Abstract
A soft viscoelastic drop has dynamics governed by the balance between surface tension, viscosity, and elasticity, with the material rheology often being frequency dependent, which are utilized in bioprinting technologies for tissue engineering and drop-deposition processes for splash suppression. We study the free and forced oscillations of a soft viscoelastic drop deriving (1) the dispersion relationship for free oscillations, and (2) the frequency response for forced oscillations, of a soft material with arbitrary rheology. We then restrict our analysis to the classical cases of a Kelvin–Voigt and Maxwell model, which are relevant to soft gels and polymer fluids, respectively. We compute the complex frequencies, which are characterized by an oscillation frequency and decay rate, as they depend upon the dimensionless elastocapillary and Deborah numbers and map the boundary between regions of underdamped and overdamped motions. We conclude by illustrating how our theoretical predictions for the frequency-response diagram could be used in conjunction with drop-oscillation experiments as a “drop vibration rheometer”, suggesting future experiments using either ultrasonic levitation or a microgravity environment.
Highlights
There is a long history of experimental studies of drop oscillations in ultrasonic levitation1 and microgravity2,3
Both polymer fluids and soft gels are viscoelastic materials with both a viscosity and elasticity, both of which can have a complex dependence on frequency defining the rheology of the material through a storage and loss
Unlike the Rayleigh drop, Eq [15] is a nonlinear dispersion equation, which admits an infinity of roots for the same polar mode l, which defines the radial mode number s
Summary
There is a long history of experimental studies of drop oscillations in ultrasonic levitation and microgravity. The advantages of the microgravity environment are the large length and timescales not accessible under terrestrial conditions, as well as the high degree of drop sphericity that can be achieved This allows for a direct comparison to the classical theory of Lord Rayleigh who showed that an inviscid spherical drop of radius R will oscillate about its equilibrium shape with characteristic frequency, ω 1⁄4 rffilffiðffiffilffiffiÀffiffiffiffiffi1ffiffiÞffiffiðffiffilffiffiþffiffiffiffiffi2ffiffiÞffiffiffiρffiffiσRffiffiffi3ffi; [1]. Note that the Rayleigh spectrum [1] is degenerate with respect to the azimuthal mode number m Extensions to this basic model include, but are not limited to, the effect of viscosity, large-amplitude deformation, and wetting. Similar viscoelastic effects are seen in soft solids like hydrogels, which have cross-linked polymer networks with tunable elasticity, and are widely used as biocompatible materials in rapid prototyping technologies and drug-delivery systems. Both polymer fluids and soft gels are viscoelastic materials with both a viscosity and elasticity, both of which can have a complex dependence on frequency defining the rheology of the material through a storage (elasticity) and loss (viscosity) modulus
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