# Colloidal Stability and Magnetic Field-Induced Ordering of Magnetorheological Fluids Studied with a Quartz Crystal Microbalance

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

^{1/2}, where η and ρ are the viscosity and the density of the fluid, respectively, and ω is the angular frequency. The depth of penetration in water at MHz frequencies is of the order of 100 nm. Importantly, the resonance parameters of a TSM resonator change when the resonator surface is brought into contact with a sample. When the sample consists of a rigid film (<100 nm), a thin viscoelastic layer (<1 µm, depending on the softness), or a fluid, the resonance frequency decreases because the sample increases the effective mass of the resonator [2]. In other cases, the acoustic impedance seen by the resonator is reactive. Under these conditions, the resonance frequency increases. One such case is the coupled resonance [3].

^{2}), which motivated the term “quartz crystal microbalance (QCM)”. While early applications were restricted to air or vacuum [4], Nomura showed in 1982 [5] that loading with a fluid is also possible. Since the system was not overdamped, one could still determine a resonance frequency. Additionally, the resonance bandwidth was shown to contain valuable information. The resonance bandwidth is often converted to an inverse Q-factor, Q

^{−1}, also called “dissipation factor”, D, or “dissipation”, for short. In Newtonian liquids, the dissipation factor is proportional to (ρη)

^{1/2}. These insights triggered the widespread application of the QCM in liquid environments [1]. Of particular importance has been the characterization of chemical and biological samples using these devices. The QCM is now used as a microgravimetric device or as a detector for adsorption to functionalized surfaces at many places [6,7,8,9,10].

## 2. Background on High-Frequency Contact Mechanics

_{0}is the fundamental frequency, Z

_{q}= 8.8 × 10

^{6}kg · m

^{−2}s

^{−1}is the shear-wave-specific acoustic impedance of AT-cut quartz, and 〈σ

_{0}〉

_{area}is the area-averaged complex amplitude of the tangential stress at the resonator surface. n is the overtone order and u

_{0}is the amplitude of oscillation. The ratio of stress and velocity (where the latter is equal to i ωu

_{0}) is the complex load impedance, Z

_{L}.

_{P}. If ω

_{P}is much larger than the resonance frequency of the QCM, inertial effects dominate [3] and Δf is negative. If ω

_{P}is much smaller than the QCM-frequency, the elastic limit holds (see above). If ω

_{P}is comparable to the QCM frequency, one finds a crossover between negative and positive Δf at an overtone order comparable in frequency to the particle resonance frequency ([24,25,26], see also Figure 1).

**Figure 1.**Shifts of frequency and bandwidth around a coupled resonance (23–25). f

_{ZC}is the frequency of zero crossing, which is similar to the particle resonance frequency.

## 3. Experimental Section

#### 3.1. Experimental Set-Up

- −
- A QCM cell containing an AT-cut quartz crystal provided by CH Instruments Inc. (Austin, TX, USA, 7.995 MHz fundamental frequency, 13.7 mm blank diameter, 5.1 mm electrode diameter, polished surface finish, 100 Å Ti and 1000 Å Au electrode material, keyhole electrode pattern).
- −
- 200 MHz (−3 dB) Panametric ultrasonic pulser/receiver, model 5900 PR (P/R mode, 2 kHz PRF, 1 μJ energy, 50 Ω damping, 1 MHz HP filer, 200 MHz LP filter, 0 dB attenuators, 40 dB gain, 0˚ RF output phase).
- −
- 500 MHz, 4 GSa/s, DS4054 Rigol, digitizing oscilloscope (150,000 acquisition points).
- −
- Personal computer connected to the oscilloscope via USB interface.

**Figure 2.**(

**a**) Scheme of the experimental set up; and (

**b**) measurement cell adapted to the rheometer (not to scale).

#### 3.2. Signal Processing

^{1/2}, with p = (n + 1)/2, where n is the overtone order), in an interval centered at the resonance frequency with a variable span depending on the harmonic (2 MHz multiplied by p

^{1/2}).

**Figure 3.**(

**a**) Time trace of the signal on the oscilloscope; and (

**b**) frequency spectrum derived from the time trace by Fourier transformation.

^{−1}was calculated from the bandwidth, B

_{W}, as D = B

_{W}/f

_{0}. The bandwidth was determined from the decay rate of the time-domain signal. Supposing a monochromatic sinusoidal signal, u(t), of maximum amplitude, u

_{0}, with an exponentially decaying envelope:

**Figure 4.**A fit of an exponential function to the envelopes of a monochromatic signal (obtained by a Hilbert transform). The decay constant is obtained from the fit.

#### 3.3. MR Fluid Characteristics

- −
- The sample “OM MRF” is the homemade suspension. It consists of carbonyl iron microparticles (OM grade from BASF SE, mean diameter 5 μm) dispersed in a highly-viscous silicone oil (487 mPa·s, Sigma-Aldrich, St, Louis, MO, USA). The particle content was 5 vol%. This formulation does not include additives. The density of the suspensions at 25 °C was 1126 ± 10 kg/m
^{3}. The viscosity of the OM MRF measured with the rotational shear rheometer at a temperature of 25 °C and a shear rate of 100 s^{−1}was 515 ± 15 mPa·s. - −
- The sample “Commercial MRF” (reference MRF-132 supplied by Lord Corporation, Carrey, NC, USA) consists of iron particles with a diameter between 5 to 10 μm suspended in a carrier fluid. The particle volume fraction is 32 vol%. In this case, (proprietary) thixotropic additives prevent short-term sedimentation. According to the manufacturer, the density of the suspension at 25 °C is 3050 ± 100 kg/m
^{3}. The viscosity of the commercial MRF measured with the rotational shear rheometer at a temperature of 25 °C and a shear rate of 100 s^{−1}was 300 ± 50 mPa·s.

MR Fluids | Density kg/m^{3} | Viscosity mPas | Vol. Fraction | Thixotropic Additives | Carrier Liquid | Particles | |
---|---|---|---|---|---|---|---|

OM_MR | ^{b} 1126 ± 10 | ^{b} 515 ± 15 | 5% | NO | Silicone oil | Grade | OM |

Coating | None | ||||||

Diameter | 5 µm | ||||||

Bulk composition | Fe(>97.8%), C(0.7%–0.9%), N(0.6%–0.9%), O(0.2%–0.4%) | ||||||

Commercial MRF | ^{a} 3050 ± 100 | ^{b} 300 ± 50 | 32% | YES | Hydrocarbon | Grade | Unknown |

Coating | YES | ||||||

Diameter | 5–10 µm | ||||||

Bulk composition | Unknown |

^{a}manufacturer data;

^{b}measured data.

## 4. Results and Discussion

#### 4.1. Sedimentation

^{−1}constant angular velocity. This homogenizes the suspension and erases all memory to the sample’s history. Subsequently, the rotation is stopped (region II in Figure 5 and Figure 6) and the system is allowed to evolve in the quiescent state.

**Figure 5.**Scheme of possible structures formed in OM MRF and commercial MRF at the pre-shear stage-region I (

**Left**), at rest -region II (

**Middle**), and under the effect of a magnetic field-region III (

**Right**).

**Figure 6.**Dissipation (full dots) and resonance frequency shift (open dots) of OM MRF (squares) and commercial MRF (circles) as a function of time at 8 MHz. In region I (pre-shear stage) the MR fluids are subjected to a constant shear rate (100 s

^{−1}). In region II, the shear is turned off, letting the structure approach equilibrium.

**Figure 7.**(

**a**) Frequency shift (open) and (

**b**) change in dissipation (solid) as function on the overtone at the end of the region II. Values shown are referenced to the end of the region I.

#### 4.2. Effects of Magnetic Fields

**Figure 8.**Dissipation (solid) and resonance frequency shift (open) of OM MRF 5% (squares) and commercial MRF 32% (circles) as function of time at 8 MHz. In the first step (region II) the shear rate is set to zero, letting the structure evolves to equilibrium after the pre-shear rate. In the second step (region III), a 15-kA/m magnetic field is applied perpendicularly to the rheometer plates.

**Figure 9.**Change in dissipation (solid) and in resonance frequency (open) as a function of the magnetic field intensity applied to the commercial MRF at 8 MHz.

**Figure 10.**Scheme of particle-surface interaction. (

**A**) A particle at rest; (

**B**) particles which can rotate; and (

**C**) a particle, the rotation of which is hindered by the presence of second particle touching it at the top (which is part of the same chain-like structure).

#### 4.3. Finite Element Modeling

^{3}). Structures formed with particle settling were modeled with G’ = 5 MPa, G’’= 5 MPa and ρ = 3000 Kg/m

^{3}. Structures formed under the effect of a magnetic field were modeled increasing the shear storage module G’ = 50 MPa maintaining G’’ = 5 MPa and ρ = 3000 Kg/m

^{3}. In both cases the modeled structures have a diameter of 50 μm, and the distance between them is 200 μm. Both situations have been modeled by changing the value of the parameter G′ following a simple series connection mechanical model between the metallic particles and the carrier fluid. In the case of settling, the particles form a non-consolidated structure placed over the sensor and, in the second case, the same particles form a more rigid structure due to the magnetic field. A strong assumption is made maintaining the same structure for the two cases, changing only the material properties of the constituent material. This structure is compatible with those observed in the OM MRF fluid by microscopy when applying a magnetic field perpendicular to the resonator plane. Most relevant results are contained in Table 2.

**Figure 11.**Images taken from the COMSOL model interface (

**a**) mesh (centered on the center of the electrode); and (

**b**) geometry.

**Table 2.**Resonant frequency shifts of the quartz resonator calculated for settling and external magnetic field conditions (Note: the changes are referenced to the case of carrier fluid placed on the resonator).

Description | Δfr FEM Results |
---|---|

Magnetic field applied | 950 Hz |

Sedimentation | −1000 Hz |

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Rodriguez-López, J.; Castro, P.; De Vicente, J.; Johannsmann, D.; Elvira, L.; Morillas, J.R.; Montero de Espinosa, F. Colloidal Stability and Magnetic Field-Induced Ordering of Magnetorheological Fluids Studied with a Quartz Crystal Microbalance. *Sensors* **2015**, *15*, 30443-30456.
https://doi.org/10.3390/s151229808

**AMA Style**

Rodriguez-López J, Castro P, De Vicente J, Johannsmann D, Elvira L, Morillas JR, Montero de Espinosa F. Colloidal Stability and Magnetic Field-Induced Ordering of Magnetorheological Fluids Studied with a Quartz Crystal Microbalance. *Sensors*. 2015; 15(12):30443-30456.
https://doi.org/10.3390/s151229808

**Chicago/Turabian Style**

Rodriguez-López, Jaime, Pedro Castro, Juan De Vicente, Diethelm Johannsmann, Luis Elvira, Jose R. Morillas, and Francisco Montero de Espinosa. 2015. "Colloidal Stability and Magnetic Field-Induced Ordering of Magnetorheological Fluids Studied with a Quartz Crystal Microbalance" *Sensors* 15, no. 12: 30443-30456.
https://doi.org/10.3390/s151229808