Predicting Complete Size Distribution of Nanoparticles Based on

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Predicting Complete Size Distribution of Nanoparticles Based on Interparticle Potential: Experiments and Simulation Nirmalya Bachhar, and Rajdip Bandyopadhyaya J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.5b11285 • Publication Date (Web): 26 Jan 2016 Downloaded from http://pubs.acs.org on February 1, 2016

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The Journal of Physical Chemistry

Predicting Complete Size Distribution of Nanoparticles based on Interparticle Potential: Experiments and Simulation Nirmalya Bachhar, Rajdip Bandyopadhyaya* Department of Chemical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India

*Corresponding Author Tel.: + 91 (22) 2576 7209; Fax: + 91 (22) 2572 6895, E-mail address: [email protected] (R. Bandyopadhyaya).

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ABSTRACT

Solution based synthesis of nanoparticles does not yield monodisperse particles, but rather a well-defined particle size distribution (PSD). There is currently no simple means to anticipate or model these size distributions, which critically affect the properties of the resulting nanomaterials. We simulate the temporal evolution of the PSD in the framework of a nucleation and growth model, with the critical postulate that the coagulation efficiency between two nanoparticles is quantitatively determined by the known, interparticle potential energy. Our simulation based on this ansatz, not only a-priori predicts experimentally obtained complete PSDs of uncoated or coated (with polyacrylic acid or dextran) iron oxide nanoparticles, but also accurately captures the influence of surface coverage of a coating agent on the resulting PSD.

KEYWORDS Kinetic Monte Carlo; nanoparticle; modeling; coprecipitation; coagulation; coating agent


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1. Introduction A plethora of wet-chemical techniques and empirical knowledge has been generated over the last three decades, in the quest to synthesize nanoparticles of controlled size and shape, to the point that, making spherical nanoparticles of different materials having very specific diameter is almost routine. Yet, there is lack of a predictive mathematical model, which can from first principles predict nanoparticle diameter of different materials. This will reduce the experimental burden of varying different synthesis parameters (reactant concentration, nature of solvent, reaction time and temperature, pH, choice of coating agent etc.), a key step in identifying the optimum synthesis conditions. Since all physicochemical properties of nanoparticles vary significantly with the particle diameter, and any experiment will produce a distribution in particle size, it is important to understand how particle size distribution (PSD) is obtained in a nanoparticle synthesis-experiment. The purpose of this research was to fill this void, by simulating a mechanistic model of nanoparticle formation, and validating it with our measurements on PSD of nanoparticles for coating agents of different molecular weights. Nanoparticle formation in a liquid medium is governed by three different steps [shown schematically in Fig. 1(a)] - namely nucleation, diffusion-growth and coagulation-growth (via interparticle Brownian collision followed by complete fusion). Particle growth eventually slows down in time, to generate a final PSD. We refer to each individual, separate nanoparticle as a primary particle.1 Sometimes, these particles over a long time-scale may also form weak, secondary aggregates, e. g., when Brownian collision of larger nanoparticles typically results in incomplete fusion. These secondary aggregates’ formation mechanism - which happens at a very different timescale - was explained by Kumar et al.2; hence it is not included in the present work. Similarly, aggregation of protein crystals was studied using numerical simulation by Kwon et


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al.3–5 However these models have not considered the effect of interparticle potential on aggregation, except van der Waals. Also these numerical simulations were not validated with experimental data and only the average particle sizes were estimated for different types of reactors. Molecular dynamics (MD) simulation has been used in some cases to estimate only nucleation rate6 or surface growth7 of pre-existing clusters, but over very small timescales (picosecond7 to microsecond6,8), which are unrealistically small compared to experimental timescales (of the order of many minutes) of complete nanoparticle formation. So these MD simulations cannot predict experimental PSD data.

Figure 1. (a) Schematic of nanoparticle formation consisting of homogeneous nucleation, diffusion- and coagulation-growth. (b) Schematic representation of two particles with radii r1 and r2 separated by a surface-to-surface distance (s) and center-to-center distance (r).

Hence, we focus on developing a complete model, considering all the above three steps, with the coagulation process being calculated based on a first principle calculation. Earlier


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models have described the coagulation rate of primary nanoparticles by the well-known Smoluchowski equation.1,9–11 The latter gives the Brownian collision frequency between two particles, having radii r1 and r2 as, q p 

2 k BT 3

 r1 r2   2    , where kB, T and μ are Boltzmann  r2 r1 

constant, temperature and viscosity, respectively. This was modified by a coagulation-efficiency term (not measurable through experiments though), to capture complete fusion. It is defined as the fraction of collisions resulting in complete fusion, a factor that attempts to include the effects of different coating agents, solvents etc. into a single, lumped parameter.12–14 This efficiency term is generally fitted to match the experimental, primary particle diameter. Other attempts have used more than one fitting parameter in an ad-hoc fashion.15,16 However, most of these works did not consider all the constituent steps from the beginning of the particle formation process, namely, nucleation, diffusion- and coagulation-growth rates of particles, which is essential to predict the final PSD. The premise of this paper is to consider all these three steps together and generate new insight into the coagulation step; in particular, quantify how complete coagulation of nanoparticles take place in the liquid phase to generate a given PSD. To this end, we explain through a kinetic Monte Carlo (kMC) simulation of the mechanistic model [shown in Fig. 1(a)], as to how nucleation and growth drives it to an equilibrium size distribution, and how by using different coating agents one can control such equilibrium size distribution of nanoparticles.

2. Experimental Method 2.1. Materials For synthesis, all chemicals used were reagent grade, without any further purification. Ferric chloride hexahydrate (FeCl3·6H2O), ferrous chloride tetrahydrate (FeCl2·4H2O), sodium


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salt of polyacrylic acid [Mw ~ 2,100 and 5,100 g/mol], dextran from leuconostoc spp. [Mw ~ 40,000, 60,000 & 100,000 g/mol] (from Sigma Aldrich) and 25% ammonium hydroxide (NH4OH) (Qualigens) were used. 2.2. Synthesis of Fe3O4 nanoparticles and characterization All synthesis experiments were done in N2 atmosphere. Fe3O4 nanoparticle synthesis description via coprecipitation route has been described elsewhere.17,18 We have used two iron chloride precursors (0.9 mmol FeCl3•6H2O and 0.45 mmol FeCl2•4H2O, maintaining a Fe3+/Fe2+ molar ratio of 2) in 100 ml water. 20 ml of 25% v/v NH4OH was added to this precursor solution, to precipitate Fe3O4 nanoparticles. For polymer coated nanoparticles, PAA (polyacrylic acid) or dextran was added in-situ. 0.4 mmol PAA (for both Mw = 2,100 and 5,100 g/mol) was added to obtain a surface coverage of 0.5 and 0.45 chains/nm2, respectively. To obtain, 0.28 chains/nm2 (for Mw = 40,000 g/mol) and 0.35 chains/nm2 (for both Mw = 60,000 and 100,000 g/mol), 50 µmol dextran was added. In case of dextran-coated nanoparticles having lower surface coverages, less amount of dextran were added in the solution. To get 0.12, 0.27 and 0.26 chains/nm2 (for Mw = 40,000, 60,000 and 100,000 g/mol, respectively), 25, 32.5 and 30.2 µmol dextran was added, in that order. The polymer coated particles cannot be separated from the mother liquor by centrifugation or magnetic separation, since they are very small. Therefore, we have diluted the dialyzed (using cellulosic membrane, having molecular cut-off 14kDa) mother liquor ~15 times to obtain the final dispersion.


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Two different molecular weights (Mw) of PAA namely 2,100 and 5,100 g/mol, referred as D2.1k and D5.1k, respectively and three different Mw of dextran namely 40,000, 60,000 and 1,00,000 g/mol, referred as D40k, D60k and D100k, respectively. All experiments were replicated three times and the nanoparticles were imaged by using FEG-TEM (Model: JEM-2100F). Representative TEM images at the end of 15 min. of synthesis are shown in Fig. S1 in Supplementary Information (SI). Approximately a total of 900-1,500 primary nanoparticles were measured for each sample to produce the PSD of the nanoparticles, shown later in section 4. 2.3. Estimation of average surface coverage of polymer on core Fe3O4 nanoparticles In case of polymer coated nanoparticles, correct estimation of steric potential is important, which bears a strong influence on nanoparticle growth. The steric repulsion potential is characterized by average surface coverage [〈

〉 and the thickness of polymeric layer (δ) on

the particle surface. Of these, the latter is approximated as the radius of gyration19,20 of the polymeric coating agent. For the former [〈

〉], we have developed an accurate way to calculate

it experimentally. We have developed an accurate technique to estimate the density of adsorbed polymer [〈

〉] on the nanoparticle surface using UV-visible spectroscopy. This procedure for estimating

〈

〉 is superior to the traditionally employed TGA analysis,2,21 which often overpredicts

〈

〉 due to the difficulties in separation of nanoparticles from the unadsorbed polymer. For this,

we prepared uncoated Fe3O4 nanoparticles and measured the surface area of particles by BET analysis of the N2-sorption data (Fig. S3, in SI). After synthesizing, the nanoparticles were washed and dried in a vacuum oven for 48 hr. at 35 °C. After drying, the particles were gently


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crushed by a mortar-pestle into fine powder; specific surface area of the powder was estimated (from BET) to be 66.2 m2/g. To measure the concentration of PAA and dextran in the solution, we have used UVvisible spectroscopy. PAA shows a plateau at 267 nm (COO- peak)22 and dextran shows a peak near 217 nm (glucose peak is near 210 nm). Also for very high molecular weight (above 1,00,000 g/mol polymer), the peak-height at 210 nm is not linear with concentration and is therefore not used for calibration. Instead a peak near 300 nm23 has been used (we have found a peak near 288 nm for D100k). All concentrations were measured in a phosphate buffer solution of pH 12, which is shown in Figs. S4 and S5 (in SI). Both of these figure shows the absorbance plot as a function of known concentration in a buffer solution of pH 12, for coating agents PAA and dextran. A linear calibration curve is fitted based on these data. Fig. S4(b) & (d) and Fig. S5(b), (d), (f) show the calibration plot of PAA (for P2.1k and P5.1k) and dextran (for D40k, D60k and D100k) respectively. Once the calibration plot is made, an adsorption study was carried out at 80 °C, for 15 minutes, in the buffer solution of pH 12. 136 mg of Fe3O4 nanoparticles (having surface area of 66.2 m2/g) were added in different solutions. For PAA 3.01 mM and for dextran 0.417 mM. After 15 minutes, the particles were separated by high speed centrifugation (18000 rpm) for 5 minutes. Since very small particles can remain in the solution even after the centrifugation, a control solution was prepared for UV-visible spectroscopy by adding the same amount of Fe3O4 nanoparticles in the buffer solution and again centrifuged for 5 minutes at 18000 rpm. The initial and final concentration for PAA and dextran are shown in both Fig. S4 and Fig. S5, respectively. Similar process has been followed for all other molecular weight of PAA and dextran coating


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agents. From these calculations we have found the surface coverage [〈

〉] of P2.1k, P5.1k,

D40k, D60k and D100k are 0.5, 0.45, 0.28, 0.35 and 0.35 chains/nm2 respectively.

3. Mechanistic Model of Nanoparticle Formation Formation of spherical nanoparticle in the aqueous medium includes nucleation, growth and coagulation, frequencies of occurrences of which are given below: 3.1. Nucleation We use classical nucleation theory24 for finding the nucleation rate. The frequency of homogeneous nucleation (fn) is therefore, 0  fn    - 16 3Vm2  VA exp  3(k T )3 ln( ) 2  l  B  

where, λl is the degree of supersaturation, l 

if, l  1 if, l  1

(1)

C (l ) . The other parameters in Eq. 1 namely, V, A, S

σ, kB, T, S, C(l) and Vm are total volume of liquid solution, pre-exponential factor, interfacial free energy between solid nuclei and surrounding liquid, Boltzmann constant, temperature, solubility of product molecule (Fe3O4),25 concentration of Fe3O4 in liquid phase and Fe3O4 molecular volume, respectively. Values of all the parameters are listed in Table 1. 3.2. Diffusion-growth The rate of nanoparticle-growth by solute transport from the bulk, aqueous solution onto the surface of the nanoparticle consists of two steps: (i) diffusional transfer of solute from the bulk solution onto the solid surface and (ii) surface-integration (bonding of molecule to solid


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surface) of solute molecules by surface-reaction. Since nanoparticles have a much higher surface defect density compared to micron-sized particles (because of high curvature and a high surface area to volume ratio), surface-integration of diffusing molecules onto the defect site is very fast. Moreover, precipitation reaction is very fast, hence it is expected that surface integration of solute molecules by surface reaction will be much faster, compared to the slow, diffusionmediated transport onto the particle surface. Therefore, we have assumed the surface integration process to be instantaneous, compared to the much slower diffusion-mediated transport of Fe3O4 molecule on the nanoparticle surface. A timescale analysis comparing surface reaction and diffusional transport has been shown in section S1 in SI, which shows that growth of nanoparticle via solute transport is diffusion-limited. Therefore, for a system with instantaneous surface-reaction, growth can be represented by equating the total flux of molecules diffusing onto the surface with the rate of growth of the particle.13,14 fg 

dr DmVm N AC (l )  dt r

(2)

In Eq. 2, Dm, C(l), NA and r are molecular diffusivity of solute molecule, concentration of solute molecule at the interface, Avogadro’s number and radius of particle, respectively. Since the surface reaction rate is instantaneous, solute concentration near the particle surface can be approximated to the solubility (S, which is the maximum concentration of solute in a solution before precipitation), which is only 1×10-6 mol m-3 for Fe3O4 in our system. However, the initial concentration [C(l)] of Fe3O4 is 3.7 mol m-3, which is six orders of magnitude higher than the solubility. Therefore, the concentration gradient term [C(l)-S] which appears in a mass transport rate expression is simplified to C(l).


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3.3. Coagulation-growth The growth rate of a particle (due to coalescence) is estimated using the Brownian collision frequency (qp) and a coagulation efficiency, 1/W which accounts for the change in diffusivity of nanoparticles in liquid medium due to interparticle interaction.1 qp is given by Smoluchowski’s equation1 qp 

2 k BT 3

 r1 r2  2    r2 r1  

(3)

where r1 and r2 are radii of colliding particles, μ is viscosity of the solvent.26 The coagulation frequency is hence calculated as27 fc 

1 1 q pVN 2 W 2

(4)

where, N is number density of particles in liquid dispersion. 1/W has been calculated based on the total interparticle potential [  ( r ) ], which is explained in the next section. In this context, by Kolmogorov microscale analysis in section S2 in SI, we have shown that shear due to stirring does not affect coagulation in our system. It proves that the length scale of the eddy is much larger than the separation distance between two nanoparticles, which implies that particle growth happens at a very small length scale compared to the eddy length-scale. 3.4. Effect of Interparticle Potential on Coagulation In the above model of coagulation, we postulate that collisions which lead to successful coagulation are induced purely by interparticle interactions.1,9,11 This effect of interparticle forces on coagulation can be captured by the equation proposed by Fuchs,11 which is estimated based on the diffusion flux (J) of particles of radius r1 (from a distance r), to the surface of particles


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having radius r2 (left-hand-side of Eq. 5). The corresponding diffusion flow rate of particles of radius r1 towards particles having radius r2 is given by1 (right hand side of Eq. 5), 4 r 2 J  

4 D12 n (r1  r2 ) W

(5)

where, D12 is diffusivity of particles r1 towards r2, n∞ is concentration of particles at r   and W is function of the total interparticle potential [  ( r ) ], where 

W  ( r1  r2 )

exp( ( r ) / k BT ) dr r2 r1  r2



(6)

Comparing the right-hand-side of Eq. 5, with the Brownian collision rate between two particles in absence of any interparticle interaction [i. e., 4 D12 n (r1  r2 ) ], we find that the Brownian collision rate in presence of interparticle interaction has been modified by a correction factor, 1/W. The latter is therefore equivalent to the coagulation efficiency, in presence of interparticle interactions. Eq. 5 is simplified by using the Stokes-Einstein equation for diffusivity (Di 

k BT ; ∀i =1, 2), and the diffusion constant for relative motion ( D12  D1  D2 ). So, Eq. 5 6 ri

becomes,1 4 r 2 J 

2k BTn 3

 r1 r2  1 2   r2 r1  W 

(7)

Right hand side of Eq. 7 gives the rate of coagulation (experienced by a particle of radius r2 with all other particles of radii r1 or vice versa), which is modified by the interparticle potential. Therefore, in order to accurately obtain the coagulation rate, 1/W has to be evaluated correctly. This would give the correct kinetics of a particle’s growth (especially important in the


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latter stages of nanoparticle synthesis), which translates into the predicted PSD from our model at any instant of time. To evaluate 1/W for a given pair of particles, integral in Eq. 6 needs to be calculated, which requires all individual interparticle potentials (Vi) to be summed up, since we know that,

 ( r )   Vi . The relevant interparticle potentials considered here are: van der Waals (Eq. 8), electrostatic (Eq. 9); steric (for polymer coated particles, Eq. 10), magnetic (for magnetic nanoparticle, Eq. 11) and solvation (to account for the solvent effect, Eq. 12). The schematic of two coagulating particles is shown in Fig. 1(b), where particles with radii r1 and r2 are separated by a center-to-center distance, r. We also define, a = r1 + r2, b = r1 - r2 and c = r1r2, with surfaceto-surface distance of particles being given by, s (= r – a). These are later used in Eqs. 8 – 12, which give the different interparticle potentials, as shown below. 3.4.1. van der Waals potential The van der Waals potential (VvdW) for a pair of particles of radii r1 and r2 is given by2,28 VvdW

Aeff  2c  r 2  a 2  2c    ln  2  2  6  r 2  a2 r 2  b2  r  b 

(8)

where, Aeff is effective Hamaker constant. 3.4.2. Electrostatic potential The electrostatic potential (Ve) between two charged particles is calculated using the expression29–31  c 2 4 a    ln 1  exp   s   Ve   2  4 cY Y   k BT  exp   s  1 2     r  e 

for 0.5a  5 (9) for 0.5a  5


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where, Yi 

 e   8 tanh    4 k BT    e  2 ri  1 1  1  tanh 2  2   ri  1  4 k BT

   

1 2

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i  1, 2 , and ε, ε0, ψ0, e, κ−1 and s are the

dielectric constant of the solvent (water), the permittivity of free space, the surface potential, the elementary charge, the Debye length and the surface to surface separation distance ( s  r  a ), respectively. The surface potential ψ0 was estimated using the ζ-potential measurement. 3.4.3. Steric potential

The steric potential (Vs) between coated particles is given as32

1 2 r  2  a  s     a  g k BT  2  ln    Vs   2    r    0 

( s / 2 )  1

(10)

( s / 2 )  1

where, < g> and δ are surface coverage (surface coverage by polymeric coating agent) and shell thickness, respectively. 3.4.4. Magnetic potential

The magnetic potential (Vm) between two magnetic (Fe3O4) particles is calculated based on saturation magnetization (Ms) of the coated particles at the maximum applied field32 Vm  

where,

0

0 a 6 M s 2 72r 3

(11)

is the permeability of free space.

3.4.5. Solvation potential

Using the Derjaguin approximation, the repulsive solvation potential (Vsol) between two spheres can be written as9,33,34


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Vsol   As ld

 s 2c exp    a  ld 

(12)

where, ld and As are the decay length and the pre-exponential factor, respectively (order of magnitude approximation reported by Israelachvili9). Since, Fe3O4 nanoparticles are very small in our experiments (e. g. mean diameter for uncoated nanoparticles is 8.11 nm), it is expected that particles will have high surface defect density, for which a monotonically decaying repulsive solvation potential is often used.9 This potential (Eq. 12) was hence added to all the other interparticle potentials (Eqs. 8 to 11) for carrying out our simulation. Values of all parameters used in simulating the model are listed in Table 2. 3.5. kMC Simulation Methodology

Our kMC simulation was conducted following the concept of interval of quiescence.14,35 Particle formation mechanism consists of three events viz., nucleation, diffusion-growth and coagulation-growth. The interval between two successive discrete random events (nucleation or coagulation in the present case), within which, other deterministic events (like continuous diffusion growth in the present case) can occur, is defined as the interval of quiescence (τQ). Shah et al.36 have shown that, τQ is a random variable with exponential distribution and can be calculated from

Q 

 ln(1  U ) ft

(13)

where, U is a uniformly distributed random variable in the range [0, 1). In Eq. 13 the total frequency ft is calculated as ft  f n  f c , where fn and fc are nucleation (Eq. 1) and coagulation


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frequencies (Eq. 4), respectively. Equation for diffusion-growth (Eq. 2) can be solved during this interval to calculate the diffusion-growth of the nanoparticles.37 At the end of τQ, a new random event is conducted, followed by calculation of the next τQ. This process is repeated until the sum of all τQ adds up to the total process time to be simulated. The choice of random events in our kMC simulation is thus based on probability of the ith event, which is

pi (t ) 

fi ft (t )

(14)

where, i is either coagulation or nucleation. During the course of the kMC simulation, to identify an event (either nucleation or coagulation) at the end of any τQ, we generate a random number from the uniform random number generator, U [0, 1). If its value is between 0 and pc (probability of coagulation being pc), the event conducted is coagulation of two particles, which are selected randomly. Otherwise, when the value of U [0, 1) falls between pc and 1 (probability of nucleation being, 1 - pc), a new particle is nucleated in the system. The simulation was performed with nm number of Fe3O4 molecules (which at t = 0 is 4×106, obtained from system-size convergence test), without any nanoparticle to start with (Np = 0). This initial high concentration of Fe3O4 molecules causes nucleation of Fe3O4 nanoparticles, which also simultaneously starts diffusion- and coagulation-growth of these particles. At the same time, nm decreases, since nucleation and diffusion growth continues. At the end of every random nucleation and coagulation event, quiescence interval (τQ) was re-calculated; radii of all particles were also updated by integrating the growth rate (Eq. 2), for the duration of quiescence interval. Nucleation and growth continued to consume the Fe3O4 molecules this manner, till


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Fe3O4 molecular concentration in liquid, C(l), dropped to the solubility limit, S. Coagulation continues even beyond this period, till the end of the synthesis process. Now, the number of Fe3O4 molecules in the liquid phase is depleted by both nucleation and diffusion-growth, simultaneously increasing Fe3O4 molecules in the solid phase by that many Fe3O4 molecules, thereby maintaining the overall mass balance between liquid and solid phases. After nucleation stops early in the process itself, the total number of Fe3O4 nanoparticles (Np) decreases by coagulation, since each coagulation event reduces total particle number by one. In order to maintain a large nanoparticle population for statistical accuracy till the end, we have used a constant number (Np = 2000) criterion in our kMC simulation. This ensures that the number of solid nanoparticles do not decrease below 2000 due to coagulation. This is achieved by addition of a ghost nanoparticle (via inverse sampling method) from the existing PSD (at any given time instant) of the Fe3O4 nanoparticle population. These new particles which were not originally part of the simulation are referred as ghost particles. These ghost particles increases the total mass of the simulation-box volume, so the latter is increased (upon addition of new particle) to maintain the volume fraction of nanoparticles in the box constant and same as that in the experiment.14,43


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Table 1. Input parameters and their values used in the kMC simulation. Parameter

Value

Source/Reference

A

1034 m-3s-1

Ref. 13,14

C(l)t=0

3.7 mol m-3

Obtained from experimental condition

Dm

5.04 × 10-9 m2s-1

Calculated using Wilke-Chang correlation38

2

Ref. 12,13,39

lc (number of molecules in a nucleus) Np nm,0

2000

Converged value used in our simulation 6

4×10

Converged value used in our simulation -1

Mw, Fe3O4

231.53 g mol

S

10-6 mol m-3

Ref. 25

T

353 K


V

1.2 × 10-4 m3


Vb

1.77×10-18 m3

Used in our simulation

Vm

7.32 × 10−29 m3

σ

0.1 J.m-2

Ref. 13,40–42

μ (T =353 K)

3.55 × 10-4 kg.m-1s-1

Ref. 26

ρ

5180 kg m-3

Ref. 25

calculated using molar mass, density, and Avogadro Number


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Table 2. Experimental and simulation parameters required to calculate interparticle potentials for uncoated and coated Fe3O4 nanoparticles. Uncoat ed Fe3O4

P2.1kFe3O4

P5.1kFe3O4

0.28

0.12

0.35

0.27

0.35

0.26

Ms (A m-1) (×10-4)

64.3

21.5

21.5

16.5

16.5

16.5

16.5

16.5

16.5

Ref. 2,21

Aeff (J) (×1020)

4

4

4

4

4

4

4

4

4

Ref. 2

κ-1 (nm)

4.67

4.67

4.67

4.67

4.67

4.67

4.67

4.67

4.67

calculated following Israelachvili 9

ζ potential (mV)

−28.93

-36.0

-32.0

-30.1

-30.1

-30.1

-30.1

-30.1

-30.1

measured at pH 11-12

δ (nm)

0

2

2.6

6.11

6.11

7.25

7.25

9.05

9.05

Ref. 19,20

0

0.50

0.45

0.24

0.12

0.35

0.27

0.35

0.26

Fig. S4 and S5

As (J/m2) (×103)

16

8

8

8

12.9

8

14.2

8

13.9

Ref. 9

ld (nm)

1

1

1

1

1

1

1

1

1

Ref. 9

Variables

〈

〉

〈

D40k-Fe3O4 〉 (chains/nm2)

〈


〈


2

(chains/nm )

Source

4. Results and Discussion Our primary focus is on predicting these size distributions; for validation, iron oxide (Fe3O4) nanoparticle was synthesized by the coprecipitation route21,44 (one uncoated and five coated samples). As coating agent, we have used polyacrylic acid (PAA) and dextran (only one of the molecular weights at a time), amongst five different samples (P2.1k, P5.1k, D40k, D60k and D100k). Fig. 2 shows that the experimental PSD during synthesis of uncoated nanoparticles do not have a statistically significant change, when the sampling time-instant is varied between 1


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Page 20 of 34

to 15 min. Similar results are obtained for coated iron oxide nanoparticles as well. Below we show, how our present model predicts these temporal size distributions.

Figure 2. Temporal evolution of particle size distribution (PSD) of uncoated Fe3O4

nanoparticles. The relative frequency is normalized by the mass of particles in each case. All experiments were triplicated and a total of approximately 900-1500 particles were measured from several TEM-images to calculate the PSD in each case. The error bar in each bin represents the standard deviation of experiments from three replicates. The process and the model starts with homogeneous nucleation24, followed by diffusionand coagulation-growth.13,27 Nucleation (Eq. 1) and diffusion-growth (Eq. 2), being monoparticulate events (involving creation and growth of a single nanoparticle), does not get influenced by the effect of any interparticle potential between two nanoparticles. In contrast, to completely coagulate two particles upon collision (Eq. 4), the potential energy barrier due to the long-range, repulsive interparticle forces needs to be overcome. We have developed a constant number kMC43 simulation scheme using these equations and the method of interval of quiescence.36


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We have estimated all our system parameters (given in Tables 1 and 2) based on conditions used in the experiments, which are either known from existing literature or based on our experimental measurements. This renders our model to be purely predictive. 4.1. Effect of Interparticle Potential on Particle Size

Before interpreting the results of the simulation, we first illustrate the interaction between a pair of 5 nm particles (which is a typical, mean experimental particle diameter). Figs. 3(a)-(c) show potential energy plots as a function surface to surface separation distance, for uncoated, P2.1k-coated and D40k-coated Fe3O4 nanoparticles, respectively (calculated directly from Eqs. 8 –12). Clearly, the total interparticle potential energy for each pair has a large potential energy barrier, primarily because of high steric and solvation repulsive forces compared to electrostatic repulsion. Similarly, van der Waals attractive forces are dominant relative to magnetic attractions. It is observed that the heights of the potential energy barrier for uncoated, P2.1k-coated and D40kcoated Fe3O4 nanoparticles increase from 27.7kBT to 35.1kBT to 45kBT, respectively, which suggests that the mean particle diameter may decrease from uncoated to PAA-coated to dextrancoated nanoparticles, respectively. This is found to be true in experiments (shown letter). Furthermore, for either an uncoated or a coated nanoparticle, as the diameter of an individual particle increases with time, the corresponding height of the potential barrier increases too. We have observed the potential barrier increases as the particle size increases. Fig. 3(d) - (f) shows the increase in potential barrier for uncoated, PAA-coated and dextran-coated, respectively.


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The latter translates into a decreased coagulation rate with time. To exactly quantify this effect, we have calculated 1/W (Eq. 6) as a function of particle diameter. Figs. 4(a)-(c) agrees with this conjecture and show that the coagulation efficiency decreases rapidly with increase in particle diameter. This also explains the fact that coagulation almost stops for uncoated Fe3O4 nanoparticles, when it reaches a mean diameter of ~8 nm (where 1/W becomes very small, namely 7×10-9), justifying the result in Fig. 2, where the mean diameter of 8 nm for uncoated particles, remains practically unchanged with time.

Figure 3. Interparticle potential energy functions plotted against surface-to-surface separation distance between two particles of 5 nm diameter: (a) uncoated (b) P2.1k-coated and (c) D40kcoated Fe3O4 nanoparticles. For increasing particle sizes (1-10 nm) total potential vs. surface to surface separation distance is plotted for (d) uncoated, (e) P2.1k-coated and (f) D40k-coated Fe3O4 nanoparticles.


22

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Figure 4. Correction factor (1/W) of coagulation frequency plotted as a function of diameter of the particle, for (a) uncoated (b) P2.1k-coated and (c) D40k-coated Fe3O4 nanoparticles.

We find that there are two size dependent terms in coagulation rate equation (Eq. 4). The first term ( r1 / r2  r2 / r1  2 ) is a weak function of particle size for a near-monodisperse PSD (513 nm in our experiments, Fig. 5). In contrast, the term 1/W in Eq. 4 is a complex function of r [through the potential function ϕ(r)]. Fig. 4(a)-(c) shows that, 1/W decreases several orders of magnitude with r. This shows that, interparticle potential strongly influences coagulation rate as the particle size increases. Fig. 5(a)-(f) show experimental PSD based on TEM images (sample TEM images are given in Fig. S1 of SI) analyzed at 15 minute of synthesis-time. Mean particle diameter decreases from 8.1 nm [uncoated Fe3O4 nanoparticles, Fig. 5(a)] to 3.0 nm [D100k-coated particles, Fig. 5(f)]. It is clear from Fig. 5(a)–(f) that, the complete range of experimental PSD for each case is predicted extremely well (a-priori and without any additional or fitting parameter), by the new model and simulation of the present work. Simulations for each cases ware replicated 10 times and standard deviation between them were negligible (not visible in Fig. 5). In addition, Figs. S2(a)–(f) (in SI) confirm that the small time-scale (at 1 min.) experimental PSD14 is also predicted accurately by the simulation of the present model. This justifies that our interparticle


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Figure 5. Comparison of PSD of Fe3O4 nanoparticles at 15 min., from both kMC simulation and experiment, for: (a) uncoated (b) P2.1k-coated, (c) P5.1k-coated, (d) D40k-coated, (e) D60kcoated and (f) D100k-coated samples. For easy viewing, simulation curves are plotted as cubic splines of discrete data-points (latter based on 10 replicates of each simulation). Experimental data plotted as the mean of 3 runs (yielding a total of 900-1,500 measured nanoparticles), with std. dev. plotted as error bar on frequency bars. Related TEM images are shown in Fig. S1 of SI.


24

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potential based model can accurately portray the overall nanoparticle formation dynamics in a precise, quantitative fashion. It demonstrates how coagulation in experiments slows down and eventually reaches its equilibrium PSD, which has not been explained in literature. 4.2.Effect of Surface Coverage on Particle Growth

To further identify the dominant interparticle interaction, we have synthesized D40k, D60k and D100k coated nanoparticles; with two different surface coverage [〈

〉, chains/nm2] values,

which is the surface density of coating agent molecules per unit nanoparticle surface area. Fig. 6(a) shows that, on decreasing 〈

〉 from 0.28 to 0.12 chains/nm2, for the D40k case, the mean of

the PSD increased from 4.5 to 6 nm, with a simultaneous increase in standard deviation from 1.0 to 1.5 nm. Similarly, for both D60k [Fig. 6(b)] and D100k [Fig. 6(c)], on decreasing 〈

〉 from

0.35 to 0.27 and 0.26 chains/nm2, respectively, the mean of the PSD increased from 3 to 4.5 nm, with a simultaneous increase in standard deviation from 0.8 to 1.2 nm. Corresponding simulations were carried out keeping all parameters same, except for the surface coverage and solvation potential. It is known that, the solvation force increases with hydrophilicity of the surface, which implies that, with increase in surface coverage, the solvation interaction will decrease. We have used a linear variation of solvation potential with surface coverage to estimate the solvation force. The simulation again matches very well with experimental PSD (Fig. 6), which confirms that the relative balance between steric and solvation forces define the final distribution in these cases. This validates the robustness of the model for different ranges of synthesis parameters, which may arise in modeling experiments with different nanoparticlesynthesis protocols.


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Page 26 of 34

Figure 6. Comparison of PSD of dextran-coated Fe3O4 nanoparticles of two different ρg at 15 min., from kMC simulation and experiment, for: (a) D40k-coated, (b) D60k-coated and (c) D100k-coated samples. Dotted line shows the PSD with higher the PSD having lower

ρg and the solid line shows

ρg . For easy viewing, simulation curves are plotted as cubic splines of

discrete data-points (latter based on 10 replicates of each simulation). Experimental data plotted as the mean of 3 runs (yielding a total of 900-1,500 measured nanoparticles), with std. dev. plotted as error bar on frequency bars.


26

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5. Conclusions To summarize, with the primary objective of explaining the complete process of nanoparticle formation in the liquid phase, we have formulated a comprehensive model, right from the starting of nucleation, combined with simultaneous diffusion- and coagulation-growth of particles. This work clearly enunciates the very important role of coagulation in influencing PSD of primary nanoparticles, both in the presence and absence of a coating agent. It has been shown that, accounting for all interparticle interactions can lead to an accurate estimation of the time-varying coagulation rate, which slows down as the mean particle size increases with time. We have also demonstrated by using the experimentally validated simulation results that, how much one can reduce the mean and standard deviation of the PSD of primary nanoparticles, by increasing the surface coverage of a polymeric coating agent on a nanoparticle surface, or by increasing the molecular weight of the polymer. Extending this work further, our framework can be utilized in principle to predict secondary aggregates too. The latter would occur when timescale of complete fusion of nanoparticles would be much larger, compared to interparticle collision timescale. On a related note, partial fusion or aggregation in both liquid phase and aerosols has been studied with limited success in literature. Some of these works have related DLVO forces with collision efficiency2,43,45,46 between secondary aggregates, while Narsimhan and Ruckenstein47,48 have used a barrier-less (LJ potential) theoretical aggregation model in the free molecular regime of aerosol. All these models,2,43,45–48 however, had to still fit experimental data to account for unknown parameters in their model. In contrast, this part of our work provides a natural method to a-priori calculate the coagulation-efficiency term, which dynamically changes with time (as


27


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Page 28 of 34

particle size, concentration etc. changes) and would hold the key in a-priori prediction of secondary aggregates too.

Acknowledgment:

We gratefully acknowledge the sophisticated analytical instrumental facility (SAIF) of the Indian Institute of Technology Bombay for providing the TEM facilities. We are very thankful to Prof. Sanat Kumar, Columbia University and Prof. Y. S. Mayya, IIT Bombay for kindly reviewing our manuscript and providing critical insight. We also thank Mr. J. Prithivi Raj, former research scholar in our group at IIT Bombay for initial experimental help on kinetic study of nanoparticle-growth.

Supporting Information Available:

Prediction of PSD of uncoated and polymer coated Fe3O4 nanoparticles at 1 min using kMC simulation, Supplementary TEM images, N2-sorption data and calibration curve of UV-visible spectroscopy for polymer are provided in the supporting information section.

References: (1)

Friedlander, S. K. Smoke, Dust, and Haze Fundamentals of Aerosol Dynamics; 2nd ed.; Oxford University Press: New York, 2000.

(2)

Kumar, S.; Ravikumar, C.; Bandyopadhyaya, R. State of Dispersion of Magnetic Nanoparticles in an Aqueous Medium: Experiments and Monte Carlo Simulation. Langmuir 2010, 26, 18320–18330.

(3)

Kwon, J. S.-I.; Nayhouse, M.; Christofides, P. D.; Orkoulas, G. Modeling and Control of


28

Page 29 of 34

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Shape Distribution of Protein Crystal Aggregates. Chem. Eng. Sci. 2013, 104, 484–497. (4)

Kwon, J. S.-I.; Nayhouse, M.; Orkoulas, G.; Christofides, P. D. Enhancing the Crystal Production Rate and Reducing Polydispersity in Continuous Protein Crystallization. Ind. Eng. Chem. Res. 2014, 53, 15538–15548.

(5)

Kwon, J. S.-I.; Nayhouse, M.; Orkoulas, G.; Christofides, P. D. Crystal Shape and Size Control Using a Plug Flow Crystallization Configuration. Chem. Eng. Sci. 2014, 119, 30– 39.

(6)

Tanaka, K. K.; Kawano, A.; Tanaka, H. Molecular Dynamics Simulations of the Nucleation of Water: Determining the Sticking Probability and Formation Energy of a Cluster. J. Chem. Phys. 2014, 140, 114302.

(7)

Cheng, Y.-T.; Liang, T.; Nie, X.; Choudhary, K.; Phillpot, S. R.; Asthagiri, A.; Sinnott, S. B. Cu Cluster Deposition on ZnO(1010): Morphology and Growth Mode Predicted from Molecular Dynamics Simulations. Surf. Sci. 2014, 621, 109–116.

(8)

Yuhara, D.; Barnes, B. C.; Suh, D.; Knott, B. C.; Beckham, G. T.; Yasuoka, K.; Wu, D. T.; Sum, A. K. Nucleation Rate Analysis of Methane Hydrate from Molecular Dynamics Simulations. Faraday Discuss. 2015.

(9)

Israelachvili, J. N. Intermolecular and Surface Forces; 2nd ed.; Academic Press, 1992.

(10) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press, 1992. (11) Fuchs, N. A. The Mechanics of Aerosols; Davies, C. N., Ed.; Pergamon Press: London, 1965; Vol. 91. (12) Jain, R.; Shukla, D.; Mehra, A. Coagulation of Nanoparticles in Reverse Micellar Systems: A Monte Carlo Model. Langmuir 2005, 21, 11528–11533.


29


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 34

(13) Layek, A.; Mishra, G.; Sharma, A.; Spasova, M.; Dhar, S.; Chowdhury, A.; Bandyopadhyaya, R. A Generalized Three-Stage Mechanism of ZnO Nanoparticle Formation in Homogeneous Liquid Medium. J. Phys. Chem. C 2012, 116, 24757–24769. (14) Bachhar, N.; Bandyopadhyaya, R. Role of Coating Agent in Iron Oxide Nanoparticle Formation in an Aqueous Dispersion: Experiments and Simulation. J. Colloid Interface Sci. 2015, 464, 254–263. (15) Vafa, E.; Shahrokhi, M.; Molaei Dehkordi, A. Population Balance Modeling of Barium Sulfate Nanoparticle Synthesis via Inverse Microemulsion Including Coagulation Effect. Ind. Eng. Chem. Res. 2014, 53, 12705–12719. (16) Shukla, D.; Mehra, A. A Model for Particle Coagulation in Reverse Micelles with a Size Dependent Coagulation Rate. Nanotechnology 2006, 17, 261–267. (17) Massart, R.; Dubois, E.; Cabuil, V.; Hasmonay, E. Preparation and Properties of Monodisperse Magnetic Fluids. J. Magn. Magn. Mater. 1995, 149, 1–5. (18) Molday, R. S.; MacKenzie, D. Immunospecific Ferromagnetic Iron-Dextran Reagents for the Labeling and Magnetic Separation of Cells. J. Immunol. Methods 1982, 52, 353–367. (19) Reith, D.; Müller, B.; Müller-Plathe, F.; Wiegand, S. How Does the Chain Extension of Poly (acrylic Acid) Scale in Aqueous Solution? A Combined Study with Light Scattering and Computer Simulation. J. Chem. Phys. 2002, 116, 9100–9106. (20) Granath, K. A. Solution Properties of Branched Dextrans. J. Colloid Sci. 1958, 13, 308– 328. (21) Ravikumar, C.; Kumar, S.; Bandyopadhyaya, R. Aggregation of Dextran Coated Magnetic Nanoparticles in Aqueous Medium: Experiments and Monte Carlo Simulation. Colloids Surfaces A Physicochem. Eng. Asp. 2012, 403, 1–6.


30

Page 31 of 34

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(22) Hajdú, A.; Szekeres, M.; Tóth, I. Y.; Bauer, R. a; Mihály, J.; Zupkó, I.; Tombácz, E. Enhanced Stability of Polyacrylate-Coated Magnetite Nanoparticles in Biorelevant Media. Colloids Surf. B. Biointerfaces 2012, 94, 242–249. (23) Baik, Y.-S.; Cheong, W.-J. Determination of Molecular Weight Distribution and Average Molecular Weights of Oligosaccharides by HPLC with a Common C18 Phase and a Mobile Phase with High Water Content. Bull. Korean Chem. Soc. 2007, 28, 847–850. (24) Randolph, A. D.; Larson, M. A. Theory of Particulate Processes: Analysis and Techniques of Continuous Crystallization; Academic Press Inc.: San, 1988. (25) Cornell, R. M.; Schwertmann, U. The Iron Oxides: Structure, Properties, Reactions, Occurrences and Uses; 2nd ed.; WileyVCH Verlag, 2003. (26) Kestin, J.; Sokolov, M.; Wakeham, W. A. Viscosity of Liquid Water in the Range −8 °C to 150 °C. J. Phys. Chem. Ref. Data 1978, 7, 941. (27) Bandyopadhyaya, R.; Kumar, R.; Gandhi, K. Simulation of Precipitation Reactions in Reverse Micelles. Langmuir 2000, 16, 7139–7149. (28) Hamaker, H. C. The London-van Der Waals Attraction between Spherical Particles. Physica 1937, IV, 1058–1072. (29) Hogg, R.; Healy, T. W.; Fuerstenau, D. W. Mutual Coagulation of Colloidal Dispersions. Trans. Faraday Soc. 1966, 62, 1638. (30) Ohshima, H. Effective Surface Potential and Double-Layer Interaction of Colloidal Particles. J. Colloid Interface Sci. 1995, 174, 45–52. (31) Lee, K.; Sathyagal, A. N.; McCormick, A. V. A Closer Look at an Aggregation Model of the Stöber Process. Colloids Surfaces A Physicochem. Eng. Asp. 1998, 144, 115–125. (32) R. E. Rosensweig. Ferrohydrodynamics; Cambridge University Press, 1985.


31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 32 of 34

(33) Levine, S.; Bowen, B. D.; Partridge, S. J. Stabilization of Emulsions by Fine Particles I . Partitioning of Particles Between Continuous Phase and Oil / Water Interface. Colloids and Surfaces 1989, 38, 325–343. (34) Bogush, G. H.; Zukoski, C. F. Uniform Silica Particle Precipitation : An Aggregative Growth Model. J. Colloid Interface Sci. 1991, 142, 19–34. (35) Ethayaraja, M.; Ravikumar, C.; Muthukumaran, D.; Dutta, K.; Bandyopadhyaya, R. CdSZnS Core-Shell Nanoparticle Formation: Experiment, Mechanism, and Simulation. J. Phys. Chem. C 2007, 111, 3246–3252. (36) Shah, B.; Ramkrishna, D.; Borwanker, J. Simulation of Particulate Systems Using the Concept of the Interval of Quiescence. AIChE J. 1977, 23, 897–904. (37) Ravikumar, C.; Ethayaraja, M.; Bandyopadhyaya, R. Discrete-Continuous Hybrid Simulation of Monodisperse Nanoparticle Formation. Int. J. Chem. Sci. 2007, 5, 1764– 1774. (38) Treybal, R. E. Mass-Transfer Operations; McGraw-Hill, 1980. (39) Ethayaraja, M.; Dutta, K.; Muthukumaran, D.; Bandyopadhyaya, R. Nanoparticle Formation in Water-in-Oil Microemulsions: Experiments, Mechanism, and Monte Carlo Simulation. Langmuir 2007, 23, 3418–3423. (40) Oskam, G.; Hu, Z.; Penn, R. L.; Pesika, N.; Searson, P. C. Coarsening of Metal Oxide Nanoparticles. Phys. Rev. E 2002, 66, 011403. (41) Kim, D. K.; Zhang, Y.; Voit, W.; Rao, K. V.; Muhammed, M. Synthesis and Characterization of Surfactant-Coated Superparamagnetic Monodispersed Iron Oxide Nanoparticles. J. Magn. Magn. Mater. 2001, 225, 30–36. (42) Klein, D. Homogeneous Nucleation of Calcium Hydroxide. Talanta 1968, 15, 229–231.


32

Page 33 of 34

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(43) Liu, H. H.; Surawanvijit, S.; Rallo, R.; Orkoulas, G.; Cohen, Y. Analysis of Nanoparticle Agglomeration in Aqueous Suspensions via Constant-Number Monte Carlo Simulation. Environ. Sci. Technol. 2011, 45, 9284–9292. (44) Ravikumar, C.; Bandyopadhyaya, R. Mechanistic Study on Magnetite Nanoparticle Formation by Thermal Decomposition and Coprecipitation Routes. J. Phys. Chem. C 2011, 115, 1380–1387.

(45) Zhang, W.; Crittenden, J.; Li, K.; Chen, Y. Attachment Efficiency of Nanoparticle Aggregation in Aqueous Dispersions: Modeling and Experimental Validation. Environ. Sci. Technol. 2012, 46, 7054–7062. (46) Petosa, A. R.; Jaisi, D. P.; Quevedo, I. R.; Elimelech, M.; Tufenkji, N. Aggregation and Deposition

of

Engineered

Nanomaterials

in

Aquatic

Environments:

Role

of

Physicochemical Interactions. Environ. Sci. Technol. 2010, 44, 6532–6549. (47) Narsimhan, G.; Ruckenstein, E. The Brownian Coagulation of Aerosols over the Entire Range of Knudsen Numbers: Connection between the Sticking Probability and the Interaction Forces. J. Colloid Interface Sci. 1985, 104, 344–369. (48) Narsimhan, G.; Ruckenstein, E. Monte Carlo Simulation of Brownian Coagulation over the Entire Range of Particle Sizes from near Molecular to Colloidal: Connection between Collision Efficiency and Interparticle Forces. J. Colloid Interface Sci. 1985, 107, 174– 193.


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Table of Content (TOC)


34

Predicting Complete Size Distribution of Nanoparticles Based on

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