# Remote Sensing of the Nano-Rheological Properties of Soft Materials Using Magnetic Nanoparticles and Magnetic AC Susceptometry

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## Abstract

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## 1. Introduction

^{12}smaller than traditional rheologic measurements. However, it is applied at a much smaller length scale (about 5 orders of magnitude smaller).

## 2. Materials and Methods

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_{4}) MNPs (BNF-dextran, micromod Partikeltechnologie GmbH, Rostock, Germany) of 80 and 100 nm in size were added to the solution. The sample was then cooled down and stored at 8 °C. The prepared viscoelastic medium had 2 wt.% gelatin content and 5 mg/mL MNPs. Deionized (DI) water was used for all experiments.

_{H}, and the viscosity of the fluid η and is given by

_{B}is the Boltzmann constant. From Equation (1), the viscosity of the Newtonian fluid can be extracted from the Brownian relaxation time of the suspended MNPs. A sensitive method to extract the relaxation time is the magnetic AC susceptibility technique. For a more complicated non-Newtonian fluid matrix, e.g., blood, there are theoretical approaches that use the complex AC susceptibility to extract frequency-dependent rheological data. Some examples are the generalized Debye model, which replaces the viscosity with complex viscosity in the dielectric Debye model [11], the Gemant–DiMarzio–Bishop (GDB) model known from dielectric spectroscopy relaxation [21,22], and the Raikher model [23]. The MNPs used in the experiments are multicore nanoparticles with median particle sizes of 80 and 100 nm, and they are thermally blocked and exhibit fully Brownian relaxation behavior. Therefore, they facilitate the investigation of the material properties of complex fluids at the nanometer scale in an optimal way.

## 3. Results and Discussion

#### 3.1. Field Amplitude and Distance

_{B}is the Boltzmann constant, and T is the temperature. At small AC field amplitudes (smaller than 1 mT), this parameter is much smaller than 1, and the field amplitude does not affect Brownian relaxation. The relaxation time of the particles thus follows the widely known Brownian relaxation equation, Equation (1). Figure 2c demonstrates the linear dependence of normalized real components at the excitation frequency of 10 Hz and maximum imaginary components at 185 Hz vs. the distance to the detection coil. Both the real and imaginary components are normalized to the amplitude of the real component at 0 mm and 10 Hz. The decrease in the signal is due to the larger distance between the detection coil and the MNP sample, as previously discussed. In addition, the magnetic excitation field decreases with the increasing distance from the excitation coil, which lowers the sample’s magnetization. Using an F71 Teslameter (Lake Shore Cryotronics Inc., Westerville, OH, USA), the amplitude of the excitation field is measured at different distances from the center of the excitation coil. This dependency is not linear and inversely scales with the cube of the distance. However, in the distance range where the MNP sample is placed (20–40 mm from the center of the excitation coil), the field amplitude linearly depends on the distance. Figure 2d shows the linear dependency of the normalized AC susceptibility signal from 100 nm size MNPs to the magnetic field amplitude at zero distance with respect to the detection coil. Both the real and imaginary components are normalized to the amplitude of the real component at a frequency and field amplitude of 10 Hz and 70 µT, respectively.

#### 3.2. Temperature Dependence

_{T}is the water viscosity at temperature T, and η

_{20}is the water viscosity at 20 Celsius (≈1.0020 mPa.s), with the constant parameters A, B, and C as A = 1.1709, B = 0.001827, and C = −183.07. There are more complicated and accurate formulations for the temperature-dependent viscosity of water; however, the above-mentioned formula is sufficiently correct for our purpose here.

_{H}). The cubic of the ratio of the two particle diameters is then related to the slope of the frequency dependence in Figure 3c as follows:

#### 3.3. Gelatin Solution

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) A photograph of the excitation coil and detection coils used for magnetic AC susceptometry. (

**b**) The schematic diagram shows the measurement setup and the connection of the coils to the lock-in amplifier. (

**c**) A water-jacketed vial connected to a heated circulating water bath is used for changing the temperature of the MNP samples and is placed above the upper detection coil.

**Figure 2.**(

**a**) The real and (

**b**) imaginary components of magnetic AC susceptibility are plotted as a function of frequency at different distances with respect to the detection coil for the 100 nm MNP tracer. (

**c**) Normalized real (at 10 Hz) and imaginary (at 185 Hz) components of the magnetic AC susceptibility of the 100 nm MNPs are plotted against the height measured from the detection coil. The lines are linear fits to the data. The inset shows the relation between the field amplitude and the distance from the center of the excitation coil and is measured using a Tesla meter. The line shows the linear dependence in the distance range where the MNP sample is placed. (

**d**) Normalized real (at 10 Hz) and imaginary (at 185 Hz) components of magnetic AC susceptibility of the 100 nm MNPs versus the field amplitude scale linearly with the field amplitude in the range of 20–70 µT.

**Figure 3.**Normalized AC susceptibility of (

**a**) 80 nm and (

**b**) 100 nm MNP tracers versus magnetic excitation frequency plotted at different temperatures. (

**c**) Frequency of peak imaginary component for both 80 nm and 100 nm MNPs at various temperatures plotted against $T/\eta \left(T\right)$ shows a linear dependence.

**Figure 4.**The real and imaginary components of AC susceptibility are plotted as a function of temperature and frequency for (

**a**,

**b**) MNPs suspended in DI water and (

**c**,

**d**) MNPs in the gelatin matrix. The color code represents the amplitude of the real and imaginary AC susceptibility signals.

**Figure 5.**(

**a**) Viscosity and (

**b**) shear modulus of 100 nm MNPs in a matrix with 2 wt.% gelatin are estimated from AC susceptibility measurements. (

**c**) Viscosity and (

**d**) storage and loss moduli of the MNP–gelatin matrix at 10 Hz, plotted as a function of temperature measured by the rheometer (blue markers) and estimated from the AC susceptibility measurements (red markers). This trend shows the melting process of the gelatin matrix.

**Table 1.**Hydrodynamic particle diameters (volume-weighted) and their geometrical standard deviations at each temperature for both 80 and 100 nm nanoparticles extracted from fitting the AC susceptibility measurement data with the generalized Debye model. The effect of temperature on the viscosity of water is estimated from the empirical formula given by Equation (4). The 80 nm and 100 nm sizes refer to number-weighted hydrodynamic particle diameters.

MNP System | 100 nm | 80 nm | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Temperature (°C) | 20.1 | 23.5 | 27.5 | 33.5 | 35.5 | 40.5 | 45.5 | 20.9 | 24.9 | 32.9 | 37.2 |

Particle median diameter (nm) | 144 | 142 | 140 | 139 | 141 | 139 | 137 | 97 | 95 | 97 | 100 |

Geometrical standard deviation (nm) | 48 | 49 | 46 | 44 | 45 | 47 | 45 | 24 | 28 | 30 | 31 |

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**MDPI and ACS Style**

Sepehri, S.; Andersson, J.; Schaller, V.; Grüttner, C.; Stading, M.; Johansson, C.
Remote Sensing of the Nano-Rheological Properties of Soft Materials Using Magnetic Nanoparticles and Magnetic AC Susceptometry. *Nanomaterials* **2023**, *13*, 67.
https://doi.org/10.3390/nano13010067

**AMA Style**

Sepehri S, Andersson J, Schaller V, Grüttner C, Stading M, Johansson C.
Remote Sensing of the Nano-Rheological Properties of Soft Materials Using Magnetic Nanoparticles and Magnetic AC Susceptometry. *Nanomaterials*. 2023; 13(1):67.
https://doi.org/10.3390/nano13010067

**Chicago/Turabian Style**

Sepehri, Sobhan, Johanna Andersson, Vincent Schaller, Cordula Grüttner, Mats Stading, and Christer Johansson.
2023. "Remote Sensing of the Nano-Rheological Properties of Soft Materials Using Magnetic Nanoparticles and Magnetic AC Susceptometry" *Nanomaterials* 13, no. 1: 67.
https://doi.org/10.3390/nano13010067