- Email: [email protected]
Dynamic magnetogranulometry of ferrofluids
Journal Pre-proofs Dynamic magnetogranulometry of ferrofluids Alexey O. Ivanov, Olga B. Kuznetsova, Philip J. Camp PII: DOI: Reference:
S0304-8853(19)31841-4 https://doi.org/10.1016/j.jmmm.2019.166153 MAGMA 166153
To appear in:
Journal of Magnetism and Magnetic Materials
Revised Date: Accepted Date:
31 October 2019 13 November 2019
Please cite this article as: A.O. Ivanov, O.B. Kuznetsova, P.J. Camp, Dynamic magnetogranulometry of ferrofluids, Journal of Magnetism and Magnetic Materials (2019), doi: https://doi.org/10.1016/j.jmmm.2019.166153
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier B.V.
Dynamic magnetogranulometry of ferrofluidsI Alexey O. Ivanova,∗, Olga B. Kuznetsovaa , Philip J. Campa,b a Department
of Theoretical and Mathematical Physics, Institute of Natural Sciences and Mathematics, Ural Federal University, 51 Lenin Avenue, Ekaterinburg 620000, Russia b School of Chemistry, University of Edinburgh, David Brewster Road, Edinburgh EH9 3FJ, Scotland
Abstract Magnetogranulometry involves analysing the magnetic properties of a material in order to determine the microscopic composition. For ferrofluids, this means determining the number of magnetic particles of particular size and magnetic dipole moment. Previous work has focused on analysing the static initial magnetic susceptibility, χ, using an accurate theory for how χ depends on the Langevin magnetic susceptibility χL , which is a function of the concentration and dipole moment of each particle fraction. Herein, the application of similar techniques to the frequency-dependent magnetic susceptibility, χ(ω), is examined with the assumption of the Brownian rotation mechanism. The usefulness of the analysis relies on the accuracy of the underlying theory. Ignoring interparticle interactions gives the Debye theory. Interactions are taken into account using a modified mean-field theory and a modified-Weiss theory. Using computer-simulation results for known compositions as model ‘experimental’ data, it is shown that it is essential to take interactions into account, and that the modified-Weiss theory provides the most accurate results. Keywords: ferrofluids, dynamic magnetic susceptibility, Fokker-Planck-Brown equation, Brownian dynamics simulations 1. Introduction
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The macroscopic magnetic properties of a ferrofluid are determined by its microscopic composition, which can be summarised in terms of the number of magnetic parti- 30 cles of a certain size, the magnetisation of the particle, the thickness of any non-magnetic layer on each particle, and the viscosity of the carrier liquid. There are several methods of determining the particle-size distribution, including direct microscopy (which is subject to poor statis- 35 tics), light scattering, and magnetogranulometric analysis [1, 2]. Magnetogranulometry involves relating properties such as the magnetisation curve M (H) – including the initial susceptibility χ and the saturation magnetisation M∞ – to the concentrations and magnetic dipole moments 40 of the constituent particles often via the Langevin susceptibility χL and magnetisation curve ML (H) for noninteracting particles. This approach requires a reliable statistical-mechanical theory linking χL and ML (H) for non-interacting particles to χ and M (H) for interacting 45 particles. Various theories have been tested against experimental and computer-simulation results, with the conclusions that: (i) it is essential to take proper account of interactions between particles; and (ii) the so-called modified mean-field theories provide a simple and accurate means 50 of determining the compositions of ferrofluids [3]. I ICMF
The aim of this work is to determine whether is it possible to perform a reliable dynamic magnetogranulometric analysis of the frequency-dependent susceptibility, χ(ω), where ω is the angular frequency of a weak ac magnetic field. This immediately introduces a new problem, namely the dominant reorientation mechanism of the magnetic dipole moments of the constituent particles [4]. For small and/or magnetically soft superparamagnetic particles, the N´eel mechanism dominates [5], while for large and/or magnetically hard ferromagnetic particles, Brownian rotation dominates [6]. Of course, both mechanisms can operate simultaneously [7]. For the Brownian-rotation mechanism, accurate theories of the frequency-dependent magnetic susceptibility χ(ω) have recently been formulated and tested against results from experiments [8, 9] and Brownian-dynamics (BD) simulations [10, 11]. The question then arises [11] – is it possible to perform a reliable dynamic magnetogranulometric analysis of χ(ω) to determine χL and the Brownian rotation time τB for each component of a ferrofluid? In this contribution, existing BD simulation data for monodisperse and bidisperse ferrofluids are analysed using the aforementioned theories, which are analysed critically in terms of the agreement between the fit parameters and the input parameters of the simulations. The theories are summarised briefly in Section 2, the results are presented in Section 3, and Section 4 concludes the article.
2019
∗ Corresponding
author Email address: [email protected] (Alexey O. Ivanov)
Preprint submitted to Journal of Magnetism and Magnetic Materials
November 14, 2019
2. Theory
Figure 1: Dynamic magnetic susceptibilities of monodisperse and bidisperse ferrofluids: (a), (c), and (e) show the real parts; (b), (d),
Consider a polydisperse ferrofluid with n components, and (f) show the imaginary parts. (a) and (b) show results for a where the dipolar coupling constant, volume fraction, Langevinmonodisperse ferrofluid (System M) with λ = 1 and ρσ3 = 0.7 (χL = 2.93). (c) and (d) show results for a bidisperse ferrofluid (System A) susceptibility, and Brownian rotation time for component (j) (j) with χL = 3.20, specified in Table 1. (e) and (f) show results for a (j) (j) (j) (j) j are λ , ϕ , χL = 8λ ϕ , and τB , respectively. bidisperse ferrofluid (System B) with χL = 2.45, specified in Table 1. 1 The ferrofluid is subjected to a weak ac magnetic field The frequency ω is shown in dimensionless form using the Brownian rotation time; in the bidisperse systems, this is the time for the with angular frequency ω. Assuming Brownian rotation (1) small-particle fraction, τB . only, the one-particle orientational distribution function is determined by solving the Fokker-Planck-Brown (FPB) equation [12, 13], from which the linear magnetisation rewas shown that the MW theory provides better predicsponse is computed. All of the technical details have been tions than both the MMF and Debye theories, as tested described in Reference 11. For non-interacting particles, 60 against accurate BD simulation results. Here, χL and τB the result corresponds to the famous Debye (D) theory are treated as fitting parameters, and the extent to which [14]. the magnetic composition can be inferred by fitting theory n (j) X χL to measurement is assessed. χD (ω) = (1) (j) j=1 1 − iωτB The total static susceptibility in the Debye theory is the Pn (j) sum of the individual Langevin values, χL = j=1 χL . The dipole-dipole interaction between particles has been 65 considered at the level of first-order modified mean-field (MMF) theory [8, 9]. This amounts to computing the additional field experienced by a single particle due to the magnetic response of the other particles in the ferrofluid, with an accuracy proportional to χL . The solution of the 70 FPB equation in this case gives the following result, expressed entirely in terms of χD (ω). 1 χMMF (ω) = χD (ω) 1 + χD (ω) (2) 75 3 At this level of theory, the static susceptibility is χMMF = χL (1+χL /3). Finally, Weiss theory involves a self-consistent calculation of the effective field experienced by each particle in the ferrofluid. The solution of the FPB equation in 80 this case is χ(ω) = χD (ω)/[1 − χD (ω)/3]. This theory predicts a divergence in χ(0) at χL = 3 which has never been observed experimentally. To remove this divergence in the dynamic case, the numerator and denominator are multiplied by (1 + χL /3), and the term proportional to χL χD in 85 the denominator is omitted, with the following modifiedWeiss (MW) result, first derived for the monodisperse case in Reference 15. χMW (ω) =
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(1 + 13 χL )χD (ω) 1 + 13 χL − 13 χD (ω)
(3) 90
Note that the static susceptibility is the same as in the MMF theory, but in terms of dynamics, the effective field shows a greater degree of self-consistency without leading 95 to a divergence in the susceptibility. In Reference 11, it 1 For
N particles of a component with dipole moment µ and diameter σ in a fluid with volume V at temperature T , λ = µ0 µ2 /4πkB T σ 3 and ϕ = N πσ 3 /6V , where µ0 is the vacuum permeability, and kB 100 is Boltzmann’s constant. For real ferrofluids, typical values of these parameters are λ ∼ 1, ϕ ∼ 0.1, and hence χL ∼ 1.
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3. Results The application of the Debye, MMF, and MW expressions to monodisperse ferrofluids is considered first. The results of two sets of BD simulations of dipolar soft-sphere particles were reported in Reference 10: λ = 1 and reduced concentration ρσ 3 = 6ϕ/π = 0.1–0.7 (χL = 0.42–2.93); and ρσ 3 = 0.2 and λ = 0.25–3.50 (χL = 0.21–2.93). The simulation results for the dynamic magnetic susceptibility are expressed as a function of the reduced frequency ωτB , hence τB can be taken as the basic unit of time. Each of Equations (1), (2), and (3) was fitted simultaneously to the real and imaginary parts of χ(ω) as measured in the BD simulations, yielding apparent values of χL and τB . An example of this fitting procedure with λ = 1 and ρσ 3 = 0.7 (χL = 2.93) is shown in Figure 1(a) and (b). If there were no interactions between the particles, then the peak in Im χ would be at ωτB = 1. In reality, the peak is shifted to lower frequency because of the interactions between particles, and the resulting positional and orientational correlations slowing down the orientational dynamics. A feature of the MMF expression is that the best-fit curve to Im χ is too sharply peaked, while the Debye and MW expressions fit the simulation data perfectly. The deviations between the fitted and input values of χL and τB are plotted as functions of χL in Figure 2. Of course, any deviations should grow with increasing χL due to the increasing strength of the interparticle interactions. Despite the high quality of the Debye fits, the fit parameters are in error by as much as ∼ 100% in the most strongly interacting systems. The MMF and MW expressions give accurate and essentially equal values of χL , confirming the well-known accuracy of the first-order MMF theory with low-to-moderate values of λ and ρσ 3 . With λ = 1, the error in the fitted value of τB is much smaller with the MW expression than with the MMF expression. With ρσ 3 = 0.2, both the MMF and MW expressions give erroneous results with high values of χL , because the values of λ 1 are beyond the range of validity for first-order
Figure 2: Percentage errors in the Langevin susceptibility [(a) and (c)] and the Brownian rotation time [(b) and (d)] for monodisperse ferrofluids with λ = 1 [(a) and (b)] and ρσ 3 = 0.2 [(c) and (d)]. The results are plotted as functions of the input Langevin susceptibility χL = 4πλρσ 3 /3.
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MMF theory. Overall, the MW theory is more reliable than the MMF theory, and both are far superior to the Debye theory. The extension to polydisperse ferrofluids is considered next. The particle-size distribution may be assumed to be of a particular functional form with a small number of155 fitting parameters. Alternatively, the effects of polydispersity may be described using a bidisperse model; the presence of ‘small’ and ‘large’ particles is often sufficient to describe experimental observations [16]. Table 1 summarises the input parameters for two model bidisperse ferrofluids160 that have been studied using theory and BD simulations [11]. In both Systems A and B, the ‘large-particle’ fraction (2) interacts more strongly than the ‘small-particle’ fraction (1), and the Brownian rotation time is larger for fraction 2 than for fraction 1. System A has been studied over a range of concentrations giving χL = 0.80–3.20, and System B has been studied with χL = 2.45. The ratio165 nale for these choices of parameters is given in Reference 11, but the essential point is that they are treated as unknown parameters to be determined by fitting the Debye, MMF, and MW expressions. Figure 1(c) and (d) show examples of the fitting pro170 cedure to System A in the worst-case scenario, χL = 3.20. It is clear that the Debye, MMF, and MW expressions all give essentially perfect fits to the simulation data. The errors in the fit parameters for System A at all concentrations are shown in Figure 3(a) and (b). The De175 bye expression massively overestimates χL for the largeparticle fraction (errors shown with unfilled symbols), and underestimates the values for the small-particle fraction (errors shown with filled symbols). The error in the total value of χL is also large, positive, and an increasing func180 tion of concentration. The MMF and MW expressions give similar, much smaller, and almost constant errors in χL , and underestimate (overestimate) the values for the small (large) particles. The error in the total value of χL is practically zero, reflecting the accuracy of the MMF theory at the first-order level. The errors in τB show different trends: the Debye and MMF expressions get progressively worse 185 with increasing concentration, while the accuracy of the MW expression remains almost constant, and superior to the others. Finally, System B is an interesting case in which χ(ω) shows distinct contributions from the two particle fractions190 (1) (2) (1) (2) because χL > χL and τB τB [11]. Figure 1(e) and (f) show the results of fitting theory to simulation. All three expressions give almost indistinguishable fits. Figure 3(c) and (d) show that the deviations of the fit-195 ted values from the true values are very different, depend3
Figure 3: Percentage errors in the Langevin susceptibility and [(a) and (c)] and the Brownian rotation times [(b) and (d)] for bidisperse ferrofluids defined in Table 1. Filled symbols are for fraction 1, and unfilled symbols are for fraction 2. (a) and (b) show results for System A plotted as a function of the Langevin susceptibility χL ; the lines in (a) show the errors in the total value of χL . (c) and (d) show results for System B; the crosses in (c) show the errors in the total value of χL .
ing on the level of theory. Firstly, the Debye expression gives the wrong value of χL for each of the fractions, as well as the total. The MMF expression gives much more accurate results, but it underestimates (overestimates) the value for fraction 1 (2); the total value is accurate. The MW expression gives much smaller deviations for each fraction, and again, the total is accurate. For the values of τB , the Debye expression gives large errors for both fractions, while the MMF and MW expressions are more accurate, with the MW expression giving errors ∼ 10% for both fractions. 4. Discussion The outcome from the fits is a set of values of χL and τB . These values do not allow a determination of the particle size, dipolar coupling constant, and numerical concentration; some additional information is required. For example, if one knows the viscosity η of the carrier liquid, and if one assumes the validity of the Stokes-Einstein-Debye relation τB = πησ 3 /2kB T , then σ could be computed. If the saturation magnetisation of the particle material and the non-magnetic layer thickness are known, then the particle dipole moment and λ can be computed. From λ and χL , the particle concentration can be determined. But these calculations come after the experimental data have been fitted, and the initial fitting was the focus of the current work. It was shown that, overall, the MW expression gives more reliable fit parameters than do the Debye and MMF expressions. Moreover, with realistic ferrofluid parameters, the errors from the MW fits do not exceed a few tens of percent. Therefore, it may be feasible to determine the composition of a real ferrofluid using the dynamic magnetogranulometry method demonstrated in this work. Acknowledgements This research was supported by the Russian Science Foundation, Grant No. 15-12-10003. [1] R. W. Chantrell, J. Popplewell, S. W. Charles, Measurements of particle size distribution parameters in ferrofluids, IEEE Trans. Magn. 14 (1978) 975–977. [2] A. F. Pshenichnikov, V. V. Mekhonoshin, A. V. Lebedev, Magneto-granulometric analysis of concentrated ferrocolloids, J. Mag. Magn. Mater. 161 (1996) 94–102. [3] A. O. Ivanov, S. S. Kantorovich, E. N. Reznikov, C. Holm, A. F. Pshenichnikov, A. V. Lebedev, A. Chremos, P. J. Camp, Magnetic properties of polydisperse ferrofluids: A critical comparison between experiment, theory, and computer simulation, Phys. Rev. E 75 (2007) 061405.
Table 1: System parameters for bidisperse ferrofluids, showing values for each fraction. χL is the total Langevin susceptibility, and χ is the value determined in BD simulations [10, 11].
System A A A A B
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λ(1) 1 1 1 1 1
ϕ(1) 0.050 0.100 0.150 0.200 0.181
(1)
χL 0.40 0.80 1.20 1.60 1.45
λ(2) 2 2 2 2 2
ϕ(2) 0.025 0.050 0.075 0.100 0.062
[4] R. E. Rosensweig, Ferrohydrodynamics, Dover Publications, Inc., New York, 1998. [5] I. S. Poperechny, Yu. L. Raikher, V. I. Stepanov, Dynamic magnetic hysteresis in single-domain particles with uniaxial anisotropy, Phys. Rev. B 82 (2010) 174423. [6] Yu. L. Raikher, V. I. Stepanov, Power losses in a suspension of magnetic dipoles under a rotating field, Phys. Rev. E 83 (2011) 021401. [7] Yu. L. Raikher, V. I. Stepanov, Physical aspects of magnetic hyperthermia: Low-frequency ac field absorption in a magnetic colloid, J. Mag. Magn. Mater. 368 (2014) 421–427. [8] A. O. Ivanov, V. S. Zverev, S. S. Kantorovich, Revealing the signature of dipolar interactions in dynamic spectra of polydisperse magnetic nanoparticles, Soft Matter 12 (2016) 3507–3513. [9] A. O. Ivanov, S. S. Kantorovich, V. S. Zverev, E. A. Elfimova, A. V. Lebedev, A. F. Pshenichnikov, Temperature-dependent dynamic correlations in suspensions of magnetic nanoparticles in a broad range of concentrations: combined experimental and theoretical study, Phys. Chem. Chem. Phys. 18 (2016) 18342– 18352. [10] J. O. Sindt, P. J. Camp, S. S. Kantorovich, E. A. Elfimova, A. O. Ivanov, The influence of dipolar interactions on the magnetic susceptibility spectra of ferrofluids, Phys. Rev. E 93 (2016) 063117. [11] A. O. Ivanov, P. J. Camp, Theory of the dynamic magnetic susceptibility of ferrofluids, Phys. Rev. E 98 (2018) 050602(R). [12] W. F. Brown, Jr., Thermal fluctuations of a singledomain particle, J. Appl. Phys. 34 (1963) 1319–1320. [13] W. F. Brown, Jr., Thermal fluctuation of fine ferromagnetic particles, IEEE Trans. Magn. 15 (1979) 1196–1208. [14] P. Debye, Polar Molecules, Chemical Catalog Company, New York, 1929. [15] A. Yu. Zubarev, A. V. Yushkov, Dynamic properties of moderately concentrated magnetic liquids, J. Exp. Theor. Phys. 87 (1998) 484–493. [16] A. O. Ivanov, S. S. Kantorovich, Chain aggregate structure and magnetic birefringence in polydisperse ferrofluids, Phys. Rev. E 70 (2004) 021401.
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(2)
χL 0.40 0.80 1.20 1.60 1.00
(2)
(1)
τB /τB 2.370 2.370 2.370 2.370 10
χL 0.80 1.60 2.40 3.20 2.45
χ 1.03 2.47 4.34 6.68 4.50
S0304-8853(19)31841-4 https://doi.org/10.1016/j.jmmm.2019.166153 MAGMA 166153
To appear in:
Journal of Magnetism and Magnetic Materials
Revised Date: Accepted Date:
31 October 2019 13 November 2019
Please cite this article as: A.O. Ivanov, O.B. Kuznetsova, P.J. Camp, Dynamic magnetogranulometry of ferrofluids, Journal of Magnetism and Magnetic Materials (2019), doi: https://doi.org/10.1016/j.jmmm.2019.166153
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier B.V.
Dynamic magnetogranulometry of ferrofluidsI Alexey O. Ivanova,∗, Olga B. Kuznetsovaa , Philip J. Campa,b a Department
of Theoretical and Mathematical Physics, Institute of Natural Sciences and Mathematics, Ural Federal University, 51 Lenin Avenue, Ekaterinburg 620000, Russia b School of Chemistry, University of Edinburgh, David Brewster Road, Edinburgh EH9 3FJ, Scotland
Abstract Magnetogranulometry involves analysing the magnetic properties of a material in order to determine the microscopic composition. For ferrofluids, this means determining the number of magnetic particles of particular size and magnetic dipole moment. Previous work has focused on analysing the static initial magnetic susceptibility, χ, using an accurate theory for how χ depends on the Langevin magnetic susceptibility χL , which is a function of the concentration and dipole moment of each particle fraction. Herein, the application of similar techniques to the frequency-dependent magnetic susceptibility, χ(ω), is examined with the assumption of the Brownian rotation mechanism. The usefulness of the analysis relies on the accuracy of the underlying theory. Ignoring interparticle interactions gives the Debye theory. Interactions are taken into account using a modified mean-field theory and a modified-Weiss theory. Using computer-simulation results for known compositions as model ‘experimental’ data, it is shown that it is essential to take interactions into account, and that the modified-Weiss theory provides the most accurate results. Keywords: ferrofluids, dynamic magnetic susceptibility, Fokker-Planck-Brown equation, Brownian dynamics simulations 1. Introduction
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The macroscopic magnetic properties of a ferrofluid are determined by its microscopic composition, which can be summarised in terms of the number of magnetic parti- 30 cles of a certain size, the magnetisation of the particle, the thickness of any non-magnetic layer on each particle, and the viscosity of the carrier liquid. There are several methods of determining the particle-size distribution, including direct microscopy (which is subject to poor statis- 35 tics), light scattering, and magnetogranulometric analysis [1, 2]. Magnetogranulometry involves relating properties such as the magnetisation curve M (H) – including the initial susceptibility χ and the saturation magnetisation M∞ – to the concentrations and magnetic dipole moments 40 of the constituent particles often via the Langevin susceptibility χL and magnetisation curve ML (H) for noninteracting particles. This approach requires a reliable statistical-mechanical theory linking χL and ML (H) for non-interacting particles to χ and M (H) for interacting 45 particles. Various theories have been tested against experimental and computer-simulation results, with the conclusions that: (i) it is essential to take proper account of interactions between particles; and (ii) the so-called modified mean-field theories provide a simple and accurate means 50 of determining the compositions of ferrofluids [3]. I ICMF
The aim of this work is to determine whether is it possible to perform a reliable dynamic magnetogranulometric analysis of the frequency-dependent susceptibility, χ(ω), where ω is the angular frequency of a weak ac magnetic field. This immediately introduces a new problem, namely the dominant reorientation mechanism of the magnetic dipole moments of the constituent particles [4]. For small and/or magnetically soft superparamagnetic particles, the N´eel mechanism dominates [5], while for large and/or magnetically hard ferromagnetic particles, Brownian rotation dominates [6]. Of course, both mechanisms can operate simultaneously [7]. For the Brownian-rotation mechanism, accurate theories of the frequency-dependent magnetic susceptibility χ(ω) have recently been formulated and tested against results from experiments [8, 9] and Brownian-dynamics (BD) simulations [10, 11]. The question then arises [11] – is it possible to perform a reliable dynamic magnetogranulometric analysis of χ(ω) to determine χL and the Brownian rotation time τB for each component of a ferrofluid? In this contribution, existing BD simulation data for monodisperse and bidisperse ferrofluids are analysed using the aforementioned theories, which are analysed critically in terms of the agreement between the fit parameters and the input parameters of the simulations. The theories are summarised briefly in Section 2, the results are presented in Section 3, and Section 4 concludes the article.
2019
∗ Corresponding
author Email address: [email protected] (Alexey O. Ivanov)
Preprint submitted to Journal of Magnetism and Magnetic Materials
November 14, 2019
2. Theory
Figure 1: Dynamic magnetic susceptibilities of monodisperse and bidisperse ferrofluids: (a), (c), and (e) show the real parts; (b), (d),
Consider a polydisperse ferrofluid with n components, and (f) show the imaginary parts. (a) and (b) show results for a where the dipolar coupling constant, volume fraction, Langevinmonodisperse ferrofluid (System M) with λ = 1 and ρσ3 = 0.7 (χL = 2.93). (c) and (d) show results for a bidisperse ferrofluid (System A) susceptibility, and Brownian rotation time for component (j) (j) with χL = 3.20, specified in Table 1. (e) and (f) show results for a (j) (j) (j) (j) j are λ , ϕ , χL = 8λ ϕ , and τB , respectively. bidisperse ferrofluid (System B) with χL = 2.45, specified in Table 1. 1 The ferrofluid is subjected to a weak ac magnetic field The frequency ω is shown in dimensionless form using the Brownian rotation time; in the bidisperse systems, this is the time for the with angular frequency ω. Assuming Brownian rotation (1) small-particle fraction, τB . only, the one-particle orientational distribution function is determined by solving the Fokker-Planck-Brown (FPB) equation [12, 13], from which the linear magnetisation rewas shown that the MW theory provides better predicsponse is computed. All of the technical details have been tions than both the MMF and Debye theories, as tested described in Reference 11. For non-interacting particles, 60 against accurate BD simulation results. Here, χL and τB the result corresponds to the famous Debye (D) theory are treated as fitting parameters, and the extent to which [14]. the magnetic composition can be inferred by fitting theory n (j) X χL to measurement is assessed. χD (ω) = (1) (j) j=1 1 − iωτB The total static susceptibility in the Debye theory is the Pn (j) sum of the individual Langevin values, χL = j=1 χL . The dipole-dipole interaction between particles has been 65 considered at the level of first-order modified mean-field (MMF) theory [8, 9]. This amounts to computing the additional field experienced by a single particle due to the magnetic response of the other particles in the ferrofluid, with an accuracy proportional to χL . The solution of the 70 FPB equation in this case gives the following result, expressed entirely in terms of χD (ω). 1 χMMF (ω) = χD (ω) 1 + χD (ω) (2) 75 3 At this level of theory, the static susceptibility is χMMF = χL (1+χL /3). Finally, Weiss theory involves a self-consistent calculation of the effective field experienced by each particle in the ferrofluid. The solution of the FPB equation in 80 this case is χ(ω) = χD (ω)/[1 − χD (ω)/3]. This theory predicts a divergence in χ(0) at χL = 3 which has never been observed experimentally. To remove this divergence in the dynamic case, the numerator and denominator are multiplied by (1 + χL /3), and the term proportional to χL χD in 85 the denominator is omitted, with the following modifiedWeiss (MW) result, first derived for the monodisperse case in Reference 15. χMW (ω) =
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(1 + 13 χL )χD (ω) 1 + 13 χL − 13 χD (ω)
(3) 90
Note that the static susceptibility is the same as in the MMF theory, but in terms of dynamics, the effective field shows a greater degree of self-consistency without leading 95 to a divergence in the susceptibility. In Reference 11, it 1 For
N particles of a component with dipole moment µ and diameter σ in a fluid with volume V at temperature T , λ = µ0 µ2 /4πkB T σ 3 and ϕ = N πσ 3 /6V , where µ0 is the vacuum permeability, and kB 100 is Boltzmann’s constant. For real ferrofluids, typical values of these parameters are λ ∼ 1, ϕ ∼ 0.1, and hence χL ∼ 1.
2
3. Results The application of the Debye, MMF, and MW expressions to monodisperse ferrofluids is considered first. The results of two sets of BD simulations of dipolar soft-sphere particles were reported in Reference 10: λ = 1 and reduced concentration ρσ 3 = 6ϕ/π = 0.1–0.7 (χL = 0.42–2.93); and ρσ 3 = 0.2 and λ = 0.25–3.50 (χL = 0.21–2.93). The simulation results for the dynamic magnetic susceptibility are expressed as a function of the reduced frequency ωτB , hence τB can be taken as the basic unit of time. Each of Equations (1), (2), and (3) was fitted simultaneously to the real and imaginary parts of χ(ω) as measured in the BD simulations, yielding apparent values of χL and τB . An example of this fitting procedure with λ = 1 and ρσ 3 = 0.7 (χL = 2.93) is shown in Figure 1(a) and (b). If there were no interactions between the particles, then the peak in Im χ would be at ωτB = 1. In reality, the peak is shifted to lower frequency because of the interactions between particles, and the resulting positional and orientational correlations slowing down the orientational dynamics. A feature of the MMF expression is that the best-fit curve to Im χ is too sharply peaked, while the Debye and MW expressions fit the simulation data perfectly. The deviations between the fitted and input values of χL and τB are plotted as functions of χL in Figure 2. Of course, any deviations should grow with increasing χL due to the increasing strength of the interparticle interactions. Despite the high quality of the Debye fits, the fit parameters are in error by as much as ∼ 100% in the most strongly interacting systems. The MMF and MW expressions give accurate and essentially equal values of χL , confirming the well-known accuracy of the first-order MMF theory with low-to-moderate values of λ and ρσ 3 . With λ = 1, the error in the fitted value of τB is much smaller with the MW expression than with the MMF expression. With ρσ 3 = 0.2, both the MMF and MW expressions give erroneous results with high values of χL , because the values of λ 1 are beyond the range of validity for first-order
Figure 2: Percentage errors in the Langevin susceptibility [(a) and (c)] and the Brownian rotation time [(b) and (d)] for monodisperse ferrofluids with λ = 1 [(a) and (b)] and ρσ 3 = 0.2 [(c) and (d)]. The results are plotted as functions of the input Langevin susceptibility χL = 4πλρσ 3 /3.
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145
150
MMF theory. Overall, the MW theory is more reliable than the MMF theory, and both are far superior to the Debye theory. The extension to polydisperse ferrofluids is considered next. The particle-size distribution may be assumed to be of a particular functional form with a small number of155 fitting parameters. Alternatively, the effects of polydispersity may be described using a bidisperse model; the presence of ‘small’ and ‘large’ particles is often sufficient to describe experimental observations [16]. Table 1 summarises the input parameters for two model bidisperse ferrofluids160 that have been studied using theory and BD simulations [11]. In both Systems A and B, the ‘large-particle’ fraction (2) interacts more strongly than the ‘small-particle’ fraction (1), and the Brownian rotation time is larger for fraction 2 than for fraction 1. System A has been studied over a range of concentrations giving χL = 0.80–3.20, and System B has been studied with χL = 2.45. The ratio165 nale for these choices of parameters is given in Reference 11, but the essential point is that they are treated as unknown parameters to be determined by fitting the Debye, MMF, and MW expressions. Figure 1(c) and (d) show examples of the fitting pro170 cedure to System A in the worst-case scenario, χL = 3.20. It is clear that the Debye, MMF, and MW expressions all give essentially perfect fits to the simulation data. The errors in the fit parameters for System A at all concentrations are shown in Figure 3(a) and (b). The De175 bye expression massively overestimates χL for the largeparticle fraction (errors shown with unfilled symbols), and underestimates the values for the small-particle fraction (errors shown with filled symbols). The error in the total value of χL is also large, positive, and an increasing func180 tion of concentration. The MMF and MW expressions give similar, much smaller, and almost constant errors in χL , and underestimate (overestimate) the values for the small (large) particles. The error in the total value of χL is practically zero, reflecting the accuracy of the MMF theory at the first-order level. The errors in τB show different trends: the Debye and MMF expressions get progressively worse 185 with increasing concentration, while the accuracy of the MW expression remains almost constant, and superior to the others. Finally, System B is an interesting case in which χ(ω) shows distinct contributions from the two particle fractions190 (1) (2) (1) (2) because χL > χL and τB τB [11]. Figure 1(e) and (f) show the results of fitting theory to simulation. All three expressions give almost indistinguishable fits. Figure 3(c) and (d) show that the deviations of the fit-195 ted values from the true values are very different, depend3
Figure 3: Percentage errors in the Langevin susceptibility and [(a) and (c)] and the Brownian rotation times [(b) and (d)] for bidisperse ferrofluids defined in Table 1. Filled symbols are for fraction 1, and unfilled symbols are for fraction 2. (a) and (b) show results for System A plotted as a function of the Langevin susceptibility χL ; the lines in (a) show the errors in the total value of χL . (c) and (d) show results for System B; the crosses in (c) show the errors in the total value of χL .
ing on the level of theory. Firstly, the Debye expression gives the wrong value of χL for each of the fractions, as well as the total. The MMF expression gives much more accurate results, but it underestimates (overestimates) the value for fraction 1 (2); the total value is accurate. The MW expression gives much smaller deviations for each fraction, and again, the total is accurate. For the values of τB , the Debye expression gives large errors for both fractions, while the MMF and MW expressions are more accurate, with the MW expression giving errors ∼ 10% for both fractions. 4. Discussion The outcome from the fits is a set of values of χL and τB . These values do not allow a determination of the particle size, dipolar coupling constant, and numerical concentration; some additional information is required. For example, if one knows the viscosity η of the carrier liquid, and if one assumes the validity of the Stokes-Einstein-Debye relation τB = πησ 3 /2kB T , then σ could be computed. If the saturation magnetisation of the particle material and the non-magnetic layer thickness are known, then the particle dipole moment and λ can be computed. From λ and χL , the particle concentration can be determined. But these calculations come after the experimental data have been fitted, and the initial fitting was the focus of the current work. It was shown that, overall, the MW expression gives more reliable fit parameters than do the Debye and MMF expressions. Moreover, with realistic ferrofluid parameters, the errors from the MW fits do not exceed a few tens of percent. Therefore, it may be feasible to determine the composition of a real ferrofluid using the dynamic magnetogranulometry method demonstrated in this work. Acknowledgements This research was supported by the Russian Science Foundation, Grant No. 15-12-10003. [1] R. W. Chantrell, J. Popplewell, S. W. Charles, Measurements of particle size distribution parameters in ferrofluids, IEEE Trans. Magn. 14 (1978) 975–977. [2] A. F. Pshenichnikov, V. V. Mekhonoshin, A. V. Lebedev, Magneto-granulometric analysis of concentrated ferrocolloids, J. Mag. Magn. Mater. 161 (1996) 94–102. [3] A. O. Ivanov, S. S. Kantorovich, E. N. Reznikov, C. Holm, A. F. Pshenichnikov, A. V. Lebedev, A. Chremos, P. J. Camp, Magnetic properties of polydisperse ferrofluids: A critical comparison between experiment, theory, and computer simulation, Phys. Rev. E 75 (2007) 061405.
Table 1: System parameters for bidisperse ferrofluids, showing values for each fraction. χL is the total Langevin susceptibility, and χ is the value determined in BD simulations [10, 11].
System A A A A B
200
205
210
215
220
225
230
235
λ(1) 1 1 1 1 1
ϕ(1) 0.050 0.100 0.150 0.200 0.181
(1)
χL 0.40 0.80 1.20 1.60 1.45
λ(2) 2 2 2 2 2
ϕ(2) 0.025 0.050 0.075 0.100 0.062
[4] R. E. Rosensweig, Ferrohydrodynamics, Dover Publications, Inc., New York, 1998. [5] I. S. Poperechny, Yu. L. Raikher, V. I. Stepanov, Dynamic magnetic hysteresis in single-domain particles with uniaxial anisotropy, Phys. Rev. B 82 (2010) 174423. [6] Yu. L. Raikher, V. I. Stepanov, Power losses in a suspension of magnetic dipoles under a rotating field, Phys. Rev. E 83 (2011) 021401. [7] Yu. L. Raikher, V. I. Stepanov, Physical aspects of magnetic hyperthermia: Low-frequency ac field absorption in a magnetic colloid, J. Mag. Magn. Mater. 368 (2014) 421–427. [8] A. O. Ivanov, V. S. Zverev, S. S. Kantorovich, Revealing the signature of dipolar interactions in dynamic spectra of polydisperse magnetic nanoparticles, Soft Matter 12 (2016) 3507–3513. [9] A. O. Ivanov, S. S. Kantorovich, V. S. Zverev, E. A. Elfimova, A. V. Lebedev, A. F. Pshenichnikov, Temperature-dependent dynamic correlations in suspensions of magnetic nanoparticles in a broad range of concentrations: combined experimental and theoretical study, Phys. Chem. Chem. Phys. 18 (2016) 18342– 18352. [10] J. O. Sindt, P. J. Camp, S. S. Kantorovich, E. A. Elfimova, A. O. Ivanov, The influence of dipolar interactions on the magnetic susceptibility spectra of ferrofluids, Phys. Rev. E 93 (2016) 063117. [11] A. O. Ivanov, P. J. Camp, Theory of the dynamic magnetic susceptibility of ferrofluids, Phys. Rev. E 98 (2018) 050602(R). [12] W. F. Brown, Jr., Thermal fluctuations of a singledomain particle, J. Appl. Phys. 34 (1963) 1319–1320. [13] W. F. Brown, Jr., Thermal fluctuation of fine ferromagnetic particles, IEEE Trans. Magn. 15 (1979) 1196–1208. [14] P. Debye, Polar Molecules, Chemical Catalog Company, New York, 1929. [15] A. Yu. Zubarev, A. V. Yushkov, Dynamic properties of moderately concentrated magnetic liquids, J. Exp. Theor. Phys. 87 (1998) 484–493. [16] A. O. Ivanov, S. S. Kantorovich, Chain aggregate structure and magnetic birefringence in polydisperse ferrofluids, Phys. Rev. E 70 (2004) 021401.
4
(2)
χL 0.40 0.80 1.20 1.60 1.00
(2)
(1)
τB /τB 2.370 2.370 2.370 2.370 10
χL 0.80 1.60 2.40 3.20 2.45
χ 1.03 2.47 4.34 6.68 4.50