# Basics of Wetting: The Young-Laplace Equation

I. Introduction

The shape of liquid drop is governed by what is known as the Young-LaPlace equation. It was derived more or less simultaneously by Thomas Young (1804) and Simon Pierre de Laplace (1805). A short derivation of this equation is presented here. Consider a small section of a curved surface with carthesian dimensions x and y. On the normal n two arcs can be constructed that support the lengths of x and y: When the radii are enlarged by a small amount dR the work dW performed equals: Where 𝛾 is the surface energy. The work is done against a pressure. According to thermodynamics this work can also be written as: So we obtain for P: From the first equation it can simply be shown that: Insertion of the results from (5) into eq (4 yields the Young-LaPlace equation: An expression for R1 for the cylindrical symmetrical can be deduced as follows: Consider a contour f(z) with derivative f’(z) equaling n1. The length of the segment dL over distance dz equals: This is equated to α.R1 . The angle α is derived from the difference in derivatives in z and z+dz: Upon completion we obtain for R1: The second radius R2 is derived as follows: take a small segment from the circumference of the profile. From the length of this segment dC we can see that: From the figure we learn that: From which it can readily be deduced that: As a final result we obtain or the Young-Laplace equation: II. Author

C. van Veen