3D-printed ultra-small Brownian viscometers

Abstract

Measuring viscosity in volumes smaller than a microliter is a challenging endeavor. A new type of microscopic viscometers is presented to assess the viscosity of Newtonian liquids. Micron-sized flexible polymer cantilevers are created by two-photon polymerization direct laser writing. Because of the low stiffness and high elasticity of the polymer material the microcantilevers exhibit pronounced Brownian motion when submerged in a liquid medium. By imaging the cantilever's spherically shaped end, these fluctuations can be tracked with high accuracy. The hydrodynamic resistance of the microviscometer is determined by fitting the power spectral density of the measured fluctuations with a theoretical frequency dependence. Validation measurements in water-glycerol mixtures with known viscosities reveal excellent linearity of the hydrodynamic resistance to viscosity, allowing for a simple linear calibration. The stand-alone viscometer structures have a characteristic size of a few tens of microns and only require a very basic external instrumentation in the form of microscopic imaging at moderate framerates (~ 100 fps). Thus, our results point to a practical and simple to use ultra-low volume viscometer that can be integrated into lab-on-a-chip devices.

Keywords: Brownian fluctuations; Power spectral density; Two-photon polymerization; Viscoelastic polymer nanowire; Viscometer.

Figure 1

The viscometer structure. (a) Brightfield image of microstructures with beads of 2 µm, 4 µm, and 6 µm radius. (b) A screenshot from the video (see the Supplementary Materials) shows the 4 µm radius bead. (c) Schematic view of the solid support with four vertical cantilevers. A spherical bead is attached to each nanowire beam. (d) The mechanical model of the viscometer with the viscoelastic beam and the microbead immersed in a Newtonian liquid.

Figure 2

Brownian fluctuations of the microbead attached to polymer nanowire. (a) Typical record of the R_d = 4 µm bead center position in water as tracked in the X–Y plane. (b) The time course of the bead center fluctuations along the Y coordinate during the 30 s data acquisition period. (c) The probability density of the bead center position fitted by a Gaussian distribution (σ is the standard deviation).

Figure 3

Power spectral densities (PSD) of the bead position fluctuations. (a) Typical PSDs measured in pure water using microstructures with designed bead radii of 2 µm, 4 µm, and 6 µm. The shaded area indicates the 20–90 Hz region used for fitting the spectra. The fitted curves are plotted as solid lines. The dashed curve shows a 1/f² dependence. (b) PSDs obtained in different water-glycerol mixtures using an R_d = 4 µm bead. The legend indicates the solution viscosities.

Figure 4

Viscosity measurements. The hydrodynamic resistance values obtained with microstructures immersed in different water-glycerol mixtures plotted against the solution viscosity. All the measurements were repeated with three different microstructures, and the error bars represent the errors of their mean γ values.