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

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In this paper, Spaepen illustrates the difference between these two quantities by considering a hemispherical liquid drop on a solid surface. | 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 <math>\gamma_{lv}</math>, <math>\gamma_{sv}</math>, and <math>\gamma_{sl}</math>, as well as the stresses <math>f_{lv}</math> (=<math>\gamma_{lv}</math>), <math>f_{sv}</math>, <math>f_{sl}</math> (corresponding to the only applicable strain, <math>\epsilon_{11}</math>, 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. | ||

+ | [[Image:capcur_dgrm.png|400px|center|]] | ||

+ | <center>Fig. 1 Diagram of the droplet on the substrate, indicating the relevant interfaces and quantities</center> | ||

+ | |||

+ | ==Equilibrium Shape of the Droplet== | ||

+ | |||

+ | Consistent with <math>\gamma_{lv}</math> 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<math>\theta</math>. The equilibrium shape of the droplet for a given volume is determined by minimizing the free energy of the system with respect to <math>\theta</math> or R. Placing the droplet on the substrate replaces the solid-vapor interface by solid-liquid interface over an area <math>A_{sl}</math>, also creating an area <math>A_{lv}</math> of liquid-vapor interface. The associated changes in free energy are: | ||

+ | |||

+ | <center><math> \Delta F = A_{sl}(\gamma_{sl} - \gamma_{sv}) + A_{lv} \gamma_{lv}\ [1]</math></center> | ||

+ | |||

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

+ | |||

+ | <center><math> \Delta F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv}\ [2]</math></center> | ||

+ | |||

+ | |||

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

+ | |||

+ | <center><math> F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv} + \lambda R^2(\theta - sin\theta cos\theta)\ [3]</math></center> |

## Revision as of 16:02, 21 April 2012

Entry by Emily Redston, AP 226, Spring 2012

Work in Progress

## Contents

## Reference

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

## Keywords

surface tension, interface stress, Young's equation, curvature

## Introduction

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

- surface (or interface) tension <math>\gamma</math>, 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 <math>f_{ij}</math>, which is a 2x2 tensor defined such that the surface work required to strain a unit surface elastically by <math>d {\epsilon}_{ij}</math> is <math>f_{ij} d {\epsilon}_{ij}</math>

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 <math>\gamma_{lv}</math>, <math>\gamma_{sv}</math>, and <math>\gamma_{sl}</math>, as well as the stresses <math>f_{lv}</math> (=<math>\gamma_{lv}</math>), <math>f_{sv}</math>, <math>f_{sl}</math> (corresponding to the only applicable strain, <math>\epsilon_{11}</math>, 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.

## Equilibrium Shape of the Droplet

Consistent with <math>\gamma_{lv}</math> 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<math>\theta</math>. The equilibrium shape of the droplet for a given volume is determined by minimizing the free energy of the system with respect to <math>\theta</math> or R. Placing the droplet on the substrate replaces the solid-vapor interface by solid-liquid interface over an area <math>A_{sl}</math>, also creating an area <math>A_{lv}</math> of liquid-vapor interface. The associated changes in free energy are:

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

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