Capillary Interaction in Wet Granular Assemblies: Part 1

Abstract

Liquid transport and capillary cohesion in partially saturated particulate materials are relevant to many areas of geosciences and process engineering. This chapter demonstrates that computations of capillary surfaces by numerical energy minimizations (NEM) can be utilized to study the formation and spreading of wetting films on rough surfaces, interfacial shapes and capillary forces of funicular liquid clusters, and the effect of contact angle hysteresis on liquid transport and capillary forces between spherical particles.

Appendix

Equivalence of Capillary Forces

By differentiation of the Grand interfacial free energy $\mathscr {G}$ Eq. (2) with respect to the coordinates $\bar{\mathbf r}$ one obtains the following identity:

$$\begin{aligned} \varvec{\nabla }_{{{\mathbf r}}_i} G\big (\bar{\mathbf r},P \big ) = \varvec{\nabla }_{{{\mathbf r}}_i} E\big (\bar{\mathbf r},\tilde{V}(\bar{\mathbf r},P)\big ) -P\,\varvec{\nabla }_{{{\mathbf r}}_i}\tilde{V}\big (\bar{\mathbf r},P\big )~. \end{aligned}$$

(25)

where

$$\begin{aligned} P=\tilde{P}(\bar{\mathbf r},V)=\partial _V E(\bar{\mathbf r},V)~. \end{aligned}$$

(26)

Employing now the chain rule of differentiation, the first term in Eq. 25) can be rewritten as:

$$\begin{aligned}&{\varvec{\nabla }_{{{\mathbf r}}_i}E\big (\bar{{\mathbf r}},\tilde{V}(\bar{{\mathbf r}},P)\big ) =\varvec{\nabla }_{{{\mathbf r}}_i} E\big (\bar{\mathbf r},V)|_{V=\tilde{V}(\bar{\mathbf r},P)}}\nonumber \\&\,\,+\partial _V E(\bar{\mathbf r},V)|_{V=\tilde{V}(\bar{\mathbf r},P)} \varvec{\nabla }_{{{\mathbf r}}_i}\tilde{V}\big (\bar{{\mathbf r}},P \big )~, \end{aligned}$$

(27)

and one arrives at:

$$\begin{aligned} \varvec{\nabla }_{{{\mathbf r}}_i}G\big (\bar{\mathbf r}, P \big )= \varvec{\nabla }_{{{\mathbf r}}_i}E\big (\bar{\mathbf r},V \big )|_{V=\tilde{V}(\bar{\mathbf r}, P)} \end{aligned}$$

(28)

which shows the equivalence of capillary forces in the volume controlled and capillary pressure controlled case.