Multiparametric Magnetic Particle Spectroscopy of CoFe2O4

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C: Physical Processes in Nanomaterials and Nanostructures

Multiparametric Magnetic Particle Spectroscopy of CoFeO Nanoparticles in Viscous Media 2

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Sebastian Draack, Niklas Lucht, Hilke Remmer, Michael Martens, Birgit Fischer, Meinhard Schilling, Frank Ludwig, and Thilo Viereck J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b10763 • Publication Date (Web): 19 Feb 2019 Downloaded from http://pubs.acs.org on February 20, 2019

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The Journal of Physical Chemistry

Multiparametric Magnetic Particle Spectroscopy of CoFe2O4 Nanoparticles in Viscous Media Sebastian Draack,† Niklas Lucht,‡ Hilke Remmer,† Michael Martens,† Birgit Fischer,‡ Meinhard Schilling,† Frank Ludwig,† and Thilo Viereck∗,† †Institute for Electrical Measurement Science and Fundamental Electrical Engineering, TU Braunschweig, D-38106 Braunschweig, Germany ‡Department of Physical Chemistry, University of Hamburg, D-20146 Hamburg, Germany E-mail: [email protected] Abstract The detailed signal generation of the magnetization response of magnetic nanoparticles (MNPs) as a result of externally applied magnetic fields with flux densities of several milliTesla is of high interest for biomedical applications like Magnetic Resonance Imaging (MRI) or Magnetic Particle Imaging (MPI). Although, MNPs are already frequently used as contrast agents or tracer materials, experimental data is rarely compared to model predictions due to distinct deviations. In this contribution, we use a customized Brownian dominated CoFe2O4 particle system to compare experimental Magnetic Particle Spectroscopy (MPS) data with Fokker-Planck simulations considering Brownian relaxation. The influences of viscosity, size distribution, excitation frequency, and field amplitude are studied. We show that the effective magnetic moment and cluster sizes can be determined using a sample viscosity series. As introduced, such particle systems can serve as model systems to evaluate mathematical expressions and to study

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dependences on physical influencing factors. Investigations of defined MNP systems and detailed characterizations enable a wide field of improved diagnosis and therapy applications, e.g. mobility MPI and magnetic hyperthermia.

Introduction Magnetic Particle Spectroscopy (MPS) has been introduced as zero-dimensional characterization method to evaluate the suitability of magnetic nanoparticles (MNPs) as tracer for the medical imaging modality Magnetic Particle Imaging (MPI). 1 Both MPS and MPI utilize sinusoidal magnetic excitation fields with large field amplitudes to force accumulations of magnetic nanoparticles into their non-linear saturation magnetization. Resulting dynamic magnetization responses, represented as higher harmonics in frequency domain, provide information about the nanoparticles’ properties. Please note that the f-space (frequency space) reconstruction in MPI or spectral decomposition in MPS 2 requires as much higher harmonics as possible to accurately reconstruct the desired variables (e.g. spatial distribution in MPI or mixture ratios of different nanoparticle types in MPS) due to well-conditioned linear systems of equations. The linear and the non-linear range of the nanoparticles’ magnetization curve can be addressed by varying the magnetic excitation field amplitude from small to large values. Hence, multiparametric MPS is an important measurement tool for the quantification of non-linear MNP magnetization dynamics. It allows investigations of specific influencing factors like the viscosity of the particles’ environment, temperature, excitation frequency and excitation field amplitude. Additionally to the non-linearity of the magnetization curve, harmonic spectra typically also scale non-linearly with these parameters. There are several publications treating magnetization (e.g. coupled to mobility) of MNPs in dynamic scenarios but most of them are limited to single ratios of lower harmonics, e.g. A5/A3 representing the magnitude ratio of the fifth harmonic to the third harmonic. 3–6 For a comprehensive nanomagnetic understanding and especially for predicting signal generation in mobility Magnetic Particle Imaging (mMPI) as an quantitative medical imaging modality, 7 it is necessary to 2


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investigate the influence of the MNPs’ mobility on the whole spectrum. It is equally important to consider sophisticated mathematical models capable of correctly representing MNP dynamics in dependence of aforementioned parameters. Such models can be validated by means of multiparametric MPS. Additionally, extracted results can be transferred to further medical applications like drug targeting or magnetic hyperthermia. Magnetic hyperthermia uses alternating magnetic fields to generate local heat close to tumor tissue to trigger apoptosis. The heating efficiency depends on frictional and magnetic properties of the MNP. Furthermore, the heating power scales with frequency and magnetic field strength. 8 High magnetic field strengths result in high specific absorption rates (SAR), which is a measure for the magnetic heating ability. The interplay between MNP and the particle environment (e.g. fluid or matrix) is coupled via friction generated by Brownian rotation. Thus, a deep understanding of viscosity dependences at high magnetic field strengths is required to predict behaviors and to optimize MNP for their corresponding applications. In the standard model for the SAR in hyperthermia, the SAR is proportional to the imaginary part of the AC susceptibility. Thus, the analysis of the AC susceptibility of the fundamental and higher harmonics is relevant not only for MPI but also for magnetic hyperthermia. This contribution deals with the investigation of the viscosity dependence of the harmonic response using a set of excitation field frequencies and amplitudes.

Mathematical background Commonly, modeling of MPS or MPI data is introduced by using the simplest available nonlinear model: the quasi-static magnetization of an accumulation of non-interacting magnetic nanoparticles as a function of the applied magnetic field can be described by the Langevin function L(ξ) where ξ =

EM ET

denotes the ratio of magnetic energy EM = mB and ther-

mal energy ET = kB T . Here, EM contains the magnetic net moment m and the externally applied magnetic flux density B, while ET is the product of the Boltzmann constant

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kB ≈ 1.38 × 10−23 J K−1 and the temperature T . 9,10 It is well known that in addition to quantitative MNP concentration, MPS and MPI measurements also hold information about the nanoparticles’ mobility or binding state. 11 To predict dynamic behavior or to deduce physical quantities from measurements, a dynamic mathematical model is required. Externally applied magnetic fields lead to reorientations of the magnetic moments in the nanoparticles. The relaxation behavior is governed by two distinct physical processes: the spatial rotatory reorientation of the magnetic moment affixed to the hydrodynamic shell, called Brownian rotation, and the internal precession-biased realignment of the magnetic moment vector, called Néel relaxation. Both relaxations can be understood as processes of magnetic decays with certain relaxation time constants.

Relaxation times The rotational diffusion coefficient of a rotating sphere following from the Stokes-EinsteinDebye equation is Dr,B =

ET ζ

=

kB T 12–14 . 6ηVh

Under specification of the dynamic viscosity η of

the surrounding Newtonian media and the hydrodynamic volume Vh = 16 πd3h of a spherical particle with hydrodynamic diameter dh , the characteristic zero-field Brownian relaxation time is given by

τB0 =

3ηVh 1 = . 2Dr,B kB T

(1)

The Brownian relaxation time depends on the temperature for two different reasons: the thermal energy barrier and the dynamic viscosity η = η(T ), which significantly decreases with increasing ambient temperature. This relationship can be described by the ArrheniusAndrade equation 15 η(T ) = η0 · exp

4

Ea kB T


(2)

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assuming a given activation energy Ea for most Newtonian fluids and must be considered in simulations. The characteristic zero-field Néel relaxation time τ

τN 0 =

z }|0 { √ π Ms 1 + α 2 KVc = q exp 2Kγα kB T c 2 kKV BT | {z }

1 2Dr,N

(3)

τ0∗

is exponentially dependent on the effective anisotropy energy density constant K including crystal and shape anisotropy, the core volume Vc and the temperature T . The argument of the exponential function

KVc kB T

is the ratio of anisotropy energy and thermal energy. It is

scaled with τ0 ≈ 1 ns, which also depends on the material-specific Gilbert damping parameter α ≈ 0.1 and the gyromagnetic ratio γ ≈ 1.76 × 1011 rad s−1 T−1 of an electron. 16 Further, Ms is the saturation magnetization of the core material. Since both zero-field relaxation times are time constants of two different processes of magnetic decays, the effective relaxation time can be calculated from a superposition of both relaxation processes. Therefore, as a first approximation, the resulting effective relaxation time is generally modeled as a parallel arrangement of both zero-field relaxation processes, in which the smaller value dominates the effective relaxation process: 17

τeff0 =

τB0 · τN 0 . τB0 + τN 0

(4)

It should be noted that equation (4) is only valid for small-field excitations, which usually is not the case for MPS or MPI. Considering the force acting on a MNP, one can well imagine that high magnetic field strengths shorten the relaxation times due to resulting torques. During dynamic excitation there is no real relaxation since the particles are dragged continuously. However, the dynamic behavior can be modeled by continuous time and field strength-dependent relaxation times.

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Field-dependent relaxation times Applied external magnetic fields Hext lead to significant reductions of both the Brownian and the Néel relaxation time depending on the magnetic field strength. Yoshida and Enpuku 18 solved the Fokker-Planck equation for Brownian particles in a sinusoidal magnetic field and derived the field-dependent Brownian relaxation time 19 τB0

τBH = p

1 + 0.126 · ξ 1.72

(5)

by fitting the simulated spectra with a Debye-like model. Similar empirical equations were derived by Fock et al. 20 and Gratz and Tschöpe, 21 which describe the influence of the fielddependent Brownian relaxation time (or characteristic frequency, which is the inverse) on AC susceptibility and optomagnetic signals. The Néel relaxation time is field-dependent as well. However, there is no proved mathematical expression for this dependence. One approach to approximate the field-dependency for a static magnetic field and easy axes aligned parallel to the magnetic field is given by 22

τN H = τ0 · exp

2 ! KVc µ0 HMs · 1− . kB T 2K

(6)

However, it should be noted that in general Brownian and Néel relaxation are coupled processes, which means that always both relaxations occur in parallel. Thus, strictly speaking, the two processes can not be considered as separable.

Magnetic moment of clusters The magnetic moment m of spherical single core magnetic nanoparticles is given by the core volume Vc and the intrinsic saturation magnetization Ms which is expected to be homoge-

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neous and therefore constant across the core volume: ZZZ m=

Ms dVc

Ms =const.

=

(7)

Ms · V c .

If the particles are clustered either synthesis-driven or caused by magnetic or chemical interaction between individual nanoparticles, the effective magnetic moment significantly changes. For low-field excitation, an analytical expression for the effective magnetic moment meff of an ensemble of k noninteracting monodisperse magnetic nanoparticles, each having a magnetic moment m, is given by 23 :

meff

" 2 # √ (k − 1) mB =m k 1+ . 2 3kB T

(8)

Furthermore, dependencies change with dispersity and particle-particle interactions.

Fokker-Planck simulation The Fokker-Planck equation describes the time evolution of the probability density distribution function W = W (mz , t) of the net moment orientation m ~ of an ensemble of magnetic nanoparticles. The one-dimensional simplification of the Fokker-Planck equation 16 for Brownian relaxation ∂ ∂W = Dr,B ∂t ∂mz

1−

m2z

·

∂W mB − W ∂mz kB T

(9)

is assumed to be cylindrically symmetric around axis z. Here, mz is the z-component of the net magnetic moment vector. The distribution function W can be expanded in terms of Legendre polynomials Pn of order n by

W (mz , t) =

∞ X

aB,n (t)Pn (mz ),

n=0

7


(10)


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with aB,n being the coefficients of the Legendre polynomials for Brownian relaxation (index B). This leads to the following set of coupled time-domain ordinary differential equations (ODEs) with the Legendre coefficients for the Brownian relaxation: 16 n(n + 1) mB aB,n−1 aB,n+1 daB,n = · −aB,n + − . dt 2τB0 kB T 2n − 1 2n + 3

(11)

The initial condition is set to aB,0 (t) = 0.5 due to the normalization condition of the probabilR∞ ity density function −∞ W (mz , t)dmz = 1 since the probability that the magnetic moment has a direction is 1. Using the molar particle concentration c with [c] = m−3 , the timedependent magnetization signal 2 M (t) = cMs Vc · aB,1 (t) 3

(12)

can be extracted from the solution of the Fokker-Planck ODEs as it is proportional to the first order moment of the Legendre coefficients.

Dynamic magnetization phases The period diagram in Fig. 1 illustrates the dynamic relationship between magnetization µ0 M of the sample and the externally applied magnetic field µ0 H by using Fokker-Planck simulation data for Brownian relaxation only. The simulation parameters are summarized in the caption of Fig. 1. In phase I, the external magnetic field aligns the magnetic moment into positive direction with increasing field strength H. Initially, the magnetic dipole moment follows the field almost instantaneously, which is due to the fast change of the applied magnetic field and the field dependence of the relaxation time. Subsequently, the sample magnetization reaches a quasi-saturated state. The relaxation time is short, but the change of the externally applied magnetic field is small. In phase II, the external magnetic field still holds the magnetic moment in positive direction, but the field strength H decreases continuously. The relaxation time becomes longer and the magnetic moment relaxes over 8


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µ0 H µ0 M

1 0.5 B / B0

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0

Phase I

Phase II

Phase III

Phase IV

−0.5 −1 1

1.2

1.4

1.6

1.8

2

t / ms

Figure 1: Simulated dynamic magnetization phases of the Brownian relaxation scaled to B0 = µ0 Hmax and B0 = µ0 Ms , respectively. For demonstration, following simulaˆ = 25 mT/µ0 , tion parameters were used: f = 1.0 kHz, Ms = 470 kA m−1 , T = 298.15 K, H η = 1.0 mPa s, dc = 15.5 nm, dh = 38.6 nm. The saturation magnetization was chosen comparably high to emphasize asymmetries of the magnetization. time due to thermal fluctuations since the external force does not suffice to hold the magnetization. These circumstances lead to asymmetric magnetization behavior over time and not mirror-, but point-symmetric M (H)-curves. This means that both branches of the rising and falling edge exhibit different curvatures. Phase III is identical to phase I but the externally applied magnetic field forces the magnetic moment into negative direction. Finally, phase IV is identical to phase III with reversed sign.

Methods MPS is used to study the influence of viscosity on the harmonic spectra. To generate evaluable higher harmonics, a comparatively high magnetic field strength of a sinusoidal magnetic excitation field H(t) as it is shown in Fig. 2 drives the magnetization M (t) of the sample periodically into saturation. The non-linear magnetization curve M (H) leads to an approximately rectangular shaped magnetization response M (t), which is typically measured by using an induction coil for advantages in bandwidth and sensitivity. Since the induced voltage Uind is linked to the magnetic flux φ(t) via Faraday’s law, the measured data are the time derivative of the magnetic flux density from the MNPs and the external excitation 9



1 M / M0

M / M0

1

0

−1 -1

0 H / H0

0

-1

1

0

1 t/T

d dt

(b)

(c) 1 U / U0

2

t/T

2

1

0 -1

0 H / H0

0

-1

1

0

1 t/T

F

(a)

2

(d)

1 S / S0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0

1

3

5

7

9 11 f / f0

13

15

17

19

(e)

Figure 2: Simulated MPS signal generation. A sinusoidal excitation signal (a) drives the magnetization of the magnetic nanoparticles (b) into saturation. The magnetization signal (c) is received by a differential pick-up coil which measures the time-derivative (d) due to Faraday’s law. The resulting magnitude spectrum (e), evaluated via Fourier-Transform, contains only odd higher harmonics since the magnetization signal is symmetric. The higher harmonics are linked via lines to better distinguish series of curves.

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field Uind = − dφ . For a static pick-up coil with w windings pointing in z-axis direction, the dt induced voltage is

Uind

d = −w dt

ZZ

µ0 Mz (Hz (t)) + Hz (t) dA.

(13)

A

The induced voltage resulting from the externally applied magnetic field represents in this case a disturbance variable since only the emitted signal from the magnetic nanoparticles is of interest. To suppress the externally applied magnetic field, a differential coil design is required which compensates the externally applied fundamental frequency. Alternatively, a sharp notch filter could be used. However, the fundamental signal from the particles could then not be detected anymore. The Fourier transform of the induced voltage is proportional to the Fourier transform of the magnetization with

S(f ) = F(U (t))(f ) ∝ F(

∂Mz (t) )(f ) ∝ jω · F(Mz (t))(f ). ∂t

Thus, the qualitative magnitude spectra of both signals are equivalent and contain only the fundamental f0 and odd higher harmonics of the fundamental in absence of DC offset fields due to the non-linearity of the magnetization curve. Please note that a correction of the spectrum by Faraday’s law leads to the spectrum of the magnetization instead of its time derivative. Of course, resulting odd higher harmonics are discrete frequency components. Nevertheless, in illustrations of MPS spectra, the higher harmonics are linked via lines to better distinguish series of curves. A set of higher harmonics, which is more or less unique for a particle system, is called fingerprint and contains information about static and dynamic particle properties. Due to particle dynamics, the MNPs undergo a frequency-dependent phase lag which leads to hysteretic magnetization curves with particle-specific curvatures.

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Magnetic Particle Spectroscopy Setup Our custom-built MPS setup 24 (Fig. 3) provides a wide range of selectable excitation frequenˆ < 30 mT/µ0 . cies 100 Hz ≤ f ≤ 25 kHz and magnetic excitation field strengths of up to H Sample temperatures can be controlled in a broad temperature range −20 ◦C < T < 120 ◦C using a Peltier element coupled to a Shapal™ ceramic housing the sample. The temperature controller enables investigations of the temperature dependence of harmonic spectra as demonstrated for determining the dominating relaxation mechanism. 25

(a) Photography

(b) CAD model

Figure 3: Photography (a) and three-quarter section view of the corresponding computeraided design (CAD) model (b) of the custom-built MPS setup. The AC excitation field is generated with the inner solenoid, which can be seen in the three-quarter section view of the computer-aided design (CAD) model (Inventor, Autodesk, Inc., USA) in Fig. 3b. Optionally, a DC offset field can be superimposed with the outermounted Helmholtz coil pair. A differential pick-up coil is used to measure the change of the magnetization over time. The induced voltage generated by the excitation field is attenuated by a factor of −70 dB. Furthermore, a blank measurement is subtracted for each measurement. Therefore, the fundamental emitted by the MNPs can be measured besides higher harmonics. An analog output device of type PCI-6733 (National Instruments, USA) is used to generate a sinusoidal excitation signal, which is amplified using a AE Techron 7224 power 12


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amplifier. The AC excitation coil reactance is compensated by a series resonant circuit to reduce the impedance to its resistance. Additionally, a low-pass filter stage attenuates the higher harmonics generated by the power amplifier. The excitation current, which is proportional to the generated magnetic field strength, is measured with a CMS3025 (Sensitec GmbH, Germany) current sensor and controlled via software written in LabVIEW and C#. The induced signal in the receive coil is preamplified by a home-built preamplification module using LMH6624MA (Texas Instruments, Inc., USA) operational amplifiers. The amplified induction signal is then acquired via a PCI-6133 (National Instruments, USA) data acquisition card, which is synchronized with the analog output device via National Instrument’s Real-Time System Integration (RTSI) cable.

Materials Measurements are performed using a tailored particle system produced in a similar manufacturing process as described in Nappini et al. 26 The core material is CoFe2 O4 , which is stabilized in water by use of a trisodium citrate shell loaded with tetramethylammonium hydroxide. The narrow core size distribution follows an almost Gaussian distribution due to its manufacturing process, which suppresses ripening of the particles due to the very fast nucleation process and short reaction time. The geometric core size of the particles was determined from transmission electron microscopy (TEM) images (FEI Company Tecnai™ G2 Spirit TWIN) operated at 120 kV with LaB6 Wehnelt cathode by selecting and evaluating 100 particles in ImageJ (National Institutes of Health, USA). 27 The growth process resulting in a distinct lognormal size distribution was interrupted by shell generation. High material-specific anisotropy energy densities K lead to an almost complete suppression of Néel relaxation. The particle system therefore serves as a Brownian-dominated model system. The saturation magnetization amounts to Ms ≈ 380 kA m−1 , which is about 87% of the bulk magnetization Mb,CoFe2 O4 ≈ 437 kA m−1 assuming the mass density of CoFe2 O4 to 13



be ρCoFe2 O4 ≈ 5.3 g cm−3 . 28 The static magnetization curve was measured using a MPMS® XL-5 (Quantum Design, Inc., USA) magnetic property measurement system. The sample was prepared as a suspension sample with a content of 145.10 mg. Measurements were performed for quasi-static flux densities in a range of −5 T ≤ B ≤ 5 T. A detailed excerpt is depicted in Fig. 4. The curve was corrected by the subtraction of an empty sample container 1 0.5 M / Ms

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0 −0.5 −1

−40 −20

0

µ0 H / T

20

40 ·10−3

Figure 4: Static magnetization curve measured with a Quantum Design MPMS® XL-5. measurement and manually by the diamagnetic contribution of water. The measurements show that only low magnetic field strengths (H ≈ 25 mT/µ0 ) are required to saturate the particles’ magnetization to approximately 63% of the saturation magnetization. Thus, MPS excitation field amplitudes reach the non-linear range of the M (H) curve. Scanning electron microscopy (SEM) images (see Fig. 5) using a Zeiss Supra™ 35 (Carl Zeiss Microscopy GmbH, Germany) operated at 10 kV were acquired to evaluate typical cluster sizes in dried diluted solutions. The distribution of the size of individual core diameters was evaluated via TEM images. The resulting histogram was fitted with a log-normal probability density function 

2  ln (d) − µ 1 . f (d, µ, σ) = √ exp  2σ 2 2πσd

(14)

The core median value was identified to µc ≈ 2.731 whereas the corresponding standard deviation was found to be σc ≈ 0.127 (cf. Fig 6). These values correspond to a mean 14


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50 nm

200 nm (a)

(b)

20 nm

24 nm (c)

(d)

Figure 5: TEM (a), CryoTEM (b) (FEI Company Tecnai™ G2 Spirit TWIN) and SEM (Zeiss Supra™ 35) (c, d) images of partly clustered CoFe2 O4 cores in a dried 1:2048 dilution of the stock suspension on a silicon substrate, clustering is probably due to drying process. core diameter of µdc = exp (µc + 0.5 · σc2 ) ≈ 15.5 nm and a standard deviation σdc = p exp (2µc + σc2 ) · (exp (σc2 ) − 1) ≈ 2.0 nm for the proper lognormal size distribution. The hydrodynamic size was evaluated by dynamic light scattering (DLS) via photon crosscorrelation spectroscopy (PCCS) using a Nanophox (Sympatec GmbH, Clausthal-Zellerfeld, Germany) particle size analyzer. A narrow log-normal size distribution was measured as can be seen in Fig. 6. The log-normal distribution with µh ≈ 3.644 and σh ≈ 0.132 corresponds to a median hydrodynamic diameter of µdh ≈ 38.6 nm and the standard deviation was determined to σdh ≈ 5.1 nm.

Dynamic viscosity of water-glycerol mixtures To study the viscosity dependence of the harmonic response of the MNPs, a viscosity series was prepared using water-glycerol mixtures. Since deionized (DI) water was used, particle suspensions are assumed to provide the viscosity valid for pure water even though this as15



TEM data Core size fit PCCS data Hydrodyn. size fit

1 0.8 n / nmax

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0.6 0.4 0.2 0

20

40

60 80 dh / nm

100

120

140

Figure 6: Measured particle size histogram of core sizes (µc = 2.731, σc = 0.127) determined from TEM images and hydrodynamic sizes (µh = 3.644, σh = 0.132) aqcuired with PCCS. sumption could lead to small deviations between presumed and real viscosities. All samples of the series should contain the same iron concentration. The assumption enables one to neglect measurement differences, which could appear due to different particle-particle interactions. Furthermore, acquired data are directly comparable with no need to be scaled. For ˆ 600 µg this reason all samples contain the same suspension content weight of 150.0 mg (≈ CoFe2 O4 particles). The viscosity of a water-glycerol mixture is composed of the viscosity of the single ingredients water and glycerol and depends on concentration as well as temperature. Cheng et al. 29 published a method to approximate the resulting mixture viscosity from the mentioned variables for temperatures given in the units [T°C ] = ◦C and [η] = cP = mPa s. The given equations can easily be converted to the temperature unit [T ] = K by substituting T°C = T + 273.15 K. The mixture viscosity was calculated according to 29 and is given by 1−α ηmix = ηHα 2 O · ηglycerol ,

(15)

where ηH2 O and ηglycerol are the temperature-dependent viscosities of the educts: (−1230 ◦C − T°C ) · T°C = 1.79 mPa s · exp , 36 100 ◦C2 + 360 ◦C · T°C (−1233 ◦C + T°C ) · T°C = 12 100 mPa s · exp 9900 ◦C2 + 70 ◦C · T°C

ηH2 O ηglycerol

16


(16) (17)

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The dependence of the glycerol concentration was fitted by the coefficient

α = 1 − cglycerol +

a · b · cglycerol · (1 − cglycerol ) , a · cglycerol + b · (1 − cglycerol )

(18)

which again depends on two coefficients: a = (0.705 ◦C − 0.0017 · T°C )/◦C

(19)

b = (4.9 ◦C + 0.036 · T°C )/◦C · a2.5

(20)

Preparation of the sample series Particle suspensions were delivered as water (M20 ) and as high-concentrated glycerol mixtures (M1 ) with 99.60 w% (weight percent) glycerol (G). Both samples contain 0.4 w% CoFe2 O4 particles. In the following it is assumed that the particle contribution is negligible for the viscosity, so the viscosity of M20 is identical to water (ηH2 O = 0.9351 mPa s @ 23.0 ◦C). A logarithmic uniformly distributed viscosity series was prepared. A detailed digest of sample mixtures is summarized in Tab. 1. To reduce errors during the preparation process of the viscosity series, the cross-mixing approach was applied in which the most viscous mixture is diluted iteratively. The last sample of water-glycerol samples in the series should have a weight of 700.0 mg to be able to prepare at least three samples of 150.0 mg and to use a higher precision of the scales. Therefore, an iterative calculation results in 8800.88 mg of the first mixture. Subsequently, the series was filled into 350 µL glass vials with a weight of 150.0 mg each, which were sealed with synthetic lids and Parafilm® (Sigma, USA).

Experimental To determine the dominating relaxation process, AC susceptometry (ACS) data were acquired using the institute’s custom-built Rotating Magnetic Field (RMF) setup, 30,31 which

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Table 1: Mixture weight percentages of the particle viscosity series and corresponding viscosity values prepared by using Cheng’s method and the cross-mixing approach for temperature T = 23.0 ◦C. Sample ID

η/mPa s

w% H2 O

w% G

M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12 M13 M14 M15 M16 M17 M18 M19 M20

989.43 686.41 475.05 329.62 228.43 158.21 109.76 76.07 52.73 36.07 25.31 17.55 12.17 8.43 5.85 4.05 2.81 1.95 1.35 0.94

0.40 1.78 1.93 2.07 2.25 2.45 2.66 2.92 3.21 3.55 3.93 4.40 4.93 5.58 6.35 7.30 8.46 9.92 11.76 100.00

99.60 97.82 95.89 95.89 91.57 89.12 86.46 83.54 80.33 76.78 72.85 68.45 63.52 57.94 51.59 44.29 35.83 25.91 14.15 0.00

provides a high sensitivity at low frequencies using fluxgate sensors to measure the magnitude and phase, respectively real and imaginary part, of the magnetic susceptibility. Afterwards, the influence of the viscosity on the harmonic response is studied using the custom-built MPS setup. 24

18


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AC susceptometry measurements AC susceptometry (ACS) measures the susceptibility χ(ωτ ) =

∂M ∂H

of magnetic materials

in response to sinusoidal excitation fields with low magnetic field strength amplitudes of typically around 100 µT/µ0 . 32 Generally, the real and the negative flipped imaginary part of the susceptibility are plotted as a function of the excitation frequency, which is swept during measurement. The imaginary part of a complex number z = Re{z} + iIm{z} is defined positively, but the imaginary part of the Debye model 32–34 χ0 χ(ωτ ) = = χ0 1 + iωτ

1 ωτ −i 2 1 + (ωτ ) 1 + (ωτ )2

,

(21)

where ω = 2πf is the angular frequency, τ defines the relaxation time and χ0 is the peak amplitude of susceptibility, is negative. However, in this contribution, real and imaginary parts of the susceptibility are plotted as usually done for ACS measurement data. Thus, the imaginary part is flipped as mentioned in corresponding figure legends. This must be considered during discussions regarding the signs of real and imaginary parts for higher harmonics in MPS measurements. ACS data of the viscosity series samples acquired with the RMF setup sweeping the excitation frequency in a frequency range 2 Hz ≤ f0 ≤ 7 kHz at ˆ = 200 µT/µ0 are depicted in Fig. 7. The real parts of the an AC magnetic field amplitude H susceptibility are plotted as dashed lines, whereas the imaginary parts are depicted as solid lines. For increasing viscosities (from sample identifier M20 to M1 , cf. Tab. 1), the real part’s inflection point and the imaginary part’s maxima shift to lower frequencies, which conforms to the Debye model for Brownian relaxation describing the magnetic susceptibility. 35 Since both the real and the imaginary part vanish to zero for high frequencies and no second viscosity-independent imaginary part peak is observed, relaxation is due to Brownian rotation only in the investigated frequency range. Obviously, the imaginary part peak value drops and the real part slope close to the inflection point decreases by trend for increasing viscosities and corresponding lower frequencies. A fit of the generalized Debye model including the 19



Figure 7: Measured ACS real parts (Re) as dashed and imaginary parts (Im) as solid lines of the viscosity series acquired with RMF setup colored from blue (M20 : 0 w%G) to violet (M1 : 99.60 w%G) in a colormap describing increasing amount of glycerol and therefore increasing viscosity. Measurements were performed at room temperature T ≈ 23 ◦C.

20


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hydrodynamic size distribution 36 results in a comparatively broad size distribution with a standard deviation of the hydrodynamic size of σdh ≈ 0.45. The change of the standard deviation for increasing viscosities is negligible. Thus, the drop of the susceptibility is based on the fact that particles of bigger sizes are not able to follow the field at high viscosities.

Magnetic Particle Spectroscopy measurements Measurements were performed for different excitation field frequencies and amplitudes, keeping all other influencing parameters constant. Thus, the experimental part is divided into frequency-dependent and field-dependent investigations. Frequency-dependent measurements MPS measurements of the viscosity series samples were acquired at a slightly different room ˆ = 25 mT/µ0 for different excitemperature T ≈ 25 ◦C and magnetic peak field strengths of H tation frequencies f0 = {100 Hz, 500 Hz, 1.0 kHz, 2.5 kHz, 5.0 kHz, 10.0 kHz}. Figure 8 shows harmonic spectra M (f ) corrected by Faraday’s law and corresponding dynamic magnetization curves M (H) for the viscosity series exemplarily for excitation frequencies of f0 = 500 Hz and f0 = 10 kHz. To be able to judge the validity of the fundamental and higher harmonics, the residual error R is shown as well, which represents the measured data of a blank measurement subtracted from another blank measurement. Note that the influence of the residual error is less than a factor of 100 for the pure water sample, but at least it is less than a factor of ten for the whole sample series regarding the fundamental frequency for each excitation frequency. As can be seen from the harmonic spectra for 500 Hz data in Fig. 8, low viscosities contain much more higher harmonics and the slope of the harmonic curve increases by trend for high viscosities. The relationship is reflected in the series of dynamic M (H) curves. Low viscosities lead to steep slopes and small hysteresis since the particles can follow the field almost instantaneously via the Brownian rotation. The particles experience an increasing friction force for higher viscosities. Therefore, the rotation process becomes 21



slower and corresponding magnetization curves show significant hysteresis due to pronounced phase lags between the excitation field and magnetization. The dynamic magnetization of the samples furthermore decreases due to their slow response to the excitation field, so the particles do not reach the equilibrium state before the excitation field changes sign. As

w%G w%G w%G w%G w%G w%G

99.60 w%G 97.82 93.82 w%G 91.57 0 10 86.46 w%G 83.54 10−2 76.78 72.85 −4 10w%G 63.52 w%G 00.511.5257.94 44.29 w%G f / ·10 Hz 435.83 14.15 w%G 00.00

M (f ) / M (f0 )

R 95.89 89.12 80.33 68.45 51.59 25.91

f0 = 500 Hz M / Mmax

M (f ) / M (f0 )

w%G w%G w%G w%G w%G w%G w%G

1

100

10−2

0

10−4 0

0.5

−1

1 1.5 2 ·104 f / Hz (a)

−2

0 2 µ0 H / T ·10−2 (b)

−2

0 2 µ0 H / T ·10−2 (d)

1

100 f0 = 10 kHz M / Mmax

M (f ) / M (f0 )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 50

10−2

0

10−4 0

0.5

1 1.5 2 ·105 f / Hz (c)

−1

Figure 8: Measured MPS magnitude spectra (left) and dynamic M (H) curves (right) of the ˆ = 25 mT/µ0 . viscosity series acquired for f0 = 500 Hz (a,b) and f0 = 10 kHz (c,d) at H Black lines correspond to the residual error R as explained in the main text. Measurements were performed at room temperature T ≈ 25 ◦C. illustrated in Fig. 8 (d), a significant change of the hysteresis is observed for an excitation frequency of f0 = 10 kHz. The distinct torque acting on the magnetic dipoles results in a comparatively steep slope of the dynamic magnetization curves for low viscosities, but as can be seen from M (H) loop apertures for high viscosities, the ability to follow the field is suppressed much more quickly. A different illustration of the measurement data is shown in Fig. 9. Real and imaginary 22


Page 23 of 50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


parts of higher harmonics are plotted as a function of viscosity for different excitation frequencies. Please note that due to a sinusoidal excitation and a cosine reference for frequency domain evaluation via Fast Fourier Transform (FFT), real and imaginary parts are initially interchanged and the imaginary part (corresponding to real part of the Debye model) is inverted compared to the Debye model. This fact has been corrected by using the correct phase reference in a digital lock-in method evaluation. Therefore, the fundamental’s real and imaginary parts correspond to the Debye model. The well-known relationship for the position of the maximum in the imaginary part of the Debye model ωτ = 1 leads to a shift of the imaginary part peak — respectively inflection point of the real part — to lower viscosities for higher excitation frequencies. A similar behavior is observed in simulated and measured data for higher harmonics, which results in a n nωτ = 1 generalization of the relationship 37 with n being a correction factor for each harmonic index n. Certainly, the generalization is only an approximation since the Debye-like behavior of higher harmonics exhibits asymmetries and overshoots in experimental data as well as in simulation data. 18,20 Fig. 9 (b) shows details for higher harmonics of the same measurements as plotted in Fig. 9 (a). In addition to the shift of the imaginary part peak and the inflection point of the real part spectrum, a gain of the imaginary part amplitude, an increasing asymmetry of the imaginary part as well as corresponding overshoots of the real part are observed for increasing frequencies. The amplitude of the imaginary part becomes smaller for higher viscosities and a broadening of the curve is observed. Similar results were found in ACS measurements (cf. Fig. 7). For lower excitation frequencies, the peak position of the imaginary part shifts to higher viscosities. The amplitude decreases because of partly Brownian blocked particle contributions, which are not able to follow the field at high viscosities. A transition to Néel relaxation is not expected due to high anisotropies. Another reason could be the broadening of the distribution of Brownian relaxation times, which may be more significant for high magnetic excitation field strengths compared to ACS measurements. This fact is particularly conspicuous at low excitation frequencies due to the shift of the imaginary part’s peak 23



Figure 9: Measured real (dashed) and imaginary (solid) parts of the Faraday’s law corrected fundamental frequency and higher harmonics as a function of the viscosity acquired with MPS at different excitation frequencies f0 = {500 Hz, 1.0 kHz, 2.5 kHz, 5.0 kHz, 10.0 kHz} for ˆ = 25 mT/µ0 with vanishing opacity for increasing frequencies (a) as indicated by the H annotated arrow direction. Additionally, a detailed excerpt is shown (b). Measurements were performed at room temperature T ≈ 25 ◦C.

24


Page 24 of 50

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and noticeable asymmetries of the imaginary part curves. Field-dependent measurements Further measurements of the samples using a constant excitation frequency and varying field amplitudes enable experimental investigations of the field dependence of the Brownian relaxation time. A set of different excitation amplitudes µ0 H = {5 mT, 10 mT, 15 mT, 20 mT, 25 mT} was used at f0 = 1 kHz for acquisition of the same viscosity series. For Fig. 10 and Fig. 11, the same legend as in Fig. 8 applies. Thus, the black data represents the residual error R, blue lines represent data of particles suspended in water and the transition to violet represents increasing glycerol concentration. As can be seen in Fig. 10(a), higher magnetic field strengths result in more saturated magnetizations and in more pronounced hysteresis of the magnetization loops. Furthermore, even particles dispersed in high-viscous media are significantly attracted for higher magnetic field strengths whereas the signal decreases for low magnetic field strengths as shown in the set of figures 10(b) to 10(f). Due to normalization, differences between magnetic field strengths seem to be weakly pronounced, but become noticeable when focusing on the slope of the dynamic M (H) curves and on the closing of the loop for high viscosities. Corresponding harmonic spectra are depicted in Fig. 11. Increasing field strengths and thus more saturated magnetization curves cause distinct higher harmonics. The dependence of the higher harmonics magnitude on magnetic field strengths is nonlinear and is plotted in Fig. 11(a) for the pure water particle suspension. Since the series of viscosity was measured at each field amplitude, the influence of the field amplitude on the harmonic spectra can be seen in Fig. 11(b) to Fig 11(f). The higher the field amplitude, the more pronounced the spreading of the harmonic response of the viscosity series. Hence, different viscosities can be better distinguished. Figure 12(a) and 12(b) show the measured real and imaginary parts of the fundamental and higher harmonics as a function of the viscosity for different excitation field strengths. Vanishing opacities represent decreasing excitation field amplitudes. Incrementing the field amplitude leads to increasing 25



1

5 mT 10 mT 15 mT 20 mT 25 mT

0

-1

M / Mmax

M / Mmax

1

−2

2 µ0 H / T ·10−2 (a)

M / Mmax

M / Mmax

−5

0 5 µ0 H / T ·10−3 (b)

1

0

−1 −1 −0.5

0

−1

0.5 1 µ0 H / T ·10−2 (c)

0

−1

0 1 µ0 H / T ·10−2 (d)

1 M / Mmax

1

0

−1 −2

0

−1

0

1

M / Mmax

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

−1

0

1 2 µ0 H / T ·10−2 (e)

0

−1

−2

0 2 µ0 H / T ·10−2 (f)

Figure 10: Measured field-dependent dynamic M (H) curves acquired at f0 = 1 kHz. The pure water suspension sample (M20 : 0 w%G) is depicted in (a) for different field amplitudes. Furthermore, the dynamic magnetization curves of the viscosity series are shown for field amplitudes of 5 mT/µ0 (b), 10 mT/µ0 (c), 15 mT/µ0 (d), 20 mT/µ0 (e) and 25 mT/µ0 (f). Measurements were performed at room temperature T ≈ 25 ◦C.

26


Page 26 of 50

Page 27 of 50

100

5 mT 10 mT 15 mT 20 mT 25 mT

10−2

M (f ) / M (f0 )

M (f ) / Mmax

100

10−4 0

1

10−4 0

0.5

1

1.5

2 ·104

1.5

2 ·104

1.5

2 ·104

f / Hz (b) 100 M (f ) / M (f0 )

M (f ) / M (f0 )

10−2

2 3 4 f / Hz ·104 (a)

100

10−2

10−4 0

0.5

1

1.5

f / Hz (c)

10−2

10−4

2 ·104

0

0.5

1 f / Hz (d)

0

0

10 M (f ) / M (f0 )

10 M (f ) / M (f0 )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


10−2

10−4 0

0.5

1 f / Hz (e)

1.5

2 ·104

10−2

10−4 0

0.5

1 f / Hz (f)

Figure 11: Measured field-dependent odd harmonic spectra M (f ) acquired at f0 = 1 kHz. The pure water suspension sample (M20 : 0 w%G) is depicted in (a) for different field amplitudes. Furthermore, the harmonic spectra of the viscosity series are shown for field amplitudes of 5 mT/µ0 (b), 10 mT/µ0 (c), 15 mT/µ0 (d), 20 mT/µ0 (e) and 25 mT/µ0 (f). Measurements were performed at room temperature T ≈ 25 ◦C.

27



Page 28 of 50

signal strength of real and imaginary parts. Furthermore, a shift of the imaginary part’s peak respectively the real part’s inflection point is observed which is due to the field-dependent relaxation time. As can be seen in Fig. 12(b), the asymmetry of the imaginary part and overshoots of the real part increase for increasing field amplitudes. While real and imaginary parts of all higher harmonics vanish to zero for high viscosities, it is obvious that for the fundamental both seem to tend to a non-zero terminal value. Due to the limited measurement window in viscosity, this statement has only limited validity. Currently, there is no satisfying explanation for this observation, but at least the terminal value of the real part indicates superimposed linear diamagnetic or paramagnetic properties of the particles or the solvent. Biases of the measuring instrument can be excluded since the terminal value is frequency-independent as can be seen from Fig. 9. Furthermore, the intra-potential energy barrier based on anisotropy could be the reason for the observation. 38

Analysis of measurement data Similar to using the Debye model in AC susceptometry measurements, the complex-valued fundamental and higher harmonics of the viscosity series can be used in MPS to determine particle properties assuming the nanoviscosity to be the same as the macrorheologically determined one. Within the Debye model, the imaginary part peak occurs for ωτ = 1. For a constant product of angular frequency ω and Brownian relaxation time τ , a change of the viscosity is equivalent to a change of frequency regarding the effects on the measured curves. Therefore, the experimental imaginary part peak of the fundamental, which occurs at η(ωτ = 1) and which is defined as η1 for further discussions, indicates a point including information about the hydrodynamic mean diameter of the particle suspension. A rearrangement of 3η1 Vh ωτ = (2πf ) kB T = 1 using the Brownian zero-field relaxation time from Eq. (1) leads to s dh0 =

3

28

kB T π 2 η1 f


(22)

Page 29 of 50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 12: Measured real (dashed) and imaginary (solid) parts of the Faraday’s law corrected fundamental frequency and higher harmonics as a function ˆ = of the viscosity acquired with MPS at different excitation amplitudes H {25 mT/µ0 , 20 mT/µ0 , 15 mT/µ0 , 10 mT/µ0 , 5 mT/µ0 } for f0 = 1 kHz with vanishing opacity for decreasing amplitudes (a) as indicated by the annotated arrow direction. Additionally, a detailed excerpt is shown (b). Measurements were performed at room temperature T ≈ 25 ◦C.

29



Page 30 of 50

for the estimated hydrodynamic mean diameter. Generally, in applications using comparatively high magnetic field strengths as it is the case for MPS, the field dependence of the relaxation time must be considered. Therefore, Eq. (5) was transposed in the same way to

dhH

v r u 1.72 u meff B u 3 kB T · 1 + 0.126 kB T t = π 2 η1 f

(23)

and results are compared to findings estimated by Eq. (22). Furthermore, the effective magnetic moment meff from Eq. (8) was inserted in Eq. (23) to determine the hydrodynamic diameter dhHk assuming a core cluster of k single crystallites. Both equations were used to determine the hydrodynamic diameter of the particles by identifying the maximum peak of the fundamental’s imaginary part of the viscosity series using a non-negative least square fit to measured data. A determination of the maximum peak related viscosity value of the fundamental’s imaginary part was done for each measurement in the set of parameters (frequency or field strength) by calculating the maximum’s position of a nonlinear least square fit of the parameter η using the imaginary part ωτ Im{η} = − =− 1 + (ωτ )2

6πf Vh kB T

1+

·η

6πf Vh kB T

·η

2 · b

(24)

of the Debye model in Eq. (24) corresponding to the fundamental’s imaginary part in experimental data. Additionally, the parameter b was used as a degree of freedom to fit the peak value. This approach neglects the effect of size distribution, but provides a better result than manual reading due to missing data points at the maximum’s position. Frequency-dependent measurement data The estimated hydrodynamic diameters shown in Tab. 2 and calculated using Eq. (22) and Eq. (23) for single cores show significant deviations from PCCS measurement results (µdh ≈ 38.6 nm), which seem to be due to clustering of particle cores. Including an effective 30


Page 31 of 50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


magnetic moment using Eq. 8 for k particles of a core cluster to calculate the hydrodynamic diameter dhHk , an agreement of simulated and measured data can be achieved by a mix of dimer and trimer structures. Corresponding calculated results can be found in Tab. 2. Here, η1 = η(ωτ = 1) represents the viscosity value which was read at the fit’s position of the fundamental’s imaginary part maximum. It should be noted that Eq. (8) is not strictly valid for this case since it describes the effective cluster moment at low applied magnetic field strengths only. To the authors’ knowledge, there is no expression correcting for the effective magnetic moment of clusters in the high-field regime. Nevertheless, the given equation shows a significant influence and represents an upper boundary since the influence of the field strength should play an even more important role in scenarios of high magnetic field strengths. Fig. 13 (a) shows the fundamental’s imaginary part maximum positions of Table 2: Estimated hydrodynamic diameters from excitation frequency-dependent experiˆ = 25 mT/µ0 . Here, f is the frequency, η1 is the viscosity of the mental data acquired at H fundamental’s imaginary part peak position, dh0 is the hydrodynamic diameter estimated from the zero-field relaxation time and dhH to dhH3 define the estimated hydrodynamic diameter for mono-, di- and trimers including the field-dependence. f /kHz

η1 /mPa s

dh0 /nm

dhH /nm

dhH2 /nm

dhH3 /nm

10.0 5.0 2.5 1.0 0.5 0.1

3.092 6.248 12.625 33.082 73.284 344.449

23.8 23.7 23.6 23.2 22.4 22.9

28.0 27.9 27.8 27.4 26.4 27.0

36.0 35.9 35.8 35.2 34.0 34.7

42.7 42.6 42.4 41.8 40.4 41.2

the viscosity series as a function of the excitation frequency. Furthermore, an interpolated curve as a result of a nonlinear least square fit of r kB T η1 =

1 + 0.126

meff B kB T

1.72

π 2 f d3h

31


(25)


implemented in MathWorks® MATLAB® , which results from a conversion of Eq. (23) is shown as a dashed line. The effective magnetic moment meff and the hydrodynamic diameter dh were used as coefficients to be fitted, whereas the frequency f and the viscosity η were chosen to be the independent and dependent parameters, respectively. The fit results in meff = (3.544 ± 0.066) aAm2 and dh = (39.4 ± 0.2) nm. Error bounds are given for 95% confidence intervals by setting the other variable to the mean value. Using the identified effective magnetic moment meff and Eq. (8), the effective number k of particle cores in clusters can be determined. The relationship is plotted in Fig. 13 (b) supposing the single core diameter to be dc = 15.5 nm as depicted in TEM images. The fit therefore results in an 100

5

exp. fit

4

10−1

k/1

η1 / Pa s

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 32 of 50

10−2

3 2 est. model

1 10−3

0

2

4 6 f / Hz (a)

8

10 3

·10

0

0

2

4

6

8

10

meff /Am2 ·10−18 (b)

Figure 13: Fit of effective magnetic moment meff and hydrodynamic diameter dh using Eq. (25) to frequency-dependent experimental (exp.) data from Tab. 2 (a) and estimated (est.) effective number k of particles in a cluster (Eq. (8)) as a function of the effective magnetic moment meff (b) assuming a single core diameter to be dc = 15.5 nm. effective number of k = 2.683 single cores in a cluster which means that an effective mixture of dimers and trimers exists. Consequently, the effective core diameter significantly differs from the single core diameter. Solving Eq. (7) for dc,eff leads to the effective core diameter r dc,eff =

3

6meff = πMs

r 3

6 · 3.544 × 10−18 Am2 = 26.1 nm. π · 380 kA m−1

(26)

Field-dependent measurement data The experimentally determined fundamental’s imaginary part peaks of the field-dependent data acquired at f0 = 1 kHz are shown in Tab. 3. Furthermore, determined hydrodynamic 32


Page 33 of 50

diameters are given using Eq. (22) as an estimate of the hydrodynamic diameter for zerofield excitation and Eq. (23) including field dependence. Additionally, the hydrodynamic diameter dhHk was evaluated for core clusters of dimers (k = 2) and trimers (k = 3) using Eq. (23) and Eq. (8). The almost linear relationship between magnetic field strength amplitude Table 3: Hydrodynamic diameter estimation from field-dependent experimental data acquired at f = 1 kHz. ˆ µ0 H/mT

η1 /mPa s

dh0 /nm

dhH /nm

dhH2 /nm

dhH3 /nm

5.0 10.0 15.0 20.0 25.0

7.716 14.532 20.658 27.371 36.266

37.7 30.5 27.2 24.7 22.5

38.4 32.1 29.7 28.1 26.5

38.9 33.8 33.1 33.6 34.1

39.5 35.6 36.6 38.7 40.5

ˆ and peak position of the fundamental’s imaginary part η1 is plotted in Fig. 14. Using µ0 H ·10−3

40

5 4

30 k/1

η1 / Pa s

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


20 10 0

exp. fit 0

10

20

3 2 est. model

1 30

ˆ / T ·10−3 µ0 H (a)

0

0

2

4

6

8

10

meff /Am2 ·10−18 (b)

Figure 14: Fit of effective magnetic moment meff and hydrodynamic diameter dh using Eq. (25) to field-dependent experimental (exp.) data from Tab. 3 (a) and estimated (est.) effective number k of particles in a cluster (Eq. (8)) as a function of the effective magnetic moment meff (b) assuming a single core diameter to be dc = 15.5 nm. the same fit as described in the previous section, the effective magnetic moment results in meff = (4.919 ± 0.061) aAm2 and the effective hydrodynamic diameter can be determined to dh = (43.5 ± 1.5) nm. Error bounds for 95% confidence intervals were determined in the same way as for frequency-dependent measurement data. As depicted in Fig. 14 (b), the corresponding effective core cluster size k = 3.318 consists of trimers and quadromers. The resulting effective core diameter dc,eff = 29.1 nm was calculated from Eq. (26), respectively. 33



Page 34 of 50

The field-dependent measurement data lead to slightly larger effective cluster sizes (k = 3.318, dc,eff = 29.1 nm) than estimated from frequency-dependent measurements (k = 2.683, dc,eff = 26.1 nm). Resulting effective core diameters deviate by 5.4% from the average of both values. Measurement data of higher harmonics The shift of the fundamental’s real and imaginary parts can be explained by the Debye model, but there is no established expression for the behavior of higher harmonics. In contrast to the fundamental described by the Debye model, real and imaginary parts of the higher harmonics exhibit overshoots and skewnesses. Thus, no equation can be used to fit the peak position. However, the peak position was determined visually for the third and fifth harmonics for those measurement parameters for which it was possible. For harmonics with orders higher than five, the imaginary part peak position is not in the measurement window for the fielddependent investigations. The viscosity position of the higher harmonics imaginary part peak for the fundamental η1 , the third harmonic η3 and the fifth harmonic η5 are summarized in Tab. 4 and 5.

As mentioned in previous sections, the relationship n nωτ = 1 can

Table 4: Frequency-dependent imaginary part peak positions for the fundamental and higher ˆ = 25 mT/µ0 . Columns containing correction factors 3 = η1 /(3η3 ) and harmonics for H 5 = η1 /(5η5 ) are highlighted. f0 /kHz

η1 /mPa s

η3 /mPa s

η5 /mPa s

η1 /(3η3 )

η1 /(5η5 )

10.0 5.0 2.5 1.0 0.5 0.1

3.092 6.248 12.625 33.082 73.284 344.449

– 1.0 1.946 5.845 17.55 109.7

– – – 2.808 5.845 25.32

– 2.08 2.16 1.89 1.39 1.05

– – – 2.36 2.51 2.72

approximately be assumed for the peak position of higher harmonics with index n. The correction factors 3 = η1 /(3η3 ) and 5 = η1 /(5η5 ) for the third and fifth harmonics are listed 34


Page 35 of 50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 5: Field-dependent imaginary part peak positions for the fundamental and higher harmonics for f0 = 1 kHz. Columns containing correction factors 3 = η1 /(3η3 ) and 5 = η1 /(5η5 ) are highlighted. ˆ µ0 H/mT

η1 /mPa s

η3 /mPa s

η5 /mPa s

η1 /(3η3 )

η1 /(5η5 )

5.0 10.0 15.0 20.0 25.0

7.716 15.59 22.16 25.32 36.54

1.0 2.504 3.317 4.71 5.845

– – 1.155 1.761 2.808

2.57 2.08 2.23 1.79 2.08

– – 3.84 2.88 2.60

in Tab. 4 and 5 for each excitation field amplitude. In average, correction factors 3 ≈ 1.93 and 5 ≈ 2.82 are obtained. However, both measurement data and simulation data show that the shift of higher harmonics is not easily explainable by a single correction factor for each harmonic. Furthermore, overshoots and skewnesses lead to much more complicate dependences.

Simulation of magnetization dynamics Simulation data are used to compare experimental data of higher harmonics to theoretical expectations using the Brownian Fokker-Planck equation model (Eq. (11)) implemented in a Python simulation environment. Harmonics are extracted and evaluated from the simulated time-dependent magnetization signal by using a digital lockin method. Since simulation data shows a transient response in the beginning, the first magnetization periods were removed to reduce significant numerical errors. The particle parameters were chosen as determined from different characterization methods and mentioned in the materials section. The core diameter, which defines the effective magnetic moment using the given saturation magnetization in the simulation environment, was adjusted to match the experimentally determined effective magnetic moment meff from measurement data. To better match the real effective particle diameter, the average core diameter dc,eff = (26.1 nm + 29.1 nm)/2 = 27.6 nm

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and the average hydrodynamic diameter dh,eff = (39.4 nm + 43.5 nm)/2 = 41.5 nm of both the frequency-dependent and the field-dependent measurements are used for the mean core diameter µdc and the mean hydrodynamic diameter µdh of the simulated lognormal size distribution of the particles. Alternatively, the saturation magnetization could be adapted accordingly, but this is physically unfounded. The distributions’ standard deviations were assumed to be the same as measured with TEM and DLS. A representation of the size distribution was implemented by using np particles with certain diameters. All simulation parameters are listed in Tab. 6. Table 6: Fokker-Planck simulation parameters. Parameter

Value

ˆ Magnetic field strength amplitude H Frequency f Temperature T Viscosity η Saturation magnetization Ms Core mean diameter µdc Core standard deviation σc Hydrodynamic mean diameter µdh Hydrodynamic standard deviation σh Number of particles np

25 mT/µ0 1.0 kHz 25 ◦C 1.0 mPa s 380 kA m−1 27.6 nm 0.127 41.5 nm 0.132 1000

Frequency-dependent simulations Real and imaginary parts as a function of viscosity for different excitation frequencies are depicted in Fig. 15. A clear horizontal shift of the real and imaginary parts of the fundamental and higher harmonics is observed for changing frequencies. Contrary to experimental data, simulation data do not show any scaling dependence of the imaginary part amplitudes, which drop for higher frequencies in measurement data. Real and imaginary parts of the higher harmonics show similar behavior in experimental and simulated data regarding asymmetry 36


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Figure 15: Simulated real (dashed) and imaginary (solid) parts of the Faraday’s law corrected fundamental frequency and higher harmonics as a function of the viscosity calculated with Brownian Fokker-Planck equation for different excitation frequencies f0 = ˆ = 25 mT/µ0 with vanishing opacity for {500 Hz, 1.0 kHz, 2.5 kHz, 5.0 kHz, 10.0 kHz} at H increasing frequencies (a) as indicated by the annotated arrow direction. Additionally, a detailed excerpt is shown (b). and overshoots. Nevertheless, these observations are more distinct in simulated data. A decrease in the effect in simulation data can be achieved by broadening the size distribution. This also causes a widening of the real and imaginary part curves themselves and thus fits better to experimentally obtained data. A numerical adaption of the size distribution would lead to a large computational effort and was therefore not performed. However, it can be 37



stated that the effective magnetically observed core and hydrodynamic size distributions are characterized by larger variances than measured by TEM image analysis and DLS.

Field-dependent simulations Real and imaginary parts as a function of viscosity for different excitation field amplitudes are depicted in Fig. 16. A field-dependent shift based on the field dependency of the Brownian relaxation time and an increase of the imaginary part peak values for increasing excitation field amplitudes support experimentally demonstrated observations. Nevertheless, compared to experimental data in Fig. 12, the increase of the imaginary part peak value is more distinct. To be able to discuss the influence of the particle size distribution, further simulation series were performed as described in the next section.

Simulated dependence on hydrodynamic size distribution Real and imaginary parts as a function of viscosity for different hydrodynamic standard deviations are depicted in Fig. 17. All other simulation parameters have been set to the values summarized in Tab. 6. Narrow hydrodynamic size distributions, i.e. small standard deviations, cause Debye-like behavior of the fundamental frequency. Furthermore, real and imaginary parts of higher harmonics exhibit significant asymmetries and overshoots, which are even more pronounced for the real parts. Increasing standard deviations of the hydrodynamic diameter results in vanishing overshoots of the higher harmonics complex values and in particularly distinct asymmetries of the imaginary parts for both the fundamental and higher harmonics. Asymmetries of the imaginary parts, which correlate with decreasing slopes of the real parts, are caused by the superposition of single responses of the particles of a certain size. The relationship furthermore results in decreasing amplitudes of the imaginary part peak values and increasing amplitudes for higher viscosities. Comparisons of experimental data (Fig. 9 and 12) with simulations of hydrodynamic size distributions (Fig. 17) support the conclusion that existing size distributions, which can be read from 38


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Figure 16: Simulated real (dashed) and imaginary (solid) parts of the Faraday’s law corrected fundamental frequency and higher harmonics as a function of the viscosity calculated with Brownian Fokker-Planck equation for different excitation field amplitudes ˆ = {5 mT/µ0 , 10 mT/µ0 , 15 mT/µ0 , 20 mT/µ0 , 25 mT/µ0 } at f0 = 1 kHz with vanishing H opacity for decreasing excitation field amplitudes (a) as indicated by the annotated arrow direction. Additionally, a detailed excerpt is shown (b). real and imaginary part ratios and real part slopes as well as normalized imaginary part peak values, do not fit to estimated size distributions from DLS measurements. Magnetic excitation fields lead to significant changes of the particle size distributions. This effect is underlined by the reduced increase of the fundamental’s imaginary part peak in Fig. 12 compared to simulated data depicted in Fig. 16. The curvature of the real part and the 39



Figure 17: Simulated real (dashed) and imaginary (solid) parts of the Faraday’s law corrected fundamental frequency and higher harmonics as a function of the viscosity calculated with Brownian Fokker-Planck equation for different standard deviations of the hydrodynamic diˆ = 25 mT/µ0 with increasing opacity ameter σdh = {0.1, 0.2, 0.3, 0.4, 0.5} at f0 = 1 kHz and H for increasing standard deviations of the size distributions (a) as indicated by the annotated arrow direction. Additionally, a detailed excerpt is shown (b). Unless otherwise stated, the simulation parameters in Tab. 6 were used. broadening of the imaginary part in experimental data corresponds to simulation data with 0.4 ≤ σdh ≤ 0.5, which matches the determined standard deviation of the hydrodynamic size distribution from ACS measurements.

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Summary and Discussion A CoFe2 O4 particle system was prepared as a viscosity series, for which the logarithmically equidistant dynamic viscosity was adjusted by water-glycerol mixing ratios to investigate the influence on the harmonic response in MPS measurements. Measurements were performed for a set of excitation frequencies f0 = {100 Hz, 500 Hz, 1.0 kHz, 2.5 kHz, ˆ = 25 mT/µ0 of a sinu5.0 kHz, 10.0 kHz} at a constant applied magnetic field amplitude H ˆ = {5 mT/µ0 , soidal excitation field, and for a set of different excitation field amplitudes H 10 mT/µ0 , 15 mT/µ0 , 20 mT/µ0 , 25 mT/µ0 } at an excitation frequency f0 = 1 kHz. Experimental observations were compared to Fokker-Planck simulations. Harmonic magnitude spectra in MPS measurements show a continuous decrease of the higher harmonics’ magnitudes for increasing dynamic viscosity values due to a soaring friction on the particles rotating via Brownian relaxation mechanism. Pure magnitude spectra would neglect half of the information since spectral data are complex-valued. Therefore, phase spectra could be consulted. 4 Although a logarithmic illustration of experimental data in the frequency domain highlights sensitive changes of the higher harmonics, a reconstruction of the dynamic magnetization curves leads to a more common representation of the data and is sufficient to understand the dynamic magnetization behavior. Thus, data representation and illustration should be adapted to the specific application. In this study, viscosity series data were investigated as complex-valued real and imaginary part due to the association with ACS data. The representation as a function of the dynamic viscosity enables a direct comparison to the Debye model for the fundamental. Higher harmonics are compared to simulations based on the Fokker-Planck equation assuming Brownian relaxation only. Experimental data of the real and imaginary part show a clear shift of the curves for increasing viscosities, which suggests a dominating influence of the Brownian relaxation. Since the signal at higher harmonics drops to approximately zero for high viscosities in the investigated frequency range, Néel relaxation has a negligible influence on signal generation for our CoFe2 O4 particle system. Therefore, parametric MPS measurements of the viscosity 41



series allow one to determine the effective magnetic moment meff and the effective hydrodynamic diameter dh,eff of the particle system. The effective core diameter differs significantly from the determined single core diameter. Thus, an effective cluster size of approximately trimers can be assumed. The result matches SEM image evaluations of diluted dried samples and cryoTEM image analyses. The estimated hydrodynamic mean size matches DLS and ACS measurements. The standard deviation of the hydrodynamic size distribution was initially estimated to match DLS measurements. Slopes of the experimental real parts and peak values as well as broadening of experimental imaginary parts indicate a more distinct widening of the size distribution as supported by simulation results. This fact might and seems to be frequency and field dependent. Future investigations will improve the detailed understanding. It should be mentioned that cluster sizes could also vary with concentration. Thus, further concentration-dependent measurements could also improve the understanding of agglomeration formation. Furthermore, the CoFe2 O4 particle system can be used as a Brownian model system to evaluate and compare mathematical expressions, e.g. field dependences, which are rarely compared to experimental data in literature. 10,39 In particular, for the evaluation of MPI as a quantitative imaging modality as well as magnetic hyperthermia and drug targeting approaches, it is of high importance to have a long-term stable and predictable particle model system. A fast-solvable mathematical model predicting the dynamic magnetization behavior of MNP in MPI scenarios, e.g. three-dimensional Lissajous-trajectory excitation, is still an unsolved challenge of research. Together with a full transfer function description of the MPI setup, a reliable model of MNPs including temperature, viscosity or binding state could lead to substantial progress regarding quantity assessment of the imaging modality. Further, a conspicuous reduction of time due to disclaiming acquisitions of system matrices would involve rapid research results and economic feasibility. The results of investigations of physical dependences of particle model systems can be transferred to commercially available particle systems, which typically consist of a wide size distribution and are approved for in vivo ap42


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plications. A description of detection signals can be attributed to certain parameter changes, which give information about the current state of the particles and surrounding fluid or tissue. Mobility information of the particles could further indicate the binding of drugs fixed to the MNP shell, which is important for magnetic drug targeting applications. In addition, magnetic hyperthermia will benefit from improvements of the dynamic heating power of MNP, which depend on cyclic magnetization losses and friction. It should be noted that commercially available particle systems typically show multicore structures. Thus, dynamic magnetization signals always show both Brownian and Néel signal components. Currently, there is no description of underlaying physical signal generation available. 1 The investigation of Brownian-only (due to high effective anisotropy) model systems in this work is a first step towards prediction of the change of detection signals in MPS or MPI. To overcome these challenges, further parameter dependences must be studied and specific signal changes must be attributed to physical relationships like coupled Brownian and Néel relaxation, multicore structures and visco-elastic matrices.

Acknowledgement Financial support by the German Research Foundation (DFG) via SPP1681 (SCHI 383/2-1, VI 892/1-1, LU 800/4-3, FI 1235/2-2) and by the Lower Saxony Ministry for Science and Culture (MWK) via "Niedersächsisches Vorab" through "Quantum- and Nano-Metrology (QUANOMET)" initiative within the project NP-2 and young researcher program is acknowledged. Furthermore, this work was supported by the Braunschweig International Graduate School of Metrology (B-IGSM) and the Laboratory for Emerging Nanometrology (LENA). The authors thank Dr. Dirk Menzel (TU Braunschweig, Institute for Physics of Condensed Matter) for static magnetization curve measurements.

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(36) Ludwig, F.; Häfeli, U.; Schütt, W.; Zborowski, M. Characterization of Magnetic CoreShell Nanoparticle Suspensions Using AC Susceptibility for Frequencies up to 1 MHz. 2010. (37) Wawrzik, T.; Yoshida, T.; Schilling, M.; Ludwig, F. Debye-based frequency-domain magnetization model for magnetic nanoparticles in Magnetic Particle Spectroscopy. IEEE Transactions on Magnetics 2015, 51 . (38) Ludwig, F.; Balceris, C.; Johansson, C. The Anisotropy of the AC Susceptibility of Immobilized Magnetic Nanoparticles—the Influence of Intra-Potential-Well Contribution on the AC Susceptibility Spectrum. IEEE Transactions on Magnetics 2017, 53, 1–4. (39) Remmer, H.; Gratz, M.; Tschope, A.; Ludwig, F. Magnetic Field Dependence of Ni Nanorod Brownian Relaxation. IEEE Transactions on Magnetics 2017, 53, 1–4.

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TOC Graphic

f0

ˆ H

η 1f0

m ~ eff dh

Magnetization M

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


3f0 5f0 7f0 9f0

Frequency f

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1 2 3 4 5 6 7 8 9 10 11 12 13

ˆ H


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η

1f0

m ~ eff dh

Magnetization M

f0

3f0 5f0 7f0 9f0


Frequency f

Multiparametric Magnetic Particle Spectroscopy of CoFe2O4

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