State of Dispersion of Magnetic Nanoparticles in an Aqueous Medium

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State of Dispersion of Magnetic Nanoparticles in an Aqueous Medium: Experiments and Monte Carlo Simulation Santosh Kumar, Chettiannan Ravikumar, and Rajdip Bandyopadhyaya* Department of Chemical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India Received April 29, 2010. Revised Manuscript Received August 19, 2010 Monte Carlo simulation results predicting the state of dispersion (single, dimer, trimer, and so on) of coated superparamagnetic iron oxide (Fe3O4) nanoparticles in an aqueous medium are compared with our experimental data for the same. Measured values of the volume percentage of particles in the dispersion, core particle diameter, coatingshell thickness, grafting density of the coating agent, saturation magnetization, and zeta potential for the citric acidcoated and poly(acrylic acid) [PAA]-coated particles have been used in our simulation. The simulation was performed by calculating the total interaction potential between two nanoparticles as a function of their interparticle distance and applying a criterion for the two particles to aggregate, with the criterion being that the minimum depth of the secondary minima in the total interaction potential must be at least equal to kBT. Simulation results successfully predicted both experimental trends;aggregates for citric acid-coated particles and an individual isolated state for PAA-coated particles. We have also investigated how this state changes for both kind of coating agents by varying the particle volume percentage from 0.01 to 25%, the particle diameter from 2 to 19 nm, the shell thickness from 1 to 14 nm, and grafting density from 1015 to 1022 molecules/m2. We find that the use of a lower shell thickness and a higher particle volume percentage leads to the formation of larger aggregates. The possible range of values of these four variables, which can be used experimentally to prepare a stable aqueous dispersion of isolated particles, is recommended on the basis of predictions from our simulation.

1. Introduction Superparamagnetic iron oxide (Fe3O4) nanoparticles in the form of a stable dispersion are widely used in applications such as biomagnetic separation,1 magnetic resonance imaging (MRI),2 magnetic biosensing,3 and hyperthermia treatment.4 The particles are coated with either surfactants,5 polymers,6 polyelectrolytes7 or block copolymers8 in order to disperse them in an aqueous or organic medium. For these applications, one needs a stable dispersion of individual magnetic nanoparticles without any particle-particle aggregation. However, particles tend to aggregate because of their large surface area to volume ratio and thereby minimize the surface energy. Therefore, a key issue is to look at parameters that control aggregation and optimize these parameters so that we can get isolated magnetic nanoparticles in the dispersion. To this end, we explore the effects of particle diameter, the overall volume percentage of particles in the dispersion, and the shell thickness of the coating agent and its grafting density (number of molecules adsorbed on the nanoparticle surface per unit surface area). *To whom correspondence should be addressed. E-mail: [email protected]. Phone: 91-22-25767209. Fax: 91-22-25726895. (1) Jing, Y.; Moore, L. R.; Williams, P. S.; Chalmers, J. J.; Farag, S. S.; Bolwell, B.; Zborowski, M. Biotechnol. Bioeng. 2006, 96, 1139–1154. (2) Bulte, J. W. M.; Douglas, T.; Witwer, B.; Zhang, S. C.; Strable, E.; Lewis, B. K.; Zywicke, H.; Miller, B.; Gelderen, P.; Moskowitz, B.; Duncan, I. D.; Frank, J. A. Nat. Biotechnol. 2001, 19, 1141–1147. (3) Hsing, I. M.; Xu, Y.; Zhao, W. Electroanalysis 2007, 19, 755–768. (4) Jordan, A.; Wust, P.; F€ahlin, H.; John, W.; Hinz, A.; Felix, R. Int. J. Hyperthermia 1993, 9, 51–68. (5) Fu., L.; Dravid, V. P.; Johnson, D. L. Appl. Surf. Sci. 2001, 181, 173–178. (6) Xu, C.; Ohno, K.; Ladmiral, V.; Composto, R. J. Polymer 2008, 49, 3568– 3577. (7) Chung, C.; Hsu, Y.; Chun, C. Int. Conf. Bioinf. Biomed. Eng. 2007, 725–728. (8) Sondjaja, R.; Hatton, A. T.; Tam, M. K. C. J. Magn. Magn. Mater. 2009, 321, 2393–2397. (9) Kruse, T.; Spanoudaki, A.; Pelster, R. Phys. Rev. B 2003, 68, 0542081– 05420812.

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Kruse et al.9 reported Monte Carlo simulation results of the microstructure of an Fe3O4 nanoparticle dispersion on the basis of a proposed center-to-center distance criterion. If the interparticle distance was less than 1.2 times the sum of their radii, then the particles were allowed to aggregate. Aggregation was found to increase with particle diameter, but they did not address the effect of any other parameter. Castro et al.10,11 simulated the aggregation of coated magnetic nanoparticles on the basis of a different criterion. If two particles had a surface-to-surface distance of less than 2.6 nm, then they would form an aggregate, which is an ad hoc condition. They concluded that aggregation increases with an increase in the particle volume percentage. Also, their model does not have the ability to predict the effect of the particle diameter, shell thickness, and grafting density. In this article, we therefore look at the overall state of the dispersion of citric acid-coated and poly(acrylic acid) [PAA]coated Fe3O4 nanoparticles in an aqueous medium. More importantly, we look at all possible combinations of the above four parameters such that they can produce a stable dispersion of coated magnetic nanoparticles in water. Our simulation was carried out on the basis of a more appropriate interparticle interaction energy criterion. This gives the final percentage concentration of single isolated particles (monomers), dimers, and aggregates (trimers and larger aggregates). We furthermore compare these predictions with our experimental results to gain predictive capabilities for a real system. Therefore, one of the objectives of this work is to explain the experimentally observed mixture of single particles and aggregates obtained for citric acidcoated particles vis-a-vis only isolated individual nanoparticles obtained by us for PAA-coated particles. (10) Castro, L. L.; da Silva, M. F.; Bakuzis, A. F.; Miotto, R. J. Magn. Magn. Mater. 2005, 293, 553–558. (11) Castro, L. L.; Goncalves, G. R. R.; Skeff Neto, K.; Morais, P. C. Phys. Rev. E 2008, 78, 0615071–06150711.

Published on Web 11/03/2010

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Figure 1. (a) Stable aqueous dispersion containing citric acid- and PAA-coated Fe3O4 nanoparticles. (b) Schematic representation of two particles with diameters of d1 and d2 separated by a surface-tosurface distance (s) and a center-to-center distance (r).

2. Experimental Section 2.1. Materials. All chemicals were reagent grade and used without further purification. Ferric chloride hexahydrate (FeCl3 3 6H2O), ferrous chloride tetrahydrate (FeCl2 3 4H2O), poly(acrylic acid) sodium salt (molecular weight = 2100), iron(III) acetylacetonate (Fe(C5H7O2)3, [Fe(acac)3]), 2-pyrrolidone (C4H7NO), and tetramethyl ammonium hydroxide (TMAOH) were purchased from Sigma-Aldrich. Ammonium hydroxide (NH4OH, 25%) and citric acid (C6H8O7) from Qualigens were used. Deionized Millipore Milli-Q water was used in all experiments.

2.2. Method. 2.2.1. Synthesis of Citric Acid-Coated Fe3O4 Nanoparticles. This has been carried out via a controlled

chemical coprecipitation method under a nitrogen atmosphere.12 In a typical experiment, 100 mL of a mixed aqueous iron solution (1:2 molar ratio of FeCl2/FeCl3) was added to a round-bottomed flask. At 80 °C and under vigorous mechanical stirring, 15 mL of a 25% (v/v) NH4OH solution was added immediately to the flask to precipitate the Fe3O4 nanoparticles. After 15 min of reaction, the flask was cooled to room temperature and the particles were magnetically separated. The separated particles were sonicated with 15 mL of TMAOH for 10 min and magnetically separated again. TMAOH-treated Fe3O4 particles (1.1 g) were then added to 60 mL of water in a round-bottomed flask and heated to 80 °C. Citric acid (500 mg) was immediately added to the flask, and coating was allowed to take place for 10 min. Because the particle coating process was carried out under vigorous mechanical stirring, the coating molecules are expected to attach uniformly at all points on the nanoparticle surface, in the form of a shell. Afterwards, the flask was cooled to room temperature and the resultant citric acid-coated Fe3O4 nanoparticle dispersion (Figure 1a) obtained was dialyzed for 3 days using a cellulose membrane (MWCO = 12 400). 2.2.2. Synthesis of PAA-Coated Fe3O4 Nanoparticles. This has been carried out following Li et al.13,14 In a typical experiment, 20 mL of 2-pyrrolidone solvent (boiling point = 245 °C) containing 0.9 mmol of Fe(acac)3 and 0.4 mmol of PAA was taken in a round-bottomed flask and purged with nitrogen for 30 min. The mixture was then held at 210 °C for 30 min, followed by 40 min of (12) Kim, D. K.; Zhan, Y.; Voit, W.; Rao, K. V.; Muhammed, M. J. Magn. Magn. Mater. 2001, 225, 30–36. (13) Li, Z.; Sun, Q.; Gao., M. Angew. Chem., Int. Ed. 2005, 44, 123–126. (14) Li, Z.; Wei, L.; Gao, M.; Lei, H. Adv. Mater. 2005, 17, 1001–1005.

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reflux at 245 °C, to yield PAA-coated Fe3O4 nanoparticles. The resultant dispersion was cooled to room temperature, and then a 5:1 volume ratio of diethyl ether/acetone was added to precipitate the particles. The particles were magnetically separated, washed three times with Millipore water, and finally dispersed in water to obtain a stable dispersion (Figure 1a). By a stable dispersion we mean that there was no visible sedimentation of the particle phase from the nanoparticle dispersion. We have performed UVvisible absorbance measurement of the nanoparticle dispersion (shown in Figure 4) to establish this. 2.2.3. Characterization Techniques. XRD (X-ray diffractometry) of coated nanoparticles was carried out using a PANalytical X’Pert Pro with a dual goniometer provided with Cu KR radiation. Transmission electron microscopy (TEM) of coated nanoparticles was carried out with a JEOL-JEM 2100F operating at 200 kV. For this, a drop of the dispersion was placed on a Formvarcoated copper grid, followed by drying the grid under an infrared lamp for 15 min. Magnetic measurements were made in a superconducting quantum interference device (SQUID) magnetometer (Quantum design) at room temperature. For this, a few milligrams of the sample was dried at 80 °C for 30 min. Thermogravimetric analysis (TGA) of coated Fe3O4 nanoparticles were carried out using a Perkin-Elmer Pyris1 instrument. The measurement was made from room temperature up to 900 °C with a heating rate of 20 °C/min and a nitrogen flow rate of 50 mL/min. The ζ potentials of particles were measured at 25 °C using a NanoS Zeta Sizer (Malvern Instruments). For this, a dry powder sample of coated particles (0.005 wt %) was dispersed in 0.1 M NaCl. The measurement duration was set to be determined automatically, and the data were averaged from at least three runs having a standard deviation of 4 mV. The potential values reported are based on the Smoluchowski equation.

3. Nature of Aggregation and Simulation Scheme In our experiments, we have observed either individual isolated Fe3O4 nanoparticles or loose aggregates (flocs) of these particles in their aqueous dispersions. The aggregates could be redispersed into individual primary particles, if required, by later ultrasonication of these dispersions. This implies that the aggregates are reversible; that is, the constituent primary particles of the aggregate are held together by weak physical forces such as van der Waals forces. In contrast, if there were strong solid-state (covalent or ionic) bonding between the individual primary particles of an aggregate, then it would not have been possible to redisperse these aggregates by simple ultrasonication. Such an aggregate would be called an irreversible or permanent aggregate. As far as simulation is concerned, we are predicting the state of the particles during their formation in the aqueous dispersion itself prior to any ultrasonication, if at all. Now, it is possible in some cases that particles may form an aggregate because of interparticle forces such as van der Waals forces. Therefore, to simulate this state of particle aggregation, once a dimer or a higher aggregate forms in the course of our simulation, we have not allowed the aggregate to break up in our simulation algorithm. The simulated structure therefore evolves toward the equilibrium aggregated structure if that is the scenario in the corresponding experiment, as in the case of citric acid-coated particles. Monte Carlo simulation is carried out to track the above. The configuration of two particles is shown schematically in Figure 1b. n number of such coated, spherical particles all having same diameter are randomly distributed in a cubic box with a periodic boundary condition (Figure 7b(i)). In the simulation, the box size is not constant. The volume percentage is kept the same through different runs with different numbers of particles so that the results can be compared with experiments conducted at a given volume percentage. Therefore, the box size for a given volume percentage DOI: 10.1021/la1017196

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Figure 2. XRD patterns of (a) citric acid-coated and PAA-coated Fe3O4 nanoparticles and (b) the (311) plane of uncoated Fe3O4

nanoparticles before oxidation (black line) and its structural phase transformation to γ-Fe2O3 (green line) obtained after oxidation at 300 °C (XRD slow scan of 0.001°/step).

Figure 3. TEM images of dispersions of (a) citric acid-coated Fe3O4 particles with j = 0.645%, (b) citric acid-coated Fe3O4 particles with j = 0.1%, (c) PAA-coated Fe3O4 particles with j = 0.387%, and (d) magnified region of Figure 3c.

increases as the number of particles is increased. The box size, the initial number of particles, and the volume percentage for both forms of coated particles used in the simulation are given in Table 1. To start, a particle i is randomly selected and allowed to interact with another randomly selected particle j, and the depth of the secondary energy minima in their pairwise interaction potential energy (PE) curve is calculated. We considered a minimum secondary

minima depth equal to one unit of kBT as a criterion (explained later) for allowing the aggregation of particles i and j.15,16 The cutoff distance from which the particles are allowed to interact with each other is considered to be the distance at which the interaction PE between two particles is very small compared to the depth of the secondary minima. If the interparticle distance between randomly selected particles i and j is less than the cutoff distance and the minimum depth of the secondary minima in the total interaction

(15) Long, J. A.; Osmond, D. W. J.; Vincent, B. J. Colloid Interface Sci. 1973, 42, 545–553.

(16) Hamley, W. I. Introduction to Soft Matter; University of Leeds: Leeds, U.K., 1988; p 151.

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Article Table 1. Experimental and Simulation Parameters for Citric Acid- and PAA-Coated Fe3O4 Nanoparticles

variable

citric acid-Fe3O4

PAA-Fe3O4

source

Ms j d δ F κ-1 ζ potential Aeff n box dimension

3.6  105 A m-1 0.645 and 0.1% 9 nm 2 nm 1018 molecules/m2 1 nm -25.3 mV 4  10-20 J 2000 490 nm

2.9  105 A m-1 0.387% 6 nm 10.5 nm 1018 molecules/m2 1 nm -27.8 mV 4  10-20 J 2000 387.92 nm

SQUID data (Figure 5) as used in experiments TEM image (Figure 3) DLS data and TEM image TGA data (Figure 5b) for 0.1 M NaCl pH 7 (Figure 5c) From ref 25 fixed on the basis of Figure 9a fixed on the basis of j

diameters of the two particles (Figure 1b), which can be of different sizes, and r is the center-to-center distance between the particles. In our simulation, whenever an aggregate of two or more particles is formed, we consider the resultant aggregate to be a spherical particle of volume equal to the sum of the constituents. We then apply pairwise interaction potentials listed here, as usual. 3.2. Steric Interaction. The steric repulsion energy between coated particles is given by18 0 1 3 2 Vs ¼

1 2 6 lþ2 B B1 þ tC C - l7 7 ðs=2δÞe1 πd F6 42 - t ln@ l A t5 2 1þ 2 Vs ¼ 0 ðs=2δÞ > 1

Figure 4. Stability of uncoated, citric acid-coated, and PAA-coated Fe3O4 nanoparticle dispersions measured through normalized UVvisible absorbance peak intensity measurements as a function of time.

potential is at least equal to kBT, then these two particles will aggregate and form a dimer. If particle j is already a dimer, then the result is a trimer and so on. This was repeated for all single particles, dimers, and other aggregates in the box and was followed until there was no further change in the percentage concentration of singles, dimers, and aggregates, where the equilibrium configuration is said to be achieved. Such a simulation run, performed with an initial random distribution of n particles in the box, constitutes one replication or realization of the system. A number of such replications were simulated, the average from all such replications was calculated, and the mean quantities were used for comparison with our experiments. The total PE between two coated magnetic nanoparticles in a solvent is governed by van der Waals (Va), magnetic (Vm), steric (Vs), and electrostatic (Ve) interactions and is discussed next. 3.1. van der Waals Interaction. The attractive van der Waals interaction energy, Va, is given by17 2 Va ¼ -

6 6 1 d1 d2 d1 d2 6 Aeff 6 (  2 ) þ (  ) 6 6kB T d þ d d1 - d2 2 1 2 42 r 2 2 2 r 2 2

8   93 > d1 þ d2 2 > > > 2 > > =7 > 5 d d > > 1 2 > >r2 ; : 2

ð1Þ

where kB is the Boltzmann constant, T is the temperature, Aeff is the effective Hamaker constant (Table 1), d1 and d2 are the core (17) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, U.K., 1987; p 135.

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ð2Þ

ð3Þ

where F is the grafting density (Table 1), s is the surface-to-surface separation distance, δ is the shell thickness (Table 1), and d is the average core diameter of the two solid particles. The steric interaction between particles of diameter d1 and d2 is calculated using the average core diameter, d = (d1 þ d2)/2.10,11 In addition, l = 2s/d and t = 2δ/d have been defined. 3.3. Magnetic Interaction. The magnetic interaction energy (Vm) between two magnetic particles is calculated on the basis of the average saturation magnetization obtained from all of the coated particles at the maximum applied field. We do not know the Ms value of each individual nanoparticle. However, if accurate experimental measurements of the distributions of Ms and d are available, then our simulation code can be run with these distributions. In the absence of such input data, we have run our simulation with measured average values of Ms and d. Hence, Vm can be calculated from18 Vm ¼ -

1 πμo d 6 Ms 2 kB T 72ðr þ dÞ3

ð4Þ

where μo is the permeability of free space and Ms is the saturation magnetization with respect to core particle only (Table 1). 3.4. Electrostatic Interaction. The electrostatic repulsion energy (Ve) between two particles is calculated using the expression below.19-21 d1 d2 εεo ψo 2 lnf1 þ expð- KsÞg for 0:5dK > 5 d1 þ d2   kB T 2 expð- KsÞ Ve ¼ πd1 d2 Y1 Y2 εεo for 0:5dK < 5 d1 þ d2 e sþ 2 ð5Þ

Ve ¼ 2π

(18) Rosensweig, R. E. Ferrohydrodynamics; Cambridge University Press: New York, 1985; p 46. (19) Hunter, R. J. Foundations of Colloid Science; Clarendon Press: Oxford, U.K., 1987. (20) Lee, K.; Sathyagal, A. N.; McCormick, A. V. Colloids Surf., A 1998, 144, 115–125. (21) Ohshima, H. J. Colloid Interface Sci. 1995, 174, 45–52.

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Figure 5. (a) Magnetization vs applied field for citric acid-coated and PAA-coated Fe3O4 core nanoparticles at room temperature. The inset shows the temperature dependence of magnetization (M vs T) for both types of coated particles at an applied field of 100 Oe. (b) Thermogravimetric analysis of uncoated, citric acid-coated and PAA-coated Fe3O4 nanoparticles. (c) ζ potential vs pH for uncoated, citric acid-coated and PAA-coated Fe3O4 nanoparticles.

ð6Þ

corresponding to each peak in Figure 2a match the standard file (JCPDF: 82-1533, Cu target) of the bulk magnetite (Fe3O4) phase. It is known that the XRD peak positions of Fe3O4 and γ-Fe2O3 (maghemite) are very close to each other. To distinguish them from one another, we oxidized our as-synthesized iron oxide nanoparticles at 300 °C. Figure 2b shows the XRD patterns for both the as-synthesized and oxidized particles. The peak position of the as-synthesized particles corresponding to the (311) plane well matches the standard peak position of bulk Fe3O4, which on oxidation transforms to the peak position of bulk γ-Fe2O3 (JCPDS file: 39-1346, Cu target). There is a clear shift of 2θ = 0.2° in Figure 2b, which proves that oxidation caused the transformation of magnetite to maghemite. This proof has been adapted by following Aslam et al.,22 who drew the same conclusion. Furthermore, the magnetization versus temperature plot (inset of Figure 5a) of both forms of coated Fe3O4 nanoparticles shows that the Curie temperature (Tc) is 580 °C, which is in agreement with that reported for Fe3O4. However, Tc of γ-Fe2O3 is around 645 °C.23 Thus,

4.1. Experimental Results. Figure 2a shows XRD patterns of citric acid- or PAA-coated Fe3O4 nanoparticles. The angles

(22) Aslam, M.; Schultz, E. A.; Meade, T. S. T.; Dravid, V. P. Cryst. Growth Des. 2007, 7, 471–475. (23) Barick, K. C.; Aslam, M.; Lin, Y., P.; Bahadur, D.; Prasad, P. V.; Dravid, V. P. J. Mater. Chem. 2009, 19, 7023–7029.

where Yi ¼

2

  eψo 8 tanh 4kB T

31=2

 7 6 Kdi þ 1 eψo 7 2 1þ6 1 tanh  2 6 7 4kB T 5 4 di K þ1 2 e is the elementary charge, ε is the dielectric constant of the medium (water), εo is the permittivity of free space, ψo is the surface potential, and κ-1 is the Debye length. The surface potential ψo was approximated by measuring the values of the ζ potential (Table 1). The net interaction potential (Vtotal) was calculated by summing all of the above individual interaction potentials between the two particles and is given as Vtotal ¼ Va þ Vm þ Vs þ Ve

4. Results and Discussion

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XRD and magnetic measurements independently prove that both citric acid- and PAA-coated particles are solely Fe3O4. An FTIR comparison of uncoated and citric acid-coated nanoparticle with pure citric confirms the chemisorption of citric acid with the surface iron (Fe2þ and Fe3þ) moieties of the Fe3O4 nanoparticles (,Supporting Information Figure S1). Figure 3a shows a TEM image of citric acid-coated Fe3O4 nanoparticles obtained from a dispersion with a volume percentage of j = 0.645%. Mostly aggregates can be seen with some individual particles. Even at a lower particle concentration corresponding to j = 0.1%, Figure 3b shows exactly the same kind of morphology for citric acid-coated particles. This implies that aggregate formation at both higher and lower particle concentrations occurs in situ in the dispersion itself and is not due to any solvent-induced evaporation during TEM grid formation. Thus, aggregate formation is a result of interparticle forces operating in the dispersion itself, which is also predicted by our simulation. Therefore, TEM images are a valid source for quantitative analysis of the state of nanoparticle dispersions. From Figure 3a,b, the core diameter of the primary particles (averaged over all individual particles and primary particles in an aggregate) is 9 nm with a standard deviation of 2.5 nm. Figure 3c shows the PAA-coated Fe3O4 particles to be individual isolated particles having a core diameter of 6 nm with a standard deviation of 1.1 nm. In Figure 3c, some particles appear to be very close to each other, which is due to the locally higher number concentration of particles in that region of the TEM grid. These are not actual aggregates, as shown in the magnified image Figure 3d, where we find individual, isolated particles. In addition to TEM, DLS (Supporting Information Figure S2) of PAA-coated particles also proves that these are very narrowly dispersed particles compared to citric acid- and dextran-coated particle samples. Qualitative agreement with respect to the trend in the aggregate size distribution from dynamic light scattering (DLS) data and TEM images of aggregates of citric acid-coated or isolated PAAcoated particle morphology is shown in Figure S2 of the Supporting Information. Figure 4 shows the stability of citric acid- or PAA-coated nanoparticle dispersions carried out using UV-visible absorbance (Perkin Elmer-Lambda 35) measurements at pH 7. The absorbance of the uncoated nanoparticle dispersion continuously decreases and becomes zero within a very short period, implying rapid coagulation and sedimentation of the particle phase under gravity. However, the coated nanoparticle dispersions show a negligible decrease in absorbance for even up to 5 days, implying that they do not coagulate and remain completely dispersed in the aqueous phase. In fact, both kinds of coated dispersions had more than a 2 month period of stability. Figure 5a shows the magnetization values of coated Fe3O4 nanoparticles against the applied magnetic field. The absence of coercivity and remanance and the zero-field cooling (ZFC) results (Supporting Information Figure S3) prove that the particles are superparamagnetic in nature. Figure 5a also shows the saturation magnetization (Ms) values of 64.44 and 56.52 emu/g for citric acid- and PAA-coated core particles, respectively. These values have been obtained after correcting for the presence of coating agents; using the weight fraction of core Fe3O4 obtained from TGA measurements. We have also used these measured Ms values of the dried coated particles to be valid for the dispersion because all of the particles in the dispersion are separated using a permanent magnet, which contains very little solvent. Figure 5b shows TGA of citric acid-coated and PAA-coated Fe3O4 nanoparticles. We see that the weight fraction of core Fe3O4 in citric acid-coated and PAA-coated Fe3O4 nanoparticles Langmuir 2010, 26(23), 18320–18330

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Figure 6. Pairwise interaction potential energy as a function of the surface-to-surface distance (s) for two citric acid-coated Fe3O4 nanoparticles with d = 9 nm, δ = 2 nm, F = 1018 molecules/m2, and ζ potential = -25.3 mV. The inset shows the secondary minima more clearly.

are 0.9 and 0.46, respectively. We used these values to calculate Ms and F in Table 1, Figure 5a and Figure S3. The charge present on the particles causing an electrostatic repulsion between them (found from ζ-potential measurements) is shown in Figure 5c. We see that at pH 7.0, the ζ potentials of uncoated particles is almost zero, whereas, both types of coated particles show a large negative value and increase with increasing pH. 4.2. Comparison of Simulation Results with Experiments for Citric Acid-Coated Particles. 4.2.1. Potential Energy Diagram of Citric Acid-Coated Particles. Figure 6 shows the individual van der Waals, magnetic, steric, electrostatic, and total interaction PE curves as a function of the surface-to-surface separation distance between two citric acid-coated Fe3O4 nanoparticles. Table 1 shows the values used to calculate these curves. Particles in Brownian motion will have energy of the order of kBT and rarely more than a few kBT. It is reported that a barrier height of 20kBT or more implies that particles and their aggegates in a dispersion will not be able to cross the energy barrier to reach the primary minima.24 In fact, we find a maximum barrier of 29.17kBT in the total interaction potential curve in Figure 6. Therefore, particles will not be able to overcome this barrier and will instead be held at the secondary minima. This is because the depth of the secondary energy minima in the total interaction energy curve is 1.63kBT (Figure 6), which is larger than the depth (kBT) required to form aggregates. Thus, citric acid-coated Fe3O4 nanoparticles will form reversible aggregates. These aggregates can be dispersed in solution in the form of isolated individual particles by ultrasonication, which was true for our samples. On the basis of Figure 6 and later comparison with experimental results (Figure 9), we used a cutoff distance of 41 nm. Such a value is also appropriate because at a 41 nm interparticle distance the interaction energy between the two particles is negligible, only 14.6% of the depth of the secondary minima, ensuring that we have accounted for all relevant pairwise interactions. Thus, it is reasonable to neglect interactions beyond 41 nm, which are of very little consequence to the overall structure of the dispersion. 4.2.2. MC Steps to Reach Equilibrium and Replications. The percentage concentrations of single isolated particles (24) Elimelech, M.; Gregory, J.; Jia, X.; Williams, R. A. Particle Deposition and Aggregation: Measurement, Modelling, and Simulation; Butterworth-Heinemann: Boston, 1995. (25) Ivanov, A. O.; Kuznetsova, O. B. Phys. Rev. E 2004, 64, 0414051–04140512.

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Figure 7. (a) Concentration of single isolated particles, dimers, and aggregates (i.e., trimers and larger aggregates) as a function of the number of MC steps for 2000 citric acid-coated Fe3O4 nanoparticles. (b) Snapshot of the simulation box with particles (i) before and (ii) after aggregation, with j = 0.645%, d = 9 nm, δ = 2 nm, F = 1018 molecules/m2, and ζ potential = -25.3 mV for 200 citric acid-coated Fe3O4 particles with one replication.

(monomers), dimers, and aggregates (trimers and larger aggregates) obtained from simulation are compared with our experimental observation. The system chosen has 0.645% (volume percentage) citric acid-coated Fe3O4 particles with a 9 nm particle diameter, 2 nm shell thickness, 1018 citric acid molecules/m2 grafting density, and -25.3 mV ζ potential. Figure 7a shows that for a single replication of a simulation run the concentrations of all reach equilibrium after 1800 MC steps. The snapshots at the start and after aggregation (at equilibrium) are given in Figure 7b, which show that many of the initially monodisperse particles have finally formed some aggregates, as depicted by larger spheres. To obtain statistically more precise mean quantities at equilibrium, 10 different replications (each starting with a different but random positional distribution of 2000 particles) were performed. Equilibrium concentrations of single isolated particles, dimers, and aggregates were then calculated to be the mean of these 10 replications. This was further compared with the mean from 20 replications to fix the minimum number of replications required for the convergence of mean quantities. It is observed that the final mean percentage composition somewhat differs for 1 to 10 replications but not for 10 to 20 replications. Therefore, we conduct 10 replications for all further results in this work and use them to compare with experiments. 4.2.3. Code Validation. We reproduced the results of Castro et al.,10 which becomes a special case of our general simulation model. Castro et al. described their ferrofluid by considering an ensemble of 200 polydisperse nanoparticles in a cubic box with 18326 DOI: 10.1021/la1017196

Figure 8. Reproduction of Castro et al.’s results from our code. Average concentrations of single isolated particles, dimers, and aggregates as a function of the particle volume percentage with d = 8.9 nm for 2000 dodecanoic acid-coated Fe3O4 particles from 10 replications. The maximum standard error for the above simulation plot is 0.56%.

periodic boundary conditions. The particles were coated with dodecanoic acid and dispersed in a hydrocarbon. Different particle concentrations were obtained by varying the box size. They had chosen a log-normal particle size distribution with an average diameter of 8.9 nm and zero magnetic field. Using our Langmuir 2010, 26(23), 18320–18330

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Figure 9. Average aggregate number density as a function of the number of particles (single isolated particles) in an aggregate with (a) j = 0.645% and (b) j = 0.1%, d = 9 nm, δ = 2 nm, F = 1018 molecules/m2, and ζ potential = -25.3 mV for 2000 citric acid-coated Fe3O4 particles from 10 replications. The maximum standard errors for the two simulation plots are 8.35 and 9.66%, respectively.

Figure 10. Average concentration of single isolated particles, dimers, and aggregates as a function of (a) the particle volume percentage, (b) the diameter, (c) the shell thickness, and (d) the grafting density with j = 0.645%, d = 9 nm, δ = 2 nm, F = 1018 molecules/m2, and ζ potential = -25.3 mV for 2000 citric acid-coated Fe3O4 particles from 10 replications. The maximum standard error for the above simulation plots is 2.78%.

code, their results were reproduced, thereby both validating our code and proving the generality of our model. The concentration of monomers, dimers, and aggregates as a function of the particle volume percentage (for 2000 initial particles) is shown in Figure 8 for 10 replications. We have compared their model results from our computer code in two ways;by using a larger number (2000) of particles with a smaller number (1, 10, or 20) of replications or a smaller number (200) of particles with a larger number (1, 20, 50, or 100) of replications. We got exactly the same converged results for both sets, implying that our simulation result (for Castro’s model) given in Figure 8 is the globally converged result. Langmuir 2010, 26(23), 18320–18330

Therefore, we conclude that Castro et al. may not have given the converged result, which is why there is a small quantitative difference between their results and ours in Figure 8. 4.2.4. Number of Particles in an Aggregate versus Number Density and Effect of the System Size. We now compare our simulation predictions of the aggregate size distribution with experimental data for two different values of j = 0.645 and 0.1% for citric acid-coated particles and with j=0.387% for PAA-coated particles. The experimental number density (number of aggregates per cubic meter) for all of these are taken from the TEM images in Figure 3. This quantity is plotted as a function of the number of DOI: 10.1021/la1017196

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particles in an aggregate in Figure 9 for the two citric acid systems. The experimental and simulation results clearly show good agreement in Figure 9. This is also true for PAA-coated particles (plot not shown because all PAA-coated nanoparticles are monomeric, hence there is no aggregate size distribution). The same cutoff distance

Figure 11. Pairwise interaction energy as a function of the surfaceto-surface distance (s) for two PAA-coated Fe3O4 nanoparticles with d = 6 nm, δ = 10.5 nm, F = 1018 molecules/m2, and ζ potential = -27.8 mV. The inset shows the secondary minima more clearly.

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value of 41 nm, obtained by fitting the data in Figure 9a, also gave very good predictions for the other two systems. Simulations were initially carried out with 200, 2000, and 5000 particles in order to capture the effect of the size of the system. At a smaller volume percentage (below 0.7%), we have observed that there is no difference in the simulation results between 200 and 5000 initial particles (Figure 9a). Hence, a moderately large number of 2000 initial particles was chosen in order to avoid a system size effect even at a larger volume percentage (i.e., above 0.7%). Furthermore, a simulation with 2000 initial particles takes only 4 min 17 s of run time (on a pentium CPU) whereas it takes substantially more time (29 min) with 5000 initial particles, without any effect on the results. Therefore, we have used 2000 initial particles for all simulations, which gives converged results for all cases within a small CPU time. To investigate the effect of the four parametes on the state of the dispersion, we vary each one of them in turn, keeping the other three fixed at the experimentally set value, namely, a 0.645% particle volume percentage, a 9 nm particle diameter, a 2 nm shell thickness, and a 1018 molecules/m2 grafting density (Table 1). The results are shown in Figure 9. We address each parameter one by one. 4.2.5. Influence of Particle Volume Percentage on Aggregates. In Figure 10a, as the particle volume percentage increases, the concentration of single isolated particles decreases, whereas that of the aggregate increases and the dimer concentration goes through a maximum. At a larger volume percentage, the interparticle distance is smaller so the total interaction energy is more

Figure 12. Average concentration of single isolated particles, dimers, and aggregates as a function of the (a) particle volume percentage, (b) diameter, (c) shell thickness, and (d) grafting density with j = 0.387%, d = 6 nm, δ = 10.5 nm, F = 1018 molecules/m2, and ζ potential = -27.8 mV for 2000 PAA-coated Fe3O4 nanoparticles from 10 replications. The maximum standard error for the above simulation plots is 1.52%. 18328 DOI: 10.1021/la1017196

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negative (attractive) (Figure 6). Therefore, aggregation increases at a larger volume percentage. 4.2.6. Influence of Particle Diameter on Aggregates. Our results predict the required size range of particles that can produce a dispersion of only isolated particles. This is seen to be below 6.5 nm particles (Figure 10b). Attractive van der Waals and magnetic interaction PE between two particles increase with particle diameter; therefore, the depth of the secondary minima also increases with particle diameter. Only particles having a diameter larger than 6.5 nm will have the depth of secondary minima of more than kBT that is required for aggregation; therefore, particles with diameters of greater than 6.5 nm will form aggregates. In our experiments, the particle volume percentage and particle diameter can be easily controlled or changed, whereas the shell thickness and grafting density are fixed on the basis of a chosen coating agent. Therefore, it is interesting to see the combined influence of the first two parameters on the nature of particle aggregation, which is shown in a 3D plot (Supporting Information Figure S4). 4.2.7. Influence of Shell Thickness on Aggregates. The concentration of aggregates decreases with an increase in shell thickness. Above 8 nm shell thickness, all of the particles will be in a state of isolation in the dispersion (Figures 9c). Repulsive steric PE increases with an increase in shell thickness. Above 8 nm shell thickness, the depth of the secondary minima becomes less than kBT; therefore, isolated particles are observed. 4.2.8. Influence of Grafting Density on Aggregates. On varying the grafting density from 1015 to 1022 molecules/m2 (Figure 10d), the concentration of aggregates is found to decrease with an increase in grafting density. Again, repulsive steric PE increases with an increase in grafting density; therefore, aggregation is found to decrease with an increase in grafting density. To achieving a stable state of isolated particles in a dispersion, we find that the following range is suitable: diameter below 6.5 nm (Figure 10b) and shell thickness above 8 nm (Figure 10c). No particular range of volume percentage (Figure 10a) and grafting density (Figure 10d) was observed in our range of parameter values, which would give a completely isolated state of dispersion. Thus, we find that for our experiment 92% of the particles are in a state of aggregation because our experimental parameters (0.645% particle volume percentage, 9 nm particle diameter, 2 nm shell thickness, 1018 molecules/m2 grafting density, and -25.3 mV ξ potential) are not in the above required ranges of parameter values for an isolated state. 4.3. Comparison of Simulation Results with Experimental Results for PAA-Coated Particles. A similar comparison was made for PAA-coated Fe3O4 nanoparticles, with a 0.387 volume percentage of particles, a 6 nm particle diameter, a 10.5 nm shell thickness, a 1018 molecules/m2 grafting density, and a ζ potential of -27.8 mV (Table 1). In Figure 11, we find a barrier of 59.36kBT in the total interaction PE. This implies that the particles will not be able to overcome this barrier. The particles will not be held at secondary minimum because the depth of the secondary energy minimum is only 0.033kBT (Figure 11), which is much less than the depth (kBT) required to start aggregation. Thus, these particles will not form aggregates. This can be easily concluded from the enegy plot (Figure 11) itself; however, to see the detailed trends in various parameters, simulation is required. Figure 3c is the TEM image of experimentally prepared PAA-coated nanoparticles, where all of the particles are in a state of isolated individual particles without any aggregation. The same was predicted by our simulation, highlighting the good predictive capability of our model as mentioned earlier. To obtain the required range of parameters to achieve stable, isolated PAA-coated particles in a dispersion, we find the following: volume percentage below 0.8% (Figure 12a), diameter below 17 nm (Figure 12b), shell thickness above 8 nm (Figure 12c), and Langmuir 2010, 26(23), 18320–18330

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grafting density above 2.3  1017 molecules/m2 (Figure 12d). In our experiment, particles were isolated without any aggregation (Figure 3c) because our experimental parameters (0.387% particle volume percentage, 6 nm particle diameter, 10.5 nm shell thickness, and 1018 molecules/m2 grafting density) all are within the above required ranges of parameter values.

5. Conclusions In the present work, predictions from our Monte Carlo simulation are compared with our own experimental results of citric acid-coated or PAA-coated Fe3O4 nanoparticles dispersed in water, giving good agreement between the two. Our experiments show that citric acid-coated particles were mostly in the form of aggregates whereas PAA-coated particles were isolated individual particles. Both of these states of dispersion are successfully predicted by our simulation. For both citric acid- and PAA-coated particles, the percentage concentration of single isolated particles decreases with an increase in the particle volume percentage as a result of the formation of dimers and various higher aggregates. Aggregation also increases with increasing particle diameter but decreases with increasing shell thickness or increasing grafting density. We observed a minimum shell thickness required for particles with a particular diameter, volume percentage, and grafting density in order for the dispersion to remain as isolated particles and in a stable state. For both coating agents, we report suitable ranges of these experimentally controllable parameter values, which can be experimentally realized to obtain a stable dispersion of individual isolated nanoparticles. Acknowledgment. We thank Mr. J. Prithivi Raj and Mrs. Kusum Saini, Ph.D. scholars in our research group in the Chemical Engineering Department, IIT Bombay for performing stability measurements. We thank the Department of Science and Technology (DST), India (no. SR/S3/CE/049/2007) for the financial support of this project. We gratefully acknowledge IIT Bombay for all instrumental facilities used in this work.

Glossary d = core particle diameter, m e = elementary charge, 1.6022  10 -19 C n = initial number of particles r = center-to-center distance between particles, m s = surface-to-surface separation distance between two particles, m T = temperature, 298 K Aeff = effective Hamaker constant, 4  10 -20 J kB = Boltzmann constant, 1.3806  10 -23 J K-1 Ms = saturation magnetization with respect to core particle, A m-1 V = interaction potential, J Va = van der Waals interaction potential, J Vm = magnetic interaction potential, J Vs = steric interaction potential, J Ve = electrostatic interactions potential, J Vtotal = net interaction potential, J

Greek Symbols ε = dielectric constant of water, 78.54 δ = shell thickness, m j = particle volume percentage F = grafting density, molecules/m2 εo = permittivity of free space, 8.854  10 -12 m-3 kg-1 s4 A2 μo = permeability of free space, 1.26  10 -6 kg s-2 A-2 DOI: 10.1021/la1017196

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ψo = surface potential, V κ-1 = Debye length, m ζ = zeta potential, V Supporting Information Available: Nature of adsorption of coated agents on Fe3O4 nanoparticles. Comparison of

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aggregation states of nanoparticles from DLS, TEM, and MC simulation. Superparamagnetism of citric acid- or PAAcoated Fe3O4 nanoparticles. Combined influence of particle volume percentage and particle diameter on aggregation. This material is available free of charge via the Internet at http://pubs.acs.org.

Langmuir 2010, 26(23), 18320–18330