pubs.acs.org/Langmuir © 2009 American Chemical Society
Primary and Secondary Bonds in Field Induced Aggregation of Electric Double Layered Magnetic Particles F. Martı´ nez-Pedrero, M. Tirado-Miranda, A. Schmitt, and J. Callejas-Fernandez* Department of Applied Physics, University of Granada, Campus de Fuentenueva, E-18071 Granada, Spain Received October 31, 2008. Revised Manuscript Received March 12, 2009 Magnetic filaments able to survive on removal of the magnetic field have led to new applications in biotechnology and microfluidics. In this work, the stability of linear field induced aggregates composed of electric double layered magnetic particles has been studied in the framework of the DLVO theory. A suitable system of differential equations is proposed in order to determine how the percentage of bonds formed in a primary or a secondary energy minimum depends on the exposure time to the magnetic field. The theoretical predictions were compared with experimental data obtained by means of dynamic light scattering. The aggregation experiments were performed at different electrolyte concentrations and exposure times to the magnetic field. Fitting the experimental results according to the proposed theory allowed us to determine the rate at which bonds formed in a secondary energy minimum turn into stable bonds.
Introduction Over the last years, bottom up techniques were proposed by several authors for obtaining more and more complex nanomaterials in a controlled manner starting from molecules.1,2 Within this framework, the preparation of linear particle arrays may be considered as an intermediate step for the design of novel nanostructured devices. Such linear nanostructures can easily be obtained by applying an uniaxial magnetic field to a suspension of magnetic nanoparticles. The field induces a magnetic moment in each bead that gives rise to an anisotropic dipolar interaction. Consequently, the particles self-organize due to the action of the field, and aggregates of particles aligned along the field direction are formed. If the dipolar interactions are strong enough, then aggregation is irreversible as long as the magnetic field is applied.3,4 Under normal circumstances, however, aggregation is reversible once the field is removed; that is, the magnetic interaction vanishes and Brownian motion or electrostatic repulsion, which controls the stability of electric double layered particles (EDLMP) suspended in aqueous media, pushes the particles away from each other. Hence, the chains break into its constituent particles and the system returns to its nonaggregated state. Depending on the interplay between the repulsive isotropic electrical interactions and the anisotropic magnetic interactions, primary or secondary minimum aggregation or stable particle suspensions may arise. Usually, aggregation takes place in a secondary energy minimum and chain rupture is observed as soon as the external magnetic field is removed.5 When the particles have aggregated in the primary energy minimum, the chains formed are permanent; that is, they do not break once the magnetic field is turned off. If the magnetic particles are linked permanently due to strong short-range attractive interactions, then the linear geometry of the aggregates is preserved even in the absence of the field. Such a strong adhesion *To whom correspondence should be addressed. E-mail:
[email protected]. (1) Mijatovic, D.; Eijkel, J. C. T.; van den Berg, A. Lab Chip 2005, 5, 492. (2) Whitesides, G. M.; Kriebel, J. K.; Mayers, B. T. Nanoscale Assembly: SelfAssembly and Nanostructured Materials; Springer: Cambridge, UK, 2005. (3) Martı´ nez-Pedrero, F.; Tirado-Miranda, M.; Schmitt, A.; Callejas-Fernandez, J. Phys. Rev. E 2007, 76, 011405. (4) Promislow, J. H. E.; Gast, A. J. Chem. Phys. 1995, 102(13), 5492. (5) Chin, C. J.; Yiacoumi, S.; Tsouris, C.; Relle, S.; Grant, S. B. Langmuir 2000, 16, 3641.
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may be induced by van der Waals attraction6,7 or by adsorbed molecules that establish a strong bridge between the particle surfaces.8 The length of the linear aggregates can be controlled by forming the chains in microchannels of a given height.9 Surfaces patterned with strongly anchored magnetic particles can serve as templates for chain growth. Since the chains start to grow from the fixed particles, migration and flow processes may be prevented.10,11 Using alternative experimental protocols, quite rigid magnetic chains were obtained by other authors. One-dimensional structures have been obtained, for example, for cobalt nanoparticles that undergo a superparamagnetic to ferromagnetic transition as their size increases during the synthesis process.12 On the other hand, if the synthesis of the magnetic nanoparticles is due to coagulation of primary units, an applied magnetic field has shown to have a dramatic effect on the morphology. In the latter case, extremely rigid rodlike particles were achieved.13 Over the last years, magnetic filaments able to survive on removal of the applied field have led to new applications. The stiffness and magnetorheological properties of such permanent chains have been the subject of several works. Biswal and Gast,14 and Goubault et al.15 performed a variety of experiments to measure the flexural rigidity of the magnetic filaments and to probe the bending rigidity at a molecular scale. Linear chains of colloidal magnetic particles, linked by DNA and attached to a red blood cell, have even been used as flexible artificial flagellum.16 (6) Chin, C. J.; Yiacoumi, S.; Tsouris, C. Langmuir 2001, 17, 6065. (7) Martı´ nez-Pedrero, F.; Tirado-Miranda, M.; Schmitt, A.; Callejas-Fernandez, J. J. Chem. Phys. 2006, 125, 084706. (8) Cohen-Tannoudji, L.; Bertrand, E.; Bressy, L.; Goubault, C.; Baudry, J.; Klein, J.; Joanny, J. F.; Bibette, J. Phys. Rev. Lett. 2005, 94, 038301. (9) Furst, E. M.; Suzuki, C.; Fermigier, M.; Gast, A. P. Langmuir 1998, 14, 7334. (10) Singh, H.; Laibinis, P. E.; Hatton, T. A. Nano Lett. 2005, 5, 2149. (11) Lyles, B. F.; Terrot, M. S.; Hammond, P. T.; Gast, A. P. Langmuir 2004, 20, 3028. (12) Salguiri~no-Maceira, V.; Correa-Duarte, M. A.; Hucht, A.; Farle, M. J. Magn. Magn. Mater. 2006, 303, 163. (13) Vereda, F.; de Vicente, J.; Hidalgo-Alvarez, R. Langmuir 2007, 23, 3581. (14) Biswal, S. L.; Gast, A. Phys. Rev. E 2004, 69, 041406. (15) Goubault, C.; Jop, P.; Fermigier, M.; Baudry, J.; Bertrand, E.; Bibette, J. Phys. Rev. Lett. 2003, 91, 260802. (16) Dreyfus, R.; Baudry, J.; Roper, M.; Fermigier, M. L.; Stone, H. A.; Bibette, J. Nature 2005, 437, 862.
Published on Web 05/19/2009
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Permanently linked linear aggregates were also employed as micromechanical sensors,15,17 as micromixers, 18 and for DNA molecule separation where they serve as obstacles that impede the convective transport of the biological species.19 Using a well characterized colloidal suspension of EDLMP, the main aim of this work is to determine how the number of primary and secondary bonds evolves in time during field induced aggregation processes. For that purpose, we designed an accurate experimental protocol that allowed these quantities to be determined at different electrolyte concentrations and exposure times to the magnetic field. The particles were always aggregated in the presence of a magnetic field for a given time. Then, the field was removed. During the entire process, the average length of the linear field induced aggregates was monitored using dynamic light scattering. Hence, the average chain length prior to and after field removal could be compared. As we will see, the degree of chain disintegration is determined by the average number of stable bonds per chain. Furthermore, we propose a set of coupled differential equations that describe the time evolution of the secondary and primary bonds as a function of the experimental conditions. These equations depend on the rate at which bonds formed in a secondary energy minimum turn into stable bonds. This rate can be determined by comparing the theoretical predictions with the experimental data obtained under different experimental conditions. Finally, we will use the results to check the consistency of our approach and to confirm that the energy barrier separating the primary from the secondary minimum is lower the higher the electrolyte concentration becomes. In diffusion limited aggregation processes, for example, it has been extensively observed that added electrolyte screens the electrostatic repulsion between double layered particles and enhances the formation of strong or primary bonds. The next section of this paper describes the theoretical background. After that, the materials and methods used in this work are presented. The paper ends with a discussion of the obtained results.
Theory Time Evolution of Primary and Secondary Bond Population. The total interaction energy potential corresponding to two approaching EDLMP features, in general, a shallow secondary and a deep primary energy minimum that are separated by an energy barrier Ea of certain height.6,20 A schematic plot of the corresponding interaction profile is shown in Figure 1. For the plot, the angle between the external magnetic field direction and the line joining the particle centers was supposed to be smaller than jc ≈ 55.3 Particle aggregation may occur in the primary minimum, where the particles are in contact with each other, or in the secondary minimum, where the neighboring particles within the linear aggregates are a short distance apart from each other. The height of the energy barrier is mainly determined by the electrostatic repulsion between the particles.20 Secondary minimum aggregation is reversible, because the attractive magnetic interaction and the secondary minimum disappear as soon as the magnetic field is turned off. Then, the electrostatic repulsion controls the stability of the system and pushes the particles away from each other, (17) Cohen-Tannoudji, L.; Bertrand, E.; Baudry, J.; Robic, C.; Goubault, C.; Pellissier, M.; Johner, A.; Thalmann, F.; Lee, N. K.; Marques, C. M.; Bibette, J. Phys. Rev. Lett. 2008, 100, 108301. (18) Biswal, S. L.; Gast, A. Anal. Chem. 2004, 76, 6448. (19) Doyle, P. S.; Bibette, J.; Bancaud, A.; Viovy, J. L. Science 2002, 295, 2237. (20) Martı´ nez-Pedrero, F.; Tirado-Miranda, M.; Schmitt, A.; Vereda, F.; Callejas-Fernandez, J. Colloids Surf., A 2007, 306, 158.
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Figure 1. Typical profile of the interaction energy between two approaching double layered magnetic particles under the influence of an external field (;). The interaction energy is the sum of the van der Waals ( 3 3 3 ), the electrostatic (- - -), and the magnetic dipole interaction (- 3 -). The curve is plotted as a function of the surface to surface distance d. np(t), ns(t), and nf(t) denote the number of primary, secondary, and open bonds, respectively.
giving rise to a complete breakup of the linear aggregates.5,7 On the other hand, aggregation in the primary energy minimum is usually irreversible, since, at close contact, the attractive London van der Waals interaction is capable to keep the particles together even when the external field is removed.6,7 Hence, linear aggregates formed in this way have an almost infinite lifetime. Figure 2 shows an example of filaments made of particles aggregated in a primary energy minimum. As stated before, magnetic particles can be found in three different states when a magnetic field is applied: (i) aggregated in the primary minimum (close contact), (ii) aggregated in the secondary minimum (short distance), and (iii) unlinked (large distance). If np(t), ns(t), and nf(t) denote the number of primary bonds, secondary bonds, and open bonds (i.e., not yet established bonds), respectively, the total number of bonds becomes n(t) = ns(t) + np(t). The number of secondary bonds increases in time when two particles or aggregates establish a new bond. It diminishes when two neighboring particles contained within a linear aggregate overcome the energy barrier and go from a metastable secondary bond to a stable primary bond. Hence, we propose the following rate equations for describing these field induced aggregation dns ðtÞ dnf ðtÞ ¼ -ksp ns ðtÞ dt dt
ð1aÞ
dnp ðtÞ ¼ ksp ns ðtÞ dt
ð1bÞ
The rate constant ksp represents the probability per unit time that a secondary bond requires for turning into a primary bond. The rate constant includes all the factors that affect the reaction rate, except for number of secondary bonds that is explicitly accounted for. The primary bonds, formed due to short-range van der Waals interactions, are usually stable enough so that the back rate constant kps may safely be neglected. Since rupture of the linear aggregates is forbidden in these equations, it adequately describes the experimental observations only as long as the magnetic field is applied. All the initially free particles that will end up bound in the primary minimum have to pass through a secondary minimum for at least a short time. Moreover, the proposed equations assume that bond formation between particles can be considered DOI: 10.1021/la803626v
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time evolution of the average chain length ÆN(t)æ, the rate constant ksp, and the boundary conditions for [ns(t)] and [np(t)] are known. The rate constant ksp depends on the temperature and the ionic strength.8 The particles are supposed to stick when they collide along the center to center line with a relative kinetic energy that exceeds Ea. At an absolute temperature T, the fraction of particles that have a kinetic energy larger than Ea should follow a Maxwell-Boltzmann distribution and thus be proportional to exp(-Ea/kBT). Consequently, the corresponding rate constant must obey an Arrhenius law:21
dnp ðtÞ ¼ ksp ns ðtÞ dt
ð2aÞ
ð2bÞ
From an experimental point of view, it is convenient to rewrite the previous equation in terms of particle and bond concentrations [n(t)], [ns(t)], and [np(t)]. d½ns ðtÞ dÆNðtÞæ -1 ¼ -c0 -ksp ½ns ðtÞ dt dt
ð3aÞ
d½np ðtÞ ¼ ksp ½ns ðtÞ dt
ð3bÞ
where c0 is the initial particle concentration. Solving these equations, the concentration of primary and secondary bonds, [ns(t)] and [np(t)], can be determined once the 6660 DOI: 10.1021/la803626v
ð4Þ
)
The units of the pre-exponential factor τ0-1 are s-1, and so it is usually referred to as a reaction attempt frequency. Average Chain Size. The aggregation behavior and the stability of our experimental system can be monitored using noninvasive dynamic light scattering (DLS). DLS determines an effective average diffusion coefficient Deff of the scattering entities contained within the scattering volume. It depends on the different diffusive modes that the scatterers may undergo. Hence, the diffusion coefficients depend on the chain length L, and DLS provides information about the average size of the linear aggregates. In this work, we assume that chains formed by N magnetic particles of radius a may be modeled as rigid cylinders of length L = Nd and diameter d = 2a. Tirado and Garcı´ a de la Torre proposed hydrodynamic equations for the rotational diffusion coefficient Dr and the diffusion coefficients D^ and D for translational motion perpendicular and parallel to the rod axes, respectively. These equations relate the different diffusion coefficients with the chain length L.22 ! kB T L ln D ¼ þ γ ðLÞ 2πηL d )
! kB T L þ γ^ ðLÞ ln D^ ¼ 4πηL d ! 3kB T L þ γr ðLÞ Dr ¼ ln πηL3 d
ð5aÞ
ð5bÞ
ð5cÞ
In the presence of an external magnetic field, the linear aggregates are forced to align in the field direction and so rotational chain diffusion is forbidden. Consequently, only the translational diffusion coefficients D and D^ will have to be considered for the theoretical analysis. In our experimental setup, however, the magnetic field is aligned perpendicular to the scattering plane. Due to this geometry, the measurements are only sensitive to the transversal motion of the linear aggregates, and the parallel diffusion coefficient D may also be neglected.23 Nevertheless, relative positional particle fluctuations inside the linear aggregates may take place due the competition between Brownian motion and the magnetic dipole-dipole interactions (see Figure 2). At the field strength employed in our experiments, )
dns ðtÞ dÆNðtÞæ -1 ¼ -N0 -ksp ns ðtÞ dt dt
1 -Ea =kB T e τ0
)
as independent events, and all the secondary bonds show the same rate constants ksp regardless of their positions inside the linear aggregates. Equations 1a and 1b allow the time evolution of the number of primary and secondary bonds to be determined as a function of the exposure time to the magnetic field if the boundary conditions, the rate constant value ksp, and the time evolution of the number of not yet established bonds nf are known. It is, however, not straightforward to assess the number of not yet established bonds nf, and consequently the term dnf/dt. In what follows, we propose to work with quantities that can be measured by means of dynamic light scattering. The average number of bonds per chain is given by Æn(t)æ ≈ n(t)/Naggr(t), where Naggr(t) ≈ N0/ÆN(t)æ is the total number of aggregates present in the suspension, N0 is the initial number of free monomeric particles dispersed in the solution, and ÆN(t)æ is the average chain length. Thus, the total number of bonds becomes n(t) ≈ (N0Æn(t)æ)/ÆN(t)æ = N0(ÆN(t)æ 1)/ÆN(t)æ. In linear aggregates, Æn(t)æ = (ÆN(t)æ - 1), and so we obtain n(t) ≈ N0(ÆN(t)æ - 1)/ÆN(t)æ. The time derivative of n(t) becomes dn(t)/dt = -N0 dÆN(t)æ-1/dt. Since an increase in the total number of bonds can only be due to a decrease of the number of not yet established bonds, it is clear that dn(t)/dt = -dnf(t)/dt. Hence, eq 1a reads:
ksp ¼
)
Figure 2. Video microscopy images of linear aggregates formed under the presence of an external magnetic field. The images show that at high field strengths the filament has relatively straight and linear form (a), while at lower field strengths it looks more bended and twisted due to the competition between Brownian motion and magnetic dipole-dipole interactions (b). At high electrolyte concentrations, some linear aggregates formed are able to survive the absence of the magnetic field (c). The degree of chain disintegration is determined by the average number of stable bonds per chain.
(21) Krammers, H. A. Physica 1940, 7, 284. (22) Tirado, M.; Garcı´ a de la Torre, J. J. Chem. Phys. 1979, 71, 2581. (23) Maeda, T.; Fujime, S. Macromolecules 1984, 17, 1157.
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however, the absence of significant internal fluctuations can be guaranteed.3 When the magnetic field is removed, rotational Brownian motion of the linear aggregates becomes possible. Experimental works using DLS usually assume that translational diffusion is isotropic and not coupled with rotational diffusive modes. This means that the diffusion coefficients for motion parallel and perpendicular to the rod axis should be similar and the rods must reorient many times while diffusing a distance comparable to q-1, where q = (4π/λm) sin(θ/2) is the scattering vector. Here, θ is the scattering angle and λm is the wavelength in the dispersion medium. For long rods, however, a strong coupling between translational and rotational modes is expected and a no-trivial coupled diffusion equation has to be solved. Neglecting relative motion of the particles inside the linear aggregates, Maeda and Fujime derived the following theoretical approximation for the effective diffusion coefficient Deff:23 "
Deff
# 1 qL L2 ql þ Dr f1 ¼ D -ΔD -f2 3 2 2 12
ð6Þ
)
)
where D = 1/3D + 2/3D^ and ΔD = D - D^. In the present work, the numerical values for f1(qL/2) and f2(qL/2) were employed as given by Maeda and Fujime. This approach allows the average chain length to be assessed in units of particles per chain rather than in terms of an average diffusion coefficient. Further details about the light scattering methods employed can be found in ref 24.
Materials and Methods Magnetic polystyrene particles are known for their almost monodisperse size distribution and perfect spherical shape. The polymer matrix stabilizes the magnetic particles and decreases the particle mass density. In some applications, an additional polymer coating is required in order to provide selective functionality and interaction with target solutes. The superparamagnetic latex particles used for the light scattering experiments were purchased from Merck Laboratories S.A. (ref: R0039). The particles are roughly monodisperse polystyrene spheres with an average diameter of (170 ( 5) nm. Their magnetic character derives from magnetite grains of approximately 10 nm in size. In transmission electron micrographs, the magnetic grains appear as dark spots (TEM) randomly distributed within the polystyrene matrix. According to the manufacturer, the magnetic polystyrene particles have a ferrite mass content of approximately 53.2% and a saturation magnetization Ms of approximately 36 kA/m. The sample was dried and the magnetization of the resulting powder was measured at 298 K in order to quantify the magnetization of the particles. The magnetization curve reveals the superparamagnetic character of the magnetic particles.24 Consequently, dipolar magnetic interactions between the particles only appear in the presence of an applied magnetic field. The particles reach the saturation magnetization for external magnetic field above 200 kA/m. The particles are dispersed in water, and the stability of the system is ensured by repulsive forces due to charged carboxylic surface groups and anionic sodium dodecyl sulfate (SDS) surfactant molecules adsorbed on the particle surface. The particle surface potential of approximately -50 mV was determinated by means of electrophoretic mobility measurements. Due to their relatively low density of 1.2 g/cm3, particle sedimentation was found to be negligible during the experiments. The intensity of the light scattered by a stable sample remains almost constant for 2 days. Prior to the aggregation experiments, (24) Martı´ nez-Pedrero, F.; Tirado-Miranda, M.; Schmitt, A.; Callejas-Fernandez, J. J. Colloid Interface Sci. 2008, 318, 23.
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the diluted samples were filtered through a 450 nm pore size membrane filter in order to eliminate any primary clusters. The particle number concentration was always adjusted to 1010 cm-3. Such a low particle concentration avoids multiple light scattering and lateral chain-chain aggregation. Initially stable samples of monodisperse superparamagnetic particles were aggregated in the presence of an external magnetic field and different amounts of electrolyte. Several series of experiments have been performed at different concentrations of an indifferent 1:1 electrolyte (KBr). The final electrolyte concentrations used for reducing the repulsive electrostatic interactions were 0.0, 0.25, 0.50, 1.0, 2.0, 5.0, 10, 20, 25, and 50 mM. The electrolyte was always added to the colloidal dispersion before exposing the samples to the magnetic field. This was achieved by mixing equal amounts of KBr solution and particle suspension through a Y-shaped mixing device directly into the measuring cell. Up to an electrolyte concentration of approximately 50 mM, the system remained stable when it was not exposed to the magnetic field. The magnetic field required to achieve field induced aggregation was applied to the sample by placing a different number of neodymium disk magnets on top of the sample cell. The magnetic flux was guided to the scattering volume by a soft iron cylinder. In order to reduce the inhomogeneity of the magnetic field and collect the magnetic flux, another iron disk was placed at the bottom of the cell. A nonuniform magnetic field would give rise to a net magnetic force, and the clusters would tend to move out of the scattering volume in direction toward the magnet, where the field strength and its divergence are strongest. Spatial field homogeneity is crucial for the experiments, since it avoids particle migration and, consequently, concentration heterogeneities within the reaction vessel. The field strength employed in this work was H = 23.9 kA/m. The field was always applied perpendicularly to the scattering plane. At the field strength employed, we are in the linear regime of the magnetization curve, where the magnetization M is proportional to the external magnetic field H, i.e. M = χH. From the curve, the magnetic susceptibility of our particles was estimated to be χ ≈ 0.6. The relative strength of the magnetic forces can be assessed through the dimensionless parameter l = πμ0a3χ2H2/9kBT which represents the ratio between the maximum attractive magnetic dipole-dipole energy and the thermal energy. In our experiments, λ ≈ 14 was achieved. The light scattering experiments were performed using a commercial Malvern 4700C DLS setup, working with a vertically polarized 632.8 nm wavelength HeNe laser. The DLS measurements were always performed at a scattering angle of θ = 60 corresponding to a scattering vector of q-1 = 75.7 nm. Data analysis was carried out using software developed by our team. The effective diffusion coefficient Deff was obtained from the first cumulant μ1 = 2Deffq2 of the intensity autocorrelation function. Since this quantity is related to the average aggregate size, it will allow the state of aggregation to be monitored reliably.
Results and Discussion At the beginning of this section, ns(t) and np(t) are calculated indirectly by solving eqs 3a and 3b. Then, the sizes of the linear aggregates just before and after removing the magnetic field are extracted from the experimental data. These data allow also the percentage of secondary ns(t) and primary bonds np(t) to be determined directly. Finally, the rate constants ksp were calculated for different electrolyte concentrations by comparing both results. Indirect Determination of the Bond Concentrations [np(t)] and [ns(t)]. As was stated above, the boundary conditions, the term ÆN(t)æ-1, and the rate constant ksp are required for solving eqs 3a and 3b. Since the aggregation process always started from monomeric conditions, it reasonable to assume the concentrations of metastable and stable bonds to be negligible at the beginning of the aggregation process. This means that the boundary conditions DOI: 10.1021/la803626v
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Figure 3. Time evolution of the average number of constituent particles per aggregate at different electrolyte concentrations. During the first 30 min, the magnetic field was applied. Thereafter, the field was turned off.
become [ns(0)] = [np(0)] = 0 at t = 0 s. The term dÆN(t)æ-1/dt can be assessed by means of light scattering techniques. According to the experimental method described in the previous section, particles were aggregated in the presence of a magnetic field at different electrolyte concentrations. During the first tc minutes of the experiments, the magnetic field was applied, and the aggregation processes were monitored. In presence of the external field, the DLS measurements are only sensitive to the transversal motion perpendicular to the chain axis, and Deff = D^. Hence, the corresponding hydrodynamic equations were employed for the size dependency of the different diffusion coefficients involved. From the data for the perpendicular diffusion coefficient, the average filament length expressed in number of particles per aggregate ÆN(t)æ was extracted. As an example, the time evolution of the mean number of particles per chain ÆN(t)æ at different electrolyte concentrations is shown in Figure 3 for tc = 30 min. The data shown are always the average of five measurements. The corresponding error bars are not shown for the sake of clarity. The electrolyte free sample remains stable. At higher electrolyte concentrations, however, field induced aggregation was observed. The results obtained confirm that the average filament size increases monotonously with the exposure time to the magnetic field. At fixed aggregation time, the filaments are on average larger the higher the electrolyte concentration becomes. The dÆN(t)æ-1/dt term can be obtained by taking the derivative of an exponential decay function A e-Bt fitted previously to the inverse of the N(t) curves. Other fitting functions could have been chosen. The choice of this arbitrary function, however, is justified by the quality of the fits obtained. The described fitting process has been done for all the electrolyte concentrations employed. Having determined the term dÆN(t)æ-1/dt and the boundary conditions, the rate constant ksp remains as the only free parameter that is required for the calculation of the time evolution of the number of primary and secondary bonds. Direct Experimental Determination of the Bond Concentrations [np(t)] and [ns(t)]. Once the field was turned off, the DLS measurements were still performed for some additional time (please, see Figure 3). In this region, Maeda-Fujime’s model (eq 6) was employed for the calculations. As before, the electrolyte free sample remained stable. At higher electrolyte concentrations, however, field induced aggregation and filament breakup is observed. Our experiments show how the relative difference between the length of the linear aggregate prior to and after field 6662 DOI: 10.1021/la803626v
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removal becomes smaller as the electrolyte concentrations increase. The linear aggregates disassemble completely at low electrolyte concentrations. At intermediate amounts of added electrolyte, a partial cluster breakup is observed. This implies that some stable bonds in the deeper primary energy minimum must exist. However, the particles still have to overcome the relatively large energy barrier caused by the electrostatic repulsion before a stable bond will be formed. Since the height of the energy barrier decreases for increasing electrolyte concentration, it is not surprising that the average length of the stable aggregates becomes larger at higher electrolyte concentrations. Only at the highest electrolyte concentration of 50 mM, all the chains withstand the absence of the magnetic field. This means that the electrostatic energy barrier that prevented the particles from aggregation in the primary minimum is now almost completely suppressed and all the bonds are stable. The fact that the observed average length does not decrease further in time after the removal of the magnetic field allows us to confirm that the chains do not suffer any additional breakup. At 50 mM, the length of the linear aggregates even continues to grow slowly when the field is turned off. The aggregation is now led by the diffusion of the linear aggregates. At high enough electrolyte concentrations, the electrolyte repulsion between the linear aggregates is almost completely screened and linear aggregates are expected to aggregate further following a reaction or diffusion limited aggregation. Linear aggregates continue to aggregate, forming more and more branched structures, and fractal aggregates made of chainlike aggregates are observed. Further details about this interesting behavior can be found in a previous work.24 Having described how to determine the average chain length ÆN(t)æ from the experimental data when the magnetic field is applied and when it is turned off, we propose a method to estimate the average number of secondary and primary bonds per chain that existed at time tc just before the moment when the magnetic field was removed. Therefore, we analyze what happens when a linear aggregate breaks. As an example, Figure 4 shows a chain formed by 14 monomers containing three (case a) or perhaps four secondary bonds (case b). When the rupture of the secondary bonds takes place, the final average lengths become 14/4 = 3.5 (case a) and 14/5 = 2.8 (case b). This implies that the final average chain length after rupture can be obtained by the ratio between the number of constituent particles prior to breakup and the number of secondary bonds plus 1. More recently formed bonds nearer the ends of the filaments have not had as long to form primary bonds. Hence, inner bonds are more likely to be primary bonds, and breaks in the filaments would occur nearer the ends. The proposed average, however, does not depend on the position of the bonds within the aggregates. Applying this rule to a set of linear chains of different lengths, we obtain on average ÆNðtcþ Þæ ¼ ÆNðtc- Þæ=ðÆns ðtc- Þæ þ 1Þ ÆN(tc )æ
ð7Þ
ÆN(t+ c )æ
where and are the average numbers of particles per chain just prior to and just after the removal of the magnetic field, respectively. Æns(tc )æ is the average number of secondary bonds per chain assessed just prior to removing the magnetic field. From our light scattering data, it is straightforward to determine ÆN(t+ c )æ and ÆN(tc )æ. When the chains continue aggregating (high electrolyte) even after removal of the field, ÆN(t+ c )æ was obtained by linear extrapolation for t f t+ c . Similarly, ÆN(tc )æ was calculated by linear extrapolation from the values measured in presence of the field. The average number of secondary bonds per chain Æns(tc )æ is obtained then according to eq 7. Langmuir 2009, 25(12), 6658–6664
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Figure 4. The length of the linear aggregates just prior to and after removal of the magnetic field is directly related to the number of secondary bonds per chain. For example, we show a chain formed by 14 monomers that would have three (case a) or perhaps four secondary bonds (case b). The final average lengths are 3.5 (case a) and 2.8 (case b). Figure 6. Fitted ksp values as a function of the electrolyte concentration [KBr].
Figure 5. Percentage of metastable (O) and stable bonds (4) assessed after 30 min of exposition to the magnetic field as a function of the electrolyte concentration [KBr].
Figure 5 shows the percentage of primary and secondary bonds prior to field removal as a function of the electrolyte concentration. The percentage of stable bonds is larger the higher the electrolyte concentration becomes. These results imply that an effective control of the filament size requires also an adequate adjustment of the relative strength of the isotropic electric and anisotropic magnetic interactions. ksp Determination. Having selected an exposure time tc, the rate constant ksp is determined via a comparison between direct and indirect measurements. For that purpose we selected those ksp values which gave theoretical predictions, calculated through eqs 3a and 3b, that were closest to the percentages of the secondary ns and primary bonds np obtained through the DLS measurements. The ksp values determined at tc = 30 min are shown in Figure 6 as a function of the electrolyte concentration. Figure 6 shows clearly that the probability to overcome the potential barrier between the primary and the secondary minimum increases as the electrolyte concentration does. This trend agrees with the well-known fact that, in aggregation of charged colloidal particles, high electrolyte concentrations give rise to branched structures, since approaching clusters stick immediately and irreversibly at first contact. At low electrolyte concentrations, however, the first contact does not necessarily lead to the formation of an irreversible primary bond, and so the aggregating clusters will have the opportunity to interpenetrate even further, forming more compact morphologies. Using eq 4, the obtained ksp values will allow the energy barrier Ea to be determined. For the calculations, we employed the estimated by Cohen-Tannoudji et al. attempt frequency τ-1 0 according to Kramer’s theory. These authors confirmed their results experimentally by studying the dependence of ksp with T.8 They worked with calibrated 800 nm emulsion droplets of an organic ferrofluid in water and found τ-1 0 to be of the order of Langmuir 2009, 25(12), 6658–6664
Figure 7. Energy barrier height Ea normalized by the thermal
energy kBT as a function of [KBr]-1/2. According to the DebyeHu. ckel theory, there is a linear relation between both magnitudes. The straight line is a guide for the eyes.
105 s-1. Figure 7 shows how Ea decreases for increasing electrolyte concentration. Such behavior agrees with the empirical fact that surface charge screening due to added electrolyte enhances the formation of strong or primary bonds. According to the DebyeHu. ckel theory, there is a linear relation between the energy barrier height Ea and [KBr]-1/2. The absolute values of the observed barrier heights Ea, however, are significantly smaller than the values predicted by the extended DLVO theory. Similar discrepancies have been reported by Tannoudji et al., when these authors studied van der Waals adhesion of field induced linear aggregates. In order to check the consistency of the methods used so far, we compare the direct and indirect results obtained at different exposure times to the magnetic field tc keeping the electrolyte concentration constant at 5 mM. As was described above, the time evolution of the average number of primary and secondary bonds per chain was assessed by measuring the average chain size just after ÆN(t+ c )æ and before ÆN(tc )æ field removal. The exposure times to the magnetic field were tc = 10, 20, 30, 40, 50, and 60 min. Figure 8 shows the time evolution of the average number of particles per chain ÆN(t)æ at 5 mM keeping the field turned on together with the average chain length that was obtained turning the field off after the corresponding exposure time (t+ c = t). From these experimental data, the percentages of primary and secondary bonds were obtained directly. The results are shown in Figure 9 together with the indirect theoretical predictions that DOI: 10.1021/la803626v
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Martı´nez-Pedrero et al.
represents the probability per unit time for a secondary bond to turn into a primary bond, it must be independent of the exposure time to the magnetic field. Two main conclusions can be extracted from Figure 9: On the one hand, formation of primary bonds is favored for long exposure times to the magnetic field. On the other hand, the theoretical predictions reproduce the measured time evolution of the percentage of primary and secondary bonds quite satisfactorily and so confirm the validity of the proposed methods.
Figure 8. Time evolution of the average number of particles per chain ÆN(t)æ (4) at 5 mM. Symbols (O) show the average chain length ÆN(t+ c )æ measured at different exposure times to the magnetic field (t = t+ c ), once the field was turned off.
Figure 9. Percentage of primary (O) and secondary (0) bonds assessed at 5 mM KBr as a function of the exposure time to the magnetic field. The theoretical predictions were calculated by using eqs 3a and 3b. For the calculations, the rate constant ksp was fixed at ksp = 7.8 10-4 s-1.
were determined by solving eqs 3a and 3b. Here, the term dÆN(t)æ-1/dt was calculated for the longest exposure time to the magnetic field (60 min). In all cases, the concentration of metastable and stable bonds was set to satisfy the monomeric boundary conditions, that is, [ns(0)] = [np(0)] = 0. The rate constant ksp was fixed at ksp = 7.8 10-4 s-1. This is the value that was previously obtained for [KBr] = 5 mM. Since ksp
6664 DOI: 10.1021/la803626v
Conclusions From the analysis of field induced aggregation processes of EDMLP and in the hypothesis that the bonds between the chainforming particles can be established in a primary or a secondary minimum of energy, the main conclusions of this work are as follows: I. A set of differential equations to describe the number of the stable bonds as a function of the exposure time to the magnetic field has been proposed. II. An experimental protocol to determine the fraction of stable bonds at different electrolyte concentrations and exposure times to the magnetic field has been designed. III. By contrasting the experimental data with the theoretical predictions, the rate at which chain-forming particles aggregated in a secondary energy minimum form stable bonds was assessed. IV. The theoretical predictions reproduce the measured time evolution of the percentage of secondary and primary bonds quite satisfactorily, confirming that the percentage of stable bonds is larger the higher the electrolyte and the exposure time to the magnetic field concentration become. Acknowledgment. Financial support from the Spanish Ministerio de Ciencia e Innovaci on (Plan Nacional de Investigacion Cientı´ fica, Desarrollo e Innovacion Tecnologica (I+D+i), Projects MAT2006-12918-C05-01, MAT2006-13646-C03-03), the European Regional Development Fund (ERDF), the Junta de Andalucı´ a (Excellency Projects P07-FQM-2496, P07-FQM02517), and the Accion Integrada (Project HF2007-0007) is gratefully acknowledged. We also wish to express gratitude to Dr. Jose Manuel Lopez-Lopez for fruitful discussions.
Langmuir 2009, 25(12), 6658–6664
