Vibrational Modes in Packings of Deformable Particles*

Computational studies of the soft particle model, where interparticle overlaps give rise to purely repulsive forces, have shown interesting behavior of the vibrational modes as a function of the packing fraction and particle shape. However, this model does not accurately describe particle deformability, which is significant in packings of foams and emulsions, as well as confluent cell monolayers. In this work, we numerically study the vibrational modes in packings of deformable particles in two dimensions, as a function of the shape parameter A = p²/(4πa), where p is the perimeter and a is the area of the particle. We find that packings of deformable particles possess low-frequency quartic modes, where the energy increases quartically with the perturbation amplitude when the system is perturbed along these modes. We show that the number of quartic modes equals the number of missing interparticle contacts required to constrain the total number of degrees of freedom in the system. We then decompose each mode into the contributions from particle translations, rotations, and shape changes. We find that nearly all of the vibrational modes contain large contributions from particle shape changes and these contributions couple strongly to particle translations and rotations. In fact, we find that there are large contributions to the vibrational modes from the shape degrees of freedom even when packings are compressed to confluence, underscoring the importance of deformability in soft particulate systems. We further show that as the particle bending rigidity is increased, the quartic modes disappear, and all of the nontrivial vibrational modes become quadratic in the perturbation amplitude. For systems with finite particle bending rigidity, we find that the shear modulus scales as a power-law in pressure with a smaller exponent compared to the shear modulus exponent for packings of particles without bending rigidity.