Macroscopic wettability based on an interfacial jump condition

Young’s equation, describing an interfacial equilibrium condition of a liquid droplet on a smooth solid surface, raises issues concerning the existence of a sine term which has not yet been resolved theoretically and continues to be discussed to the present day. From a thermodynamics viewpoint, the equilibrium condition arises by minimizing the total free energy of the system while intensive parameters are kept constant. In the derivation, variations in the virtual work in both horizontal and vertical directions of the droplet on the smooth solid are considered. From a hydrodynamics viewpoint, there is a momentum jump condition at the gas-liquid interface that is derived based on a mechanical balance. Using standard mathematical procedures such as Stokes’ theorem and differential geometry, a test volume is considered across the interface between two continuous phases from which the jump condition is derived. In the present paper, Young’s equation is revisited from the point of view of the momentum jump condition at the two-phase interface and a modified Young’s equation is derived. The analytical solution derived from the modified Young’s equation is then used to compare theory with experimental data. The line tension and contact angle for a lens droplet are also discussed on the basis of this model.

Figure 1

Interfacial coordinates at the gas-liquid interface.

Figure 2

Contact angle: solid/liquid/gas.

Figure 3

Contact angle of a three-dimensional droplet on a solid substrate.

Figure 4

Definition of base vector on a three phase line. (a) Base vector on $s_a$ axis ($P\text{−}a$ line). (b) Base vector on $s_b$ axis ($P\text{−}b$ line). (c) Base vector on $s_c$ axis ($P\text{−}c$ line).

Figure 5

Small area on $s_a$ axis.

Figure 6

Schematic of the term ${\sigma }_b\text{sin}\theta$ at the contact line. [Decomposition of surface tension in the (a) normal and (b) tangential direction with respect to the gas-liquid interface.]

Figure 7

Ideal droplet on a solid substrate.

Figure 8

Experimental data with respect to droplet radius and contact angle (adapted from Ref. [26]). (a) Droplet radius vs contact angle of heptane $\left(C_2=1.53\times {10}^{-2}\left[{\text{m}}^2/\text{N}\right],{\theta }_{\infty }=4.0°\right)$. (b) Droplet radius vs contact angle of octane $\left(C_2=1.04\times {10}^{-2}\left[{\text{m}}^2/\text{N}\right],{\theta }_{\infty }=6.7°\right)$.

Figure 9

Droplet radius vs contact angle of heptane $\left(C_2=1.04\times {10}^{-2}\left[{\text{m}}^2/\text{N}\right],{\theta }_{\infty }=4.8°\right)$.

Figure 10

Droplet radius vs contact angle of dodecane $\left(C_2=7.0\times {10}^{-5}\left[{\text{m}}^2/\text{N}\right],{\theta }_{\infty }=69.65°\right)$ (adapted from Ref. [27]).

Figure 11

Analytical solution of Line tension of octane and heptane in Figs. 8, 9 (solid line: heptane, dashed line: octane).

Figure 12

Definition of base vector on three-phase line of a lens droplet. (a) Base vector on $s_{a^{\prime }}$ axis ($O\text{−}{a}^{\prime }$ line). (b) Base vector on $s_{b^{\prime }}$ axis ($O\text{−}{b}^{\prime }$ line). (c) Base vector on $s_{c^{\prime }}$ axis ($O\text{−}{c}^{\prime }$ line).