# When is the angle of contact between a liquid and solid interface equal to 90 degrees?

Dec 12, 2017

Angle of Contact for a liquid-solid interface is the angle between the tangent to the liquid surface at the point of contact and the plane of the solid (inside the liquid).

Cohesive Force: Attractive force between two liquid molecules,
Adhesive Force: Attractive force between a liquid molecule and the solid molecule at the interface.

There are two cases to consider -
[Case 1] A column of liquid inside a capillary tube: In this case the plane of the solid is vertical and so gravity acts parallel to the solid surface. The liquid-air interface may be curved up, curved down or flat, depending on the relative strengths of cohesive and adhesive forces.

In this case the ratio of the cohesive force (${F}_{c}$) to the adhesive force (${F}_{a}$) determines the contact angle ($\setminus \theta$).

(a) Convex Meniscus : $\setminus q \quad \setminus \theta > {90}^{o} , \setminus \quad$ if $\setminus \quad {F}_{a} / {F}_{c} > \frac{1}{\setminus} \sqrt{2} ,$
(b) Flat Meniscus : $\setminus \theta = {90}^{o} , \setminus \quad$ if $\setminus \quad {F}_{a} / {F}_{c} = \frac{1}{\setminus} \sqrt{2} ,$
(c) Concave Meniscus : $\setminus \theta < {90}^{o} , \setminus \quad$ if $\setminus \quad {F}_{a} / {F}_{c} < \frac{1}{\sqrt{2}} .$

Why is it so?: Requires quite a lot of explanation and illustration. You may ask that as a separate question.

[Case 2] A liquid droplet resting on a horizontal solid plane: In this case the plane of the solid is horizontal and gravity acts perpendicular to it.

This case was first discussed by Thomas Young (1805) who postulates the existence of three surface tensions,
(i) solid-liquid interface tension, $\setminus {\gamma}_{s l}$
(ii) solid-air interface tension, $\setminus {\gamma}_{s a}$
(iii) liquid-air interface tension, $\setminus {\gamma}_{l a}$

At the point of interface these three must be at equilibrium. While the solid-liquid and solid-air tensions are horizontal and opposite to each other, the liquid-air tension makes an angle $\theta$ to the horizontal. This is the contact angle.

Young's Equation: \qquad \gamma_{sa} - \gamma_{sl} + \gamma_{la}\cos\theta = 0;

This equation shows that when $\setminus {\gamma}_{s a} = \setminus {\gamma}_{s l}$, the contact angle $\setminus \theta$ is ${90}^{o}$.