Propulsion Efficiency of a Dynamic Self-Assembled Helical Ribbon

Abstract

We study the dynamic self-assembly and propulsion of a ribbon formed from paramagnetic colloids in a dynamic magnetic field. The sedimented ribbon assembles due to time averaged dipolar interactions between the beads. The time dependence of the dipolar interactions together with hydrodynamic interactions cause a twisted ribbon conformation. Domain walls of high twist connect domains of nearly constant orientation and negligible twist and travel through the ribbon. The particular form of the domain walls can be controlled via the frequency and the eccentricity of the modulation. The flux of twist wallsa true ribbon property absent in slender bodies-provides the thrust onto the surrounding liquid that propels this biomimetic flagellum into the opposite direction. The propulsion efficiency increases with frequency and ceases abruptly at a critical frequency where the conformation changes discontinuously to a flat standing ribbon conformation.

Paramagnetic colloidal ribbons in a precessing magnetic field

“…Using the least-squares method for the data represented in the upper panel of Fig. 4 of [5], one obtains the expression v = 314.91/ω 0.58 with a correlation coefficient equal to 0.9197. For the sake of simplicity in the calculations, we approximate the exponent 0.58 as 1/2 or 3/5.…”

“…Our next step is to use the experimental data reported in [5] to obtain the functional dependence of χ on ω.…”

“…(1.3) that models the dynamics of paramagnetic colloidal ribbons in a precessing magnetic field [5], i.e., Eq. (2.1) with the parameters (2.3).…”

“…Details of the preparation and manipulation of the colloidal ribbon have been published elsewhere [5]. In short, the ribbon consists of two colloidal chains of paramagnetic beads of diameter 2.8 μm immersed in water and prepared such that they lie side by side on a glass substrate.…”

“…In [5], the following equation was proposed to describe the dynamics of the orientation angle field φ b (X,τ ) of the cross section of the paramagnetic colloidal ribbons in a precessing magnetic field along the backbone (characterized by the coordinate X) of the ribbon as a function of the dimensionless time τ ,…”