# Onset of Buckling in Drying Droplets of Colloidal Suspensions

Entry: Chia Wei Hsu, AP 225, Fall 2010

N. Tsapis, E. R. Dufresne, S. S. Sinha, C.S. Riera, J. W. Hutchinson, L. Mahadevan, and D. A. Weitz, Phys Rev Lett **94**, 018302 (2005)

## Contents

## Summary

This work studies the drying of freely suspended colloidal droplets. These droplets initially behave like pure liquids and shrink isotropically. Eventually, they buckle like elastic shells. The authors conclude the mechanism: as droplets dry, a viscoelastic shell of densely packed particles forms at its surface. Initially, the shell yields and thickens as the droplet shrinks. Eventually, the capillary forces that drive deformation of the shell overcome the electrostatic forces stabilizing the particles. At this point, the shell undergoes a sol-gel transition, becomes elastic, and buckles.

## Background

Minute concentrations of suspended particles can dramatically alter the behavior of a drying droplet. After a period of isotropic shrinkage, similar to droplets of a pure liquid, these droplets suddenly buckle like an elastic shell. While linear elasticity is able to describe the morphology of the buckled droplets, it fails to predict the onset of buckling. Also, drying suspensions can be viscoelastic, depending on the experimental time scale. Thus, dyring droplets pose a number of intriguing fundamental questions.

## Experiment and observation

The authors study drying droplets of aqueous suspensions of monodisperse carboxylate-modified polystyrene colloids, suspended through the Leidenfrost effect (ie, fluid droplets float on a thin layer of their own vapor above 150 degree C). The radii range from 0.8 to 2.2 mm (They have to be smaller than the capillary length 2.5 mm in order to remain spherical). The dynamics of drying is monitored with high-speed camera. Fig 1 shows the transition from isotropically shrinking to the sudden buckling.

## Explanation

A shell on the surface of the droplet forms, because particles pile up just inside the drying droplet's receding air-water interface. The time for the fluid to evaporate (<math>\tau_{\mathrm{dry}} \approx 60 </math> s) is much shorter than the time required for homogenizing the fluid ((<math>\tau_{\mathrm{dry}} \approx 10^3 </math> s), so the boundary between the shell and the bulk droplet is sharp. With mass conservation, the shell thickness <math>T</math> (with volume fraction <math>\phi_c</math>) can be related to the droplet radius <math>R</math> (with volume fraction <math>\phi_i</math>) by

<math>\frac{T}{R} = 1 - \left[ \frac{\phi_c - \phi_i (R_i / R)^3}{\phi_c - \phi_i} \right] ^{1/3}</math>

Capillary forces drive the deformation of the shell, and the shell response viscoelastically. During the isotropic shrinking, the viscous nature of the shell dominates. However, buckling requires the shell to become elastic. So the onset of buckling correspond to a crossover from viscous to elastic regimes of the shell's rheology.

The ratio between the buckling radius <math>R_B</math> and the initial radius <math>R_i</math> is a constant with respect to <math>R_i</math> (fig 2a inset), and scales as <math>\phi_i^{1/3}</math> (fig 2a). Meanwhile, the ratio between the shell thickness and the droplet radius at buckling seems to be a constant (fig 2b). Furthermore, the pressure drop through the shell at buckling <math>(\Delta P)_B</math> also seems to be a constant with respect to <math>\phi_i</math>.

This "critical pressure" suggests that the crossover from viscous to elastic behavior is initiated by stress. This critical pressure corresponds to a critical force <math>F_B \approx \pi a^2 (\Delta P)_B</math>. The authors found that this critical force agrees with the maximum repulsive force due to electrostatic double-layers (estimated from the DLVO theory). Therefore, they hypothesize that buckling occurs when the capillary forces driving the deformation and the flow of a shell overcome the electrostatic forces stabilizing the particles against aggregation.

## Connection to Soft Matter

This work utilizes several things we just learned in this soft matter class, including calculation of the capillary length, the Laplace pressure, and the DLVO forces. It is nice to see how these simple theory are used in real research.

This work suggests that the shell formed on the surface of a drying droplet behaves both as a viscous and a elastic material, and that the onset of buckling marks the transition between the two. The transition is an interplay between many forces. Thus even phenomenon as simple as a drying droplet can be quite complex when we examine its mechanism.