time evolution of size and polydispersity of an ensemble of nanoparticles growing in the confined...

Colloids and Surfaces A: Physicochem. Eng. Aspects 259 (2005) 7–13

Time evolution of size and polydispersity of an ensembleof nanoparticles growing in the confined space of AOT

reversed micelles by computer simulations

Francesco Ferrante∗, Vincenzo Turco LiveriDipartimento di Chimica Fisica “F. Accascina”, Universit`a degli Studi di Palermo,

Viale delle Scienze, 90128 Palermo, Italy

Received 30 July 2004; accepted 10 February 2005Available online 2 March 2005

Abstract

The time dependence of size and polydispersity of an ensemble of nanoparticles growing in the confined space of water-containing AOTreversed micelles has been investigated by computer simulations. It has been found that, in a wide time range, the mean nanoparticle size canb s while then ternal ande ocedure area©

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1

irsmcrafdsomns

ex-ater-e, i.e.lo-

d that

tol ex-thethe

n the

andrsityetic

ouslynsturalout

0d

e described by power laws whose exponent is critically dependent on the efficiency of the intermicellar material exchange procesanoparticle polydispersity increases with time. From the analysis of all the disentangled effects arising from the variation of inxternal parameters provided by simulations, useful suggestions for a better and rationale control of the nanoparticle synthetic prchieved.2005 Elsevier B.V. All rights reserved.

eywords:Reversed micelles; Nanoparticle growth; Computer simulations

. Introduction

Solutions of water-containing reversed micelles dispersedn an apolar solvent are currently employed as suitableeaction media for the synthesis and time stabilization ofize-controlled nanoparticles. This is generally achieved byixing two micellar systems carrying inside the aqueous

ore of reversed micelles the appropriate hydrophiliceactants. Then, as a consequence of micellar diffusionnd coalescence, reactants can come in contact and react

orming the precursors of nanoparticles. Driven by the sameynamic processes, these precursors are accumulated inome ‘lucky’ micelles leading to the formation and growthf stable nuclei whereas the closed structure of reversedicelles and their dispersion in the apolar medium preventanoparticle unlimited growth and precipitation providingize and, sometimes, shape control.

∗ Corresponding author. Tel.: +39 0916459844; fax: +39 091590015.E-mail address:[email protected] (F. Ferrante).

Empirically, it has been found that the most relevantternal parameters allowing size modulation are: (i) the wto-surfactant molar ratio (R), which controls the size of thaqueous micellar core; (ii) the surfactant concentrationthe number density of reversed micelles; (iii) the initialcal concentration of reactants. It has been also arguea pivotal role is played by the internal parameterβ whichdescribes the fraction of intermicellar collisions leadingeffective micellar coalescence and intermicellar materiachange processes[1,2]. This parameter is a measure ofpermeability of the oriented surfactant layer surroundingaqueous nanodroplets and its value mainly depends osurfactant nature.

Aiming to understand the influence of each externalinternal parameter on the nanoparticle size and polydispeand to have a rational control of the nanoparticle synthprocedure, some computer simulations have been previreported in literature[3–8]. However, all these calculatiohave been directed to predict the nanoparticle strucproperties at the end of the precipitation reaction with

927-7757/$ – see front matter © 2005 Elsevier B.V. All rights reserved.oi:10.1016/j.colsurfa.2005.02.005

8 F. Ferrante, V. Turco Liveri / Colloids and Surfaces A: Physicochem. Eng. Aspects 259 (2005) 7–13

a detailed investigation of the kinetics of the nanoparticlegrowing process. It must be stressed that the knowledge ofthe time dependence of the nanoparticle structural propertiesis of particular interest not only to rationalize experimentalresults but also in view of the possibility to stop quiteinstantaneously the nanoparticle growth by the addition ofspecific coating agents[9–11].

In the present work, in order to shed more light on thetime evolution of nanoparticle size and polydispersity andon the influence of the various external parameters, we haveused a suitable model and carried out computer simulationson nanoparticle synthesis in solutions of water-containingreversed micelles. Calculations have been performedtaking as paradigm the synthesis of CdS nanoparticles insolutions of AOT reversed micelles dispersed in isooc-tane.

2. Simulation procedure

The algorithm for the simulations reported in this workwas originally conceived by Li and Park[3] and developedin a more realistic form by Bandyopadhyaya et al.[4]later, some changes were introduced by Singh et al.[5]allowing less time demanding simulations without loss ofa

ndt on-fi in anam op-u ndN eredq er ofm non-n ctmt e be-t ord-i rentc

•• e A,

• typeis

•

m p-p :1 y be-t r

andNj is calculated using the Brownian-like expressions:

Fij = β8

3

kBT

ηNmic

NiNj

N;

Fii = β8

3

kBT

ηNmic

Ni(Ni − 1)

2N(1)

whereβ is the coalescence efficiency of intermicellar colli-sions, i.e., the fraction of micellar encounters leading to in-termicellar fusion and material exchange,kB the Boltzman’sconstant,T the temperature in Kelvin,Nmic the total numberdensity of reversed micelles andη the viscosity of the contin-uous organic medium in which the micelles are dispersed. Itis worth to note that the parameterβ is constant during eachsimulation and it is mainly controlled by the permeability ofthe micellar surfactant layer. The total coalescence frequencyis given by:

Fcoal =∑i–j

Fij, i–j = 1–2; 1–3; 2–2; 2–3; 3–3; 3–4 (2)

and the probability of a coalescence between micelles of classi andj to occur is:

Pij = Fij

Fcoal(3)

T wingw osena

k

w ndkk .( (ii) am hosenr s oft 2 ares ged.T ; (iv)t ac-c

• y to

• o

• par-to

( utedi tri-b mi-c

inga tain

ccuracy.The algorithm allows to simulate the precipitation a

he growth of nanoparticles from reagents A and B cned in the aqueous pool of reversed micelles dispersedpolar medium. Reagents are distributed amongN reversedicelles according to a Poisson law; the initial micellar plation consisting inN/2 micelles containing reactant A a/2 containing B. To every micelle is assigned an orduadruple of integers, whose components are the numbolecules A, the number of molecules B, the number ofucleated product molecules Cm, and the number of produolecules forming the nanoparticle Cn, respectively. Two

ypes of event can occur in the simulation: coalescencween two micelles and nucleation inside a micelle. Accng to their population, micelles are assigned to four diffelasses:

class 1: micelles containing only a nanoparticle;class 2: micelles containing only one molecule, of typB or Cm;class 3: micelles where the number of molecules ofA or B and/or Cm (plus 1 if a nanoparticle is present)greater than 1;class 4: empty micelles.

A coalescence event involving a micelle from classi and aicelle from classj is considered fruitful, and allowed to haen in the simulation, only ifi–j is one of the following pairs–2; 1–3; 2–2; 2–3; 3–3; 3–4. The coalescence frequenc

ween micelles belonging to class with population numbeNi

he algorithm simulates a coalescence event in the folloay: (i) the classes involved in the coalescence are chccording to:

i ≤ rnd1 ≤ ki+1 (4)

here rnd1 is a random number in the interval (0, 1] a0 = 0,k1 =P1–2, k2 =k1 +P1–3, k3 =k2 +P2–2, k4 =k3 +P2–3,5 =k4 +P3–3, k6 =k5 +P3–4= 1. The indexi satisfying Eq4) defines the couple of classes participating the event;icelle from each of the classes defined in pass 1 are c

andomly; (iii) the first three corresponding componenthe quadruple assigned to the micelles chosen in passummed, while the fourth components remain unchanhis simulates the formation of a coalescence product

he quadruple describing the micellar dimer is modifiedording to one or all of the following rules:

A and B, if present, react immediately and completelform Cm, leaving an excess of reactant, if any;if a nanoparticle is present, Cm will add instantaneously tit;if each of the two coalesced micelles contains a nanoticle, Cm will distribute between them proportionallytheir size.

v) at last, the quadruple resulting from pass 4 is redistribn two ‘daughter’ quadruples according to a binomial disution to simulate the separation of the two coalescedelles.

A nucleation event can occur only in a micelle containnumber of Cm molecule equal or greater than a cer

F. Ferrante, V. Turco Liveri / Colloids and Surfaces A: Physicochem. Eng. Aspects 259 (2005) 7–13 9

nucleation critical numberkc. The total nucleation frequencyis the sum of nucleation rates,K(n), over all micelles:

Fnuc =N∑

i=1

K[n(i)]

=N∑

i=1

ξn(i),kcn(i)γ exp

[− 16πσ3V 2

m

3(kBT )3(ln λn(i))2

](5)

wheren(i) is the number of Cm molecules in the micellecore i; γ the preexponential factor;σ the interfacial tensionbetween nucleus and micellar core liquid;Vm the volume ofa molecule C;λn(i) the supersaturation in the liquid phase:

λn(i) = n(i)

NAvVmicS(6)

NAv being the Avogadro’s number,Vcore the volume of theacqueous core andSthe solubility of C in water; further

ξn(i),kc ={

0 n(i) < kc

1 n(i) ≥ kc(7)

A nucleation event is simulated by randomly choosing a mi-celle among all that containing a number of molecules Cmequal or greater thankc and calculating their nucleation rates;then, an unitary segment is divided in many little segmentsw anda al.Nr

K

T

F

a otherr ndii stab-l

τ

w het valso ta .

: them issond puten ken aar nen-tts

we have used the value of 100 dyn cm−1, typical for solids.As reference point, the product of solubility (KS) of CdS inwater is 3.6× 10−29 mol2 l−2. The value of the viscosityηis that typical for organic non polar media, 0.01 g cm−1 s−1.The value of the critical nucleation numberkc has been setequal to 2 to account for the very low solubility of CdS inwater[4] and for the occurrence of a marked cage effect inreversed micelles.

All of the results here reported are obtained as the averageof five simulation runs onN= 20,000 micelles per system.The parameters varying in the simulations are the concentra-tion of surfactant and of the deficient reactant A, the excessof B as the ratioX= [B]/[A], the coalescence efficiencyβ andthe effective solubility of CdS in the micellar acqueous core.All simulation runs were considered completed when >99%of the reactant in defect was converted to nanoparticle.

Even if the ‘nanosystems’ of the simulations are simplifiedmodels of real micelles and nanoparticles, here for simplicity,they will be called micelles and nanoparticles. It must bealso emphasized that the timing of the nanoparticle growingprocess results from the evaluation of the total frequency ofthe class of events taken into account by the simulation sothat the accuracy of the time dependence of nanoparticle sizeand polydispersity must be considered strictly entangled withthe validity of the model.

3d

-t g-a oft es,t them s andt od).D nt.I ev r-t

F ,(

hose lengths are proportional to the nucleation ratesrandom number, rnd2, is generated in the [0, 1] intervucleation occurs in the micelles of indexi satisfying the

elation:

[n(i)] ≤ rnd2 < K[n(i + 1)] (8)

he total frequency of the events is:

tot = Fcoal + Fnuc (9)

nd the next event to occur is chosen by generating anandom number, rnd3, in the (0, 1] interval: the event at has that whose probability value is nearer to rnd3. The timenterval,τ, between an event and the successive one is eished according to the interval of quiescience method[12]:

= − ln(1 − rnd4)

Ftot(10)

here rnd4 is a random number in the [0, 1) interval. Total time of the overall process is the sum of all interf quiescence. According to[3], the completion time is noffected by the lack of unfruitful events in the simulation

The present simulations use the following parametersicellar aggregation numbers, needed to calculate Poistribution, and the aqueous pool volume, needed to comucleation rates. The values of these parameters are tafunction ofR= [water]/[surfactant] from Ref.[13] and are

elative to the water/AOT/isooctane system. The preexpoial factorγ for nucleation rates is taken equal to 278.42 s−1;he volume of a product molecule Cm is that of cadmiumulfide, 5.24× 10−23 cm3, and for the interfacial tensionσ

s

. Time evolution of particle size and particle sizeistribution

Typical trends of the mean nanoparticle size (d) as a funcion of time (t) at variousβ values are shown as double lorithmic plots inFig. 1. It can be noted the occurrence

hree distinct physical regimes. At sufficiently small timhe formation of Cm, i.e. the nanoparticle precursors inonomeric state, their accumulation in reversed micelle

he formation of embryos predominate (induction periuring this periodd is low and practically time independe

t can be noted that the induction time is triggered by thβ

alue; a largerβ involves a smaller induction time. In paicular, at sufficiently highβ values (β > 0.1) the induction

ig. 1. Time dependence of the nanoparticle diameter (d) at variousβ values�, β = 1;©, β = 10−1;�, β = 10−2;�, β = 10−3;�, β = 10−4;♦, β = 10−5).


Table 1Mean particle diameter (d) at completion time, power law exponent (b), maximum Cm time (tC) and completion time (tS) at variousR and [AOT] constant(series 1), at variousRand [mic] constant (series 2) and at various [AOT] andRconstant (series 3), in all simulationsβ = 10−3 andX= 1

Series 1 R [mic] × 103 (mol l−1) d (A) b (s−s) tC × 104 (s) tS × 104 (s) tC/tS

5 4.2 14.30 0.226 1.7 10.9 0.1568 2.6 14.32 0.194 1.9 12.8 0.148

13 1.7 15.26 0.148 2.3 16.1 0.14315 1.5 15.50 0.135 2.3 17.7 0.130

Series 2 R [AOT] mol l−1

5 0.126 14.74 0.166 2.0 13.9 0.1448 0.202 14.82 0.167 2.2 14.1 0.156

13 0.315 14.90 0.179 2.0 14.1 0.14215 0.357 14.94 0.167 2.2 14.3 0.154

Series 3 [AOT] mol l−1 [mic] × 103 (mol l−1)

0.126 1.05 16.30 0.099 2.3 23.0 0.1000.176 1.47 15.44 0.131 2.2 17.6 0.1250.315 2.63 14.64 0.194 2.0 12.9 0.1550.357 2.98 14.54 0.198 2.0 12.2 0.164

Reference (R= 10, [AOT] = 0.252 (mol l−1),[mic] = 2.10× 10−3(mol l−1))

14.88 0.166 2.1 14.1 0.149

region practically disappears. This is because a large inter-micellar material exchange frequency determines a high rateof formation of nanoparticle precursors and nuclei, leadingto an immediate start up of the nanoparticle growing process.

In order to investigate the influence of the size and con-centration of reversed micelles upon the induction time, threeseries of simulations have been carried out:

• series 1:R is a variable in the range 5–15 at constant sur-factant concentration (the micellar concentration changesaccordingly);

• series 2:Ris a variable in the range 5–15 while the micellarconcentration is costant. The concentration of the surfac-tant is settled to give the desired micellar concentration;

• series 3:R is mantained constant at the value of 10while the surfactant concentration is variable in the range0.126–0.357 mol l−1 (the concentration of the micelleschanges accordingly).

The results of these simulations are collected inTable 1. Inall simulationsβ = 10−3 andX= 1. The simulation atR= 10,[AOT] = 0.25 mol l−1, [A] = 0.05 mol l−1, is taken as a refer-ence. The time,tC, at which the maximun of the monomericproduct Cm is reached can be considered as a misure of the

induction time, being the accumulation of Cm a trigger to thenucleation. One can observe that, in the investigated range,the value oftC is substantially independent on micellar con-centration andR value. This finding enphasizes that theseparameters scarcely influence the nucleation process.

Taking into account that equilibrium constants in confinedspace can be significantly changed with respect to that in bulkmedium, the effect ofKS smaller and larger than the valuerelative to CdS in water (3.6× 10−29 mol2 l−2) on the in-duction time was also studied. The simulation using param-etersR= 10, [AOT] = 0.25 mol l−1, Ca = 0.05 mol l−1, X= 1and β = 10−3 was taken as a reference. The results of thesimulations, reported inTable 2, show that the timetC ofmaximum accumulation of Cm is longer at greaterKS, whilethe relative time,tC/tS reaches a maximum and then dimin-ishes at the higher values ofKS. This result is quite obviousbecause the nucleation process is affected by the effectivesolubility of the precursor.

At intermediate times, the formation of nuclei through theaggregation of Cm within the core of reversed micelles andtheir growth occur; corresponding to the quite linear trendof the curves shown inFig. 1. The linear trends of ln(d) ver-sus ln(t) found by the simulations are in full agreement with

Table 2Mean particle diameter (d) at completion time, power law exponent (b), number of nanoparticle (NNP), maximumCm time (tC) and completion time (tS) atv

K NNP

3 79873 76563 71473 65313 48293 16973 350

ariousKS values

S (mol2 l−2) d (A) b (s−1)

.6× 10−38 14.34 0.151 (1)

.6× 10−32 14.54 0.158 (1)

.6× 10−29 14.88 0.166 (1)

.6× 10−23 15.32 0.182 (1)

.6× 10−17 16.96 0.215 (1)

.6× 10−11 24.02 0.291 (2)

.6× 10−8 40.66 0.311 (2)

tC × 104 (s) tS × 104 (s) tC/tS

1.9 13.5 0.1412.0 13.8 0.1452.1 14.1 0.1492.4 14.9 0.1613.1 18.3 0.1696.4 44.8 0.141

13.0 198.0 0.066


Fig. 2. β-dependence of the power law exponentb (continuous line is onlya guide for the eye).

previous experimental findings[10], suggesting that the timedependence of nanoparticle diameter can be well-describedby power laws:

d = atb (11)

Moreover, it was empirically observed that the exponentb takes a value of about 0.07 for the growing process of bothCdS and ZnS nanoparticles in AOT reversed micelles[11,14].On the other hand, theb values obtained by the simulationsshow a marked dependence on the coalescence frequency pa-rameterβ (seeFig. 2). It is of interest that at smallβ values,i.e. for growing process controlled by the intermicellar coa-lescence, theb values agree with those found experimentallywhile atβ approaching the unityb tends to 0.31, a value cor-responding to a pure diffusional controlled growing process[15].

An inspection ofTable 1shows that the value of the ex-ponentb increases with the micellar concentration due tothe parallel increase of the encounter frequency. On the otherhand, it is quite independent on the micellar radius. This hap-pens as a consequence of the concurrence of two oppositeeffects: in fact, aR increase involves a decrease of the en-counter frequency and an increase of the Poisson-averagednumber of reactants in micelle (seeTables 1 and 3).

TheKS dependence ofb is summarized inTable 2. It isw

TM so tio(

[

0

0

0

Fig. 3. Time evolution of the particle size distributions atR= 5 (top), thereference simulation (center) and [AOT] = 0.126 mol l−1 (bottom).

the nucleation rates (Eqs.(5) and (6)), so the greater values ofb result from the smaller number of nucleated nanoparticlesin the simulations at higherKS. This finding underlines thatthe nanoparticle growing process can be triggered not onlyby intermicellar coalescence or pure micellar diffusion butalso by the nucleation frequency.

The time evolution of the particle size distribution (PSD)relative to three simulations carried out varying theR value(3a) and the concentration of the surfactant (3c) with respectto the reference simulation (3b) is shown inFig. 3. For eachPSD, the dimensionless numeric density of particles (Npart)

orth to note that the value of the solubility of Cm only affects

able 3ean particle diameter (d) at completion time andbvalues at various valuef the concentration of the deficient reactant ([A]) and of the molar raXX= [B]/[A])

A] (mol l −1) X d (A) b (s−1)

.01 1 11.74 0.2042 11.56 0.1835 11.50 0.162

20 11.28 0.125

.05 1 14.88 0.1662 14.78 0.1395 14.68 0.100

.1 1 17.26 0.1352 17.24 0.121


Fig. 4. Particle size distributions at completion time at variousR (a, solid line:R= 5; dash-dotted line:R= 10; dotted line:R= 15), [AOT] (b, solid line:[AOT] = 0.126; dash-dotted line: [AOT] = 0.256; dotted line: [AOT] = 0.357 mol l−1), β (c, solid line:β = 10−5; dash-dotted line:β = 5× 10−2; dotted line:β = 1) andKS (d, solid line:KS = 3.6× 10−35; dash-dotted line: 3.6× 10−21; dotted line: 3.6× 10−8 mol2 l−2). The parameters of the reference simulation are:R= 10, [AOT] = 0.256 mol l−1, β = 10−3, 3.6× 10−29 mol2 l−2.

was obtained by partitioning the total size interval (�S) in 200bins, counting the number of particles (Nb) per bin accordingto [4], and normalizing these data with respect to the totalnumber of particles and total size range:

Npart = Ni

�S∑Sj

j=1Nj

(12)

In the graphs,Npart is reported as a function of the meannanoparticle diameter. After calculation of the reaction time,it was partitioned in 20 intervals and the PSD was cal-culated at the end of each interval. Plots ofFig. 3 showthat PSD is narrower at small time, becoming broader andshifted at higher nanoparticle diameters with time. This sug-gests that smaller and monodisperse nanoparticles can be ob-tained by inhibiting their growth at the early stage of theprocess.

At longer times the total depletion of Cm determiningthe end of nanoparticle population growth is observed (seeFig. 1). It is worth to note that the actual model does notconsider the coalescence of two nanoparticle containingreversed micelles which could lead to the formation of biggernanoparticles and an unlimited nanoparticle growth process.This is consistent with the hypothesis that the coalescenceof reversed micelles involves the opening of a intercon-n sizes

4. Particle size and size distribution at completiontime

According with experimental findings[16], the diameterslightly increases withR and decrease with the surfactantconcentration, while shows a stronger dependence on the co-alescence efficiency and on the effective solubility of Cm atlow β and relatively highKS values, respectively.

The trends of the averaged nanoparticle diameter are mir-rored by the PSD at completion time. While the PSD seemsto be unaffected by the micellar radius (Fig. 4a), the max-imum shifts toward smaller values of the diameter and aslight sharpening of the curve is observed when the micel-lar concentration decreases (Fig. 4b). Moreover, the PSDshape is strongly influenced byβ and the effective solubil-ity (Fig. 4c and d). As expected, the completion time de-creases withβ (seeFig. 1) while at highKS it becomeslonger because the number of nanoparticle to growth issmaller.

5. Conclusions

Computer simulations of the growing process of CdSnanoparticles in solutions of water-containing reversed mi-c mee evo-

ecting channel allowing only the exchange of smallpecies.

elles of AOT have shown the pivotal role played by soxternal and internal parameters in controlling the time


lution and the final mean size and polydispersity of nanoparti-cle ensemble, allowing to rationalize previous experimentalresults concerning the law describing the time dependenceof the mean nanoparticle diameter as well as the changesof the final nanoparticle diameter induced by increasingR,local concentration of reactants and surfactant concentra-tion.

Among the parameters modulating the nanoparticle sizeand polydispersity the most critical is the coalescence effi-ciencyβ. Being a measure of the micellar surfactant layer per-meability, its value could be varied by changing the surfactantnature or by adding an appropriate cosurfactant. Simulationresults show that, in order to obtain smaller and monodis-perse nanoparticles, systems characterized by lowβ valuesare preferable.

More generally, taking into account the present simula-tion results, a better comprehension of experimental find-ings can be achieved or from a judicious selection ofthe parameters controlling the nanoparticle growth and theaddiction at selected time of suitable coating agents, afiner nanoparticle size and polydispersity control can bereached.

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