Effective modulus of particle-packing containing hard and soft particles

Drying films of colloidal dispersion containing hard particles crack while fast drying drops of similar dispersion buckle, both caused by the capillary pressure exerted by the liquid menisci between particles. For drying films on substrates, there exists a maximum crack-free thickness while for fast drying drops, there is a maximum buckle-free shell size, with both quantities depending on particle size, elastic modulus of the particles and nature of particle packing. Here, we measure the critical cracking thickness for a drying colloidal film made of a mixture of hard and soft elastic particles to extract the effective modulus of the film for various ratios of hard and soft particles. Scanning electron images of the cross section of the film reveal the spatial distribution of the hard and soft particles. The measured effective modulus exhibits a trend that is identical to that obtained using the nanoindentation technique indicating that measurement of the critical cracking thickness gives a quantitative measure of the effective modulus of the packing. The values of effective modulus from the two techniques fall between the two limits obtained from the simple rule of mixtures for composite materials. The deviation from either limit is attributed to particle segregation caused by sedimentation of the heavier particles.