A Generalized Model for Nano- and Submicron Particle Formation in

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C: Physical Processes in Nanomaterials and Nanostructures

A Generalized Model for Nano- and Submicron Particle Formation in Liquid Phase, Incorporating Reaction Kinetics and Hydrodynamic Interaction: Experiment, Modeling and Simulation Vivekananda Bal, and Rajdip Bandyopadhyaya J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b03521 • Publication Date (Web): 13 Aug 2018 Downloaded from http://pubs.acs.org on August 27, 2018

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The Journal of Physical Chemistry

A Generalized Model for Nano- and Submicron Particle Formation in Liquid Phase, Incorporating Reaction Kinetics and Hydrodynamic Interaction: Experiment, Modeling and Simulation

Vivekananda Bal, Rajdip Bandyopadhyaya * Department of Chemical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India.

* Corresponding authors ([email protected]) 1 ACS Paragon Plus Environment

The Journal of Physical Chemistry 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

Abstract

A generalized model of particle formation has been developed, including reaction, nucleation and simultaneous particle growth by diffusion, coagulation and Ostwald ripening (OR). This enables a priori prediction of both nano-and submicron size SiO2, from a few nm, to at least 400 nm. This was possible, on including: (i) a finite reaction kinetics, (ii) a hydrodynamic correction in Brownian coagulation and (iii) the coupled effect of reaction kinetics with OR; the latter two for the first time. We show that, hydrodynamic interaction can reduce the coagulation frequency up to a factor of about 8.5. Consequently, without it, a large prediction error (up to 25% with respect to the experimental particle size) is observed. For the final particle size, synthesis has been conducted up to ~4 h. This imparts accounting of finite reaction kinetics to be critical, which is substantiated by accurate prediction of temporal particle diameter, throughout the synthesis period. We thus avoid erroneous under-prediction or over-prediction of size, for moderately fast or slow reaction, respectively, which is true of existing models based on instantaneous reaction kinetics. Our model is further validated by quantitative prediction of SiO 2 particle size trends with different reactant concentration and temperature.

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1 Introduction Nano- and submicron particle formation in the liquid phase is driven by five different steps, namely reaction to form growth units, nucleation, diffusion-growth, coagulation-growth by complete fusion, and finally Ostwald ripening (OR) driven-growth, the latter occurring over a long time scale. Growth of primary particles stop when the equilibrium particle size distribution is reached. However, further aggregation of primary particles is possible afterwards and may lead to the formation of loose secondary aggregates (formed by incomplete fusion of primary particles) or fractal structures. In the present work, we solely deal with primary particle formation and its size, which is a result of coagulation by complete fusion of smaller primary particles.

Earlier models concerning the formation of nanoparticles in the liquid phase batch synthesis1-4 mostly dealt with chemically precipitating systems (like BaSO4, ZnO, CaCO3, Fe3O4 etc.), wherein reaction leading to the formation of growth units was instantaneous in nature. Their models for nanoparticle formation were based on elementary events viz. nucleation, diffusion- and coagulation-growth and OR. In case of continuous synthesis too, particle formation models mainly concentrated on systems with instantaneous reaction.5-9 However, in case of a slow reaction kinetics, all subsequent elementary events responsible for particle growth are strongly affected. Therefore, current models fail to predict the particle size for systems with a slow or finite reaction kinetics, as is true in many inorganic systems at low temperature and for almost all organic systems.10 Although there have been some attempts to include reaction kinetics in the particle formation model in case of continuous, hydrothermal, supercritical synthesis of nanoparticles,11-12 the effect of the former is insignificant as the kinetics is extremely fast (reaction time < 2s ), due to an extremely high operating temperature. However, the rate of formation of growth units in their8,11-12 case was controlled by the slow mixing dynamics, which plays a similar role as that of the intrinsic reaction kinetics. In our present work, intrinsic rate kinetics itself is very slow, due to the slow nature of the silica formation step. Therefore, in the present work, we explicitely study the quantitiave effect of a slow reaction kinetcs, which has not been done earlier. To the best of our knowledge, this is probably the first time that, effect of reaction kinetics on particle formation process has been studied thoroughly, with simulation



results (shown latter) for both with and without reaction kinetics. Although the continuous synthesis of nanoparticles is beyond the scope of this paper, the model presented here is equally applicable in continuous synthesis too, given that the effect of convection on particle formation is negligible. As far as growth by coagulation is concerned, majority of previous works5-6,11,13-15 incorporated the classical coagulation kernel, based on the simple expression provided by Smoluchowski.16-17 The latter assumed that, all collisions lead to complete fusion. There have been some attempts2-4 to include coagulation efficiency as a lumped fitting parameter to consider only successful collisions. Earlier, there had also been an attempt18 to include the modified form of the coagulation kernel, proposed by Fuchs,19 incorporating only the effects of inter-particle interactions to capture successful collisions resulting in complete fusion. This had eliminated the need to explain particle formation by using coagulation efficiency as a fitting parameter. However, it still did not account for hydrodynamic interactions affecting collisions. Therefore, so far, there has not been any attempt to investigate the effect of hydrodynamic interaction on particle formation.

All the previously mentioned collision frequency functions were derived based on the premises that the particle motion follows the classical theory of Brownian diffusion, originally proposed by Einstein.20-21 The latter says that the random motion of colloidal particles, due to collision with suspending fluid molecules, is independent of each other. Hence, the classical theory is applicable to extremely dilute suspensions only, where each particle is virtually isolated in an infinite medium. When the suspension is not very dilute, particles are closer to each other and start interacting hydrodynamically.22 Hence, the motion of particles no longer remain independent of each other. This requires a correction in the relative diffusivity of particles to incorporate the effect of hydrodynamic interactions.22 Derjaguin23 was the first who observed that mobility of particles do not follow the Stokes law at short distances. There has been very little work to study the effect of hydrodynamic interaction on coagulation. Initial effort to understand theoretically the effect of hydrodynamics on coagulation was made by Spielman,24 for the case of both rapid and slow coagulation, with 4 ACS Paragon Plus Environment

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the incorporation of a resistance function for hydrodynamic correction. These functions were deduced by Maude,25 for the motion of two unequal spheres, moving in opposite directions along the centerline, with unequal velocities. Although the resistance functions are more general in nature, their complexity, and unavailability of an asymptotic form for very small separations, limit their use. Later on, Honig and co-workers26 investigated the hydrodynamic effect, based solely on the rate of coagulation of hydrophobic colloids, including a simple algebraic expression as a hydrodynamic correction factor. This expression was derived by Brenner27 for the motion of a rigid sphere towards a flat surface, using the same method as Stimson and Jeffery.28 Though this function is simple in nature, it is only applicable for coagulation of spheres with equal radii. In the present work, we incorporate the hydrodynamic corrections to relative diffusivity, as proposed by Batchelor,22,29 assuming that the relative diffusive flux due to Brownian motion is a result of equal and opposite forces acting on two spheres. The mobility functions suggested by their work can be suitably applied for the collision of spheres with arbitrary radii. Furthermore, their asymptotic forms for very small interparticle separation are easily obtainable. In fact, while calculating coagulation rates, previous works24,26 entirely ignored the effect of the presence of structured water around particles.30 In this paper, we qualitatively include them into the coagulation rate calculation, solely to demonstrate the effect of hydrodynamic interaction on coagulation, and for the first time, on the whole particle formation process, without use of any fitting parameter. To the best of our knowledge, there has not been any prior attempt of such kind.

In the end, there is no model in literature for the formation of nanoparticles in the liquid phase, which can predict particles of submicron size, on considering all mechanistic details. Previous models1-4, 9,11 for the liquid phase synthesis of nanoparticles, can predict particles of size less than only about 40 nm. Though there are few models for the continuous crystallization process or continuous synthesis of nanoparticle,5-7,12,15,31-32 which are well capable of predicting particles of mean size of several micrometers, their models incorporate either a power-law, diffusion-growth model with Brownian coagulation only or without talking about the coagulation process at all in their model. Moreover, these models for continuous synthesis do not mention anything regarding coagulation efficiency in their models.



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The principal purpose of the present paper is three folds. Firstly, inclusion of the finite reaction kinetics into the particle formation model, to thoroughly investigate the effect of reaction rate, for the case of either a slow or a fast reaction. Secondly, modification of coagulation kernel, by incorporating particle mobility functions to relative diffusivity and thereby study the effect of hydrodynamic interaction on the overall particle size, without any fitting parameter. At the end, inclusion of all the five elementary steps together in particle formation, for a priori prediction of both nano-and submicron size particles, without use of any fitting parameter, for several different regimes and values of operating parameters.

2 Model formulation

Formation of nanoparticles from a homogeneous medium represents a multiphase system, where the continuous liquid phase constitutes the primary phase and the dispersed solid phase containing the particles represents the secondary phase. In the primary phase, the species balance equation is used to model the change in the mass of the species, whereas, in the secondary phase, due to the presence of only one species (SiO2), a volume fraction equation is employed to model change in mass of the secondary phase. Time evolution of particle size in the secondary phase has been obtained through the classical population balance equation (PBE).

2.1 Species balance equation

Primary phase species balance equation is given by  p p p   Yi  . p ˆjip   p Rip  m ps t





(1)

where, p and s represent the primary and secondary phases, respectively. ρ p is the primary phase density, α p is the primary phase volume fraction, Yi p is the mass fraction of species i in phase p, ˆjip is the diffusion flux of species i in phase p, Rip is the loss or gain of species i due to reaction in phase p, and mps is the mass transfer rate from phase p to s. The above equation is solved using the initial condition: Yi t  0  Yi ,0 , where, Yi , 0 is the initial mass fraction of ith species, and the 6 ACS Paragon Plus Environment

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boundary condition: N i .nˆ  0 , where N i is the total flux of species i in the primary phase and

nˆ is the unit normal vector.

2.1.1 Reaction kinetics

Reaction kinetics for SiO2 nanoparticle synthesis from tetraethyl orthosilicate has been taken, based on the work of Giesche33. Our preliminary estimation of rate kinetics is also found to be in good agreement with Giesche.33 Giesche’s rate expression33 is given as:

dcSiO2 dt

 kc1H.18 c 0.97 c 2 O NH 3 TEOS

(2)

where, c SiO2 , c H 2O , c NH 3 and cTEOS are bulk concentrations of SiO2, water, ammonia and tetraethyl ortho-silicate, respectively. k is the reaction rate constant (see supporting information, S1).

2.2 Volume fraction equation

Volume fraction equation for secondary phase is given by





d s s    m ps  S q dt

(3)

where, ρ s is the secondary phase density, α s is the secondary phase volume fraction, Sq is the source term in the secondary phase. In this work, there is no source term and hence Sq is zero. The above equation is solved using following initial condition:  s t  0  0 and boundary condition: N .nˆ  0 , where N is the total flux of secondary phase species.

2.3 Population balance equation The PBE given in Randolph34 for a quiescent medium is:



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1 n(v, x, t )   v .[Gv n(v, x, t )]  q (v  v ' , v ' )n (v  v ' , t )n (v ' , t )dv ' 20 t v







 q (v , v )n (v , t )n (v , t )dv '

'

(4)

'

0

where, n(v, x, t) is the number density function of particles,  v represents the operator in particle volume co-ordinate (internal co-ordinate), x represents the external co-ordinate system, v is particle volume, q is the rate of coagulation and Gv is the volumetric growth rate of particles (obtained from linear growth rate). In this case, the reactor is well mixed and hence, there is no spatial variation of number density. The above equation is solved using the following initial and boundary conditions: IC: nv, t  0  0 and BC: nv  0, t   n0

2.3.1 Nucleation Classical theory of homogeneous nucleation35-36 has been used to model the nucleation rate as      



 16 3vm2  n0  A exp   3 2 3(k BT ) (ln S ) 

(5)

where, A, σ, vm, are pre-exponential factor, interfacial tension of silica-alcohol system and molecular volume of SiO2, respectively.37 S  cSiO2 cSiO2 ,s is the supersaturation of SiO2 in the liquid phase, where, cSiO2 ,s is the solubility of SiO2. In the present simulation, the nuclei size (rcn) is assumed to be formed by a cluster of two molecules (see supporting information, S2).

2.3.2 Diffusion-growth Once nuclei are formed, they start growing by addition of monomeric growth units from bulk of the medium. The rate of diffusion-controlled growth is given as

dr Dm v m N A c SiO2  dt r 8 ACS Paragon Plus Environment

(6)

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where, Dm, NA, r are the Avogadro’s number, molecular diffusion coefficient of SiO2 in liquid medium and radius of particle, respectively. Radius of the concerned particle is obtained from solving the PBE at any given time (see supporting information, S3).

2.3.3 Brownian coagulation and hydrodynamic correction The expression of coagulation kernel proposed by Fuchs19 (after modification of von Smoluchowski’s classical kernel16) can be used to model the coagulation event. The expression for coagulation frequency q  is given as

  1 1  2k B T  1 1  3 q (v i , v j )   1  1  vi  v 3j  W 3  3  3   vi v j 

(7a)

where, μ is the viscosity of the medium, and vi and vj are volumes of colliding particles. W is the Fuchs’s stability ratio and is given as W  r1  r2 





r1  r2

expU / k BT  dr r2

(7b)

where, U is the sum of all interparticle potentials (see supporting information, S4.1). In the present work, effect of interparticle forces like van der Waal’s,38-40 electrical double layer41-42 and solvation force43 have been considered into the calculation. The hydrodynamic correction factor has been included into the coagulation kernel by incorporating the hydrodynamic mobility tensor 22,29,44 into the relative translational diffusivity D12 r  tensor. This has been done by taking the normal diffusion flux J, at steady state, for the

general scenario of a particle of radius r2 , diffusing to a particle of radius r1 at rest. The normal flux is given as





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

  n  D12 r , r1 , r2 .( Fˆ ) n  4r 2 J   N  4r 2  D12 r , r1 , r2 . er    k BT  r   

(8)

where, N is the number of particles (crossing through each spherical surface, concentric with the central sphere at steady state), Fˆ is the interparticle interaction force, and er is the radial unit vector. D12 r  is the relative diffusion tensor given by Bachelor as  rr rr   D12 r , r1 , r2   D120 G r , r1 , r2  2  H r , r1 , r2  I  2  r r   

(9)

where, D120 is the relative diffusivity of particles in the absence of hydrodynamic interactions, Gr  and H r  are the scalar diffusion functions (hydrodynamic correction functions), for

diffusion along the centerline and tangential direction.

Solution of the flux equation with proper boundary condition results in the following modified stability ratio, rest of the expression remaining same as that of Fuchs: W  r1  r2 





r1  r2

expU / k B T  dr G r , r1 , r2 r 2

(10)

where, Gr , r1 , r2  is given by the following expression

G r , r1 , r2  

A11  A22 4A12  1  1   2

(11)

Here, λ is the particle size ratio and A represents the mobility function (see supporting information, S4.2).

When colliding particles are very close to each other (nearly touching condition, r  r1  r2 ), G assumes an asymptotic expression with the mobility functions, following the

relationship shown below 10 ACS Paragon Plus Environment

A11 

Hydrodynamic correction function(G)

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


2 1 A12  A22 1  

0

10

-1

10

Batchelor, far field solution Batchelor, near field solution Honig et al.

-2

10

2

3

4

5

Dimensionless interparticle separation distance

Hydrodynamic correction function(G)

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10

(12)

0

10

-1

10

-2

10

-3

=0.01 =0.1 =0.5 =1

5

10

Dimensionless interparticle separation distance

(a)

(b)

Fig.1: (a) Comparison of far field and near field hydrodynamic function with mobility function of Honig and co-workers26 (see supporting information, S4.3) for monodisperse system and (b) Variation of hydrodynamic function with size ratio. Asymptotic function (near-field solution of Batchelor’s two particle hydrodynamics29 is found to be more accurate to represent the mobility of particles when they are very close to each other. Hence, in this work, near-field and far-field solution of Batchelor29 have been used for computations at very close and large interparticle separations, respectively. A quantitative comparison (Fig.1a) with the expression proposed by Honig and co-workers26 for hydrodynamic correction for the motion of two equal spheres (λ=1) shows that, this approach complies with their limiting values and limiting slopes, both for small as well as large interparticle separations. A comparison (Fig.1b) of scalar diffusion functions for different size ratios reveals that, hydrodynamic resistance is observably significant, for the case of equal-sized colliding particles, reducing the translational diffusion motion by order >2, close to the nearly touching condition. Hydrodynamic resistance gradually becomes less significant, as the size difference of colliding particles gets prominent.



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Table 1: Comparison of stability ratio with and without hydrodynamic interactions between spheres of arbitrary radii r1 (nm)

1 10 50 100 400 2 10 100 200 10 100 500

r λ 2  r1

r2 (nm)

1 10 50 100 400 1 5 50 100 1 10 50

  

1

0.5

0.1

W

Whydrodynamic

W hydrodynamic W

1.329 11

2.92 63.8

1.536  10 9 1.5  10 20 4.5  1087

1.2  1010 1.216  10 21 3.7  1088

1.389 3.649

2.4 8.4

5.86  1012 3.564  10 27

1.66  1013 1.013  10 28

1.25

1.426

1.1  10 4.785  1017

1.341  10 2 5.84  1017

2

2.2 5.8 7.8 8.1 8.22 1.73 2.3 2.83 2.84 1.14 1.22 1.22

Table 1 shows the comparison of stability coefficients, with or without hydrodynamic interactions, in presence of all three interparticle forces. It is seen that, hydrodynamic interaction reduces the coagulation frequency function and hence the coagulation rate, by a factor of about 1.5 to 8.5, for λ 0.15 to 1. This implies that hydrodynamic interaction effects are far from negligible. When λ ≪0.1, only for those selective collisions, effects of hydrodynamic interaction can be negligible. Therefore, neglecting the hydrodynamic resistance is greatly erroneous.

2.3.4 Ostwald ripening (OR) Literature45 suggests that, amorphous silica shows Ostwald ripening because of its higher solubility, compared to other oxides.2,4 In water and alcoholic solvents, amorphous silica shows a higher rate of Ostwald ripening. Since, Ostwald ripening is essentially a diffusion-limited process,46-47 therefore, the dissolution or growth rate of particles can be derived by secondary phase mass balance, as in the diffusion growth process, resulting in the following equation48-52


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

dr Dm vm N A cSiO2  cr  dt r



(13)

where, c r is the equilibrium surface concentration of a spherical particle of radius r and is given by Gibbs-Thomson equation as follws  2Vmol 1  cr  cSiO2 ,s exp   RT r 

(14)

where, Vmol is the molar volume of species (see supporting information, S5).

3. Time scale Analysis

Time scale analysis was performed to understand which of the above mentioned five events are the slowest and hence rate limiting, during different stages (time periods) of the particle formation process. Since, rate of all the events mentioned above changes with process time, time scales in two different stages, namely initial and final period of synthesis have been discussed here (Tables 2, 3).  r , n , dg and  OR are time scales for reaction, nucleation, diffusion growth and OR, respectively.  cog , wr and  cog , sp are coagulation time scale for whole reactor and single particle, respectively. (see supporting information, S6, Table S3).

Table 2: Conclusions from time scale analysis (experiment set 2 mentioned in section 5) Time scales comparison i. τn  τr

Conclusions (initial stage of synthesis) Nucleation process is controlled by reaction and this shows the importance of modeling finite reaction kinetics.

ii.

 cog , wr   n

Nucleated particles start coagulating as soon as they are formed, since coagulation is faster than nucleation. So, polydisperse particles are expected.

iii.

 OR   cog , sp

OR is faster and so dominates over coagulation-growth.

iv.

 dg   OR

Diffusion growth dominates over OR for a short period of time and then OR leads the growth process as the supersaturation level



comes down. Conclusions (final stage of synthesis) i.

τ n  τ r

Nucleation process is controlled by reaction and this shows the importance of modeling finite reaction kinetics.

ii.

 OR   cog ,sp

OR is much faster than coagulation-growth. So it dominates the growth dynamics and small number of large particles are expected.

Table 3: Conclusions from the time scale analysis (experiment set 1 mentioned in section 5) Time scales comparison i. τ n  τ r

Conclusions (initial stage of synthesis) Nucleation process is controlled by reaction and this shows the importance of modeling finite reaction kinetics.

ii.

 n   cog ,wr

Coagulation is slower than nucleation. So, monodisperse particles are expected.

iii.

 cog ,sp   OR

Coagulation rate is faster than OR. So it dominates the growth process.

iv.

 dg   OR

Diffusion growth dominates over OR for a short period of time and then OR leads the growth process as the supersaturation level comes down. Conclusions (final stage of synthesis)

i.

τ n  τ r

Nucleation process is controlled by reaction and this shows the importance of modeling finite reaction kinetics.

ii.

 OR   cog , sp

OR is slightly faster than coagulation-growth. So it dominates the growth process only towards the end.

4. Method of computation

Governing equations 1, 3 and 4 were solved numerically using the conventional finite volume based commercial solver Fluent 14.5. The whole computation domain was discretized into finite volumes, using tetrahedral elements. The parameter values used in the simulation have been listed in S7 (Table S2) of supporting information. It was found that the solution is 14 ACS Paragon Plus Environment

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independent of control volume, with further reduction in grid size. Time-step size was varied between 0.1- 0.000001 sec, which on further reduction did not bring any further improvement in solutions, ensuring step-size independence of final results. The solution converged when scaled absolute residuals were less than 0.00001 for all variables. Mass balance check in the system confirmed that the system’s total mass remains conserved with a mass balance error well below 1.12 % in all cases, except for the case of simulation with experimental data set 2 (mentioned in section 5 below) in the figure 3b and 4a. In case of simulation with data set 2b, mass balance error was found to be 2.7 % at the end of simulation. (see supporting information, S8).

5. Experimental section Silica nanoparticles were synthesized following the very well-known Stober’s sol-gel method.53-57 SEM images were then taken to obtain particle size distribution (see supporting information, S9). Two sets of experiments (with different pH and water concentrations) were performed to study the effect of hydrodynamic interaction, reaction kinetics and OR. Within each set, two experiments (with different precursor and catalyst concentrations) were performed. Set 1 (low pH and low water content): (a) TEOS= 0.080 (M); NH 4OH= 0.0273 (M); H2O= 1.622 (M) and (b) TEOS=0.0834; NH4OH=0.102; H2O= 3.14 (M). Here, rate of reaction and OR are slow and coagulation drives the growth mechanism. Set 2 (high pH and high water content): (a) TEOS= 0.192 (M); NH 4OH= 0.857 (M); H2O= 15.4 (M) and (b) TEOS= 0.158 (M); NH4OH= 3.8 (M); H2O= 22.55 (M). Here, both reaction and OR rates are fast and their coupled effect drives the growth mechanism.

6. Results and discussion

Each of the experimental runs were triplicated to check the reproducibility of the experiment. Diameters of approximately 500 particles were measured from the corresponding



SEM images for each run, comprising a total of around 1500 particles being measured for each synthesis condition, ensuring statistically reliable mean and standard deviation of particle diameter.

6.1 Effect of hydrodynamic interaction, reaction kinetics, OR 6.1.1 Effect of hydrodynamic interaction

It has been already shown in Table 1 that, hydrodynamic interaction between colliding particles plays a very significant role by diminishing the coagulation rate substantially, about 1.5 to 8.5 times, depending on the ratio of diameter of colliding particles. 60

60

Particle diameter (nm)


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

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40

20 Experimental With hydrodynamic correction Without hydrodynamic correction 0

1

2

3 Time (hr)

4

40

20

5

(a)

Experimental data With hydrodynamic correction Without hydrodynamic correction 0.5

1.0 Time (hr)

1.5

2.0

(b)

Fig. 2: Comparison of simulation results with experiments. Experimental conditions: (a) set 1a and (b) set 1b

As discussed earlier, silica has a strong effect by OR, so one set of experiment (set 1, fig. 2a and fig. 2b) was devised such that effect of OR is minimized (  OR   cog ). This is ensured by the time scale analysis (Table 3), which shows that particle growth dynamics in these experimental conditions is dominated by the coagulation step ( cog   OR ) throughout the larger section of synthesis period, before it slows down and OR dominates over coagualtion (  cog   OR ). Fig. 2a and fig. 2b show that simulation including hydrodynamic interaction nicely 16 ACS Paragon Plus Environment

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predicts the experimental data. However, when hydrodynamic interaction is ignored, the model fails to predict experimental data accurately and the absolute difference between two predictions increases with time. Exclusion of hydrodynamic correction generates an overprediction with approximately 27 % relative prediction error with respect to experimentally calculated particle size at the beginning, and approximately 19% at the end of experiment. This reduction in prediction error is mainly because of gradual decrease of coagulation rate and very slow rise of dissolution growth rate. Hence, the effect of reduced coagulation rate is preserved in the overall particle size prediction. This indicates that effect of hydrodynamic interaction on particle formation process is not negligible.

Experimental With hydrodynamic correction Without hydrodynamic correction

400

100

Experimental With hydrodynamci interaction Without hydrdynamic interaction 0

2

4 6 Time (min)

8

300 Particle diameter (nm)

200


300


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


200

100

0

10

(a)

2

4

40 30 20 10 0.05

6 8 Time (min)

0.06

Time (min)

10

0.07

12

(b)

Fig. 3: Comparison of simulation results with experiments. Experimental conditions: (a) set 2a and (b) set 2b

On the other hand, another set of experiments (set 2, fig. 3) was conducted such that OR

is the dominant mechanism in the particle growth process  OR   cog , which has been proved by time scale analysis (Table 2). Fig. 3a and fig. 3b show the comparison of corresponding experimental and simulation results. Here, it is observed that, both types of predictions from the model - either including or excluding the hydrodynamic correction - match very well with the mean particle size from experimental data. Though, there still exists a large absolute deviation of 17 ACS Paragon Plus Environment


around 11-13 nm between two predictions towards the end of the experiment, the deviation is small compared to the error bar. The prediction excluding the hydrodynamic interaction also shows an insignificant relative error of about 3.5%, relative to the experimental final particle size.

However, closer inspection of the simulation result (inset in Fig. 3b) at a short time scale, [immediately after nucleation, when coagulation dominates over OR (Table 2)] reveals that, hydrodynamic correction brings significant changes in particle size prediction and here, the size prediction error can be as high as 25%. This is also confirmed by the time scale analysis that, coagulation lasts for a very short period of time and then dissolution growth τ OR  τ cog  takes full control of the growth process. Hence, we observe a reduction in the prediction error as soon as OR starts, because of the smoothening effect of OR. Since majority3-4,11-12 of nanoparticles show OR effect at a much larger time scale, dynamics of these nanoparticles are completely governed by the reaction, nucleation, diffusion growth and coagulation mechanisms during their synthesis. This highlights the importance of including hydrodynamic correction in explaining particle formation.

6.1.2 Effect of reaction kinetics

To explicitly make out the effect of reaction kinetics on the particle formation process, simulation was carried out with the same experimental conditions (set 2b, fig. 4a and set 1a, fig. 4b), but assuming the reaction to be instantaneous, without altering any of the other mechanisms used in the previous case. In case of fig. 4a, it is seen that prediction of instantaneous reaction model differs widely from the prediction of the complete model, which captured the exact particle size. Underprediction of particle size by the instantaneous reaction model can be effectively explained by time scale analysis. In case of instantaneous reaction, at a very short time period τ r  0 itself, the complete amount of growth units are available. Therefore, extremely short nucleation time scale τ n  4.64  10 20 s  generates extremely large number of nuclei almost instantaneously, leaving very small amount of material for diffusion growth to take 18 ACS Paragon Plus Environment

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



place. These nuclei coagulate quickly τ cog  8.1  10 21 s . Once coagulation slows down, no other





growth mechanism operates. As τ n  τ cog and τ n  τ dg τ dg  2  10 10 , particles produced have very low polydispersity and hence negligible OR effect is observed after coagulation stops, and overall particle size remains small (fig. 4a). On the other hand, when reaction is moderately fast (fig. 4a) τ n  τ r  , relatively less number of nuclei are formed (at a low supersaturation level); these nuclei initially growing mainly by coagulation and to some extent by diffusion growth (Table 2). Immediately after coagulation stops, OR takes over as the dominant growth mechanism (Table 2). Since reaction produces a significant amount of growth material at every instant, this helps in building up of the bulk concentration and consequently, OR is facilitated even more. These two processes combined together gives rise to a high growth rate and a very

80

Particle diamter (nm)

Experimental data With moderately fast reaction With instantaneous reaction

400


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


300

200

100

0

0

2

4 6 8 Time (min)

10

12

60

40

20

Experimental With slow reaction kinetics With instantaneous reaction 0

(a)

1

2 3 Time (hr)

4

5

(b)

Fig. 4: Profile of particle evolution dynamics. Experimental conditions: (a) set 2b and (b) set 1a large particle size (fig. 4a). This implies that reaction rate affects both growth processes coagulation and OR - quite remarkably.

In contrast, for fig. 4b, it is observed that, instantaneous reaction model overpredicts the particle size compared to the full scale model. Here, a very slow reaction rate produces a low supersaturation level and hence, a vey less number of nuclei (Table 3) is formed. Initially, these 19 ACS Paragon Plus Environment


nuclei grow by coagulation, which slows down very significantly. Afterwards, particles grow by both coagulation and OR (Table 3). Since both these processes are very slow, the final particle size is small. Hence, one can conclude that, when both the reaction and OR processes are very slow, particle formation is mainly driven by coagulation, and prediction of instantaneous reaction model will give a higher particle size, since instantaneous reaction starts with a large number of nuclei and consequent very fast coagulation rate, as explained in case of fig. 4a.

Hence, we conclude that, reaction kinetics plays an important role in the particle formation process, by acting as a medium of controlled release of growth materials.

6.1.3 Effect of OR 60



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200

100 Experimental With OR Without OR 0

2

4

6 Time (min)

8

10

40

20

Experimental With OR Without OR

0.5

12

(a)

1.0 Time(hr)

1.5

2.0

(b)

Fig. 5: Profile of particle evolution dynamics. Experimental conditions: (a) set 2a and (b) set 1b

To study the effect of OR, simulation results with both curvature dependent and curvature independent solubility have been compared with experimental results as shown in fig. 5a (set 2a) and fig. 5b (set 1b). In case of fig. 5a, it is observed that, a model without curvature dependent solubility (i. e. without OR) shows a strong deviation from the prediction of a model with curvature dependent solubility (i. e. with OR). In case of former, all particles in the system grow by the simple diffusion growth process, which slows down the overall growth rate of particles 20 ACS Paragon Plus Environment

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and consequently particle size is smaller. Whereas, in presence of a strong curvature dependent solubility (higher bulk solubility) as in fig. 5a, due to higher dissolution growth rate (OR), particles grow at a much faster rate (smaller particles dissolve and larger particles grow, as whole small number of particles grow at a time) and therefore larger particles are formed. This is observed by the large difference in prediction between two cases in fig. 5a. In case of fig. 5b, since the bulk solubility is extremely low, effect of curvature dependent solubility (OR) is also insignificant and hence no remarkable difference is observed between the two predictions.

6.1.4 Coupled effect of OR and reaction kinetics

In the previous two cases, it is found that, exclusion of either finite reaction kinetics or OR, cannot explain the experimental data. This implies that a coupled effect is essential to predict the formation dynamics of particles of sub-micron size. In the conventional case of curvature dependent solubility, when there is no extraneous addition of growth material (reaction is either complete or extremely slow), particles smaller than critical particle radius dissolves and thus a left-skewed particle size distribution profile is obtained. Most interesting fact about this particular case under study is that there is a finite reaction rate and when this is coupled with the Ostwald ripening phenomena, because of the extra addition of material from reaction (fig. 6), larger particles grow more than the rate predicted solely by Ostwald ripening. This is observed by the right-skewed particle size distribution profile (fig. 7). In this case, because of the buildup of bulk concentration due to reaction kinetics, critical radius is lowered (fig. 6 and see supporting information, S10) and hence, on an average, large number of particles grow and small number of particles dissolve, compared to what would have been predicted by OR alone in absence of reaction kinetics. This effect seems to be quite consistent with the experimental observation by Talapin and co-workers58 for the synthesis of CdSe, where focusing of size distribution occurred and this lasted for a long duration, producing a narrow particle size distribution. This might be due to the presence of a large excess of growth material, as a result of a fast reaction rate and consequent drastic decrease of critical particle radius for OR, strongly suppressing the OR (i. e. defocusing). As long as higher bulk concentration is maintained, defocusing is suppressed and therefore, narrow size distribution is maintained for a long duration.



Fig. 6: Coupled effect of reaction and Ostwald ripening

Particle number density (number/m3)

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

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1.5x10

18

1.0x10

18

5.0x10

17

0.0

t = 6.85 s t = 18.86 s

0

100

200 300 Particle dia (nm)

400

Fig. 7: Particle size distribution at different times. Experimental conditions: TEOS= 0.158 (M); NH4OH=3.8 (M); H2O = 22.55 (M). Depending on the operating condition, five different cases may arise, which are described as follows:

Case 1: rate of change of OR rate is higher than reaction rate Case 2: rate of change of reaction rate is higher than OR rate Case 3: OR rate is constant, but reaction rate increasing Case 4: negligible OR rate, but a fast reaction rate Case 5: negligible reaction rate, but a fast OR rate


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6.2 Parametric variation 6.2.1 Effect of ammonia concentration From the experimental data, it is observed that, as ammonia concentration increases, particle size increases too (Fig. 8). Increase in ammonia concentration causes enhancement of both the reaction rate and the pH of the medium. As discussed in the previous section (see reaction kinetics effect, section 6.1.2), very fast reactions are expected to produce smaller particles. However, the sharp dependence of solubility on the pH 45 and consequent increase in OR rate dominates the increase in reaction rate. This leads to a faster particle-dissolution driven growth rate, which overshadows the adverse effect of enhanced reaction rate. Higher solubility causes large number of particles to dissolve by increasing the critical particle radius for OR, thereby increasing the bulk concentration even more (fig. 6). So, small number of particles grow with large amounts of growth material available (fig. 6). This makes the average particle to grow much faster and hence, particle size is found to increase with ammonia concentration (case 1).

400 Experimental Simulation Particle diameter (nm)

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


300

200

100

0

1 2 3 Ammonia concentration (M)

4

Fig. 8: Effect of ammonia concentration. Experimental conditions: TEOS= 0.158 (M); H2O= 22.55 (M).

6.2.2 Effect of temperature



Here, it is observed that, particle size decreases with increase in temperature (Fig. 9). Similar trend was also observed by Giesche. 34 This is against the usual trend of increasing particle size with temperature because of increasing coagualtion rate with temperature. Anomalous trend in the present case can be explained through the coupling effect of OR and reaction kinetics. Almost all processes - reaction, nucleation, coagulation and OR are enhanced at higher temperature. In the initial phase of synthesis, significantly large number of nuclei is formed due to the combined effect of increased reaction and nucleation rate at higher temperature. These nuclei grow faster due to the higher coagulation rate (due to large number of nuclei) and higher coagulation efficiency. Gradually, coagulation-induced growth rate decreases because of reduced efficiency of coagulation as well as number density of particles. In the next phase of synthesis, particle growth is controlled by the reaction and dissolution growth. Increased temperature also causes increase in solubility, and hence the dissolution-driven growth rate.


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300

200 Experimental Simulation 100 290

300 Temperature (K)

310

Fig. 9: Effect of temperature. Experimental conditions: TEOS= 0.158 (M); NH4OH= 3.8 (M); H2O= 22.55 (M). However, solubility of silica is not as sensitive45,59 as reaction rate, in the temperature range of concern in the present work. Therefore, effect of fast reaction kinetics is more pronounced as compared to that of OR (see coupled effect, section 6.1.4), and the consequent presence of large number of growth units in the medium (fig. 6) brings down the critical particle 24 ACS Paragon Plus Environment

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radius for OR. This results in less number of particles to dissolve and larger number of particles to grow at the same time, as opposed to the dissolution- growth rate dominated process, as in the previous case. This causes the growth material to be shared by a large number of particles obtained inherently by faster reaction and nucleation rate. So the average particle size increases at a slower pace. Hence, particle size decreases with increase in temperature (case 2).

6.2.3 Effect of water addition


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


300

200

100

Experimental Simulation 5

10 Water concnetration (M)

15

Fig. 10: Effect of water addition. Experimental conditions: TEOS= 0.158 (M); NH4OH= 3.8 (M). Similar to the previous two cases, particle size variation in this case can be explained thorough competitive rates between OR and reaction kinetics. Solubility of silica is highly dependent on the percentage of water in the medium.60 While solubility of silica in pure water is of the order of 100 mg/l at 25 oC, solubility in 90% ethanol reduces to only about 0.1 mg/l. Thus, higher percentage of water leads to increase in bulk solubility of silica (see OR effect, section 6.1.3) and this results in a faster curvature dependent growth rate (OR), compared to the effect of increased reaction rate, as in the case of ammonia addition. Hence, water addition results in larger particles (case 1).

6.2.4 Effect of TEOS concentration



Effect of TEOS is observed only through variation of reaction rates. Here, rate of dissolution growth is constant. Higher concentration of TEOS increases the production rate of growth units, which leads to a faster increase in bulk concentration of silica. Hence, large number of nuclei are formed in the initial phase of synthesis. Here too, faster coagulation produces relatively larger size particles, but the growth rate is slowly taken over by the combined effect of OR and reaction rate, as obtained from time scale analysis. As in the case of temperature variation, higher bulk concentration of silica in this case lowers the critical particle radius for OR (fig.6) and large numbers of particles grow at the same time, with growth materials provided mostly by reaction kinetics. This results in smaller particles (case 3).

400


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

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300

200 Experimemental Simulaiton 100

0.2

0.3 0.4 TEOS concentration (M)

0.5

Fig.11: Effect of TEOS concentration. Experimental conditions: NH4OH= 3.8 (M); H2O= 22.55 (M).

7. Conclusions

To elucidate the liquid phase particle formation process, a generalized model has been proposed here, including simultaneous reaction, nucleation, diffusion growth, coagulation, and Ostwald ripening steps. More importantly, this work incorporates the effect of the previously ignored hydrodynamic interaction in the coagulation process of polydisperse particles, by way of modification of the diffusion coefficient. This helped us explain the importance of the hydrodynamic correction factor on the coagulation process, by calculating the colloid stability 26 ACS Paragon Plus Environment

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coefficient. Thus for the first time, it brings out the role of hydrodynamics on the overall particle formation process, by modulating coagulation. In fact, coagulation is found to be diminished by a factor of about 1.5 to 8.5, for particle size ratios from 0.15 to 1, respectively. This reduced coagulation rate is observed to have a substantial effect on the whole particle evolution dynamics, specifically in case of a coagulation-dominated particle formation regime. Thereby, it is found that, particle size is over-predicted in absence of hydrodynamic interaction with approximately 25% relative prediction error at the beginning and 15% relative prediction error towards the end of the synthesis, relative to the experimental particle size. Even in case of systems, where coagulation is not the dominant mechanism throughout the synthesis period, absolute prediction error is found to be as high as 12-13 nm, towards the end of the synthesis and relative prediction error can be as high as 25% with respect to experimental particle size during the coagualtion dominant period. This signifies that hydrodynamic interaction is far from negligible.

To the best of our knowledge, this is the first time that the present work shows both reaction and OR to have very significant effects on particle formation. In case of moderately fast reactions, effect of reaction kinetics is more pronounced, by effectively controlling the nucleation step via limiting the saturation level, while continuously pumping-in growth material for the growth process. In contrast, for very slow reactions, where availability of growth material is limited by the slow reaction kinetics, this is not possible. Earlier models considering only instantaneous reaction cannot capture this variation and consequently fails to explain the experimental data. This is observed by under-prediction of experimental particle size in case of moderately fast reactions and over-prediction of size in case of very slow reactions, as obtained by the conventional, instantaneous reaction model. Finally, in case of very high curvaturedependent solubility, OR driven growth dominates the overall growth process. Reaction kinetics and OR, coupled together, provides a rapid growth rate, which earlier models (those including instantaneous reaction, coagulation, and diffusion growth or OR) could not predict. Earlier research results on particle size prediction mainly focused on nanometer size particles. The present comprehensive model however could very effectively predict the whole range of particle sizes, ranging from nanometer to submicron size particles and their temporal evolution throughout the synthesis period. This model has also been able to accurately capture the 27 ACS Paragon Plus Environment


anomalous trends observed in the parametric variation study. All these have not been reported so far, from calculations of any single model, because of limitations of earlier models.

Although, the present work demonstrates and validates the synthesis of SiO2 nanoparticle via the model developed, the model itself is well-posed to be useful for a host of other material systems, including both organic, as well as inorganic. In addition to these systems, the present model is very suitable in the low temperature inorganic synthesis and almost all of the organic synthesis reactions, wherein the chemical reaction kinetics is comparatively slow, rendering the much more prevalent and current instantaneous reaction kinetics models to fail.

Associated content Supporting information Detailed description of reaction kinetics, nucleation, diffusion-driven growth, coagulation, Ostwald ripening, time scale analysis, parameter values used in simulation, method of computation, experimental method and critical radius variation as a function of solubility are given in the supporting information. This material is available free of charge via the Internet at http://pubs.acs.org.

Acknowledgement The authors acknowledge Cryo-SEM facility at the Department of Chemical Engineering, IIT Bombay for SEM imaging of a large number of samples.


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