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Characterization of aggregate state of polydisperse ferrofluids: Some aspects of anisotropy analysis of 2D SAXS in magnetic field
Accepted Manuscript Characterization of aggregate state of polydisperse ferrofluids: some aspects of anisotropy analysis of 2D SAXS in magnetic field A.A. Veligzhanin, D.I. Frey, A.V. Shulenina, A.Yu. Gruzinov, Ya.V. Zubavichus, M.V. Avdeev PII: DOI: Reference:
S0304-8853(17)32068-1 https://doi.org/10.1016/j.jmmm.2017.10.052 MAGMA 63266
To appear in:
Journal of Magnetism and Magnetic Materials
Received Date: Revised Date: Accepted Date:
5 July 2017 10 October 2017 13 October 2017
Please cite this article as: A.A. Veligzhanin, D.I. Frey, A.V. Shulenina, A.Yu. Gruzinov, Ya.V. Zubavichus, M.V. Avdeev, Characterization of aggregate state of polydisperse ferrofluids: some aspects of anisotropy analysis of 2D SAXS in magnetic field, Journal of Magnetism and Magnetic Materials (2017), doi: https://doi.org/10.1016/ j.jmmm.2017.10.052
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. 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.
Characterization of aggregate state of polydisperse ferrofluids: some aspects of anisotropy analysis of 2D SAXS in magnetic field A.A. Veligzhanin1, D.I. Frey2, A.V. Shulenina3,1, A.Yu. Gruzinov4, Ya.V. Zubavichus1, M.V. Avdeev5,3 * 1
National Research Centre ‘Kurchatov Institute’, Moscow, Russia 2
3
Shirshov Institute of Oceanology RAS, Moscow, Russia
Faculty of Physics, Lomonosov Moscow State University, Moscow, Russia 4 5
EMBL, Hamburg, Germany
Joint Institute for Nuclear Research, Dubna, Moscow Reg., Russia *
Corresponding author, e-mail [email protected].
Keywords: ferrofluids; magnetic fluids; small-angle scattering; dipole-dipole interaction; dipolar fluids; aggregation
Abstract 2D small-angle X-ray scattering (SAXS) patterns for dilute ferrofluids with different degree of particle interaction under external magnetic field are analyzed to obtain structural characteristics of aggregates formed by polydisperse particles. Classical ferrofluids based on organic low-polarity solvents (toluene, decalin) with dispersed nanoparticles of magnetite and cobalt coated with oleic acid and water-based ferrofluid with dispersed magnetite nanoparticles coated with double layer of sodium oleate, are studied. All systems show the Langevin type behavior regarding the orientation of the anisotropic aggregates in them with the saturation at a relatively weak magnetic field strength. The field-induced anisotropy in the SAXS patterns is considered in the frame of a simple approximation.
1. Introduction The description and regulation of interaction effects in ferrofluids, or magnetic fluids (MFs) where magnetic dipole-dipole interaction is usually suppressed by various coatings of magnetic nanoparticles in liquid carriers are of current interest for developing advanced magnetically sensitive liquid systems in wide range of practical applications [1]. One of the most significant effects in the physics of MFs is the formation of chain-like particle aggregates characterized by a strong shape anisotropy. The behavior of such aggregates can be regulated by external magnetic fields, however, there is a variety of scenarios (chain orientation and growth, chain interaction, formation of secondary 1
structures like bundles or hexagonal columns, as well as 3D colloidal crystals) depending on the level of particle interaction, particle and chain concentrations, as well as the strength of magnetic field. Anisotropic aggregates partially or fully oriented along the field has a specific anisotropic contribution to 2D patterns obtained by scattering methods including small-angle X-ray scattering (SAXS) (e.g. [210]), small-angle neutron scattering (SANS) (e.g. [11-13]) and even light scattering [14, 15]. In most cases, its detailed analysis faces difficulties because of particle polydispersity. Here, we consider anysotropic aggregates in polydisperse ferrofluids with particle concentration and interaction sufficiently low to avoid complex interaction effects in the systems. For such systems which are close to thermodynamic equilibrium the theory of dipolar fluids predicts [16, 17] the formation of comparatively small and stable chain-like aggregates already in the absence of external magnetic field. They are just oriented when placed in moderate magnetic fields, the effect which we monitor and use in the structure characterization of the aggregates by analyzing anisotropic 2D SAXS patterns. The application of X-rays for this purpose is justified by two factors which simplify the data interpretation. First, due to the contrast the coatings around particles are non-visible in SAXS, so magnetic cores can be considered as quasi-spherical particles (as compared to more complicated core-shell structure in the case of SANS application). Second, there is no magnetic scattering contribution like in SANS which also becomes anisotropic under magnetic fields and competes with the orientation effect. Stable ferrofluids with different characteristic coupling parameters determined by the relation between the particle magnetic moments, particle size distribution and thickness of the surfactant coating, are studied. The anisotropy of the 2D experimental SAXS patterns as a function of the external magnetic field strength is followed to show that the changes in the scattering are determined by the orientation of anisotropic in shape aggregates along the field. It tends to saturation at a relatively weak magnetic field strength satisfying the Langevin type behavior. A simple model approximation for the scattering structure-factor of the oriented chain-like aggregates formed by highly polydisperse particles is considered to obtain structural characteristics of the aggregates.
2. Experimental Three stable in time ferrofluids were studied including samples MF1 (0.8 vol. % magnetite coated with double layer of sodium oleate in water), MF2 (1.5 vol. % magnetite coated with oleic acid in decalin), MF3 (0.5 vol. % cobalt coated with oleic acid in toluene). The samples were provided by the Institute of Experimental Physics of the Slovak Academy of Sciences, Kosice, Slovakia (sample MF1, for preparation details see [18]) and the Center of Fundamental and Advanced Technical Research of the Timisoara Branch of the Romanian Academy of Sciences, Timisoara, Romania 2
(samples MF2, MF3, for preparation details see [19]). The numeration of the samples corresponds to the growth in the magnetic dipolar interaction between particles in the solutions. The small-angle synchrotron X-ray scattering (SAXS) was measured at the DIXI beamline [20] at the Kurchatov Centre of Synchrotron Radiation, Moscow, Russia. The beamline utilizes X-ray wavelength of 0.16 nm and vacuum chamber length of 2.4 m, which allows one to cover the q-range of 0.07 - 1.1 nm-1. The silver behenate standard was utilized to calibrate the scattering vector interval. Each sample was placed in a quartz capillary of 1 mm diameter and wall thickness of 10 µm. The scattering was detected by the MarCCD165 2D detector with 600 seconds exposition, and then treated by the Fit2D software [21] to obtain the so-called 1D scattering intensity versus scattering vector module, I(q). The measurements were performed at room temperature. The homogeneous external magnetic field (strength up to 0.2 T) was induced at the sample by dipole electric coils and was directed perpendicular to the synchrotron radiation beam.
3. Results and discussion The effect of an external magnetic field on the experimental 2D SAXS patterns from MFs under study is demonstrated in Fig. 1. Initially isotropic over radial ϕ-angle on the (qx, q y) detector plane in the ‘no field’ state of all MFs, the 2D scattered patterns become anisotropic in the ‘in field’ state with respect to the dependence on the ϕ-angle. The 1D SAXS curves obtained by the averaging of the isotropic patterns over ϕ-angle (Fig. 2) exhibit no fringes typical for particle form-factors thus indicating that smearing because of high particle polydispersity takes place. There is a significant difference in the curves which is determined by the different aggregate organizations as it is followed from the comparison (see inset to Fig. 2) of the pair distance distribution functions, p(r), obtained by the indirect Fourier transform procedure [22]. Despite the smallest particle interaction, the MF1 sample is characterized by strong aggregation (size ~20 nm) which is consistent with the previous SANS experiments for similar samples [18]. The scattering curve from the MF2 sample is close to the case of independent non-aggregated particles with some size polydispersity observed many times in previous SAXS and SANS experiments [23]. A small band at the right end of the p(r) function around 20 nm shows some aggregation which is, however, significantly smaller as compared to the previous case. The scattering from the MF3 sample gives the p(r) function differing much from other two samples. Together with the aggregation state (size up 100 nm) there are modulations which indicate that some kind of structuring takes place. The following analysis of the anisotropic 2D scattering patterns makes it possible to conclude about the nature of the revealed aggregates in non-magnetized MFs.
3
Fig. 1. Illustration of evolution of experimental 2D SAXS scattering patterns (logarithmic scale) in external magnetic field for magnetic fluids with different particle interaction. All patterns are very similar for the ‘field off” state; as an example, the pattern for the MF2 fluid is shown. The patterns for the MF1 and MF2 fluids are very similar; as an example, the pattern for the MF2 fluid is shown.
Fig. 2. Experimental SAXS curves for the samples under study after averaging over ϕ-angle of the isotropic scattering patterns corresponding to the ‘no field’ state. Insets shows the normalized pair distance distribution functions obtained from the experimental curves by indirect Fourier transform. First, an anisotropy degree in the scattering from MFs under magnetic field can be analyzed when considering the evolution of the so-called ϕ-cuts, dependences of the intensity as a function of the radial ϕ-angle at a constant module of the scattering vector, q, on the detector plane starting from the direction of the magnetic field (Fig. 3a, b). As a numerical characteristic of this effect, we introduce the parameter of anisotropy, P, based on the periodical behavior of the ϕ -cuts, as a ratio: P = (Imax – Imin)/Imean.
(1)
4
where Imax and Imin are the intensities in the minima and maxima of the ϕ-cut, respectively, and Imean and the mean intensity over the whole ϕ-range. This parameter follows the Langevin-type dependence (Fig. 3c) with respect to the growth of the external magnetic field strength, B:
B W P( B ) = Pmax ⋅ coth − , W B
(2)
where Pmax is the maximal P at the saturation and W = (MB/kT) is the Langevin parameter. The obtained parameters of Eq. 2 for MFs under study are given in the caption to Fig. 3. The Langevin character of the P(B) dependences in Fig. 3c corresponds to a paramagnetic behavior of the studied fluids which, besides single magnetic nanoparticles, can also be extended to the aggregate state. The difference in the saturation rate testifies that a strong contribution from anisotropic aggregates to magnetization process and consequent changes in the scattering anisotropy takes place. Thus, for all samples under study the P(B) dependences approach the saturation already at the magnetic field strength just above 100 mT, which is a result of aggregation. Among three fluids, the maximal anisotropy parameter Pmax is significantly higher for MF3, which correlates with the stronger coupling of Co nanoparticles and the formation of rather long chain-like aggregates. It is very important that the P-values at a fixed B do not show any variation in time, which means that the aggregates in all samples are stable and do not change under magnetic field; the anisotropy in the scattering is related to the extended shapes of the aggregates, and the only process taking place in the solutions under magnetic field is the aggregate orientation due to interaction with the field.
5
Fig. 3. Evolutions of ϕ-cuts in 2D patterns (scattered intensity as a function of the radial angle ϕ at a fixed q) for MFs with weak (a) and strong (b) particle interaction with increasing external magnetic field strength. For convenient comparison of the cuts for different B-values, the data are divided by the mean intensities over the cuts. (с) Anisotropy parameter (1) as a function of the external magnetic field strength for MFs under study. Lines follow the best fits of Eq. 2; parameters are Pmax = 0.15 ± 0.05, W = (9 ± 2) mT (MF2); Pmax = 1.02 ± 0.03, W = (23 ± 1) mT (MF3). To make quantitative estimates, we consider the equilibrium chain formation to be a preferable aggregation type. Also, small total particle concentration in the solutions makes it possible to admit that the chains do not interact with each other. Under saturation field, all chains are oriented along the field. In the case when the aggregates are formed by monodisperse particles, the scattering structurefactor responsible for the anisotropy corresponds to the oriented non-interacting chains of Nagg particles, and 2D scattered intensity from such chains takes the form:
2 I (q) = nV 2 (∆ρ )2 Fsphere ( q) S ( q ) ,
(3)
where n is the total particle number density in the solution, V is the particle volume and ∆ρ is the scattering contrast (the difference between the scattering length densities of the particles and the solvent), which is determined here by the scattering of the magnetic cores in the particles [23]; 2 Fsphere (q ) is the form-factor of a spherical particle (magnetic core); and S (q ) is the structure-factor of the chain: S (q ) = [1 − cos( qelN )] /[1 − cos(qel )] = [1 − cos( qlN cos ϕ )] /[1 − cos( ql cos ϕ )] . 6
(4)
Here, e is the unit vector along the chain, and l is the distance between the neighboring particle centers in the chain. The latter differs from the diameter of magnetite nanopariticles by the double thickness of the surfactant shell. To take into account the particle polydispersity in the scattered intensity from correlated particles is a non-trivial problem. For polydisperse systems with isotropically interacting particles there are several well-proven approximations for orientation-averaged structure factor S(q)= < S (q ) >Ω [24]. The case of anisotropic interaction between polydisperse particles in solutions is not well studied; attempts for dipolar ferrofluids in bimodal approximation can be mentioned [25]. Here, we probe the simplest approach, when the particles are polydisperse in the calculations of the isotropic form-factor, while the structure-factor is considered in the monodisperse approximation with some characteristic correlation length l:
2 I (q ) = n(∆ρ ) 2 < V 2 Fsphere ( q) > S ( q ) .
(5)
The brackets <…> denote the averaging over some particle size distribution function, Dn(R), for which we used the typical [23] log-normal form:
(
)
Dn ( R) = 1/ 2π SR exp ( − ln 2 ( R / R0 ) / 2S 2 )
(6)
with the characteristic parameters R0 and S determining the most probable particle radius and width of the distribution, respectively. Like in (3), in Eq. 5 one deals with the separated form- and structurefactors, which drastically simplifies the calculations and makes it possible at some extent to fit the corresponding model to experimental data. In addition, a temperature de-orientation factor in accordance with the Boltzmann energy distribution for the chains in the magnetic field was taken into account in the model calculations of the average scattered intensity. We have found that the developed model works only for the organic magnetic fluids (MF2 and MF3) where non-equilibrium colloidal aggregation during preparation is suppressed [19]. The waterbased ferrofluid MF1 with double stabilization of magnetic particles represents a specific class characterized by the presence of slightly anisotropic aggregates formed during the synthesis. Their concentration and structural parameters of the aggregates (size, polydispersity) are mostly determined by the preparation procedure. In the cases of MF2 and MF3 samples, the comparison of the experimental and model scattering patterns were made using the q-cuts of the I (q ) function along two specific directions in the (qx, qy) detector plane − parallel (ϕ = 0) and perpendicular (ϕ = π/2) to the direction of the magnetic field strength (Fig. 4). In the second case, there is no specific correlations between particles, and, according to (4), S⊥ (q) = 1. The intensity averaged in a small sector around this direction can be used to analyze the particle size distribution function. The structure-factor, S ( q ) , in the intensity 7
distribution along the first direction contains the information about the correlation length in the chain aggregate, which is equal to the distance l in (4). These q-cuts were analyzed by varying parameters in Eq. 3 including the parameters of the particles size distribution function (mean radius Rmean = R0 exp(S2/2) and S), as well as the mean aggregate number, Na, and interparticle distance in the chain, l. Additionally, the isotropic background from non-aggregated particles ( S (q ) =1) was introduced which made it possible to estimate the distribution of the particles between non-aggregated and aggregates states in terms of the fraction of particles in the chains, η. The best modelling curves corresponding to the q-cuts along the specific directions are compared with the experimental dependences in Fig. 4. The corresponding parameters are given in the Figure caption. Full 2D patterns calculated with the same parameters are presented in Fig. 5. Figures 4, 5 show that the model used reflects well the basic features of the scattering. The disagreement between experimental and model curves in Fig. 4 can be related to a polydispersity effect, which was considered using the simplest approximation. The chain parameters in the fluids are in qualitative agreement with the theoretical predictions for the considered MFs [17]. Thus, the equilibrium aggregate number determined by the coupling energy is rather small (~ 2) in MF2, and is significantly higher (~20) in MF3. This correlates with the fractions of particles in the aggregates, ~50% in MF2 and ~97% in MF2. Also, the interparticle distance in the chain in both cases is higher (~10% in MF2 and ~40% in MF3) as compared to the full (mean particle diameter plus double thickness of the surfactant shell) particle diameter. This observation can be explained by the fact that the chains are somewhat preferably formed by the largest particles in the polydisperse ensemble (polydispersity indexes are about 40% and 30% for the two fluids, respectively) because of stronger dipole-dipole interaction while the smallest are left unbound. The effect naturally correlates with the stronger interaction in the MF3 fluid and is consistent with the predictions of simplified thermodynamic approaches [26, 27]. It should be, however, mentioned that the scattering from the aggregates is strongly size dependent (I ~ V2). For this reason, the monodisperse approximation used for the structure-factor can give interparticle distances in the chains effectively biased to larger values.
4. Conclusions
The analysis of the aggregate state of ferrofluids using the anisotropy in 2D SAXS patterns in magnetic field shows that highly stable samples based on organic solvents (single surfactant stabilization) behaves in full agreement with the theoretical predictions for dipolar fluids with respect to the formation of chain-like particle associates. The experimentally estimated chain length and the aggregation rate correlate with the coupling parameter for magnetic interaction between nanoparticles in the systems. In water-based ferrofluids (double surfactant stabilization) the anisotropy in the scattering from the samples under magnetic field is caused by the presence of compact and slightly anisotropic aggregates. This effect is hardly can be considered in the framework of the thermodynamic 8
approximation, and is mostly determined by the preparation procedure. So, it has been shown that the presented type of the small-angle scattering analysis is rather useful in clarifying aspects concerning the aggregate formation in ferrofluids including the type and time stability of the aggregates when applying magnetic fields.
Fig. 4. q-cuts of 2D experimental patterns along and perpendicular to the direction of the saturating magnetic field for two MFs under study; comparison with the calculations in accordance with model Eqs. 3, 4. Parameters of the model are (a) Rmean = 3.5 nm, S = 0.37, Nagg = 2, l = 14 nm, η = 50 %; (b) Rmean = 7.5 nm, S = 0.25, Nagg = 20, l = 26.3 nm, η = 97 %. MF2
MF3
Fig. 5. Model 2D SAXS patterns (logarithmic scale) in accordance with Eqs. 3, 4 for two MFs under study. 9
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A. Ivanov, A. Zubarev, Yu. Raikher, S. Kantorovich, M. Lopez-Lopez (Eds.), Proceedings of the 14th International Conference on Magnetic Fluids, J. Magn. Magn. Mater. 431 (2017) pp. 296.
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[10] Z. Rozynek, et al., Eur. Phys. J. E 34 (2011) 28–35. [11] A. Wiedenmann, A. Heinemann, J. Magn. Magn. Mater. 289 (2005) 58-61. [12] L.M.Pop, S.Odenbach, J. Phys. Cond. Matt. 18 (2006) S2785–S2802. [13] M. Klokkenburg, B.H. Erné, A. Wiedenmann, A.V. Petukhov, A.P. Philipse, Phys. Rev. E 75 (2007) 051408 [14] J.M. Laskar, J. Philip, B. Raj, Phys. Rev. E 78 (2008) 031404. [15] J.M. Laskar, S. Brojabasi, B. Raj, J. Philip, Opt. Comm. 285 (2012) 1242–1247. [16] E. Pyanzina, S. Kantorovich, J.J. Cerda, A. Ivanov, C. Holm, Mol. Phys. 107 (2009) 571−590. [17] J.J. Cerdà, E. Elfimova, V. Ballenegger, E. Krutikova, A. Ivanov, Ch. Holm, Phys. Rev. E 81 (2010) 011501. [18] M.V. Avdeev, A.V. Feoktystov, P. Kopcansky, G. Lancz, et al., J. Appl. Cryst. 43 (2010) 959– 969. [19] L. Vekas, D. Bica, M.V. Avdeev, China Particuology 5 (2007) 43–49. [20] V.N. Korneev, P.M. Sergienko, A.M. Matyushin, V.A. Shlektarev, et al., Nucl. Instr. Meth. A 543 (2005) 368–374. [21] A.P. Hammersley, S.O. Svensson, M. Hanfland, A.N. Fitch, D. Hausermann, J. High Press. Res. 14 (1996) 235–248. [22] D.I. Svergun J. Appl. Cryst. 25 (1992) 495–503. [23] M.V. Avdeev, V.L. Aksenov, Phys. Uspekhi 180 (2010) 1009–1034. [24] J. Skov Pedersen, Adv. Coll. Inter. Sci. 70 (1997) 171–210.
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Highlights
• • •
Chain-like aggregates of polydisperse particles in ferrofluids are characterized. Anisotropic 2D patterns of small-angle X-ray scattering from ferrofluids under magnetic field are analyzed Structural characteristics of aggregates are obtained using simple approximation.
12
S0304-8853(17)32068-1 https://doi.org/10.1016/j.jmmm.2017.10.052 MAGMA 63266
To appear in:
Journal of Magnetism and Magnetic Materials
Received Date: Revised Date: Accepted Date:
5 July 2017 10 October 2017 13 October 2017
Please cite this article as: A.A. Veligzhanin, D.I. Frey, A.V. Shulenina, A.Yu. Gruzinov, Ya.V. Zubavichus, M.V. Avdeev, Characterization of aggregate state of polydisperse ferrofluids: some aspects of anisotropy analysis of 2D SAXS in magnetic field, Journal of Magnetism and Magnetic Materials (2017), doi: https://doi.org/10.1016/ j.jmmm.2017.10.052
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. 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.
Characterization of aggregate state of polydisperse ferrofluids: some aspects of anisotropy analysis of 2D SAXS in magnetic field A.A. Veligzhanin1, D.I. Frey2, A.V. Shulenina3,1, A.Yu. Gruzinov4, Ya.V. Zubavichus1, M.V. Avdeev5,3 * 1
National Research Centre ‘Kurchatov Institute’, Moscow, Russia 2
3
Shirshov Institute of Oceanology RAS, Moscow, Russia
Faculty of Physics, Lomonosov Moscow State University, Moscow, Russia 4 5
EMBL, Hamburg, Germany
Joint Institute for Nuclear Research, Dubna, Moscow Reg., Russia *
Corresponding author, e-mail [email protected].
Keywords: ferrofluids; magnetic fluids; small-angle scattering; dipole-dipole interaction; dipolar fluids; aggregation
Abstract 2D small-angle X-ray scattering (SAXS) patterns for dilute ferrofluids with different degree of particle interaction under external magnetic field are analyzed to obtain structural characteristics of aggregates formed by polydisperse particles. Classical ferrofluids based on organic low-polarity solvents (toluene, decalin) with dispersed nanoparticles of magnetite and cobalt coated with oleic acid and water-based ferrofluid with dispersed magnetite nanoparticles coated with double layer of sodium oleate, are studied. All systems show the Langevin type behavior regarding the orientation of the anisotropic aggregates in them with the saturation at a relatively weak magnetic field strength. The field-induced anisotropy in the SAXS patterns is considered in the frame of a simple approximation.
1. Introduction The description and regulation of interaction effects in ferrofluids, or magnetic fluids (MFs) where magnetic dipole-dipole interaction is usually suppressed by various coatings of magnetic nanoparticles in liquid carriers are of current interest for developing advanced magnetically sensitive liquid systems in wide range of practical applications [1]. One of the most significant effects in the physics of MFs is the formation of chain-like particle aggregates characterized by a strong shape anisotropy. The behavior of such aggregates can be regulated by external magnetic fields, however, there is a variety of scenarios (chain orientation and growth, chain interaction, formation of secondary 1
structures like bundles or hexagonal columns, as well as 3D colloidal crystals) depending on the level of particle interaction, particle and chain concentrations, as well as the strength of magnetic field. Anisotropic aggregates partially or fully oriented along the field has a specific anisotropic contribution to 2D patterns obtained by scattering methods including small-angle X-ray scattering (SAXS) (e.g. [210]), small-angle neutron scattering (SANS) (e.g. [11-13]) and even light scattering [14, 15]. In most cases, its detailed analysis faces difficulties because of particle polydispersity. Here, we consider anysotropic aggregates in polydisperse ferrofluids with particle concentration and interaction sufficiently low to avoid complex interaction effects in the systems. For such systems which are close to thermodynamic equilibrium the theory of dipolar fluids predicts [16, 17] the formation of comparatively small and stable chain-like aggregates already in the absence of external magnetic field. They are just oriented when placed in moderate magnetic fields, the effect which we monitor and use in the structure characterization of the aggregates by analyzing anisotropic 2D SAXS patterns. The application of X-rays for this purpose is justified by two factors which simplify the data interpretation. First, due to the contrast the coatings around particles are non-visible in SAXS, so magnetic cores can be considered as quasi-spherical particles (as compared to more complicated core-shell structure in the case of SANS application). Second, there is no magnetic scattering contribution like in SANS which also becomes anisotropic under magnetic fields and competes with the orientation effect. Stable ferrofluids with different characteristic coupling parameters determined by the relation between the particle magnetic moments, particle size distribution and thickness of the surfactant coating, are studied. The anisotropy of the 2D experimental SAXS patterns as a function of the external magnetic field strength is followed to show that the changes in the scattering are determined by the orientation of anisotropic in shape aggregates along the field. It tends to saturation at a relatively weak magnetic field strength satisfying the Langevin type behavior. A simple model approximation for the scattering structure-factor of the oriented chain-like aggregates formed by highly polydisperse particles is considered to obtain structural characteristics of the aggregates.
2. Experimental Three stable in time ferrofluids were studied including samples MF1 (0.8 vol. % magnetite coated with double layer of sodium oleate in water), MF2 (1.5 vol. % magnetite coated with oleic acid in decalin), MF3 (0.5 vol. % cobalt coated with oleic acid in toluene). The samples were provided by the Institute of Experimental Physics of the Slovak Academy of Sciences, Kosice, Slovakia (sample MF1, for preparation details see [18]) and the Center of Fundamental and Advanced Technical Research of the Timisoara Branch of the Romanian Academy of Sciences, Timisoara, Romania 2
(samples MF2, MF3, for preparation details see [19]). The numeration of the samples corresponds to the growth in the magnetic dipolar interaction between particles in the solutions. The small-angle synchrotron X-ray scattering (SAXS) was measured at the DIXI beamline [20] at the Kurchatov Centre of Synchrotron Radiation, Moscow, Russia. The beamline utilizes X-ray wavelength of 0.16 nm and vacuum chamber length of 2.4 m, which allows one to cover the q-range of 0.07 - 1.1 nm-1. The silver behenate standard was utilized to calibrate the scattering vector interval. Each sample was placed in a quartz capillary of 1 mm diameter and wall thickness of 10 µm. The scattering was detected by the MarCCD165 2D detector with 600 seconds exposition, and then treated by the Fit2D software [21] to obtain the so-called 1D scattering intensity versus scattering vector module, I(q). The measurements were performed at room temperature. The homogeneous external magnetic field (strength up to 0.2 T) was induced at the sample by dipole electric coils and was directed perpendicular to the synchrotron radiation beam.
3. Results and discussion The effect of an external magnetic field on the experimental 2D SAXS patterns from MFs under study is demonstrated in Fig. 1. Initially isotropic over radial ϕ-angle on the (qx, q y) detector plane in the ‘no field’ state of all MFs, the 2D scattered patterns become anisotropic in the ‘in field’ state with respect to the dependence on the ϕ-angle. The 1D SAXS curves obtained by the averaging of the isotropic patterns over ϕ-angle (Fig. 2) exhibit no fringes typical for particle form-factors thus indicating that smearing because of high particle polydispersity takes place. There is a significant difference in the curves which is determined by the different aggregate organizations as it is followed from the comparison (see inset to Fig. 2) of the pair distance distribution functions, p(r), obtained by the indirect Fourier transform procedure [22]. Despite the smallest particle interaction, the MF1 sample is characterized by strong aggregation (size ~20 nm) which is consistent with the previous SANS experiments for similar samples [18]. The scattering curve from the MF2 sample is close to the case of independent non-aggregated particles with some size polydispersity observed many times in previous SAXS and SANS experiments [23]. A small band at the right end of the p(r) function around 20 nm shows some aggregation which is, however, significantly smaller as compared to the previous case. The scattering from the MF3 sample gives the p(r) function differing much from other two samples. Together with the aggregation state (size up 100 nm) there are modulations which indicate that some kind of structuring takes place. The following analysis of the anisotropic 2D scattering patterns makes it possible to conclude about the nature of the revealed aggregates in non-magnetized MFs.
3
Fig. 1. Illustration of evolution of experimental 2D SAXS scattering patterns (logarithmic scale) in external magnetic field for magnetic fluids with different particle interaction. All patterns are very similar for the ‘field off” state; as an example, the pattern for the MF2 fluid is shown. The patterns for the MF1 and MF2 fluids are very similar; as an example, the pattern for the MF2 fluid is shown.
Fig. 2. Experimental SAXS curves for the samples under study after averaging over ϕ-angle of the isotropic scattering patterns corresponding to the ‘no field’ state. Insets shows the normalized pair distance distribution functions obtained from the experimental curves by indirect Fourier transform. First, an anisotropy degree in the scattering from MFs under magnetic field can be analyzed when considering the evolution of the so-called ϕ-cuts, dependences of the intensity as a function of the radial ϕ-angle at a constant module of the scattering vector, q, on the detector plane starting from the direction of the magnetic field (Fig. 3a, b). As a numerical characteristic of this effect, we introduce the parameter of anisotropy, P, based on the periodical behavior of the ϕ -cuts, as a ratio: P = (Imax – Imin)/Imean.
(1)
4
where Imax and Imin are the intensities in the minima and maxima of the ϕ-cut, respectively, and Imean and the mean intensity over the whole ϕ-range. This parameter follows the Langevin-type dependence (Fig. 3c) with respect to the growth of the external magnetic field strength, B:
B W P( B ) = Pmax ⋅ coth − , W B
(2)
where Pmax is the maximal P at the saturation and W = (MB/kT) is the Langevin parameter. The obtained parameters of Eq. 2 for MFs under study are given in the caption to Fig. 3. The Langevin character of the P(B) dependences in Fig. 3c corresponds to a paramagnetic behavior of the studied fluids which, besides single magnetic nanoparticles, can also be extended to the aggregate state. The difference in the saturation rate testifies that a strong contribution from anisotropic aggregates to magnetization process and consequent changes in the scattering anisotropy takes place. Thus, for all samples under study the P(B) dependences approach the saturation already at the magnetic field strength just above 100 mT, which is a result of aggregation. Among three fluids, the maximal anisotropy parameter Pmax is significantly higher for MF3, which correlates with the stronger coupling of Co nanoparticles and the formation of rather long chain-like aggregates. It is very important that the P-values at a fixed B do not show any variation in time, which means that the aggregates in all samples are stable and do not change under magnetic field; the anisotropy in the scattering is related to the extended shapes of the aggregates, and the only process taking place in the solutions under magnetic field is the aggregate orientation due to interaction with the field.
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Fig. 3. Evolutions of ϕ-cuts in 2D patterns (scattered intensity as a function of the radial angle ϕ at a fixed q) for MFs with weak (a) and strong (b) particle interaction with increasing external magnetic field strength. For convenient comparison of the cuts for different B-values, the data are divided by the mean intensities over the cuts. (с) Anisotropy parameter (1) as a function of the external magnetic field strength for MFs under study. Lines follow the best fits of Eq. 2; parameters are Pmax = 0.15 ± 0.05, W = (9 ± 2) mT (MF2); Pmax = 1.02 ± 0.03, W = (23 ± 1) mT (MF3). To make quantitative estimates, we consider the equilibrium chain formation to be a preferable aggregation type. Also, small total particle concentration in the solutions makes it possible to admit that the chains do not interact with each other. Under saturation field, all chains are oriented along the field. In the case when the aggregates are formed by monodisperse particles, the scattering structurefactor responsible for the anisotropy corresponds to the oriented non-interacting chains of Nagg particles, and 2D scattered intensity from such chains takes the form:
2 I (q) = nV 2 (∆ρ )2 Fsphere ( q) S ( q ) ,
(3)
where n is the total particle number density in the solution, V is the particle volume and ∆ρ is the scattering contrast (the difference between the scattering length densities of the particles and the solvent), which is determined here by the scattering of the magnetic cores in the particles [23]; 2 Fsphere (q ) is the form-factor of a spherical particle (magnetic core); and S (q ) is the structure-factor of the chain: S (q ) = [1 − cos( qelN )] /[1 − cos(qel )] = [1 − cos( qlN cos ϕ )] /[1 − cos( ql cos ϕ )] . 6
(4)
Here, e is the unit vector along the chain, and l is the distance between the neighboring particle centers in the chain. The latter differs from the diameter of magnetite nanopariticles by the double thickness of the surfactant shell. To take into account the particle polydispersity in the scattered intensity from correlated particles is a non-trivial problem. For polydisperse systems with isotropically interacting particles there are several well-proven approximations for orientation-averaged structure factor S(q)= < S (q ) >Ω [24]. The case of anisotropic interaction between polydisperse particles in solutions is not well studied; attempts for dipolar ferrofluids in bimodal approximation can be mentioned [25]. Here, we probe the simplest approach, when the particles are polydisperse in the calculations of the isotropic form-factor, while the structure-factor is considered in the monodisperse approximation with some characteristic correlation length l:
2 I (q ) = n(∆ρ ) 2 < V 2 Fsphere ( q) > S ( q ) .
(5)
The brackets <…> denote the averaging over some particle size distribution function, Dn(R), for which we used the typical [23] log-normal form:
(
)
Dn ( R) = 1/ 2π SR exp ( − ln 2 ( R / R0 ) / 2S 2 )
(6)
with the characteristic parameters R0 and S determining the most probable particle radius and width of the distribution, respectively. Like in (3), in Eq. 5 one deals with the separated form- and structurefactors, which drastically simplifies the calculations and makes it possible at some extent to fit the corresponding model to experimental data. In addition, a temperature de-orientation factor in accordance with the Boltzmann energy distribution for the chains in the magnetic field was taken into account in the model calculations of the average scattered intensity. We have found that the developed model works only for the organic magnetic fluids (MF2 and MF3) where non-equilibrium colloidal aggregation during preparation is suppressed [19]. The waterbased ferrofluid MF1 with double stabilization of magnetic particles represents a specific class characterized by the presence of slightly anisotropic aggregates formed during the synthesis. Their concentration and structural parameters of the aggregates (size, polydispersity) are mostly determined by the preparation procedure. In the cases of MF2 and MF3 samples, the comparison of the experimental and model scattering patterns were made using the q-cuts of the I (q ) function along two specific directions in the (qx, qy) detector plane − parallel (ϕ = 0) and perpendicular (ϕ = π/2) to the direction of the magnetic field strength (Fig. 4). In the second case, there is no specific correlations between particles, and, according to (4), S⊥ (q) = 1. The intensity averaged in a small sector around this direction can be used to analyze the particle size distribution function. The structure-factor, S ( q ) , in the intensity 7
distribution along the first direction contains the information about the correlation length in the chain aggregate, which is equal to the distance l in (4). These q-cuts were analyzed by varying parameters in Eq. 3 including the parameters of the particles size distribution function (mean radius Rmean = R0 exp(S2/2) and S), as well as the mean aggregate number, Na, and interparticle distance in the chain, l. Additionally, the isotropic background from non-aggregated particles ( S (q ) =1) was introduced which made it possible to estimate the distribution of the particles between non-aggregated and aggregates states in terms of the fraction of particles in the chains, η. The best modelling curves corresponding to the q-cuts along the specific directions are compared with the experimental dependences in Fig. 4. The corresponding parameters are given in the Figure caption. Full 2D patterns calculated with the same parameters are presented in Fig. 5. Figures 4, 5 show that the model used reflects well the basic features of the scattering. The disagreement between experimental and model curves in Fig. 4 can be related to a polydispersity effect, which was considered using the simplest approximation. The chain parameters in the fluids are in qualitative agreement with the theoretical predictions for the considered MFs [17]. Thus, the equilibrium aggregate number determined by the coupling energy is rather small (~ 2) in MF2, and is significantly higher (~20) in MF3. This correlates with the fractions of particles in the aggregates, ~50% in MF2 and ~97% in MF2. Also, the interparticle distance in the chain in both cases is higher (~10% in MF2 and ~40% in MF3) as compared to the full (mean particle diameter plus double thickness of the surfactant shell) particle diameter. This observation can be explained by the fact that the chains are somewhat preferably formed by the largest particles in the polydisperse ensemble (polydispersity indexes are about 40% and 30% for the two fluids, respectively) because of stronger dipole-dipole interaction while the smallest are left unbound. The effect naturally correlates with the stronger interaction in the MF3 fluid and is consistent with the predictions of simplified thermodynamic approaches [26, 27]. It should be, however, mentioned that the scattering from the aggregates is strongly size dependent (I ~ V2). For this reason, the monodisperse approximation used for the structure-factor can give interparticle distances in the chains effectively biased to larger values.
4. Conclusions
The analysis of the aggregate state of ferrofluids using the anisotropy in 2D SAXS patterns in magnetic field shows that highly stable samples based on organic solvents (single surfactant stabilization) behaves in full agreement with the theoretical predictions for dipolar fluids with respect to the formation of chain-like particle associates. The experimentally estimated chain length and the aggregation rate correlate with the coupling parameter for magnetic interaction between nanoparticles in the systems. In water-based ferrofluids (double surfactant stabilization) the anisotropy in the scattering from the samples under magnetic field is caused by the presence of compact and slightly anisotropic aggregates. This effect is hardly can be considered in the framework of the thermodynamic 8
approximation, and is mostly determined by the preparation procedure. So, it has been shown that the presented type of the small-angle scattering analysis is rather useful in clarifying aspects concerning the aggregate formation in ferrofluids including the type and time stability of the aggregates when applying magnetic fields.
Fig. 4. q-cuts of 2D experimental patterns along and perpendicular to the direction of the saturating magnetic field for two MFs under study; comparison with the calculations in accordance with model Eqs. 3, 4. Parameters of the model are (a) Rmean = 3.5 nm, S = 0.37, Nagg = 2, l = 14 nm, η = 50 %; (b) Rmean = 7.5 nm, S = 0.25, Nagg = 20, l = 26.3 nm, η = 97 %. MF2
MF3
Fig. 5. Model 2D SAXS patterns (logarithmic scale) in accordance with Eqs. 3, 4 for two MFs under study. 9
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Highlights
• • •
Chain-like aggregates of polydisperse particles in ferrofluids are characterized. Anisotropic 2D patterns of small-angle X-ray scattering from ferrofluids under magnetic field are analyzed Structural characteristics of aggregates are obtained using simple approximation.
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