, 1089
–1093
(2011
). 21.
T.
Lpez-Arias
, L. M.
Gratton
, and S.
Oss
, “High precision pressure measurement with a funnel
,” Eur. J. Phys.
29
, 1235
–1241
(2008
). 22.
F.
Behroozi
, “Surface tension in soap films: revisiting a classic demonstration
,” Eur. J. Phys.
31
, L31
–L35
(2010
). 23.
L.
Gao
and T. J.
McCarthy
, “Contact angle hysteresis explained
,” Langmuir
22
, 6234
–6237
(2006
). 24.
C. W.
Extrand
and S. I.
Moon
, “Contact angles of liquid drops on super hydrophobic surfaces: Understanding the role of flattening of drops by gravity
,” Langmuir
26
, 17090
–17099
(2010
). 25.
T.
Cubaud
, M.
Fermigier
, and P.
Jenffer
, “Spreading of large drops on patterned surfaces
,” Oil Gas Sci. Technol.
56
, 23
–31
(2001
). 26.
L.
Gao
and T. J.
McCarthy
, “Wetting 101°
,” Langmuir
25
, 14105
–14115
(2009
). © 2012 American Association of Physics Teachers.
2012
American Association of Physics Teachers
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.
