Langmuir 1997, 13, 7299-7300
7299
Notes A Very Simple Derivation of Young’s Law with Gravity Using a Cylindrical Meniscus Yves Larher SCM-DRECAM-DSM-CEA/Saclay, Baˆ t. 522, 91191 Gif-sur-Yvette Cedex, France Received July 14, 1997. In Final Form: September 30, 1997
The question whether Young’s law is rigorously valid within a gravitational field has been long debated. In a recent paper, Blokhuis, Shilkrot, and Widom,1 after a brief review of the controversy to which we refer the reader, have presented a decisive demonstration that the law is unaffected by gravity. For their proof they consider an axially symmetric meniscus. What we intend to do here is to present an alternative demonstration using a cylindrical meniscus. Clearly this does not bring any new physical insight but has the advantage of requiring much simpler mathematics, avoiding in particular the calculus of variations, which is not necessarily well-mastered by the colloid and surface scientists who are the prime users of this equation. Consequently, our contribution has mainly a pedagogical interest. However, it can be added that our demonstration is more complete, since we show that the extremum of the thermodynamic potential we are using is a minimum, which Blockhuis et al. do not: for an axially symmetric meniscus this would involve quite intricate mathematics. For a cylindrical meniscus the Laplace equation (see for instance ref 2) writes
sin(φ) dφ ∆Fgz ) σLV dz
(1)
z is the ordinate above the horizontal plane of the liquidvapor interface of surface tension σLV, ∆F ) F′ - F′′ is the density difference between these two phases (here and in the following the liquid and vapor will be designated respectively by the upper indexes ′ and ′′), g is the acceleration of free fall, φ is the angle of the tangent plane to the meniscus with the horizontal plane, defined as tan φ ) dz/dx, -π/2 e φ e π/2, x being the abscissa, for which the origin may remain arbitrary. Let us consider an open system of fixed volume (see Figure 1), for simplicity a rectangular parallelepiped limited at the right by the vertical solid surface in contact with the liquid-vapor interface. This solid surface is supposed to be rigid; let x2 be its abscissa. The other mathematical planes limiting the rectangular parallelepiped are a plane parallel to the first at the left whose abscissa we call x1, two horizontal planes well above or below the meniscus, and two planes parallel to the x and z axes and distant by the unit length. The equilibrium conditions for that system are obtained by minimizing the grand potential Ω at constant temperature and chemical potentials. The virtual change we consider for the minimization is a horizontal translation of the meniscus by a length δl (1) Blokhuis, E. M.; Shilkrot, Y.; Widom, B. Mol. Phys. 1995, 86, 891. (2) Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity; Clarendon Press: Oxford, U.K., 1982; Chapter 1.2.
S0743-7463(97)00786-5 CCC: $14.00
Figure 1. Normal section of the cylindrical meniscus. The vertical heavy line at the right represents the solid surface.
toward the left. This makes particularly easy the calculation of the change of surface area of the vapor-liquid interface. The corresponding vertical displacement is δz) δl tan(φ), so that the resulting variation of the surface contribution to Ω writes
[
[
δΩA ) [σSL - σSV] tan(φ2) + σLV
1 cos(φ2) 1 cos(φ1)
]]
δl (2)
σSL and σSV are respectively the surface density of the grand potential of the solid-liquid and solid-vapor interfaces (note that it is more usual to say free energy rather than grand potential, but this is correct only in the case of a one-component liquid). To establish the volume contribution to Ω, we consider an element of thickness dx, for which we can write, p and V being respectively the pressure and the volume,
δ(dΩV) ) -p′δ(dV′) - p′′δ(dV′′) Since δ(dV′) ) -δ(dV′′) ) dx δz and, according to hydrostatics, p′′ - p′ ) ∆Fgz, we will have
δ(dΩV) ) ∆Fgz tan(φ) dx δl This is the gravity contribution to energy. Taking Laplace law (eq 1) into account, we obtain
δ(dΩV) ) σLV sin(φ) dφ δl Integration over the volume yields
δΩV ) σLV[cos(φ1) - cos(φ2)] δl
(3)
Summing eqs 2 and 3, we get
δΩ ) [[σSL - σSV] tan(φ2) + σLV[sin(φ2) tan(φ2) sin(φ1) tan(φ1)]] δl (4) From this equation the condition for an extremum of Ω is © 1997 American Chemical Society
7300 Langmuir, Vol. 13, No. 26, 1997
Notes
Knowing additionally that eq 1 can be integrated to give z ) ax2 sin(φ/2), in which a ) x2σLV/g∆F is the capillary length, we easily get
tan(φ1) σSV - σSL ) σLV sin(φ2) - σLV sin(φ1) tan(φ2) Letting x1 f -∞ w φ1 f 0 yields
σSV - σSL ) σLV sin(φ2)
(5)
The contact angle θ being related to φ2 by the relation φ2 ) (π/2) - θ, eq 5 expresses Young’s law
σSV - σSL ) σLV cos(θ) We still have to show that this corresponds to a minimum of Ω, i.e., that δ2Ω > 0. To do this, we write eq 4 under a form in which the first partial derivatives with respect to z1 and z2 appear:
δΩ ) -σLV sin(φ1) δz1 + [σSL - σSV + σLV sin(φ2)] δz2
δ2Ω )
[ ()
( )
]
φ2 φ1 σLVx2 sin tan(φ2) - sin tan(φ1) δl2 a 2 2
which is positive, sin (φ/2) tan(φ) is an increasing function of |φ| and |φ2| > |φ1|. When we had finished the writing of this note, an additional bibliographic search led us to find out that the subject we were dealing with had already been solved by McNutt and Andes3 almost 40 years ago. However, we feel our contribution still keeps its pedagogical value, since it is much shorter and it involves definitely simpler mathematics. LA970786N (3) McNutt, J. E.; Andes, J. M. J. Chem. Phys. 1959, 30, 597.
