Nanorheology of viscoelastic shells: Applications to viral capsids

We study the microrheology of nanoparticle shells [Dinsmore et al. Science 298, 1006 (2002)] and viral capsids [Ivanovska et al. PNAS 101, 7600 (2004)] by computing the mechanical response function and thermal fluctuation spectrum of a viscoelastic spherical shell that is permeable to the surrounding solvent. We determine analytically the damped dynamics of the shear, bend, and compression modes of the shell coupled to the solvent both inside and outside the sphere in the zero Reynolds number limit. We identify fundamental length and time scales in the system, and compute the thermal correlation function of displacements of antipodal points on the sphere and the mechanical response to pinching forces applied at these points. We describe how such a frequency-dependent antipodal correlation and/or response function, which should be measurable in new AFM-based microrheology experiments, can probe the viscoelasticity of these synthetic and biological shells constructed of nanoparticles.

💡 Research Summary

This paper presents a comprehensive theoretical framework for the microrheology of thin, viscoelastic spherical shells that are permeable to the surrounding fluid, with particular emphasis on synthetic nanoparticle shells and biological viral capsids. Building on earlier experimental observations of nanoparticle shells (Dinsmore et al., Science 2002) and viral capsids (Ivanovska et al., PNAS 2004), the authors derive the full frequency‑dependent mechanical response and thermal fluctuation spectra of such shells in the low‑Reynolds‑number (Stokes) limit.

The model treats the shell as a continuum characterized by shear modulus G(ω), bending rigidity κ_b, and bulk (compressional) modulus K, while the interior and exterior fluids are Newtonian with viscosity η. Permeability is introduced via a scalar coefficient κ that controls the normal fluid flux through the membrane. By expanding the displacement field in spherical harmonics Yℓm(θ,φ) and solving the coupled Stokes–elastic equations, the authors obtain analytic expressions for the complex eigenfrequencies ω_ℓ of three families of modes: (i) shear (ℓ ≥ 2), (ii) bending (ℓ ≥ 2), and (iii) pure compression (ℓ = 0). Each mode exhibits two distinct damping mechanisms: intrinsic viscoelastic dissipation within the shell and extrinsic viscous screening arising from fluid flow through the permeable wall.

A central result is the identification of characteristic length and time scales that govern the crossover between shell‑dominated and fluid‑dominated dynamics. The length scale ℓ_c ≈ (κ_b/η κ)^{1/4} and the time scale τ_c ≈ (η/κ_b)^{1/2} separate a low‑frequency regime (ωτ_c ≪ 1) where the shell’s complex modulus dictates the response, from a high‑frequency regime (ωτ_c ≫ 1) where fluid‑induced damping dominates. The authors also derive the thermal correlation function C(ω) of antipodal points on the sphere and the corresponding linear response function χ(ω). These satisfy the fluctuation‑dissipation relation C(ω) = 2k_BT Im