Concentration-dependent zero-field magnetic dynamic response of polydisperse ferrofluids

Journal of Magnetism and Magnetic Materials 459 (2018) 252–255

Contents lists available at ScienceDirect

Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm

Research articles

Concentration-dependent zero-field magnetic dynamic response of polydisperse ferrofluids A.O. Ivanov a,b,⇑, S.S. Kantorovich a,c, V.S. Zverev a, A.V. Lebedev d, A.F. Pshenichnikov d, P.J. Camp a,e a

Ural Federal University, Lenin Av. 51, Ekaterinburg 620000, Russian Federation M.N. Mikheev Institute of Metal Physics of UB RAS, S. Kovalevskaya Str. 18, Ekaterinburg 620137, Russian Federation c University of Vienna, Sensengasse 8, 1090 Vienna, Austria d Institute of Continuous Media Mechanics of UB RAS, Academician Korolev Street, 1, Perm 614013, Russian Federation e School of Chemistry, University of Edinburgh, David Brewster Road, Edinburgh EH9 3FJ, Scotland, United Kingdom b

a r t i c l e

i n f o

Article history: Received 20 June 2017 Received in revised form 9 October 2017 Accepted 23 October 2017 Available online 25 October 2017 Keywords: Dynamic initial magnetic susceptibility Dynamic spectra Dipolar interaction Polydispersity Magnetic nanoparticles

a b s t r a c t In this contribution, we discuss the experimental results obtained for seven ferrofluid samples with the same particle size distribution that differ only in concentration of magnetic material. The dynamic response to a weak linearly-polarised probing AC field is measured for each sample at five different temperatures. We investigate Cole-Cole diagrams and phase shifts in order to describe the impact of ferroparticle concentration on the initial magnetic susceptibility. Our findings show that the main contribution comes from the increasing effective viscosity of the ferrofluid that results in the growth of the Brownian relaxation time. This mechanism is very important for the systems containing even a small portion of large magnetic colloids. The influence of dipolar correlations was separately analysed using computer simulations. Ó 2017 Elsevier B.V. All rights reserved.

1. Introduction Nowadays, magnetic nanocolloids are widely used in medical, technological and even art-related applications. For the suspensions of magnetic particles in liquid magnetopassive carriers (known as ferrofluids), the most important feature is the ability to actively react to the applied magnetic field, preserving the fluidity. Thus, any practical or scientific application of a ferrofluid relies on its magnetic properties, both static and dynamic. For the latter, magnetic AC susceptometry was shown to be an adequate technique for its analysis [1–6]. In this paper we focus on the influence of particle concentration on the zero-field magnetic dynamic response of magnetic nanocolloids. Assuming that the temperature is fixed in the experiment, one can name at least three different interrelated mechanisms, through which the ferroparticle concentration manifests itself. The first one, is simply the amplitude of the response, reflected also in the simplest Debye approach [7]. In the ideal gas approximation of the latter model, the real ðv0D Þ and imaginary ðv00D Þ parts of the dynamic initial susceptibility vD ¼ v0D  iv00D are given by the following expressions: ⇑ Corresponding author at: Ural Federal University, Lenin Av. 51, Ekaterinburg 620000, Russian Federation. E-mail address: [email protected] (A.O. Ivanov). https://doi.org/10.1016/j.jmmm.2017.10.089 0304-8853/Ó 2017 Elsevier B.V. All rights reserved.

v0D ðf Þ ¼ v00D ðf Þ ¼

l0 n 3kB T

l0 n 3kB T

Z

1

1 þ ½2pf sðxÞ

0

Z 0

1

1

2

m2 ðxÞpðxÞdx;

ð1Þ

2

m2 ðxÞpðxÞdx:

ð2Þ

2pf sðxÞ 1 þ ½2pf sðxÞ

here, l0 stands for the vacuum magnetic permeability; kB T is the thermal energy; mðxÞ and sðxÞ are the magnetic moment and the characteristic relaxation time of the randomly chosen ferroparticle having the diameter x of the magnetic core; f has a meaning of the probing AC field frequency. These expressions linearly depend on the number density n. However, entering the equations as a prefactor, the impact of n is usually significantly blurred by the intrinsic polydispersity of magnetic nanocolloids here described through the size-distribution pðxÞ. Moreover, the amplitude of the response cancels out in the Debye formalism when applying to the experimental measurements of the phase shift D/ ¼ arctan ðv00 =v0 Þ. The second mechanism stems from the fact that ferrofluids never behave as an ideal superparamagnetic gas. The particles interact via magnetic dipole–dipole interaction, and the interparticle correlations are known to slow down the dynamic response [8–12]. The growth of the magnetic material concentration inevitably leads to the increase of these correlations as confirmed by Brownian Dynamics simulations [13]. Recently, we made the first attempt to analytically describe the contribution of dipolar interactions in the

A.O. Ivanov et al. / Journal of Magnetism and Magnetic Materials 459 (2018) 252–255

253

dynamic spectra of polydisperse magnetic nanocolloids, using perturbation theory [11]:

v0 ðf Þ ¼ v0D ðf Þ þ 

   o 1 n 0 vD ðf Þ 2  v00D ðf Þ 2 ; 3 2 3

ð3Þ



v00 ðf Þ ¼ v00D ðf Þ 1 þ v0D ðf Þ ;

ð4Þ

where both the real ðv0 Þ and imaginary ðv00 Þ parts of the susceptibility are expressed in terms of Debye susceptibilities (1) and (2). Importantly, the expressions (3) and (4) have the quadratic precision in ferroparticle concentration n. The third mechanism is especially pronounced for the systems with a significant part of large particles, exhibiting Brownian relaxation. The latter is determined by the effective viscosity g of the media in which the relaxation takes place. The changes in the effective viscosity take place due to the perturbation in hydrodynamic fields caused by the colloids dispersed: once the concentration gets high enough for hydrodynamic fields of individual colloids to interact, instead of the single-particle relaxation, one observes collective effects. The growth of g with n leads to the overall slow-down of the dynamic response as shown, for example, in Refs. [14,15]. As mentioned above, all the three mechanisms are highly interconnected. To analyse their roles separately at various temperatures is an experimental challenge, related, in the first place, to the polydispersity of regular magnetic nanocolloids. Previously, we proposed to avoid the uncertainties caused by the polydispersity by using an accurate dilution procedure [16]. Here, we analyse the experimental results obtained for a standard magnetite-inkerosene ferrofluid, with nanoparticles in it being stabilised by oleic acid. The polydispersity of our system can be described by the maximum of 9.3 nm and the standard relative width of 0.29. The fraction of particles with x P 18 nm is approximately 1 per cent, i.e. there are not many particles with Brownian relaxation in our samples. Of course, 18 nm is just an approximate border between Brownian and Neél relaxation for magnetite particles, and this value can change depending on temperature. The basic sample was diluted in order to obtain seven samples S1–S7 with the same granulometric composition and different only in n (for experimental details, see, Ref. [12]). At room temperature, the values of the static saturation magnetisation measured in kA/m are: 22.6 (S1); 30.9 (S2); 43.2 (S3); 59.9 (S4); 75.9 (S5); 82.2 (S6); 88.7 (S7). Corresponding values for the volume fraction of magnetic material are 0.047 (S1), 0.065 (S2), 0.09 (S3), 0.125 (S4), 0.159 (S5), 0.172 (S6) and 0.186 (S7). Dynamic susceptibility was measured using thermostated mutual induction bridge [17] in the frequency range 100 Hz –100 kHz. In the experiment, the phase shift was measured at five different temperatures T 1 ¼ 232 K; T 2 ¼ 252 K; T 3 ¼ 273 K; T 4 ¼ 300 K and T 5 ¼ 337 K. Below we provide a detailed analysis of phase shifts and Cole-Cole diagrams. We employ computer simulations to understand the contribution of dipolar correlations to the phase shift. 2. Results and discussions In Fig. 1 we plot Cole-Cole diagrams for experimental samples S1–S7. For high temperature (a) and (b) all the samples are exhibiting qualitatively similar behaviour. The shift here is mainly caused by the growing nanocolloids’ concentration and is realised through the first mechanism described above. Importantly, similar behaviour of Cole-Cole diagrams in this T-range once again confirms the preservation of the granulometric composition on dilution. The kinking behaviour can be seen at any temperature, at T ¼ 273 K, the signs of folding become totally evident and are

Fig. 1. Cole-Cole diagram for experimental samples S1–S7 for the high and intermediate values of T, provided in the figures. The legend is the same for (a)–(d).

especially pronounced for the most diluted samples S1 and S2. On further cooling (d), samples S1–S4 are showing a clear tendency to fold, whereas it is much less pronounced for samples S5–S7. The fact, that these differences occur at the fixed temperature, indicates the concentration-dependent evolution of the relaxation spectrum: the characteristic time of the Neél relaxation could be considered basically unchanged at fixed T, thus, what changes is the typical time for the rotational (Brownian) relaxation that depends on ferrofluid viscosity, and significantly increases with n. Indeed, the results of viscosity measurements for samples S1, S2 and S3 are provided in Table 1 and show the tripling of its value even though the concentration of S1 is almost half that of S3. The folding behaviour is fully observed in Fig. 2 for the lowest temperature. Slight deviations in the folding behaviour for samples S6 and S7 again confirm the dependence of the relaxation spectra on ferroparticle concentration. It is worth mentioning that the linear scaling found for the high-frequency region of the Cole-Cole diagram is evidence of a similar decay for real and imaginary parts of the dynamic susceptibility with f. This scaling becomes evident especially in the log–log representation used in Fig. 2. This scaling, and not the one that could be obtained from Debye approximations Eqs. (1) and (2), underlines again the deviations of our samples from ideal superparamagnetic gas. The impact of growing viscosity can be seen even more clearly in Fig. 3, where we plot the experimental data for the phase shift D/ as a function of frequency f in log scale. Here, we only show the results for samples S1, S3, S5 and S7 as the results for S2 are very close to those for S1; S4 looks similar to S5; and, finally S6 is basically indistinguishable from S7. At T ¼ 337 K (a) and 300 K (b) the curves can be divided into two groups. The value of D/ for low concentrated samples grows monotonically with f, whereas high concentrated ferrofluids exhibit a clear plateau for the intermediate frequencies (f  103 Hz). Similar plateaux can be observed for S5 and S7 at zero degrees Celsius (c). At the lowest temperature (d), the behaviour of D/ changes also for low-concentration samples, and the curves develop a characteristic flat part, but shifted towards higher f, as compared to the high-concentration samples. The inflection point of D/ for S7 is situated lower than those for other samples. The overall trend observed in Fig. 3(a)–(d), namely the shift of the plateaux down and to the left with growing n can be attributed mainly to an increase of the effective viscosity, as the

254

A.O. Ivanov et al. / Journal of Magnetism and Magnetic Materials 459 (2018) 252–255

Table 1 Effective viscosity of low concentrated samples as measured at zero-field condition using capillary viscosimeter. This technique has concentration limitations, so the data for sample S4–S7 is not provided as unreliable.

g, mPas

S1 S2 S3

T1

T2

T3

T4

T5

6.1 9.3 21.1

4.0 5.7 12.6

2.7 3.7 8.0

1.7 2.3 4.7

1.1 1.4 2.9

Fig. 2. Cole-Cole diagram for experimental samples S1–S7 for the lowest value of T in log–log scale. The solid line shows the linear fit v00  v0 .

Fig. 4. Phase shift obtained in Brownian Dynamics simulations for samples Sim 1– Sim 4 (symbols). Theoretical expressions Eqs. (3) and (4) are used to plot the phase shift with solid lines. The dashed line is calculated using Debye approximation Eqs. (1) and (2). We use logarithmic scale for the reduced frequency axis.

tions are not playing any part, and the viscosity of the carrier liquid is scaled out by using a reference time scale of Debye relaxation. In this approach, any changes of the spectra at fixed T are caused by the dipolar correlations in the system and linearly depend on n, i.e. there are only two, out of three, mechanisms left for the concentration to affect the dynamic magnetic response. In order to exclude the first mechanism (Debye linear in n contribution) and to focus on dipolar correlations only, it is convenient to look at the phase shift for monodisperse systems. In Fig. 4, we plot simulation results (symbols) for four different monodisperse dipolar hard sphere fluids with the same intensity of magnetic interaction

Fig. 3. Experimentally measured phase shift for samples S1, S3, S5, S7. Temperatures are provided in the figures. The other samples are not shown for the sake of clarity. Lines are guides for the eye.

frequency range f  103 Hz, corresponds to the Brownian relaxation process. The polydispersity of the experimental samples makes it very difficult to distinguish between the mechanism related to viscosity growth and to dipolar correlations. The latter lead to local fluctuations in particle density and, as such, also slow down the collective relaxation. The presence of this effect can be shown analytically, using Eqs. (3) and (4), by finding the peak frequency of v00 . This maximum of v00 shifts toward lower frequencies with growing n [11]. For a model monodisperse dipolar soft sphere fluid this effect was confirmed by Brownian Dynamics simulations [13]. In the simulations, in contrast to experiment, the hydrodynamic interac-

(l0 m2 =4pkB Tx3 ¼ 1) different only in concentration: in Sim 1 pnx3 =6 ¼ 0:052; in Sim 2 – 0.105; in Sim 3 – 0.210; and in Sim 4 – 0.314. The only trend that can be observed here is the following: the inflection point shifts to the left with growing concentration. To quantify this evolution we use expressions from Eqs. (3) and (4) and plot them with solid lines for low concentrated systems. For higher n the assumptions of this theory are violated, so we do not show the comparison. To underline the importance of dipolar correlations in the behaviour of D/, we also plot the nindependent Debye approximation, which is clearly lower than any of the simulation results. Note that there is no plateau developing in this case, no matter how dense is the system. The reason is that the evolution of the relaxation spectra in simulations is only characterised by the slowdown of a single time inherent to a single particle due to the local correlated microstructures intensified by the growing n.

3. Conclusion In the present contribution we discussed three possible mechanism through which particle concentration can affect the dynamic zero-field magnetic response of ferrocolloids. These mechanisms

A.O. Ivanov et al. / Journal of Magnetism and Magnetic Materials 459 (2018) 252–255

are: (i) linear dependence of the initial dynamic susceptibility on ferroparticle number density, which however cancels out in case of measuring a phase shift; (ii) the growth in the collective relaxation time related to dipolar correlations intensified by the increase in concentration; (iii) the overall slowdown of the particle rotational diffusion due to the concentration growth of the system effective viscosity. We tried to separate the latter highly interweaved mechanisms by analyzing the combination of experimental data for samples obtained on dilution with fixed polydispersity, analytical model based on the perturbation theory and Brownian Dynamics simulations of monodisperse dipolar soft sphere fluids. We showed that the strongest impact of the concentration, for the range of parameters studied here, comes through the growing effective viscosity. The latter leads to a complex transformation of the relaxation time spectra and gives rise to the qualitative changes in Cole-Cole diagrams and phase shifts at fixed temperature. In other words, we report a very strong dependence of the relaxations on the magnetic particle concentration. In general, if one considers the distribution of relaxation times for a ferrofluid with a given polydispersity, its evolution is defined by the changes in temperature and density and related to the redistribution of particles between Néel and Brownian mechanisms. The relaxation time spectra per se as a function of temperature and concentration and its transformations deserve a separate investigation, which we plan to perform in the future. Acknowledgements The research was carried out in the frame work of the grant of the Russian Science Foundation, Grant No 15-12-10003. S.S.K. also acknowledges partial financial support of the Austrian Research Fund (FWF): START-Projekt Y 627-N27 and ETN-COLLDENSE (H2020-MSCA-ITN-2014, Grant No. 642774). References [1] P.C. Fannin, S.W. Charles, The study of a ferrofluid exhibiting both Brownian and Néel relaxation, J. Phys. D: Appl. Phys. 22 (1) (1989) 187.

255

[2] W. Coffey, P. Cregg, D. Crothers, J. Waldron, A. Wickstead, Simple approximate formulae for the magnetic relaxation time of single domain ferromagnetic particles with uniaxial anisotropy, J. Magn. Magn. Mater. 131 (3) (1994) L301– L303. [3] E. Blums, A. Cebers, M.M. Maiorov, Magnetic Fluids, Walter de Gruyter, 1997. [4] Y.L. Raikher, V. Stepanov, Physical aspects of magnetic hyperthermia: lowfrequency ac field absorption in a magnetic colloid, J. Magn. Magn. Mater. 368 (2014) 421. [5] L. Maldonado-Camargo, I. Torres-Daz, A. Chiu-Lam, M. Hernandez, C. Rinaldi, Estimating the contribution of brownian and nel relaxation in a magnetic fluid through dynamic magnetic susceptibility measurements, J. Magn. Magn. Mater. 412 (2016) 223. [6] S.B. Trisnanto, Y. Kitamoto, Nonlinearity of dynamic magnetization in a superparamagnetic clustered-particle suspension with regard to particle rotatability under oscillatory field, J. Magn. Magn. Mater. 400 (2016) 361. [7] P. Debye, Reprinted 1954 in collected papers of Peter J. W. Debye Interscience, New York., Ver. Deut. Phys. Gesell. 15 (1913) 777. [8] A.Y. Zubarev, A.V. Yushkov, Dynamic properties of moderately concentrated magnetic liquids, JETP 87 (1998) 484. [9] B.U. Felderhof, R.B. Jones, Mean field theory of the nonlinear response of an interacting dipolar system with rotational diffusion to an oscillating field, J. Phys.: Condens. Matter 15 (23) (2003) 4011. [10] P. Ilg, S. Hess, Nonequilibrium dynamics and magnetoviscosity of moderately concentrated magnetic liquids: A dynamic mean-field study, Zeitschrift für Naturforschung A 58 (2003) 589–600. [11] 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. [12] 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: a combined experimental and theoretical study, Phys. Chem. Chem. Phys. 18 (2016) 18342–18352. [13] J.O. Sindt, P.J. Camp, S.S. Kantorovich, E.A. Elfimova, A.O. Ivanov, Influence of dipolar interactions on the magnetic susceptibility spectra of ferrofluids, Phys. Rev. E 93 (2016) 063117. [14] S. Odenbach, Magnetoviscous Effects in Ferrofluids, Lect. Notes Phys., Vol. 71, Springer, Berlin, Heidelberg, 2002. [15] A. Lebedev, Viscosity of magnetic fluids must be modified in calculations of dynamic susceptibility, Journal of Magnetism and Magnetic Materials 431 (2017) 30–32, proceedings of the fourteenth International Conference on Magnetic Fluids (ICMF14). [16] 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. [17] A.F. Pshenichnikov, A mutual-inductance bridge for analysis of magnetic fluids, Instrum. Exp. Tech. 50 (4) (2007) 509–514.