Onset of Buckling in Drying Droplets of Colloidal Suspension
Original entry: Tom Kodger, APPHY 226, Spring 2009
Authors: N. Tsapis, E. R. Dufresne, S. S. Sinha, C. S. Riera, J.W. Hutchinson, L. Mahadevan, and D. A. Weitz
Physical Review Letters 94, 018302 (2005)
Soft matter keywords
buckling, elastic shell, drying, sol-gel, Leidenfrost effect
Abstract from the original paper
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. Instead, we find that buckling is coincident with a stress-induced fluid to solid transition in a shell of particles at a droplet’s surface, occurring when attractive capillary forces overcome stabilizing electrostatic forces between particles.
Practical Application of Research
Rapidly dried droplets which contain suspended colloids are found in several industrial areas, and probably in your home office (unless you use a laser printer). Spray drying where fine powders are made by the evaporation of aerosols have been used in the manufacture of foodstuffs, pharmaceuticals, polymers, and detergents.
Capillarity at Work
The authors use a water droplet with suspended colloidal particles, with no added ions; therefore the capillary length l= √(γ/ρ*g) ≈ 2.5mm at 100°C. To induce the evaporation the authors heat the droplet on a stainless steel surface uniformly heated to 200°C using the Leidenfrost effect (cerca 1756). This effect is familiar to anyone who has sprinkled water on a hot griddle to check the temperature where the fluid droplets do not wet the surfaces above about 150°C; rather they float on a thin layer of their own vapor (Fig. 1).
The droplets used are always less than the capillary length for water which ensures that they are spherical. Using the well known Surface Evolver mathematical modelling program, the simulated buckling droplets nicely resemble the experimental drops (Fig. 2).
Capillary forces drive the buckling to the shell when menisci form between the particles at the surface and the pressure inside becomes, 2*γ/rM. For nearly three orders of magnitude, the thickness of shell, T, when buckling occurs (T/R)B scaled by the droplet radius is constant. The thickness of the shell can be determined using mass conservation; possibly a handly equation to remember (Eq. 1). But here the shell response in viscoelastic. Therefore the authors note that the transition into the buckling regime must correspond to a crossover from the viscous to elastic regimes of the shell's rheology
The authors find a critical pressure for buckling by first determining the shell thickness. During the initial isotropic shrinking, the capillary forces are about ≈πa2ΔP. There force the force for buckling FB is ≈37pN and ≈ 120pN for 85nm and 1000nm suspended colloids. Most interestly, these forces are on the correct order of the electrostatic repulsion between suspended colloids. The electrostatic repulsion can be calculated using a well known equation from Israelachvili (Eq. 2).
The author conclude that the buckling occurs when the capillary forces driving the deformation and flow of the shell overcome the electrostatic forces stabilizing the particles against aggregation.
By Tom Kodger