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Aggregation of magnetic polystyrene particles: A light scattering study
Colloids and Surfaces A: Physicochem. Eng. Aspects 270–271 (2005) 317–322
Aggregation of magnetic polystyrene particles: A light scattering study Fernando Mart´ınez-Pedrero, Mar´ıa Tirado-Miranda, Artur Schmitt, Jos´e Callejas-Fern´andez ∗ Departamento de F´ısica Aplicada, Universidad de Granada, Campus de Fuentenueva, E-18071 Granada, Spain Available online 7 October 2005
Abstract The aggregation behavior of dilute dispersions of magnetic polystyrene particles was studied experimentally by means of light scattering techniques. In absence of external magnetic fields, the particles are stable against flocculation due to charged carboxylic surface groups. Aggregation was induced applying an external magnetic field. Static light scattering (SLS) was used to study the geometry of the fractally shaped aggregates. Fractal dimensions close to 1.2 were found for the aggregates. Transmission electron microscopy micrographs show that the formed aggregates are chain-like, as predicted by the SLS results. Dynamic light scattering (DLS) was employed to monitor the time evolution of cluster size and to study the underlying growth kinetics. The observed results could be explained qualitatively in the framework of the classical DLVO theory when magnetic dipole–dipole interactions are taken into account. © 2005 Elsevier B.V. All rights reserved. Keywords: Polystyrene; Static light scattering; Dynamic light scattering; Magnetic polystyrene
1. Introduction A detailed analysis of colloidal aggregation involves two main aspects: the cluster morphology and the kinetics of aggregate formation. The cluster morphology is usually characterized by means of the fractal dimension, df , that is understood as a measure on how the particles fill the three-dimensional space. The kinetics of the processes is frequently studied by means of the temporal evolution of the mean aggregate size. Studies concerning aggregation phenomena in magnetic fluids have been of great interest over the past decades. However, papers that consider simultaneously both aspects of colloidal aggregation are scarce in bibliography [1–16]. Static (SLS) and dynamic (DLS) light scattering are suitable techniques to study the structure and the kinetics of colloidal aggregates. In the literature, however, there are only a few papers that apply these techniques for studying aggregation phenomena in magnetic fluids. In general, high volume fractions are used and so, both adsorption and multiple scattering impede light scattering studies to be performed reliably. Furthermore, the relatively small size and large polydispersity of the employed samples make the interpretation of light scattering data a difficult task. In order to overcome these difficulties, optical microscopy, neu-
∗
Corresponding author. E-mail address: [email protected] (J. Callejas-Fern´andez).
0927-7757/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2005.09.001
tron and X-rays scattering techniques are most commonly used [1–3,12–14]. One of the aim of this work is to demonstrate that visible light can be used to get valuable data on magnetic colloidal aggregation if the systems is sufficiently dilute and the size of magnetic particles is large. In the literature, several papers study the spatial structure of magnetic particle clusters by means of video microscopy or cryogenic electron microscopy [17,18]. Nevertheless, the measurement of the cluster fractal dimension in magnetic fluid has not been the focus of a great number of works, so far. Moreover, most papers that report measurements of the cluster fractal dimension do not address it as the central question. They usually omit a deeper discussion of its meaning and do not relate it with the cluster morphology and the underlying aggregation mechanisms [1,5,9,12–19]. As was mentioned above, small particles that lie in lower limit of the colloidal domain and high particles concentrations have been used for the experiments. Under these conditions, an adequate analysis of the cluster morphology becomes a difficult task. For larger particles and diluted systems, however, this becomes quite straightforward. Over the last years, optical microscopy techniques have been widely used [1,6,15] for monitoring the time evolution of the mean cluster size through a detailed analysis of sequences of photographs. In situ DLS measurements, however, are scarce in the literature. Nevertheless, the latter technique allows the average cluster diffusion coefficient and the corresponding mean cluster size to be determined continuously without having to
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handle and alter the sample. To the best of our knowledge, only Hagenb¨uchle and Liu studied the kinetics of the magnetic particle aggregation in ferrofluid emulsion of about 500 nm in diameter [4] and Tsouris and Scott worked on ferric oxide [11] using DLS as main experimental technique. In this paper, field-induced aggregation processes arising in well-characterized suspensions of monodisperse magnetic polystyrene microspheres will be studied. For this purpose, the fractal dimension of the formed clusters and the temporal evolution of the mean cluster size will be measured by means of SLS and DLS. Only diluted samples will be used in order to guarantee that hazardous multiple scattering can be neglected. In a previous work [20], we compared the scattering pattern of these kind of particles and similar non-magnetic particles. We found that light adsorption in the magnetic particles does not substantially modify the obtained results. 2. Experimental 2.1. Materials Aqueous suspensions of magnetic polystyrene microspheres were used for the experiments. The samples were provided by Merck (Estapor Microspheres, Ref.: R 00-39). The essentially monodisperse particles (see Fig. 1) are charge stabilized due to carboxylic surface groups. Their mean particle diameter was 160 ± 5 nm as determined by means of photocorrelation measurements. Their magnetic behavior arises from small ferromagnetic grains embedded in the polymer matrix. The diameter of these magnetic grains is approximately 10 nm. According to the manufacturer, the ferrite mass content was 53.2% and the saturation magnetization (Ms ) was approximately 36 kA/m. The mean particle mass density was 1.2 g/ml. The magnetic polystyrene particle suspensions were initially stable. When exposed to a
sufficiently strong external magnetic field, however, the particles start to aggregate. Due to their small size, the magnetic grains contain a single magnetic domain. Since no long-range order exists between domains, the dipoles are randomly oriented. The anisotropy energy, Ea , of each domain is of the order of the thermal energy kT, and its magnetization vector fluctuates around the main axis of magnetization with a characteristic Neel relaxation time given by [21]: Ea τ = τ0 exp (1) kT When an external magnetic field is applied, one of the possible orientations in each grain is favored and the entire particle acquires a net magnetization. For this reason, the particles are called superparamagnetic. They are paramagnetic on the colloidal scale. Their magnetization is completely reversible and there is no magnetic remanence. Nevertheless, the net magnetic moment of a magnetic colloidal particle is much larger than the moments in a pure paramagnet. 2.2. Methods The final particle number concentrations in the reaction vessel was 7.2 × 1017 p/m3 . The temperature was set to 298.0 ± 0.5 K. The magnetic field needed to achieve field-induced aggregation was applied to the sample by means of a neodymium disk magnet (Hadle Gac, Barcelona, Spain). The magnet was placed directly above the measuring cell containing the colloidal dispersion. This implies, of course, that the field strength is not uniform and depends on the position within the cell. A homogeneous magnetic field would be desirable but is not essential for the experiments since the goal of this work is to ensure that the dipoles induced in the magnetic particles are strong enough to give rise to aggregation. The DLS experiments were performed during at least 20 min. For the SLS experiment, the magnet was removed from the measuring cell after 30 min and the sample was vigorously shaken and returned to the SLS instrument. Hence, all the SLS measurements were performed without any applied external magnetic field. The magnetic particles were also aggregated in the absence of an external magnetic field by adding KBr as indifferent electrolyte. The electrolyte-induced aggregation processes were started by mixing equal amounts of electrolyte solution and particle suspension through a Y-shaped mixing device. A final electrolyte concentration of 0.9 M was used in order to compress the electric double layer and to screen the electrostatic repulsion efficiently. 2.3. Light scattering measurements
Fig. 1. Transmission electron microscopy micrograph of: (a) several individual magnetic polystyrene particles and (b) a sample aggregate formed due to an applied magnetic field.
The light scattering experiments were performed using a slightly modified Malvern 4700 System (UK) working with a 488 nm wavelength argon laser. The photomultiplier arm was previously located at the reference position in order to set the 0◦ angle. The morphology of the aggregates was assessed by means of SLS. The mean scattered intensity was obtained for
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different angles in the range from 20◦ to 140◦ . SLS allows the fractal dimension, df , to be measured using the following relationship between the mean scattered light intensity, I(q), and the θ scattering wave vector, q = 4π sin λ 2 , where λ is the medium wavelength and θ is the scattering angle [22]: I(q) ≈ S(q) ≈ q−df
(2)
The aggregate structure factor S(q) is related to the spatial distribution of the particles contained within the aggregates. The mean scattered intensity contains information about the cluster structure in the q range, Rh −1 < q < a−1 where a is the particle radius and Rh is the average hydrodynamic radius of the formed aggregates. For higher q values, the length scale corresponds to individual particles, and thus, the intensity is related to their form factor. In the lower q region, topological length correlations between clusters could be studied. After a sufficiently long time, the mean intensity I(q) showed the expected asymptotic time-independent behavior for all experiments reported in this paper. This means that the final fractal structure of the clusters was then totally established. From the slope of the asymptotic curves, the fractal dimensions were finally determined. DLS assess the scattered intensity autocorrelation function as the product of the photon counts at times t and t + τ such that G(τ) = I(t)I(t + τ). There from the normalized intensity autocorrelation function gint (τ) =
I(t)I(t + τ) I(t)I(t)
(3)
was calculated and converted into the scattered field autocorrelation function by the aid of the Siegert relationship: gint (τ) = 1 + C|gfield (τ)|2
(4)
where C is a constant which depends on the optics of the instrument [23]. For our experiments, the scattering angle was set to 60◦ and the scattered light intensity autocorrelation functions were determined at different times during aggregation. Data analysis was performed using our own programs. Information on the cluster size distribution was obtained from the fitting coefficients of the expanded logarithm of the field autocorrelation function [24]: 2 3 τ τ field ln g (τ) = −µ1 τ + µ2 − µ3 + ··· (5) 2 3! This method is known as cumulant analysis and has been widely used for dynamic light scattering data analysis. For polydisperse systems, the first cumulant, µ1 , is related to the effective mean particle diffusion coefficient, Deff , by µ1 = Deff q2 . From the mean particle diffusion coefficient, the mean hydrodynamic size may be calculated using the well-known Stokes–Einstein relationship. 3. Results and discussion Fig. 2 shows the I(q) curves for the magnetic polystyrene particles aggregated due to 0.9 M KBr of added electrolyte and due to an applied external magnetic field. The curves exhibit a linear
Fig. 2. Scattered light intensity vs. scattering vector for the magnetic polystyrene particles aggregated at two different experimental conditions: () in the presence of a magnetic field and () at 0.9 M KBr.
behavior in the scattering wave vector range Rh −1 < q < a−1 indicated in the plots by vertical lines and so, the cluster fractal dimension df can be determined from the slope in both cases. At high electrolyte concentration, a fractal dimension of 1.81 ± 0.05 was obtained. This value is quite close to 1.75 that is frequently reported in the literature for freely diffusing sticky particles [25]. This means that our superparamagnetic particles show the same typical experimental behavior than other monodisperse non-magnetic particles. Therefore, light absorption and magnetic permeability changes due to the magnetic grains embedded in the latex matrix of these particles do not significantly alter the aggregate structure factor at the employed wavelength. Moreover, the previous results are in agreement with the superparamagnetic character of the particles, i.e. they behave as standard non-magnetic particles when they are not in a magnetic field. In this case, the aggregation behavior of the samples is controlled by the isotropic electrostatic and van der Waals interactions. In the presence of an external magnetic field, however, the particles aggregate due to anisotropic magnetic interactions. In this case, a very low fractal dimension of 1.21 ± 0.14 was found for the magnetic polystyrene particles. The given values are the average of six measurements with a linear regression coefficient of at least 0.99. The given error intervals are the corresponding standard deviation. It should be emphasized that similar fractal dimensions were reported by other authors for magnetic liposomes [7], magnetite particles coated by two layers of different surfactants [19] and simulations [26]. We interpret the experimentally obtained fractal dimensions close to 1.2 as a result of more or less linear structures present in the sample. In order to corroborate the results obtained by means of SLS, we performed also transmission electron microscopy (TEM). Fig. 1b shows an image of a typical cluster contained in a sample of aggregated magnetic latex particles. This sample was handled according to the procedure described in the previous section when the particles aggregated only due to an applied magnetic field. In the micrograph, the above-mentioned linear chain-like structures together with individual particles can clearly be observed. This confirms that latex particle chains
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form. These chains are so stable that they do not necessarily disassemble when the magnetic field is turned off. Some are even strong enough to survive the necessary drying step prior to TEM imaging. DLS was employed for measuring the average diffusion coefficient of the aggregates. In a dilute and monodisperse system of spherical particles, Deff is given by the Stokes–Einstein equation: D0 =
kT 6πηa
(6)
As was shown by means of SLS, the magnetic latex particles start forming chains when the magnetic field is turned on. It is well established by theoretical and experimental works that the diffusion coefficient depends on the relative orientation between the applied magnetic field and the observation axis. In other words, Deff,|| reflects the translational motion of the chains along their axis if the observation axis is parallel to the applied field. If the observation axis is perpendicular to the applied field, then the lateral motion of chains is characterized by Deff,⊥ . Theoretical, simulation and experimental works show that Deff,|| is larger than Deff,⊥ , i.e. the diffusive motion of the chains perpendicular to their axis is slower than the diffusive motion parallel to their axis [14,27]. In our light scattering instrument, the field is perpendicular to the scattering plane and so, we mainly assess the lateral diffusive motion of the chains. Fig. 3 shows the time dependence of the effective diffusion coefficient Deff = Deff,⊥ when aggregates are growing in the presence of the applied magnetic field. As expected, Deff decreases with increasing time due to the slower motion of larger aggregates. Since the formed clusters are chain-like structures, the increasing aggregates size indicates that the average number of particles per chain grows in time. Taking into account the friction coefficient for a chain moving perpendicular to its axis, some authors propose the following relationship between Deff and the number of chain-forming particles N [4,27]: Deff = D0
3 ln N 2N
(7)
where D0 is the diffusion coefficient of the individual particles.
Fig. 3 shows the time evolution of the effective diffusion coefficient Deff for the field-induced aggregation process. As can be seen, Deff decreases in time and so, the average chain length increases due to aggregation. Equation (7) relates the effective diffusion coefficient Deff to the chain length. The corresponding chain lengths are indicated in the plot as horizontal lines. This allows to estimate the time at which chains of an average length N are formed. In the experiment, a neodymium disk magnet was placed directly over the cell containing the colloidal dispersion. This implies, of course, that the field strength is not uniform and depends on the position within the cell. A measurement of the magnetic field strength versus distance from the tip of the magnet indicates that, after an initial rapid decreased very close to the magnet’s surface, the field changes slowly at moderate distances. A non-uniform magnetic field gives rise to a magnetic force. The magnetic force on the aggregates contained in the scattering volume was estimated to be mainly parallel to the field direction and consequently to the chain axis. This means that the clusters tend to move out of the scattering volume in direction towards the magnet, where the field strength and its divergence are strongest. This effect, however, may affect the diffusive motion of the chains perpendicular to their axis and explains the relatively large fluctuations of the measured average diffusion coefficient. Moreover, one should bear in mind that the effective diffusion coefficient of the chains may also be affected by thermal shape fluctuations [28] and other hazardous effects. This means that the obtained results should be taken only as a rough estimation. Despite of these uncertainties, our DLS results provide clear experimental evidence of fieldinduced aggregation and show that the chains grow due to the applied magnetic field. In order to give an at least qualitative explanation for our experimental results, we estimated the interaction energy between two magnetic polystyrene particles, Et ,using the extended Derjaguin–Landau–Verwey–Overbeek (DLVO) theory. In this case, van der Waals attraction, the electrostatic repulsion and the long-range dipolar magnetic interaction have to be taken into account and so, the total interaction energy becomes Et (r) = Eel (r) + EvdW (r) + Edd (r). Using the constant potential solution for κa 1 and moderate potential, a reasonable expression for the repulsive electrostatic term, Eel (r) is given by: Eel (r) = 2πaεε0 ψδ2 ln[1 + exp(−κr)]
Fig. 3. Time dependence of the effective diffusion coefficient Deff = Deff,⊥ during field-induced aggregation.
(8)
where ε is the dielectric constant of the medium, κ the wellknown Debye parameter and ψδ is the Stern potential. As frequently found in colloid stability, the Stern potential is identified with the experimental available zeta potential, which can be directly obtained from electrophoretic mobility measurements. The expression for the van der Waals attraction is: A 2a2 4a2 2a2 EvdW (r) = − (9) + 2 + ln 1 − 2 6 r 2 − 4a2 r r
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where A is the Hamaker constant of the particles within the medium. In this work, a typical value of 1 × 10−20 J for the Hamaker constant of aqueous suspensions of polystyrene particles was used. In order to estimate the long-range magnetic interactions, the magnetized spheres are usually approximated as point dipoles of a well-defined magnetic moment. A measurement of the magnetic field strength versus distance from the magnet indicates that the magnetic field strength in the scattering volume is approximately 100 mT. At these field strengths, the effective magnetic moment m may be estimated using the relationship m =
4πa3 χH 3
(10)
is the external magnetic field and χ = 2.20 is the magwhere H netic susceptibility of the particles as obtained from the expression proposed in Ref. [1]. Neglecting higher order interactions, the potential energy for two aligned identical dipoles m is given by [1,11]: µ0 m2 1 − 3 cos2 θ Edd (r, θ) = (11) 4π r3 where θ is the angle between the magnetic moments and the distance vector. The proposed method for calculating the magnetic moment is only an approximation since we cannot ensure to be still in the linear regime of the magnetization curves where H and M are proportional. Furthermore, the non-uniformity of the applied magnetic field does not allow us to determine an exact value for the field strength at the place where the chains are formed. Fig. 4a shows the total particle–particle interaction energy Et (r) for the magnetic polystyrene particles when the magnetic particles are stable, i.e. when no external field is applied and no additional electrolyte is added. In this case, only electrostatic and van der Waals terms had to be included. Fig. 4b shows the same curves at high salt concentration. Fig. 4c, finally, plots the total interaction energy when a magnetic field of 60 mT is applied.
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The total particle–particle interaction energy Et (r) shown in Fig. 4b was determined for the magnetic polystyrene particles at a monovalent electrolyte concentration of 0.9 M. The Stern potential was estimated from electrophoretic mobility measurements obtaining −35 mV at 100 mM. At higher electrolyte concentration, the sample was not stable enough for electrophoretic mobility measurements and so, the same value was used for the Stern potential. The remaining parameters were chosen according to the experimental conditions. The magnetic dipole–dipole interaction was supposed to be negligible since the particles are expected to have no remanent magnetization at zero field. At high electrolyte concentration, the repulsive electrostatic interactions are almost entirely shielded and the predominant term in the total interaction energy is the attractive van der Waals energy. Hence, two approaching particles feel an attractive force as soon as they come sufficiently close, and finally, will adhere upon contact due to the presence of the deep minimum at zero distance. This explains why relatively open and ramified structures characterized by a fractal dimension close to 1.75 were found experimentally. The particle–particle interaction energy curve shown in Fig. 4c was calculated for magnetic particles in the presence of an external magnetic field without added electrolyte. Therefore, the magnetic moments were supposed to be aligned head to tail (θ = 0◦ ) along the field direction. A value of −70 mV was established for the Stern potential. The latter value was estimated from electrophoretic mobility measurements at low electrolyte concentrations. The electrolyte concentration used for the calculations was set to 1 mM. The total interaction energy exhibits a deep primary minimum at close contact, followed by a maximum and a shallower secondary minimum. Aggregation is, therefore, in principal, allowed either in the primary or secondary minimum. The most important fact observed in Fig. 4c is that the magnetic interactions at not too short distances are at least one order of magnitude stronger as the electrostatic and van der Waals terms. Consequently, they control the aggregation behavior completely. Hence, aggregation in head to tail configuration is strongly favored and so, chain-like structures will form. According to the figure, aggregation in this configuration takes clearly place in the primary minimum. Since the primary minimum is principally due to the attractive van der Waals interaction, it will not disappear when the magnetic field is turned off. This means that the formed chain-like structures cannot disassemble and remain in the system. 4. Conclusions
Fig. 4. Total particle–particle interaction energy Et (r) for the magnetic polystyrene particles: (a) without an applied magnetic field at an electrolyte concentration of 1 mM; (b) without an applied magnetic field at an electrolyte concentration of 0.9 M; (c) in presence of an applied magnetic field when the induced magnetic moments are aligned head to tail.
In a sufficiently dilute magnetic colloidal system, light scattering methods are suitable to study both the cluster structure and the kinetics of the cluster formation. Static light scattering shows that the magnetic particles aggregate in a chain-like manner which is characterized by a fractal dimension close to one. This prediction is corroborated via transmission electron microscopy. Dynamic light scattering measurements allows to monitor the time evolution of chain size. The extended DLVO theory gives an at least qualitative explanation for the experimental findings.
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Acknowledgments Financial support from the Spanish Ministerio de Ciencia y Tecnolog´ıa (Plan Nacional de Investigaci´on Cient´ıfica, Desarrollo e Innovaci´on Tecnol´ogica (I + D + I), Project MAT200308356-C04-01) and from the European Regional Development Fund (ERDF) is gratefully acknowledged. References [1] J.H.E. Promislow, A.P. Gast, J. Chem. Phys. 102 (13) (1995) 5942. [2] J. Liu, E.M. Lawrence, A. Wu, M.L. Ivey, G.A. Flores, K. Javier, J. Bibette, J. Richard, Phys. Rev. Lett. 74 (1995) 2828. [3] G. Hegelsen, A.T. Skjeltorp, P.M. Mors, R. Botet, R. Jullien, Phys. Rev. Lett. 61 (1988) 1736. [4] M. Hagenb¨uchle, J. Liu, Appl. Opt. 36 (1997) 7664. [5] A.V. Teixeira, I. Morfin, F. Ehrburger-Dolle, C. Rochas, E. Geissler, Phys. Rev. E 67 (2003) 215047. [6] S. Relle, S.B. Grant, C. Tsouris, Physica A 270 (1999) 427. [7] P. Licinio, F. Fr´ezard, Braz. J. Phys. 31 (3) (2001) 356. [8] P.J. Camp, G.N. Patey, Phys. Rev. E 62 (2000) 5403. [9] J.L. Carrillo, F. Donado, M.E. Mendoza, Phys. Rev. E 68 (2003) 61509. [10] S. Melle, M.A. Rubio, G.G. Fuller, Phys. Rev. Lett. 87 (2001) 115501. [11] C. Tsouris, T.C. Scott, J. Colloid Interface Sci. 171 (1995) 319. [12] F. Cousin, E. Dubois, V. Cabuil, Phys. Rev. E 68 (2003) 21405.
[13] E. Dubois, V. Cabuil, F. Bou´e, R. Perzynski, J. Chem. Phys. 111 (1999) 7147. [14] J. Lal, D. Abernathy, L. Auvray, O. Diat, G. Gr¨ubel, Eur. Phys. J. E 4 (2001) 263. [15] A.K. Vuppu, A.A. Garc´ıa, M.A. Hayes, Langmuir 19 (21) (2003) 8646. [16] M. Fermigier, A.P. Gast, J. Colloid Interface Sci. 154 (1992) 522. [17] K. Butter, P.H. Bomans, P.M. Frederik, G.J. Vroege, A.P. Philipse, J. Phys. Condens. Matter 15 (2003) S1451. [18] L.N. Donselaar, P.M. Frederik, P. Bomans, P.A. Buining, B.M. Humbel, A.P. Philipse, J. Magn. Magn. Mater. 201 (1999) 58. [19] L. Shen, A. Stachowiak, S.E.K. Fateen, P.E. Laibinis, T.A. Hatton, Langmuir 17 (2001) 288. [20] F. Mart´ınez-Pedrero, Estructura Fractal de Agregados de Part´ıculas Magn´eticas Coloidales, Diploma de Estudios Avanzados, Universidad de Granada (2004). [21] V. Cabuil, Preparation and properties of magnetic nanoparticles, in: A. Hubbard (Ed.), Encyclopedia of Surface and Colloid Science, Marcel Dekker, New York, 2002. [22] R. Julllien, R. Botet, Aggregation and Fractal Aggregation, World Scientific, Singapore, 1987. [23] R. Pecora, Dynamic Light Scattering, Plenum Press, London, 1985. [24] D.E. Koppel, J. Chem. Phys. 57 (11) (1972) 4814. [25] G. Odriozola, et al., Colloidal aggregation processes, in: A. Hubbard (Ed.), Encyclopedia of Surface and Colloid Science, Marcel Dekker, New York, 2004. [26] R. Pastor-Satorras, J.M. Rub´ı, J. Magn. Magn. Mater. 221 (2000) 124. [27] M.C. Miguel y, P. Satorras, Phys. Rev. E 59 (1999) 826. [28] S. Cutillas, J. Liu, Int. J. Mod. Phys. B 15 (2001) 803.
Aggregation of magnetic polystyrene particles: A light scattering study Fernando Mart´ınez-Pedrero, Mar´ıa Tirado-Miranda, Artur Schmitt, Jos´e Callejas-Fern´andez ∗ Departamento de F´ısica Aplicada, Universidad de Granada, Campus de Fuentenueva, E-18071 Granada, Spain Available online 7 October 2005
Abstract The aggregation behavior of dilute dispersions of magnetic polystyrene particles was studied experimentally by means of light scattering techniques. In absence of external magnetic fields, the particles are stable against flocculation due to charged carboxylic surface groups. Aggregation was induced applying an external magnetic field. Static light scattering (SLS) was used to study the geometry of the fractally shaped aggregates. Fractal dimensions close to 1.2 were found for the aggregates. Transmission electron microscopy micrographs show that the formed aggregates are chain-like, as predicted by the SLS results. Dynamic light scattering (DLS) was employed to monitor the time evolution of cluster size and to study the underlying growth kinetics. The observed results could be explained qualitatively in the framework of the classical DLVO theory when magnetic dipole–dipole interactions are taken into account. © 2005 Elsevier B.V. All rights reserved. Keywords: Polystyrene; Static light scattering; Dynamic light scattering; Magnetic polystyrene
1. Introduction A detailed analysis of colloidal aggregation involves two main aspects: the cluster morphology and the kinetics of aggregate formation. The cluster morphology is usually characterized by means of the fractal dimension, df , that is understood as a measure on how the particles fill the three-dimensional space. The kinetics of the processes is frequently studied by means of the temporal evolution of the mean aggregate size. Studies concerning aggregation phenomena in magnetic fluids have been of great interest over the past decades. However, papers that consider simultaneously both aspects of colloidal aggregation are scarce in bibliography [1–16]. Static (SLS) and dynamic (DLS) light scattering are suitable techniques to study the structure and the kinetics of colloidal aggregates. In the literature, however, there are only a few papers that apply these techniques for studying aggregation phenomena in magnetic fluids. In general, high volume fractions are used and so, both adsorption and multiple scattering impede light scattering studies to be performed reliably. Furthermore, the relatively small size and large polydispersity of the employed samples make the interpretation of light scattering data a difficult task. In order to overcome these difficulties, optical microscopy, neu-
∗
Corresponding author. E-mail address: [email protected] (J. Callejas-Fern´andez).
0927-7757/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2005.09.001
tron and X-rays scattering techniques are most commonly used [1–3,12–14]. One of the aim of this work is to demonstrate that visible light can be used to get valuable data on magnetic colloidal aggregation if the systems is sufficiently dilute and the size of magnetic particles is large. In the literature, several papers study the spatial structure of magnetic particle clusters by means of video microscopy or cryogenic electron microscopy [17,18]. Nevertheless, the measurement of the cluster fractal dimension in magnetic fluid has not been the focus of a great number of works, so far. Moreover, most papers that report measurements of the cluster fractal dimension do not address it as the central question. They usually omit a deeper discussion of its meaning and do not relate it with the cluster morphology and the underlying aggregation mechanisms [1,5,9,12–19]. As was mentioned above, small particles that lie in lower limit of the colloidal domain and high particles concentrations have been used for the experiments. Under these conditions, an adequate analysis of the cluster morphology becomes a difficult task. For larger particles and diluted systems, however, this becomes quite straightforward. Over the last years, optical microscopy techniques have been widely used [1,6,15] for monitoring the time evolution of the mean cluster size through a detailed analysis of sequences of photographs. In situ DLS measurements, however, are scarce in the literature. Nevertheless, the latter technique allows the average cluster diffusion coefficient and the corresponding mean cluster size to be determined continuously without having to
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handle and alter the sample. To the best of our knowledge, only Hagenb¨uchle and Liu studied the kinetics of the magnetic particle aggregation in ferrofluid emulsion of about 500 nm in diameter [4] and Tsouris and Scott worked on ferric oxide [11] using DLS as main experimental technique. In this paper, field-induced aggregation processes arising in well-characterized suspensions of monodisperse magnetic polystyrene microspheres will be studied. For this purpose, the fractal dimension of the formed clusters and the temporal evolution of the mean cluster size will be measured by means of SLS and DLS. Only diluted samples will be used in order to guarantee that hazardous multiple scattering can be neglected. In a previous work [20], we compared the scattering pattern of these kind of particles and similar non-magnetic particles. We found that light adsorption in the magnetic particles does not substantially modify the obtained results. 2. Experimental 2.1. Materials Aqueous suspensions of magnetic polystyrene microspheres were used for the experiments. The samples were provided by Merck (Estapor Microspheres, Ref.: R 00-39). The essentially monodisperse particles (see Fig. 1) are charge stabilized due to carboxylic surface groups. Their mean particle diameter was 160 ± 5 nm as determined by means of photocorrelation measurements. Their magnetic behavior arises from small ferromagnetic grains embedded in the polymer matrix. The diameter of these magnetic grains is approximately 10 nm. According to the manufacturer, the ferrite mass content was 53.2% and the saturation magnetization (Ms ) was approximately 36 kA/m. The mean particle mass density was 1.2 g/ml. The magnetic polystyrene particle suspensions were initially stable. When exposed to a
sufficiently strong external magnetic field, however, the particles start to aggregate. Due to their small size, the magnetic grains contain a single magnetic domain. Since no long-range order exists between domains, the dipoles are randomly oriented. The anisotropy energy, Ea , of each domain is of the order of the thermal energy kT, and its magnetization vector fluctuates around the main axis of magnetization with a characteristic Neel relaxation time given by [21]: Ea τ = τ0 exp (1) kT When an external magnetic field is applied, one of the possible orientations in each grain is favored and the entire particle acquires a net magnetization. For this reason, the particles are called superparamagnetic. They are paramagnetic on the colloidal scale. Their magnetization is completely reversible and there is no magnetic remanence. Nevertheless, the net magnetic moment of a magnetic colloidal particle is much larger than the moments in a pure paramagnet. 2.2. Methods The final particle number concentrations in the reaction vessel was 7.2 × 1017 p/m3 . The temperature was set to 298.0 ± 0.5 K. The magnetic field needed to achieve field-induced aggregation was applied to the sample by means of a neodymium disk magnet (Hadle Gac, Barcelona, Spain). The magnet was placed directly above the measuring cell containing the colloidal dispersion. This implies, of course, that the field strength is not uniform and depends on the position within the cell. A homogeneous magnetic field would be desirable but is not essential for the experiments since the goal of this work is to ensure that the dipoles induced in the magnetic particles are strong enough to give rise to aggregation. The DLS experiments were performed during at least 20 min. For the SLS experiment, the magnet was removed from the measuring cell after 30 min and the sample was vigorously shaken and returned to the SLS instrument. Hence, all the SLS measurements were performed without any applied external magnetic field. The magnetic particles were also aggregated in the absence of an external magnetic field by adding KBr as indifferent electrolyte. The electrolyte-induced aggregation processes were started by mixing equal amounts of electrolyte solution and particle suspension through a Y-shaped mixing device. A final electrolyte concentration of 0.9 M was used in order to compress the electric double layer and to screen the electrostatic repulsion efficiently. 2.3. Light scattering measurements
Fig. 1. Transmission electron microscopy micrograph of: (a) several individual magnetic polystyrene particles and (b) a sample aggregate formed due to an applied magnetic field.
The light scattering experiments were performed using a slightly modified Malvern 4700 System (UK) working with a 488 nm wavelength argon laser. The photomultiplier arm was previously located at the reference position in order to set the 0◦ angle. The morphology of the aggregates was assessed by means of SLS. The mean scattered intensity was obtained for
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different angles in the range from 20◦ to 140◦ . SLS allows the fractal dimension, df , to be measured using the following relationship between the mean scattered light intensity, I(q), and the θ scattering wave vector, q = 4π sin λ 2 , where λ is the medium wavelength and θ is the scattering angle [22]: I(q) ≈ S(q) ≈ q−df
(2)
The aggregate structure factor S(q) is related to the spatial distribution of the particles contained within the aggregates. The mean scattered intensity contains information about the cluster structure in the q range, Rh −1 < q < a−1 where a is the particle radius and Rh is the average hydrodynamic radius of the formed aggregates. For higher q values, the length scale corresponds to individual particles, and thus, the intensity is related to their form factor. In the lower q region, topological length correlations between clusters could be studied. After a sufficiently long time, the mean intensity I(q) showed the expected asymptotic time-independent behavior for all experiments reported in this paper. This means that the final fractal structure of the clusters was then totally established. From the slope of the asymptotic curves, the fractal dimensions were finally determined. DLS assess the scattered intensity autocorrelation function as the product of the photon counts at times t and t + τ such that G(τ) = I(t)I(t + τ). There from the normalized intensity autocorrelation function gint (τ) =
I(t)I(t + τ) I(t)I(t)
(3)
was calculated and converted into the scattered field autocorrelation function by the aid of the Siegert relationship: gint (τ) = 1 + C|gfield (τ)|2
(4)
where C is a constant which depends on the optics of the instrument [23]. For our experiments, the scattering angle was set to 60◦ and the scattered light intensity autocorrelation functions were determined at different times during aggregation. Data analysis was performed using our own programs. Information on the cluster size distribution was obtained from the fitting coefficients of the expanded logarithm of the field autocorrelation function [24]: 2 3 τ τ field ln g (τ) = −µ1 τ + µ2 − µ3 + ··· (5) 2 3! This method is known as cumulant analysis and has been widely used for dynamic light scattering data analysis. For polydisperse systems, the first cumulant, µ1 , is related to the effective mean particle diffusion coefficient, Deff , by µ1 = Deff q2 . From the mean particle diffusion coefficient, the mean hydrodynamic size may be calculated using the well-known Stokes–Einstein relationship. 3. Results and discussion Fig. 2 shows the I(q) curves for the magnetic polystyrene particles aggregated due to 0.9 M KBr of added electrolyte and due to an applied external magnetic field. The curves exhibit a linear
Fig. 2. Scattered light intensity vs. scattering vector for the magnetic polystyrene particles aggregated at two different experimental conditions: () in the presence of a magnetic field and () at 0.9 M KBr.
behavior in the scattering wave vector range Rh −1 < q < a−1 indicated in the plots by vertical lines and so, the cluster fractal dimension df can be determined from the slope in both cases. At high electrolyte concentration, a fractal dimension of 1.81 ± 0.05 was obtained. This value is quite close to 1.75 that is frequently reported in the literature for freely diffusing sticky particles [25]. This means that our superparamagnetic particles show the same typical experimental behavior than other monodisperse non-magnetic particles. Therefore, light absorption and magnetic permeability changes due to the magnetic grains embedded in the latex matrix of these particles do not significantly alter the aggregate structure factor at the employed wavelength. Moreover, the previous results are in agreement with the superparamagnetic character of the particles, i.e. they behave as standard non-magnetic particles when they are not in a magnetic field. In this case, the aggregation behavior of the samples is controlled by the isotropic electrostatic and van der Waals interactions. In the presence of an external magnetic field, however, the particles aggregate due to anisotropic magnetic interactions. In this case, a very low fractal dimension of 1.21 ± 0.14 was found for the magnetic polystyrene particles. The given values are the average of six measurements with a linear regression coefficient of at least 0.99. The given error intervals are the corresponding standard deviation. It should be emphasized that similar fractal dimensions were reported by other authors for magnetic liposomes [7], magnetite particles coated by two layers of different surfactants [19] and simulations [26]. We interpret the experimentally obtained fractal dimensions close to 1.2 as a result of more or less linear structures present in the sample. In order to corroborate the results obtained by means of SLS, we performed also transmission electron microscopy (TEM). Fig. 1b shows an image of a typical cluster contained in a sample of aggregated magnetic latex particles. This sample was handled according to the procedure described in the previous section when the particles aggregated only due to an applied magnetic field. In the micrograph, the above-mentioned linear chain-like structures together with individual particles can clearly be observed. This confirms that latex particle chains
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form. These chains are so stable that they do not necessarily disassemble when the magnetic field is turned off. Some are even strong enough to survive the necessary drying step prior to TEM imaging. DLS was employed for measuring the average diffusion coefficient of the aggregates. In a dilute and monodisperse system of spherical particles, Deff is given by the Stokes–Einstein equation: D0 =
kT 6πηa
(6)
As was shown by means of SLS, the magnetic latex particles start forming chains when the magnetic field is turned on. It is well established by theoretical and experimental works that the diffusion coefficient depends on the relative orientation between the applied magnetic field and the observation axis. In other words, Deff,|| reflects the translational motion of the chains along their axis if the observation axis is parallel to the applied field. If the observation axis is perpendicular to the applied field, then the lateral motion of chains is characterized by Deff,⊥ . Theoretical, simulation and experimental works show that Deff,|| is larger than Deff,⊥ , i.e. the diffusive motion of the chains perpendicular to their axis is slower than the diffusive motion parallel to their axis [14,27]. In our light scattering instrument, the field is perpendicular to the scattering plane and so, we mainly assess the lateral diffusive motion of the chains. Fig. 3 shows the time dependence of the effective diffusion coefficient Deff = Deff,⊥ when aggregates are growing in the presence of the applied magnetic field. As expected, Deff decreases with increasing time due to the slower motion of larger aggregates. Since the formed clusters are chain-like structures, the increasing aggregates size indicates that the average number of particles per chain grows in time. Taking into account the friction coefficient for a chain moving perpendicular to its axis, some authors propose the following relationship between Deff and the number of chain-forming particles N [4,27]: Deff = D0
3 ln N 2N
(7)
where D0 is the diffusion coefficient of the individual particles.
Fig. 3 shows the time evolution of the effective diffusion coefficient Deff for the field-induced aggregation process. As can be seen, Deff decreases in time and so, the average chain length increases due to aggregation. Equation (7) relates the effective diffusion coefficient Deff to the chain length. The corresponding chain lengths are indicated in the plot as horizontal lines. This allows to estimate the time at which chains of an average length N are formed. In the experiment, a neodymium disk magnet was placed directly over the cell containing the colloidal dispersion. This implies, of course, that the field strength is not uniform and depends on the position within the cell. A measurement of the magnetic field strength versus distance from the tip of the magnet indicates that, after an initial rapid decreased very close to the magnet’s surface, the field changes slowly at moderate distances. A non-uniform magnetic field gives rise to a magnetic force. The magnetic force on the aggregates contained in the scattering volume was estimated to be mainly parallel to the field direction and consequently to the chain axis. This means that the clusters tend to move out of the scattering volume in direction towards the magnet, where the field strength and its divergence are strongest. This effect, however, may affect the diffusive motion of the chains perpendicular to their axis and explains the relatively large fluctuations of the measured average diffusion coefficient. Moreover, one should bear in mind that the effective diffusion coefficient of the chains may also be affected by thermal shape fluctuations [28] and other hazardous effects. This means that the obtained results should be taken only as a rough estimation. Despite of these uncertainties, our DLS results provide clear experimental evidence of fieldinduced aggregation and show that the chains grow due to the applied magnetic field. In order to give an at least qualitative explanation for our experimental results, we estimated the interaction energy between two magnetic polystyrene particles, Et ,using the extended Derjaguin–Landau–Verwey–Overbeek (DLVO) theory. In this case, van der Waals attraction, the electrostatic repulsion and the long-range dipolar magnetic interaction have to be taken into account and so, the total interaction energy becomes Et (r) = Eel (r) + EvdW (r) + Edd (r). Using the constant potential solution for κa 1 and moderate potential, a reasonable expression for the repulsive electrostatic term, Eel (r) is given by: Eel (r) = 2πaεε0 ψδ2 ln[1 + exp(−κr)]
Fig. 3. Time dependence of the effective diffusion coefficient Deff = Deff,⊥ during field-induced aggregation.
(8)
where ε is the dielectric constant of the medium, κ the wellknown Debye parameter and ψδ is the Stern potential. As frequently found in colloid stability, the Stern potential is identified with the experimental available zeta potential, which can be directly obtained from electrophoretic mobility measurements. The expression for the van der Waals attraction is: A 2a2 4a2 2a2 EvdW (r) = − (9) + 2 + ln 1 − 2 6 r 2 − 4a2 r r
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where A is the Hamaker constant of the particles within the medium. In this work, a typical value of 1 × 10−20 J for the Hamaker constant of aqueous suspensions of polystyrene particles was used. In order to estimate the long-range magnetic interactions, the magnetized spheres are usually approximated as point dipoles of a well-defined magnetic moment. A measurement of the magnetic field strength versus distance from the magnet indicates that the magnetic field strength in the scattering volume is approximately 100 mT. At these field strengths, the effective magnetic moment m may be estimated using the relationship m =
4πa3 χH 3
(10)
is the external magnetic field and χ = 2.20 is the magwhere H netic susceptibility of the particles as obtained from the expression proposed in Ref. [1]. Neglecting higher order interactions, the potential energy for two aligned identical dipoles m is given by [1,11]: µ0 m2 1 − 3 cos2 θ Edd (r, θ) = (11) 4π r3 where θ is the angle between the magnetic moments and the distance vector. The proposed method for calculating the magnetic moment is only an approximation since we cannot ensure to be still in the linear regime of the magnetization curves where H and M are proportional. Furthermore, the non-uniformity of the applied magnetic field does not allow us to determine an exact value for the field strength at the place where the chains are formed. Fig. 4a shows the total particle–particle interaction energy Et (r) for the magnetic polystyrene particles when the magnetic particles are stable, i.e. when no external field is applied and no additional electrolyte is added. In this case, only electrostatic and van der Waals terms had to be included. Fig. 4b shows the same curves at high salt concentration. Fig. 4c, finally, plots the total interaction energy when a magnetic field of 60 mT is applied.
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The total particle–particle interaction energy Et (r) shown in Fig. 4b was determined for the magnetic polystyrene particles at a monovalent electrolyte concentration of 0.9 M. The Stern potential was estimated from electrophoretic mobility measurements obtaining −35 mV at 100 mM. At higher electrolyte concentration, the sample was not stable enough for electrophoretic mobility measurements and so, the same value was used for the Stern potential. The remaining parameters were chosen according to the experimental conditions. The magnetic dipole–dipole interaction was supposed to be negligible since the particles are expected to have no remanent magnetization at zero field. At high electrolyte concentration, the repulsive electrostatic interactions are almost entirely shielded and the predominant term in the total interaction energy is the attractive van der Waals energy. Hence, two approaching particles feel an attractive force as soon as they come sufficiently close, and finally, will adhere upon contact due to the presence of the deep minimum at zero distance. This explains why relatively open and ramified structures characterized by a fractal dimension close to 1.75 were found experimentally. The particle–particle interaction energy curve shown in Fig. 4c was calculated for magnetic particles in the presence of an external magnetic field without added electrolyte. Therefore, the magnetic moments were supposed to be aligned head to tail (θ = 0◦ ) along the field direction. A value of −70 mV was established for the Stern potential. The latter value was estimated from electrophoretic mobility measurements at low electrolyte concentrations. The electrolyte concentration used for the calculations was set to 1 mM. The total interaction energy exhibits a deep primary minimum at close contact, followed by a maximum and a shallower secondary minimum. Aggregation is, therefore, in principal, allowed either in the primary or secondary minimum. The most important fact observed in Fig. 4c is that the magnetic interactions at not too short distances are at least one order of magnitude stronger as the electrostatic and van der Waals terms. Consequently, they control the aggregation behavior completely. Hence, aggregation in head to tail configuration is strongly favored and so, chain-like structures will form. According to the figure, aggregation in this configuration takes clearly place in the primary minimum. Since the primary minimum is principally due to the attractive van der Waals interaction, it will not disappear when the magnetic field is turned off. This means that the formed chain-like structures cannot disassemble and remain in the system. 4. Conclusions
Fig. 4. Total particle–particle interaction energy Et (r) for the magnetic polystyrene particles: (a) without an applied magnetic field at an electrolyte concentration of 1 mM; (b) without an applied magnetic field at an electrolyte concentration of 0.9 M; (c) in presence of an applied magnetic field when the induced magnetic moments are aligned head to tail.
In a sufficiently dilute magnetic colloidal system, light scattering methods are suitable to study both the cluster structure and the kinetics of the cluster formation. Static light scattering shows that the magnetic particles aggregate in a chain-like manner which is characterized by a fractal dimension close to one. This prediction is corroborated via transmission electron microscopy. Dynamic light scattering measurements allows to monitor the time evolution of chain size. The extended DLVO theory gives an at least qualitative explanation for the experimental findings.
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Acknowledgments Financial support from the Spanish Ministerio de Ciencia y Tecnolog´ıa (Plan Nacional de Investigaci´on Cient´ıfica, Desarrollo e Innovaci´on Tecnol´ogica (I + D + I), Project MAT200308356-C04-01) and from the European Regional Development Fund (ERDF) is gratefully acknowledged. References [1] J.H.E. Promislow, A.P. Gast, J. Chem. Phys. 102 (13) (1995) 5942. [2] J. Liu, E.M. Lawrence, A. Wu, M.L. Ivey, G.A. Flores, K. Javier, J. Bibette, J. Richard, Phys. Rev. Lett. 74 (1995) 2828. [3] G. Hegelsen, A.T. Skjeltorp, P.M. Mors, R. Botet, R. Jullien, Phys. Rev. Lett. 61 (1988) 1736. [4] M. Hagenb¨uchle, J. Liu, Appl. Opt. 36 (1997) 7664. [5] A.V. Teixeira, I. Morfin, F. Ehrburger-Dolle, C. Rochas, E. Geissler, Phys. Rev. E 67 (2003) 215047. [6] S. Relle, S.B. Grant, C. Tsouris, Physica A 270 (1999) 427. [7] P. Licinio, F. Fr´ezard, Braz. J. Phys. 31 (3) (2001) 356. [8] P.J. Camp, G.N. Patey, Phys. Rev. E 62 (2000) 5403. [9] J.L. Carrillo, F. Donado, M.E. Mendoza, Phys. Rev. E 68 (2003) 61509. [10] S. Melle, M.A. Rubio, G.G. Fuller, Phys. Rev. Lett. 87 (2001) 115501. [11] C. Tsouris, T.C. Scott, J. Colloid Interface Sci. 171 (1995) 319. [12] F. Cousin, E. Dubois, V. Cabuil, Phys. Rev. E 68 (2003) 21405.
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