Moduli and modes in the Mikado model

• Published in: Soft matter 2021-11, Vol.17 (45), p.1286-1293
• Main Authors: Baumgarten, Karsten, Tighe, Brian P
• Format: Article
• Language: English
• Subjects: Bending modulus; Chemistry; Mechanical properties; Modulus of elasticity; Networks; Shear modulus
• ISSN: 1744-683X; 1744-6848
• DOI: 10.1039/d1sm00551k

We determine how low frequency vibrational modes control the elastic shear modulus of Mikado networks, a minimal mechanical model for semi-flexible fiber networks. From prior work it is known that when the fiber bending modulus is sufficiently small, (i) the shear modulus of 2D Mikado networks scales as a power law in the fiber line density, G ∼ ρ α +1 , and (ii) the networks also possess an anomalous abundance of soft (low-frequency) vibrational modes with a characteristic frequency ω κ ∼ ρ β /2 . While it has been suggested that α and β are identical, the preponderance of evidence indicates that α is larger than theoretical predictions for β . We resolve this inconsistency by measuring the vibrational density of states in Mikado networks for the first time. Supported by these results, we then demonstrate analytically that α = β + 1. In so doing, we uncover new insights into the coupling between soft modes and shear, as well as the origin of the crossover from bending- to stretching-dominated response. We determine how low frequency vibrational modes control the elastic shear modulus of Mikado networks, a minimal mechanical model for semi-flexible fiber networks.