A2.4 - A Suspended Plate In-Plane Resonator for Rheological Measurements at Tunable Frequencisies

Abstract

A resonating sensor for viscosity facilitating measurements at tunable frequencies is presented. A sample liquid is subjected to time harmonic shear stress induced by a vibrating plate. By measuring the resonant behavior of this fluid-structure interaction the liquid’s rheological properties can be deduced. The resonating plate is affixed to two parallel wires, placed in an external static magnetic field. One of these wires is used to excite lateral vibrations of the plate by means of Lorentz forces while the second acts as pick-up providing an induced voltage representing the movement of the plate. The recorded frequency response is used to determine the liquid’s physical properties. This setup, based on the mechanical coupling of two wires with a small plate and inducing time harmonic shear stresses with the latter, allows examining the test liquid at adjustable frequencies. In first experiments resonance frequencies from 820 Hz to 4040 Hz were achieved for a given geometry. Introduction The focus of our work is on tunable resonating rheometers. One advantage of resonating rheometers in comparison to conventional shear flow based rheometers is the possibility to characterize the physical properties of very complex non-newtonian liquids such as polymeric fluids, suspensions, emulsions etc. Vibrating viscometers are mechanically simpler than other types and the volume of fluid required for their use is much smaller making operation at high pressures and temperatures easier [1]. An overview of conventional principles for measuring viscosity is given in [2]. Piezoelectric or quartz crystal based oscillators proved to be well suitable for viscosity sensing and are commonly reported in literature, see e.g., [3], [4] and [5]. However, there is an increasing interest for vibrating resonators in the low kilohertz range to increase the penetration depth of the shear waves imposed into the liquid by the resonator [6], [7]. To determine the physical properties of a liquid such as viscosity or mass density using a resonator based principle, the change of the oscillator’s quality factor and the shift of its resonant frequency (which are both known in vacuum or air) are recorded when it is immersed into the test liquid. For the investigation of some liquids it can either be important to analyze the liquid’s physical behavior at one particular frequency or at several frequencies in a certain bandwidth. In both cases the capability of tuning the sensor’s resonance frequency is required. In this contribution a rheometer based on a suspended in-plane vibrating plate with tunable resonance frequencies is presented. The principle of this sensor is depicted in Fig. 1(a). A similar setup has been presented in [7] aiming to improve the quality factor of micro-machined sensors. There, the suspended plate has been implemented in silicon technology. The sensor presented in this work consists of a non-conductive plate and two conductive wires placed in a constant magnetic field. The plate (for first prototypes PET and glass have been used) is affixed to the wires using thermal fusing or gluing, thus yielding a mechanical coupling of the two wires. One of these wires is used to deflect the plate in lateral direction by means of Lorentz forces on AC-currents flowing through the wire. The other wire, following the movement of the first, is used as pick-up representing the in-plane movement of the plate by an induced voltage. By varying the normal stresses within both wires by an appropriate tensioning mechanism the desired resonance frequency can be adjusted. Theory and Modeling One-dimensional shear waves in viscous liquids To analytically describe the propagation of shear waves in a viscous liquid induced by a plate resonating at an angular frequency ω and amplitude x and considering a no-slip boundary condition at the solid-liquid interface, see Fig. 2(a), the stress tensor for liquids [8] T = −pI+ 2μE+ λ∇ · u I (1) S E N S O R + T E S T C o n f e r e n c e s 2 0 1 1 S E N S O R P r o c e e d i n g s 6 1 (a) Sensor Principle Fundamental Harmonic Second Harmonic Third Harmonic