On the Mechanism of Metal Nanoparticle Synthesis in the Brust−Schiffrin MethodSiva Rama Krishna Perala and Sanjeev Kumar*

Department of Chemical Engineering, Indian Institute of Science, Bangalore, India

*S Supporting Information

ABSTRACT: Brust−Schiffrin synthesis (BSS) of metal nano-particles has emerged as a major breakthrough in the field forits ability to produce highly stable thiol functionalizednanoparticles. In this work, we use a detailed populationbalance model to conclude that particle formation in BSS iscontrolled by a new synthesis route: continuous nucleation,growth, and capping of particles throughout the synthesisprocess. The new mechanism, quite different from the othersknown in the literature (classical LaMer mechanism, sequentialnucleation−growth-capping, and thermodynamic mechanism),successfully explains key features of BSS, including size tuningby varying the amount of capping agent instead of the widelyused approach of varying the amount of reducing agent. The new mechanism captures a large body of experimental observationsquantitatively, including size tuning and only a marginal effect of the parameters otherwise known to affect particle synthesissensitively. The new mechanism predicts that, in a constant synthesis environment, continuous nucleation−growth-cappingmechanism leads to complete capping of particles (no more growth) at the same size, while the new ones are born continuously,in principle leading to synthesis of more monodisperse particles. This prediction is validated through new experimentalmeasurements.

■ INTRODUCTION

Gold nanoparticles find applications in a number of fields, suchas nanoelectronics, nonlinear optics, biophysics, theranostics,and so forth.1,2 Their unique properties and synthesis methodsare extensively reviewed in the literature.3−8 The synthesismethods differ from one another in the use of reducing agent,synthesis temperature, and their ability to produce water-soluble vs organic phase soluble, relatively monodispersed vspolydispersed, and tunable vs fixed mean size particles. Citratereduction9,10 and citrate-tannic acid reduction11,12 of chlor-oauric acid are in wide use to produce relatively mono-dispersed, water-soluble, spherical gold nanoparticles, withmean size in the range of 4 to ∼150 nm. The mean particle sizeis tuned by controlling the amount of reducing agent added tothe metal precursor solution, in agreement with the LaMer13

mechanism.The Brust−Schiffrin method14 (BSM) facilitated room

temperature synthesis of highly stable functionalized nano-particles of small sizes (2−2.5 nm), with 10 times larger particleloading. The method has impacted the pace of the subsequentdevelopments quite substantially.15 A great variety of function-alized nanoparticles of noble metals have been synthesized overthe years16,17 using this method. The extraordinary stability ofthe synthesized particles is attributed to alkanethiols whichform a strong bond with the particle surface14,18−20 andpassivate it. Instead of following the widely used approach ofadding different amount of reducing agent, the mean particle

size in BSM is tuned by adding different amounts ofalkanethiol18,19,21−23 to the solution. Figure 1 brings together

the data available in the literature. The mean particle sizedecreases from 8 nm at 0:1 mol ratio of alkanethiol to goldprecursors to about 2 nm at a ratio of 2:1, and a marginaldecrease thereafter.Although BSM is in extensive use and is studied

widely,18,19,22,24−28 the synthesis process and the features it

Received: April 27, 2013Revised: July 13, 2013

Figure 1. Experimentally measured mean particle size for variousvalues of thiol to gold ratio [*, data of Brust et al.14 and #, data ofBrust et al.30].

Article

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manifests are poorly understood. For example, the reasons forthe addition of alkanethiols instead of the reducing agent forsize control, high polydispersity19 even though the alkanethiolsare quite effective at preventing coagulation, and less than theexpected effect of variables such as temperature and the rate ofaddition, etc are not clear. The absence of a mechanisticunderstanding of the synthesis process is the main impedi-ment.29

The present work aims to probe the mechanism of particlesynthesis in BSM using a mathematical model. The synthesismechanism should not only explain size tuning and the variousfeatures of BSM, but also rationalize observations such as thepresence of a shallow maximum in time variation of the meanparticle size.24 The improved understanding will hopefully leadto modification of BSM for large scale synthesis of relativelymonodispersed nanoparticles.In the next section, we put together the findings of a large

body experimental investigations reported in the literature,some of them apparently quite unrelated, and develop a generalmodel of the synthesis process. The framework of populationbalances31 is used next to quantify it. The model is then used toexplore the mechanism of the synthesis process and to developan understanding of its many unexpected features. One modelprompted variation of BSM is tried experimentally for itsimpact on the synthesis of gold nanoparticles.

■ CURRENT UNDERSTANDING OFBRUST−SCHIFFRIN SYNTHESIS

Briefly, the Brust−Schiffrin protocol is as follows. First,chloroauric acid (Au3+) is phase transferred into toluene froman aqueous phase using a phase transfer catalyst. This isfollowed by the separation of the two phases and the additionof a desired amount of dodecanethiol to the separated organicphase, which reduces Au3+ to Au1+. Another aqueous solutioncontaining sodium borohydride, the main reducing agent, iscontacted with the organic phase next. The particle formation isindicated by the change of color of the organic phase. Anoverview of the protocol is presented in Figure 2.The phase transfer of chloroauric acid into toluene using

tetraoctylammonium bromide (TOAB), a phase transfercatalyst (PTC), occurs through the following reaction:14

+

→ +

+ − + −

+ −

N

N X X

H AuCl (aq) (R ) Br (org)

(R ) Au (org) H (aq)4 8 4

8 4 4 (1)

Here, X stands for both Cl and Br as the extent of substitutionof Cl− by Br− is not known. The reduction of Au3+ to Au1+ stateby alkanethiol (RSH) in BSM was believed to occur throughthe following reaction14,22,25

+

→ − − + + + +

+ −

+ − +

N X

N X

(R ) Au 3RSH

(AuSR) RSSR (R ) 4 3Hn

8 4 4

8 4(2)

The organic phase at this stage thus consisted of TOAB, dialkyldisulfide (RSSR), Au1+ in the form of −(AuSR)n− polymer,and either AuX4

− complexed with TOA ion or excess RSH. Thereduction of Au3+ and Au1+ to Au0 by sodium borohydride wasconsidered to occur through the following reaction:22,25

− − + + + →−(AuSR) BH RSH RSSR Au (SR)nk

x y42

(3)

+ + + →+ − −R N X( ) Au BH RSH RSSR Au (SR)k

x y8 4 4 43

(4)

These reactions for the biphasic BSM were initiallyconsidered14,32 to occur at the organic−aqueous interface.The robust nature of the protocol14,18,19,24 and a number ofvariables that can influence interfacial area, such as the intensityof mixing, vessel size, and so forth, suggesting that either theabove reactions do not occur on the interface or they have noeffect on particle synthesis. Corbierre and Lennox26 eventuallycontacted toluene containing −(AuSR)n− polymer withsodium borohydride solution. The overnight stirring of thetwo phases did not produce any reaction. The addition of PTCbrought about a rapid formation of particles, though. The PTCthus facilitates particle synthesis in the bulk of the organicphase, in the presence of alkanethiols or dialkyl disulfide. Sincethe aqueous phase remains colorless throughout the synthesis,the nucleation and the growth of the particles must be confinedto the organic phase.The role of −(AuSR)n− polymer in BSM has interested

investigators from the beginning itself. Porter et al.33

substituted octadecanethiols by dioctadecyl disulfides toeliminate the formation of −(AuSR)n− polymer whilemaintaining capping agents in the system. Their measurementsshowed that the mean particle size and the breadth of the sizedistribution remained unaffected for both gold and silvernanoparticles, provided the ratio of the concentration of Sgroup to metal precursor was kept constant. Shon et al.22

substituted dodecanethiol by didodecyl disulfide, and foundthat both the synthesis routes lead to formation of particles ofnearly similar sizes and distributions. The experimental data isshown in Figure 1 through entries marked RSH and RSSR (1mol of RSSR is equivalent to 2 mol of RSH in the plot). Theauthors also formed mixed gold(I) thiolate polymers of threedifferent compositions and found that their reduction led to thesynthesis of identical size particles. A complete breakup of−(AuSR)n− polymer upon the addition of sodium borohydridewas proposed to explain the observation.A number of changes designed to influence the kinetic steps

of the synthesis had only limited influence on the end results.

Figure 2. Overview of the synthesis of thiol capped gold nanoparticles by the method of Brust et al.14

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This pointed to the possibility of thermodynamic control ofparticle size.18,22 Leff et al.18 considered surface excess energy ofgold atoms, energy of adsorption of thiol molecules on goldparticles, and the entropy of mixing of thiol molecules and goldnanoparticles, and proposed that particle size corresponds tothe minimum in free energy at equilibrium. Their predictionsare in reasonable agreement with the experimental data at thiolto gold ratios larger than 1/3. The model predicts a steepincrease in particle size to infinity as the thiol concentrationapproaches zero. The experimental measurements on the otherhand show a gradual increase in particle size to 8 nm in theabsence of thiols19,30 (Figure 1). Hostetler et al.19 conductedinitial reaction at different temperatures, followed by mixing thecontents at room temperature for about 3 h. The final particlesize in all the cases depended on the path taken. The mainadvantage of the thiol capped nanoparticles is that they can bestored as a powder and resuspended in an organic solvent withzero amount of thiol in it. This cannot be realized if theparticles were to be in thermodynamic equilibrium.Murray and co-workers,16,19,35 who studied BSM most

extensively, have proposed particle formation to be kineticallycontrolled through nucleation−growth−passivation sequence.The larger the concentration of alkanethiols, the faster thepassivation of growing particles and therefore the smaller theparticle size. The sequential mechanism indicates16 that a fasterrate of addition of reducing agent should produce smallerparticles, through a burst of nuclei formation in the beginning.The experimental observations34 suggest that both the meanparticle size and the breadth of size distribution are affectedonly marginally.The recent studies of Goulet and Lennox27 and Li et al.28

show that gold(I) thiolate polymer −(AuSR)n− is not formedin the original BSM. This finding reconciles a number of earlierconjectures related to the role of −(AuSR)n− in BSM. Thereduction of AuX4

− by alkanethiol is instead found to produce anew intermediate, AuX2

− through

+

→ + +

+ −

+ −

R N X

R N X X

( ) Au 2RSH

( ) Au RSSR 2Hk

8 4 4

8 4 21

(5)

Two moles of RSH reduce one mole of Au3+ to Au1+. For thiolto gold ratios smaller than 2, the gold precursor is present asAu3+ and Au1+ before the addition of sodium borohydride.Goulet and Lennox27 suggest that the ratio of precursors, alongwith the ratio of thiol to gold, affects the kinetics of particleformation. The size tuning known for BSM can thus beexplained through different rates of reduction of Au3+ and Au1+

to Au0. This mechanism is similar to the classical mechanismdue to LaMer and Dinegar13 in which the reactivity ofprecursors determines the number of nuclei born.Goulet and Lennox27 and Li et al.28 show that −(AuSR)n− is

formed when water is not removed before the addition of thiol.The formation of −(AuSR)n− as an intermediate precur-sor36−38 also plays a critical role in single phase syntheses ofgold nanoparticles in polar media such as tetrahydrofuran andalcohols, a method that has recently gained importance to makethermodynamically stable magic clusters with less than 150 goldatoms.39,40 The etching of large size particles in this method tomake small size clusters during the aging process is known tobe influenced by Br−, H+, O2, TOA

+, and so forth in interestingways.39,41 The formation of AuSR from gold nanoparticles inthe presence of alkanethiols under reflux conditions isimplicated in digestive ripening.42 Given that size focusing to

form clusters by aging43 and digestive ripening at hightemperature involve long synthesis times,42 we consider thatBrust−Schiffrin synthesis carried out over 1−3 h time scale atroom temperature in tolune, a nonpolar medium, is notsignificantly influenced by the above-mentioned processes.Tong and co-workers28,44 have recently concluded, based on

their NMR studies, that tetraoctylammonium bromide withfour octyl chains forms inverse micelles with water core inthem. They propose that the ions are transferred from theaqueous to the organic phase by their incorporation intoinverse micelles. The water cores are suggested as reaction sites.Particle formation through the fusion of inverse micelles45,46

containing precursors in them however brings up a number ofunexplainable situations. The alkanethiols must reduce Au3+ toAu1+ inside the water core of the inverse micelles. We contactedtoluene containing alkanethiol with an aqueous solutioncontaining Au3+ and found no reduction. Further, the particlesconfined in water cores of inverse micelles should equilibratewith the external aqueous phase and get partitioned betweenthe two aqueous phases. The aqueous phase however remainscolorless all through, indicating that nanoparticles do notpartition between the two phases. The capping of ananoparticle present in an aqueous phase by organic phasesoluble thiol is known to require solvating agents such asethanol or acetone47 which are not present in BSM. Giventhese unresolved issues, we carried out a detailed inves-tigations48 into the presence of inverse micelles using SAXS,light scattering, and variation of interfacial tension, viscosity,and water content at equilibrium as a function of PTCconcentration. All the measurements point to the absence ofany structure in the organic phase. These results and the NMRstudies are consistently explained through water of hydrationaround polar groups of PTCs.49 In this work, we have thereforeconsidered, similar to the earlier investigators,14,19,27 that ionsare transferred from the aqueous phase into the organic phasethrough complexation with the phase transfer catalyst.Liu et al.24 have measured the time variation of mean size of

gold nanoparticles stabilized by butanethiolate in place ofdodecanethiol in BSM. The experimental data interestinglyshow the presence of a weak maximum in particle size withtime. The authors have assumed the number of particles toremain constant during their growth. The nucleation phase isthus assumed to be complete in a short burst. The number ofparticles born is obtained by dividing the total gold in thesystem with the experimentally measured mean particle size.The growth rate was considered to be proportional to (i) therate of reduction of gold ions and (ii) the product of the rate ofreduction of gold ions and the surface area of the particles. Theagreement of the fitted growth kinetics with the experimentaldata for the two cases is not conclusive. The presence of a weakmaximum in the particles size is attributed to the etching of theparticles. A mechanism that facilitates the gold atoms to firstdeposit on the particles to grow them and then overcome thelattice energy to leave them on the time scale of particlesynthesis is not apparent.The mechanisms proposed in the literature are thus quite

different. To summarize, Brust et al.14 proposed that theparticle size is controlled by surface coverage by alkanethiols.Murray and co-workers16 developed it further into a sequentialnucleation−growth−passivation model; the particles stopgrowing after their surface is fully passivated. Shimmin et al.50

have shown that the competition between growth andpassivation of particles fails to explain the size tuning observed

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for BSM for the initial burst of nucleation to be independent ofthe concentration of alkanethiols, and also when alkanethiol isreplaced by polymeric thiol. Leff et al.18 proposed athermodynamic model which drew its support from theinsensitivity of BSM to a number of parameters that arenormally known to influence particle synthesis. Goulet andLennox,27 based on their finding that the reduction of the mainprecursor Au3+ by alkanethiols produces another reactiveprecursor Au1+, suggested that the ratio of precursors Au3+

and Au1+ may affect the kinetics of formation of nanoparticles.The observed size tuning may be explained through theclassical approachprecursor reactivity controlled nucleiformation as per the LaMer mechanism. The steep decreasein mean particle size for alkanethiol to gold ratio from 0:1 to2:1 over which the composition dependent reactivity changeslends support to this mechanism. Liu et al.24 considerednucleation−growth mechanism (LaMer mechanism), withnucleation and particle growth to occur at separate time scales,to fit the variation of mean particle size with time. Clearly, theformation of nanoparticles in BSM is not well understood.In the next section, we take up development of a

comprehensive model that encompasses various kinetic path-ways for the synthesis of gold nanoparticles, and is consistentwith the experimental findings discussed above. We then use itto explore the synthesis mechanism and the other relatedfindings quantitatively.

■ MODEL

The general kinetic model considers (i) PTC to transfer water-soluble precursors by complexing with them, (ii) all thereactions to occur in the organic phase, and (iii) the reductionof Au3+ to Au1+ by alkanethiols to go to completion before theaddition of the main reductant. We assume ripening andetching to have negligible effect on this room temperaturesynthesis. Since TOAB and alkanethiols can stabilize particlesindividually, we assume that the particles do not coagulateduring the synthesis. The starting point for the model is theaddition of sodium borohydride solution which produces Au0.At some stage, the nuclei are born. The nuclei offer surface areafor simultaneous assimilation of Au0 atoms and capping. Figure3 shows a schematic of the synthesis process. With thedominance of one process over the others, the general modelcan lead to size control through (i) change in precursor

reactivity effected by the ratio of Au3+ and Au1+, and/or by (ii)the capping of the growing particles by alkanethiols and dialkyldisulfides.

Reduction of Au3+ to Gold Atoms. As discussed before,AuX4

− can also be directly reduced by phase transferred BH4−

to produce gold atoms through

+ →− −XAu 3BH Auk

4 403

(6)

AuX4− can be reduced by alkanethiol to produce AuX2

−, whichis further reduced by BH4

− to produce gold atoms through

+ →− −XAu BH Auk

2 402

(7)

These reactions (eqs 5−7) can be concisely written in terms ofAu3+, Au1+, Au0, RSH, RSSR, and BH4

− as

+ → ++ +Au 2RSH Au RSSRk3 11 (8)

+ → + ···+ −Au BH Auk1

402

(9)

+ → + ···+ −Au 3BH Auk3

403

(10)

The disproportionation of Au ions (3Au1+ → 2Au0 + Au3+)leading to the formation of gold atoms, which occurs at hightemperatures,9,31 is not considered in this room temperature,organic phase synthesis. The notation A for Au3+, B for Au1+, Cfor Au0, L for RSH, L′ for RSSR, and R for BH4

− is used in thefollowing model equations.

Nucleation. The nucleation of metal nanoparticles withclose to zero solubility of metal atoms continues to challengeresearchers.51−56 The birth of similar size particles from AuX4

−

and gold(I) thiolate polymer22 as precursors made thenucleation process even more perplexing. The identificationof AuX2

− as a new intermediate has bypassed the need for anydirect role of alkanethiols in nucleation in BSM. Abecassis etal.57 have recently investigated nanoparticles synthesis intoluene in the presence of slightly different PTC and stabilizingagents using X-ray absorption near edge spectroscopy(XANES) and small-angle X-ray scattering (SAXS). Theirmeasurements reveal significant concentration of Au0 over ashort time, coinciding with a burst of nuclei formation. Theresults of Saunders et al.32 are shown to be similar to thoseobtained for systems obeying classical homogeneous nucleation

Figure 3. Schematic representation of the over all model.

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theory.13 Both these findings point to a burst of nucleiformation through homogeneous nucleation like mechanism.We have assumed the mechanism to represent nucleation in thepresence of thiols also. The synthesis of particles forming a nearcontinuum of sizes18,19,24 has led us to ignore the formation ofparticles with close shells (magic clusters) in BSM. The rate ofparticle formation for homogeneous nucleation mechanism isgiven by58

= −⎛⎝⎜

⎞⎠⎟N k C A

kS

[ ] exp(log )n

n1 v

22

(11)

where

πγ= =k

v

k TS

CC

16

3( ),

[ ][ ]n2

3g

2

B3

s (12)

Here, kB is the Boltzmann’s constant, T is temperature, kn1 andkn2 are nucleation rate constants, [C] is concentration of goldatoms, Av is Avogadro number, vg is atomic volume of gold, γ isspecific surface energy, [Cs] is the solubility of gold atoms inthe solution, and S is the extent of supersaturation (Table 1).

The nucleus is assumed to consist of 10 gold atoms (the valueof m), similar to the smallest size particle synthesized byBSM.20 The values of S and m are similar to those consideredearlier by Abecassis et al.57

Growth and Capping. The particle growth and thecorresponding increase in free surface area occur throughassimilation of new gold atoms. The simultaneous adsorption ofsulfur group containing molecules increases the capped(passivated) surface area. Saunders et al.32 earlier found thatwhile the mean particle size increased with time, the breadth ofthe size distribution did not change, which suggests surfaceprocess controlled assimilation of new gold atoms. We assumehere the same mechanism for the adsorption of larger sizealkanethiols and dialkyl disulfides.22 The particles are assumedto be spherical at all times. Since the diffusion is not the ratecontrolling step, the species concentrations are the sameeverywhere. Thus,

= = −G v avt

k v a a C( , )dd

( )[ ]p g mg (13)

and

= = + ′ −′L v aat

k L a k L a a a( , )dd

( [ ] [ ]2 )( )L Lp ms ms (14)

Here, (a − a) is the free surface area of a particle of volume vwith capped surface area a. The total surface area a is equal to(36π)1/3v2/3. [C], [L], and [L′] represent concentrations ofgold atoms, alkanethiols, and dialkyl disulfides, respectively.The surface areas covered by one mole of L and L′, denoted byams and 2ams, respectively, are taken to be independent of theparticle size in the absence of more precise information. The

estimate of ams provided by Leff et al.18 is used in the presentmodel. The values of rate constants kg, kL, and kL′ are discussedin a later section.

Number Density. A particle needs its volume v and cappedsurface area a to be specified for its identification. Such acollection of particles is best characterized using a bivariatenumber density p(v,a,t); p(v,a,t) dv da represents the numberof particles in size range v to v + dv with thiol covered surfacearea in range a to a + da at time t per unit volume of thereaction mixture. The time evolution of number density p inthe framework of population balances31,59,60 is given by

δ δ

∂∂

+ ∂∂

+ ∂∂

= − −

p v at v

G v a p v aa

L v a p v a

N t v v a a

( , )[ ( , ) ( , )] [ ( , ) ( , )]

( ) ( ) ( )

p p

c c (15)

where N(t) is the nucleation rate, and vc and ac are the volumeand capped surface area of a nucleus, respectively. As theprocess of capping begins after a nucleus is born, ac = 0.Equation 15 can be integrated after multiplying with viaj toobtain time variation of a general moment Mi,j. The latter isdefined as

∫ ∫=∞ ∞

M t v v a p v a t a( ) d ( , , ) di ji j

,0 0 (16)

Model Equations. The model equations for the reactionsproposed in eqs 8−10 can be written as

= − −At

k A L k A Rd[ ]

d[ ][ ] [ ][ ]1 3 (17)

= −Bt

k A L k B Rd[ ]

d[ ][ ] [ ][ ]1 2 (18)

∫ ∫

π

= + −

− −

= + −

− −

∞ ∞

Ct

k B R k A RmNA

k C a a p v a dv da

k B R k A RNmA

k C M M

d[ ]d

[ ][ ] [ ][ ]

[ ] ( ) ( , )

[ ][ ] [ ][ ]

[ ][(36 ) ]

2 3v

g0 0

2 3v

g1/3

2/3,0 0,1 (19)

= − −Rt

k B R k A Rd[ ]

d[ ][ ] 3 [ ][ ]2 3 (20)

∫ ∫π

= − − −

= − −

∞ ∞Lt

k A L k L a a p v a v a

k A L k L M M

d[ ]d

2 [ ][ ] [ ] ( ) ( , ) d d

2 [ ][ ] [ ][(36 ) ]

L

L

10 0

11/3

2/3,0 0,1 (21)

∫ ∫π

′ = − ′ −

= − ′ −

′

∞ ∞

′

Lt

k A L k L a a p v a dv da

k A L k L M M

d[ ]d

[ ][ ] [ ] ( ) ( , )

[ ][ ] [ ][(36 ) ]

L

L

10 0

11/3

2/3,0 0,1 (22)

π= + −M

tNv k v M M C

d[(36 ) ][ ]1,0

c g mg1/3

2/3,0 0,1 (23)

π= + ′ −′M

tk a L k a L M M

d( [ ] 2 [ ])[(36 ) ]L L

0,1ms ms

1/32/3,0 0,1

(24)

Table 1. Values of Constants and Parameters for Nucleation

parameter value

γ 0.25 J/m2

kB 1.38 × 10−23 J/KT 300 Kvg 1.7 × 10−29 m3

m 10

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∫ ∫ π

= +

− ·∞ ∞ ⎡

⎣⎢⎤⎦⎥

M

tNv k v C

a v av

v p v a v

d[ ]

d (36 ) ( , ) d

2/3,0c

2/3g mg

0 0

1/3 1/32/3

1/3

(25)

In order to keep the model equations simple so as to solvethem using the standard ODE integrators, we introduce aclosure by assuming that a/(36π)1/3v2/3, the ratio of cappedarea of a particle to its surface area, is equal to fraction f thatevolves with time. Thus,

∫

∫

π

π

π

= +

−

= + −

∞

∞ ⎡⎣⎢

⎤⎦⎥

M

tNv k v C a

va

vv p v a v

Nv k v C f M

d[ ](36 ) d

(36 )( , ) d

23

(36 ) [ ](1 )

2/3,0c

2/3g mg

1/3

0

0

1/31/3 2/3

1/3

c2/3 1/3

g mg 1/3,0

(26)

Further,

π= + −M

tNv k v C f M

d13

(36 ) [ ](1 )1/3,0c

1/3 1/3g mg 0,0 (27)

= M

tN

d0,0

(28)

Here, f is taken to be equal to the average fractional surfacecoverage, defined as

π=f

M t

M

( )

(36 )0,1

1/32/3,0 (29)

The above simplified system of model equations is complete.The model equations were solved using LSODE solver of GNUOctave, version 3.2.4 and ODE15S solver of MATALB7.7.0.471 (R2008b).

■ RESULTS

Simulation Approach and Initial Conditions. Thesimulations are started with the addition of the reducingagent. The initial concentrations of Au3+, Au1+, Au, BH4

−, RSH,and RSSR are specified accordingly, depending on the amountof alkanethiols added. As there are no particles present initially,all the moments Mi,j and fraction f are set to zero. The PTCremaining after transferring Au+3 transports BH4

− to theorganic phase. When BH4

− is added at a finite rate, it istransported to the organic phase at the same rate until themaximum amount of BH4

− that can be transported by theremaining PTC is reached. If BH4

− is added instantaneouslyand is stoichiometrically larger than what the remaining PTCcan transport, then PTC transports BH4

− to the organic phaseat the maximum level and maintains it at this level.The governing equations are nondimensionalized using the

appropriate maximum values. The model parameters and initialconditions for room temperature synthesis for thiol to goldmolar ratio of 1:1 are presented in nondimensionalized form inTable 2. As the TEM measurements report average value ofparticle diameter, the number average diameter (Dp) definedbelow is considered to be the equivalent mean diameter.

π= ⎜ ⎟

⎡⎣⎢⎢⎛⎝

⎞⎠

⎤⎦⎥⎥D

M

M6

p

1/31/3,0

0,0 (30)

Parameter Estimation/Values. The parameter valuesneed to be specified in order to test a synthesis mechanismfor its ability to capture the size tuning and the other aspects ofthe synthesis protocol. The experiments show that the solutionchanges its color almost immediately after the reducing agent isadded. The particle formation must therefore commence over atime scale of seconds. Saunders et al.32 synthesized goldnanoparticles in the absence of thiols. They added the reducingagent slowly and found that color of the solution changes afterconversion of a significant fraction of gold ions into gold atoms.These observations together suggest that the reduction of thegold ions and the formation of nuclei occur on time scales ofseconds. The values of reduction rate constants (k2 and k3) andnucleation rate constants (kn1 and Cs) were fitted accordingly.The rate constant kn2 can be calculated using the physicalproperties alone. The rate constant k1 for reduction of Au3+ byalkanethiols, completed before the addition of sodiumborohydride, does not play a role in the synthesis. The rateconstant for particle growth, kg, is fitted based on theexperiments carried out by Brust et al.30 and Li et al.28 in theabsence of alkanethiols. Finally, the value of kL is fitted tocorrectly predict the experimental data of Hostetler et al.19 whoused thiol for capping. The rate constants k2, and k3 correspondto AuXn

− precursors, and may vary to some extent if theprecursor composition (X in terms of actual Cl and Br) isvaried.

Reactivity Controlled Particle Size. We test in thissection the hypothesis that different reactivities of AuX4

− andAuX2

− can explain the size tuning obtained by varying thiol togold ratio. We set the rate constants kL and kL′ close to zero sothat thiols and disulfides do not cap the growth of particles, butget adsorbed on them slowly to provide long-term stability. Thesimulations were carried out for thiol to gold ratio ranging from0 to 5 for the reducing agent addition time of 10 s.19 Figure 4shows the model predictions for the parameter values presentedin Table 3. The rate constant k2 for reduction of AuX2

− to Au0 isrequired to be 2 orders of magnitude larger than the rateconstant for reduction of AuX4

− to Au0 to fit the experimentaldata of Hostetler et al.19 The experimental data of Leff et al.18

shows much larger sensitivity of particle size to thiol to goldratio and the synthesis of larger size particles in the presence ofthiol than in its absence,19,30 as shown in Figure 1. The X-rayinduced aggregation of particles during the measurement isheld responsible for these observations.19 We have thereforenot compared the model predictions with their experimentaldata.

Table 2. Model Constants and Initial Conditions for Thiol/Gold::1:1 for Experiments of Hostetler et al.19b

constants and initial conditions value

vmg 1.02 × 10−5 m3 mol−1

ams 1.2887 × 105 m2 mol−1

B*(0)a 0.333A*(0)a 0.667R*(0)a 1

aAsterisk (∗) denotes nondimensionalized variable. bAll the remaininginitial conditions not mentioned here are equal to zero. [C0 = 0.00983M, R0 = 0.0244 M, L0 =0.00983 M.]

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Figure 4 clearly shows that the “reactivity based control”mechanism captures the experimental observations quite well.An increase in the thiol concentration increases the proportionof AuX2

−, which, due to its high reactivity, leads to more intenseburst of nucleation, hence a decrease in the mean particle size.For the thiol to gold ratio in excess of two, the initialcomposition of the reactive precursors becomes independent,hence the prediction for the mean particle size also becomesindependent of the thiol to gold ratio. While, the experimentaldata of Hostetler et al.19 (Figure 4) is in agreement with thisprediction, Shon et al.22 who later used the same protocol forslightly different conditions (HAuCl4: 0.8 mM vs 0.9 mM;tetraoctylammonium bromide: 3 mM vs 4 mM; sodiumborohydride: 8 mM vs 10 mM) reported the decrease inmean particle size to continue to high thiol concentration rangeas well, separately for both dodecanethiol and dodecyl disulfideas the capping agents. If the more recent measurements of thesame group are more accurate, clearly the reactivity basedcontrol cannot explain them. The predicted monotonic increasein mean particle size with time for this mechanism also doesnot offer any explanation for the weak maximum reported inthe literature.24

We next simulated the experiments of Porter et al.33 andShon et al.22 in which they substituted alkanethiols by dialkyldisulfides while keeping the concentration of sulfur groups thesame. The model predictions, indicated by the dotted line inFigure 4, show that the particle size should increase by severaltimes. The experimental data, as pointed out earlier, show nochange in particle size. The reason why the “reactivity basedcontrol” mechanism fails to capture these findings is that itrequires reduction of AuX2

− to proceed at a 2 orders ofmagnitude faster rate than the reduction of AuX4

−. Setting theinitial concentration of AuX2

− to zero, because of the directaddition of dialkyl disulfide, decreases the intensity of thenucleation burst, hence a substantial increase in the meanparticle size.We next evaluate the “capping/passivation based control”

mechanism.

Capping Rate Controlled Particle Size. Brust et al.14

suggested that the particle size in their protocol is controlled bythe surface coverage, not by the reduction kinetics ofprecursors. We test this proposal by increasing kL and kL′ tolarger value to enable surface coverage to compete with particlegrowth, for similar levels of reactivity for the two precursors.The parameter values leading to the predictions shown inFigure 5 are presented in Table 4. The figure shows that the

“capping based control” can capture the experimental data inthe entire range of thiol to gold ratio: rapid decrease in particlesize at low values and a slow decrease at larger values. We alsouse the model to test if the kinetics of reduction indeed has noeffect on the synthesis process, as conjectured above. The effectof 5-fold increase and decrease in the values of rate constantsfor the reduction reactions is shown by the dotted and thedashed lines in the same figure. The predicted particle sizes atvery low thiol to gold ratios are quite different. The differencesvanish as the thiol to gold ratio is increased. Also, the effect is inopposite direction in the two concentration ranges. The modelpredictions thus validate the insightful claim of Brust et al.14

that reduction kinetics of precursor does not play a controllingrole. The particle size is determined by the competitionbetween the capping and the growth of a particle. Larger therate of capping, smaller is the particle size to which it can growbefore its growth is capped. For thiol to gold ratios smaller than1/4, the thiol available is less than that required to cap theparticles. The predicted increase in particle size in this rangeand its dependence on the reduction kinetics is largely onaccount of the precursor reactivity, discussed earlier in thecontext of Figure 4. The “capping based control” mechanism isactivated at thiol to gold ratios larger than 1/4. It is stillintriguing why an increase in the concentration of the reducedgold atoms, led by an increase in the reactivity of theprecursors, should not favor particle growth over capping andthereby produce larger size particles. A slight under-predictionat high thiol concentrations is attributed to the inability of the

Figure 4. Comparison of model predictions for “reactivity basedcontrol” mechanism with the experimental data.

Table 3. Model Parameters of Reactivity Modela

parameter value

Cs 1.0 × 10−7 Mk2 6.10 × 10−3 dm3 mol−1 s−1

k3 7.20 × 10−5 dm3 mol−1 s−1

kn1 7.0016 × 103 s−1

kg 4.80 × 10−5 m s−1

aAll other values are same as in Table 2.

Figure 5. Comparison of model predictions for “capping basedcontrol” mechanism with the experimental data. *, Brust et al.;14 #,Brust et al.30

Table 4. Model Parameters Used Only in the Capping Model

parameter value

Cs 4.226 × 10−8 Mk2 4.0022 × 10−1 dm3 mol−1 s−1

k3 1.0006 × 10−1 dm3 mol−1 s−1

kn1 1.147 × 104 s−1

kg 1.062 × 10−4 m s−1

kL 1.9337 × 10−7 m s−1

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measurement technique to pickup the small size particlescontinuously introduced into the system (addressed in the nextsection).We now revert back to the experiments of Porter et al.33 and

Shon et al.22 The effect of substitution of alkanethiols by dialkyldisulfides, for the marginal role of precursor reactivity in thepresence of thiols, impacts particle size through the kinetics ofadsorption of capping agents. Shon et al.22 show that particlesize for sulfur in dodecanethiol to gold ratio of 4:1 and sulfur indidodecyl disulfides to gold ratio of 4:1 produces mean particlesizes of 2.0 and 2.1 nm. The former, before the addition ofsodium borohydride, has an equimolar mixture of sulfur groupsin dodecanethiol and didodecyl disulfides while the latter has allthe sulfur groups present in didodecyl disulfides. The formationof similar size particles for the two synthesis routes points tosimilar rates of capping for alkanethiols and dialkyl disulfides.The latter requires the respective rate constants kL and kL′ to besimilar. Li et al.61 recently carried out similar experiments forsulfur to gold ratio of 3:1, and found slightly different particlesizes (1.70 ± 0.22 and 2.24 ± 0.26 nm), which might requirethe use of slightly different values for kL and kL′.Zhang et al.62 used BSM to synthesize AuNP in the presence

of n-tetradecanethioacetate (C14SAc) as the capping agent.C14SAc does not reduce Au

3+ to produce disulfides during thesynthesis. The reducing agent was added over a period of 5min. The synthesis at Au:C14SAc ratio of 1:1 yielded particles ofmean size of 4.93 nm, much larger than those obtained withalkanethiols at similar ratio. The increase in size wasattributed62 to thioacetate being a weak ligating agent. Themodel predicts the same value for the mean particle size whenthe rate constant kL is reduced by 12 times while keeping all theother constants shown in Table 4 unchanged. The modelpredicts for Au:C14SAc ratio of 1/3:1 a mean size of 9.07 nm,which is in good agreement with the measured value of 7.23nm.Dynamics of Gold Nanoparticle Synthesis. The

dynamics of particle synthesis is discussed for the absence ofthiol [case (a)] and thiol to gold ratio of 1:1 [case (b)]. Figure6 shows time variation of the concentration of gold ions, goldatoms, the rate of nucleation, and the number of particles. The

time variation of only Au3+ is shown for case (a) (Figure 6A) asAu1+ is not formed for this case. For case (b), the concentrationof Au1+ approaches zero faster than that of Au3+ because of theslightly higher reactivity of the former (k2 = 4k3). Figure 6Bshows time variation of the concentration of gold atoms. Theconcentration of Au0 rises rapidly for case (a), goes through amaximum, and rapidly decreases to zero (the time is plotted onlog scale). The concentration of Au0 for case (b) rises at a fasterrate (due to the higher rate of reduction of Au1+), goes througha maximum, rapidly decreases like for case (a), but instead ofdecreasing to extremely low values and remaining low, risesagain and maintains significant levels over a prolonged timeinterval. Figure 6C compares the nucleation rate profiles for thetwo cases. The figure shows, that for case (a), as expected, asingle short burst of nucleation is realized. The rate ofnucleation drops to zero as the growing particles consume goldatoms. The rate of nucleation for case (b) shows quite aninteresting behavior. After an initial burst of nucleation, the rateof nucleation rises again and the new nuclei continue to formover a prolonged time, almost until the end of synthesisprocess. This is more clearly shown in Figure 6D; most of thenuclei for case (a) are formed quite early whereas for case (b)they continue to be formed throughout the synthesis.The continued nucleation after an initial burst finds its

imprint on time variation of the mean particle size. Figure 7

shows a characteristic increase in particle size in the absence ofthiol: rapid increase in particle size as the nuclei consume thesupersaturation available at their birth, followed by a slow andsustained increase matched with the rate of formation of goldatoms. The increase in particle size in the presence of thiolsfollows a different path. The increase at short time is similar tothat in the absence of thiol, but it soon gives way to a slowerincrease due to the capping of particles. The supersaturationbegins to build up again and so does the rate of nucleation. Thecontinued birth of new particles brings the average particle sizeslightly down as time proceeds. The “capping based control”mechanism thus predicts a shallow maximum on account ofsustained nucleation and depleting levels of gold precursors. Itis noteworthy that Liu et al.,24 who followed a similar protocolwith butanethiol as the capping agent, indeed found the particlesize to go through a weak maximum in time. While the authorsattributed it to the etching of particles, continued nucleationpredicted by the present model offers an alternativeinterpretation.

Figure 6. Dynamics of nanoparticle synthesis in the absence of thiol(thiol/gold = 0:1) and presence of thiol (thiol/gold = 1:1). In panel C,the rate of nucleation is in the nondimensionalized form. [Subscript adenotes case (a) where thiol is absent and b denotes case (b) wherethe ratio of thiol to gold is 1:1.]

Figure 7. Comparison of time evolution of mean particle size forparticle synthesis in the absence (thiol/gold = 0:1) and presence(thiol/gold = 1:1) of thiol.

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■ DISCUSSION

Figures S2−S4 in the Supporting Information show theindividual effect of 2-fold increase and decrease in rateconstants k2, k3, and kn1. The figures show negligible effect ofthese parameters on particle size for thiol to gold ratios largerthan 1/4. The effect of similar changes in the solubility of goldatoms, growth rate constant kg, adsorption rate constant kL, andthe number of atoms in the nuclei m is shown in Figures S5−S8. The figures suggest marginal to significant effect of theseparameters for all the values of thiol to gold ratios. Figure S9 inthe Supporting Information shows the effect of adding reducingagent over 10, 120, and 900 s. For thiol to gold ratios largerthan 1/4, the rate of addition does not impact particle size.Clearly, the capping based control mechanism appears as a

far more robust way to control particle size than the reactivitybased mechanism. If the concentration of gold atoms and thecapping agents is kept constant with time, the “capping basedcontrol” mechanism predicts (through eqs 13 and 14) that theparticles whose growth has been arrested by capping (a = a )should all be of the same size. The continued nucleation underthese conditions produces monodispersed particles. One of thereasons for the observed polydispersity in BSM must beattributed to the falling concentrations of the reduced goldatoms and thiols as the gold precursor is exhausted and cappingagent is utilized to arrest particle growth.Synthesis of Monodisperse Particles Using Modified

BSM. The classical LaMer mechanism is invoked universally toexplain particle formation. According to this mechanism, thesynthesis of monodispersed particles requires a burst of nucleiformation followed by the pure growth of particles. Continuednucleation is considered as the main reason for poor control onparticle size distributions. The continuous nucleation−growth−capping mechanism unraveled in the present work on the otherhand suggests that if the conditions in the reaction system aremaintained constant, the particles born at different timesthrough continued nucleation will grow to the same size beforegetting fully capped. We can use this alternative route toattempt synthesis of more uniform size particles.In the modified strategy, detailed in the Supporting

Information, we use the same precursors as in BSM, in the

same quantities. The precursors are only contacted differentlyto realize constant reaction conditions. The experiments werecarried out for total thiol to gold ratio of 3:1. Instead of addingaqueous solution of reducing agent to the organic phasecontaining gold precursors and alkanethiols in 10 s,19 we addedphase transferred gold precursor in toluene containing 2/3 ofthe total 1-dodecanethiol to another toluene phase containingphase transferred sodium borohydride and the remaining 1-dodecanethiol. The former was added slowly over a total timeof 17 min to maintain the concentration of reduced gold atomsat a low level but constant over a period that almost covers theentire synthesis process. The capping agent is added to boththe phases to minimize the effect of finite rate of mixing. Theparticles synthesized using both the protocols were analyzedusing spectroscopy and TEM. Figure 8 shows a comparison ofthe TEM images (panels A and B), size distributions (panels Cand E), and absorption spectra (panels D) for the modified andthe original BSM. It is clear that the modified BSM has resultedin the formation of far more monodispersed particles than theoriginal BSM. A slight decrease in particle size with a decreasein addition rate, supported by both TEM images andabsorption spectra with a weaker shoulder, is consistent withthe earlier findings,19 and is irreconcilable with LaMer typemechanisms for particle synthesis or its variations.

■ CONCLUSIONS

We have brought together in this work a large body of earlierinvestigations on Brust−Schiffrin method (BSM) to show thatthe mechanism of particle synthesis and size tuning is not wellunderstood. After concluding that a thermodynamic mecha-nism is untenable, two competing mechanisms for size control,variation in precursors reactivity and by arresting growth ofparticles at different stages, are tested using a populationbalance based mathematical model. The model establishes thatthe reactivity based control mechanism with a short burst ofnuclei formation, similar to the classical mechanism, does notexplain all the experimental findings when taken together.The continuous nucleation−growth−passivation based mech-

anism explains (i) the size tuning obtained with BSM, (ii) nosignificant change in particle size with substitution of

Figure 8. Comparison of size distributions obtained by the modified and the original BSM for total thiol to gold precursor ratio of 3:1. Panels A andB are the TEM micrographs of BSM and the modified BSM, respectively. Histograms C and E are plotted by analyzing 181 particles from TEMmicrographs of BSM and the modified BSM, respectively. Panel D presents the UV−visible spectroscopy of both the samples.

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alkanethiols by dialkyl disulfides, (iii) the presence of a weakmaximum in time variation of mean particle size, (iv)insensitivity of particle size to the rate of addition of reducingagent, and most importantly (v) the need to use thiol to goldratio to tune the particle size instead of the conventionalapproach of varying the amount of reducing agent. Based onthe model driven insights, a modified protocol is testedexperimentally for the synthesis of more monodispersedparticles. The TEM measurements show significant improve-ment in size distribution in comparison with the originalprotocol.From the viewpoint of the theory of colloids, the continuous

nucleation−growth−passivation based mechanism is quiteinteresting. It does away with the notion that formation ofmonodispersed particles requires a short and an intense burst ofnucleation. The new mechanism shows for the first time thatslow and sustained nucleation with particle growth containedby capping of particle surface can also leads to the synthesis ofuniform size particles, in a far more robust way than is possiblewith the other mechanisms.

■ ASSOCIATED CONTENT*S Supporting InformationExperimental section covering gold nanoparticle synthesis usingthe original and the modified Brust−Schiffrin synthesis alongwith the representative TEM images. Results showing (i) thesensitivity of model predictions to the parameter values and (ii)the effect of rate of addition of reducing agent on particle size.This material is available free of charge via the Internet athttp://pubs.acs.org.

■ AUTHOR INFORMATIONCorresponding Author*E-mail: sanjeev@chemeng.iisc.ernet.in.NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSWe thank Prof. K S Gandhi and Dr. S Venugopal of the IndianInstitute of Science (IISc), Bangalore, and Prof. G U Kulkarniof Jawaharlal Nehru Centre for Advanced Scientific Research(JNCASR), Bangalore, for their intriguing discussions duringthe model development. We also thank Intensification ofResearch in High Priority Areas-Department of Science andTechnology (DST-IRHPA), India for extending financialsupport to carry out this work.

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dx.doi.org/10.1021/la401604q | Langmuir XXXX, XXX, XXX−XXXK