H.2 Euler-Maclaur in Sum Formula
13.3 Interface Junctions and Contact Angles
In this section we investigate briefly the mechanical conditions that must be satisfied at the junctions where several fluid phases meet. We begin by considering the two- dimensional problem of a triple junction where three phases,α,β, andηmeet along a line, as illustrated inFigure 13–4. The line where the phases meet is known as atriple lineand is perpendicular to the plane of the figure. Our objective is to determine the dihedral angles θα,θβ, andθηwhere these phases meet. By studying a simple variation of the position of the triple junction, we shall see that the three tensions satisfy a simple force balance law of the form15
γαβτˆαβ+γβητˆβη+γηατˆηα=0, (13.83)
15This equation also follows from a more general result of Gibbs [3, equation 615, p. 281] that holds for curved contact lines and was obtained as part of a complete variation of the system.
β
η
α θβ
θη
θα π−θα
π−θη π−θβ γβη
γηα γαβ
FIGURE 13–4 Three phasesα,β, andηthat meet at a triple line (left) where they make dihedral anglesθα,θβ, and θη. On the right is the corresponding force triangle with forces of magnitudesγαβ,γβη, andγηαthat are directed away from the triple junction, so their vector sum is zero.
whereτˆαβ is a unit vector perpendicular to the line of intersection of the three phases, locally tangent to theαβinterface at the line of intersection, and pointing away from theη phase. The other unit vectorsτˆβηandτˆηαare similarly defined with respect to their phases.
One can interpret Eq. (13.83) by regarding the quantityγαβτˆαβto be a force per unit length that acts on the triple line along theα−β interface, and similarlyγβητˆβη, andγηατˆηαare forces per unit length that act on the triple line along their respective interfaces. Thus, Eq. (13.83) is the condition for zero force acting on the triple line.
It also follows that the vectorsγαβτˆαβ,γβητˆβη, andγηατˆηαform a triangle, as shown in Figure 13–4, whose internal angles areπ −θα,π−θβ, andπ−θη. This triangle is known as aNeumann triangleand is often drawn in an orientation such that each of its sides is perpendicular to the respective interface. This can be seen by taking the cross product of Eq. (13.83) with a unit vector along the triple line. From the law of sines, and the fact that sin(π−θ)=sinθ, it follows that
sinθα
γβη =sinθβ
γηα =sinθη
γαβ . (13.84)
For such a Neumann triangle to exist, it is necessary for each of its sides to be less than the sum of the other two sides, for exampleγβη < γηα+γαβ.
Equation (13.83) can also be generalized to junctions where more than three phases meet, but such configurations might not be stable [3, p. 287]. If crystalline solids are involved, we shall see thatγ is anisotropic so Eq. (13.83) must be modified to account for torque terms.
To derive Eq. (13.83) from a variational principle, we suppose that all interfaces are pinned at distances that are far from the triple line and vary the position of the triple line by moving it parallel to itself in the direction of a small vector. Ifαβis the pinning distance from theα−βinterface to the original triple line, the distance from the varied triple line is
|αβταβ−| =
(αβ)2−2ã ˆταβαβ+2=αβ−ã ˆταβ (13.85) to first order in /αβ. The corresponding change in distance is therefore − ã ˆταβ. By treating the other interfaces in a similar way, we see that the total change in energy per unit length for such a variation is
204 THERMAL PHYSICS
−ã(τˆαβγαβ+ ˆτβηγβη+ ˆτηαγηα)=0, (13.86) which has been equated to zero as a condition for equilibrium. For arbitrary, the quantity in parentheses must vanish, resulting in Eq. (13.83).
It is important to recognize that knowledge of the anglesθα,θβ, andθηwill allow one to determine only the ratios of the quantitiesγαβ,γβη, andγηα. This can be seen by noting that multiplication of each of these interfacial energies by some positive number would result in a triangle similar, but different in size, to that depicted in Figure 13–4, so the angles would be unchanged. However, if the ratios ofγαβ,γβη, andγηαare specified, all three angles are determined uniquely. This can be seen analytically by applying the law of cosines to the triangle inFigure 13–4to obtain
cosθα= (γβη)2−(γηα)2−(γαβ)2
γηαγαβ (13.87)
and similar expressions for cosβand cosη.
13.3.1 Contact Angle
The variational derivation that underlies the force balances represented by Eq. (13.83) must be modified for anisotropic interfaces because the orientation of interfaces can change locally when the position of the triple line is varied. This results in additional torque terms. Nevertheless, the concept of force balances can be used to understand contact angles made by fluids with a rigid amorphous solid. Figure 13–5shows a triple junction between a liquidLand a gasgon a solid substratesunder conditions for which we assume thatγg,γsg, andγscan be defined.16Then a variation that involves sliding the triple line along the solid results in the equilibrium condition
γsg=γgcosθ+γs, (13.88)
Amorphous solid
Gas Liquid Gas Liquid
θ
γsg γsl
γlg
θ
γsg γsl
γlg
Amorphous solid
FIGURE 13–5 Contact angleθfor two fluids in contact with a rigid inert amorphous solid. On the left,θis acute and the liquid is said to wet the solid. On the right,θis obtuse and the liquid does not wet the solid.
16These conditions could deviate considerably from the global equilibrium conditions discussed previously.
The solid should behave as if chemically inert, with no solubility of the substances of the liquid or the gas. The gas could contain a substance insoluble in the liquid, and the vapor of the liquid can be in local equilibrium at the solid-liquid interface provided there is negligible evaporation during some period of observation. See Gibbs [3, p. 326] for further discussion.
which may be solved to yield
cosθ=γsg−γs
γg . (13.89)
Equation (13.89) is known asYoung’s equationfor the contact angleθ and represents a balance of horizontal forces. Real values ofθ only exist when the right-hand side has a magnitude less than or equal to one, which requires|γsg−γs| ≤γg. Ifθexists andγsg− γs > 0,θ ≤ π/2 and the liquid is said to wet the solid. For kerosene on glass, one has θ ≈26◦. Complete wetting occurs forθ=0, which is approximately the case for water on clean glass. Ifθ exists andγsg −γs < 0,θ > π/2 and the liquid does not wet the solid.
For mercury on glass, one hasθ ≈ 140◦. As long as the solid remains rigid and inert, no vertical variation of the contact line is possible, although it is generally supposed that the solid provides a force of adhesion equal toγgsinθto prevent theginterface from pulling away.
Although Young’s equation helps us understand the origin of the contact angle, its derivation suffers from a lack of rigor. Moreover, experimentally measured contact angles are difficult to reproduce and can depend sensitively on impurities as well as surface conditions of the solid. Nevertheless, the use of an empirically measured contact angle can enable one to model liquid shapes in situations of practical importance.