# Difference between revisions of "Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop"

Entry by Emily Redston, AP 226, Spring 2012

Work in Progress

## Reference

Substrate curvature resulting from the capillary forces of a liquid drop by F. Spaepen. J. Mech. Phys. Solids 44, 675 – 681 (1996)

## Introduction

We typically characterize the surface of solids using two thermodynamic quantities:

• surface (or interface) tension $\gamma$, which is a scalar quantity equal to the work required to create a unit area of new interface at constant strain in the solid
• surface (or interface) stress $f_{ij}$, which is a 2x2 tensor defined such that the surface work required to strain a unit surface elastically by $d {\epsilon}_{ij}$ is $f_{ij} d {\epsilon}_{ij}$

In this paper, Spaepen illustrates the difference between these two quantities by considering a hemispherical liquid drop on a solid surface.

## Geometry of the Droplet

Figure 1 shows a droplet surrounded by a vapor phase on a solid surface. The problem is kept two-dimensional for simplicity. Associated with the three types of interfaces are the tension $\gamma_{lv}$, $\gamma_{sv}$, and $\gamma_{sl}$, as well as the stresses $f_{lv}$ (=$\gamma_{lv}$), $f_{sv}$, $f_{sl}$ (corresponding to the only applicable strain, $\epsilon_{11}$, in the direction of the interface). Spaepen also assumes that the three phases consist of the same single element to avoid complications due to surface segregation.

Fig. 1 Diagram of the droplet on the substrate, indicating the relevant interfaces and quantities

## Equilibrium Shape of the Droplet

Consistent with $\gamma_{lv}$ being isotropic, the liquid-vapor interface is considered semi-circular and has a radius of curvature R. The angle between the radii to the end points (OA and OB) is taken to be 2$\theta$. The equilibrium shape of the droplet for a given volume is determined by minimizing the free energy of the system with respect to $\theta$ or R. Placing the droplet on the substrate replaces the solid-vapor interface by solid-liquid interface over an area $A_{sl}$, also creating an area $A_{lv}$ of liquid-vapor interface. The associated changes in free energy are:

$\Delta F = A_{sl}(\gamma_{sl} - \gamma_{sv}) + A_{lv} \gamma_{lv}\ [1]$

Taking the geometry of Fig. 1 into account, we can also write this as:

$\Delta F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv}\ [2]$

To minimize this free energy at constant volume $V = R^2(\theta - sin\theta cos\theta)$, a Lagrange multiplier, $\lambda$, is introduced:

$F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv} + \lambda R^2(\theta - sin\theta cos\theta)\ [3]$

Minimization gives the conditions:

${\partial F \over \partial R} = 2sin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta \gamma_{lv} + 2\lambda R(\theta - sin\theta cos\theta) = 0\ [4a]$

${\partial F \over \partial \theta} = 2Rcos\theta(\gamma_{sl} - \gamma_{sv}) + 2R\gamma_{lv} + \lambda R^2(1 - cos^2\theta + sin^2\theta) = 0\ [4b]$