Modeling of Formation of Gold Nanoparticles by Citrate Method†
Sanjeev Kumar,*K.S.Gandhi,*and R.Kumar
Department of Chemical Engineering,Indian Institute of Science,Bangalore560012,India
Properties of nanoparticles are size dependent,and a model to predict particle size is of importance.Gold
nanoparticles are commonly synthesized by reducing tetrachloroauric acid with trisodium citrate,a method
pioneered by Turkevich et al(Discuss.Faraday Soc.1951,11,55).Data from veral investigators that ud
this method show that when the ratio of initial concentrations of citrate to gold is varied from0.4to∼2,the
final mean size of the particles formed varies by a factor of7,while subquent increas in the ratio hardly
have any effect on the size.In this paper,a model is developed to explain this widely varying dependence.
The steps that lead to the formation of particles are as follows:reduction of Au3+in solution,disproportionation
of Au+to gold atoms and their nucleation,growth by disproportionation on particle surface,and coagulation.
Oxidation of citrate results in the formation of dicarboxy acetone,which aids nucleation but also decompos
into side products.A detailed kinetic model is developed on the basis of the steps and is combined with
population balance to predict particle-size distribution.The model shows that,unlike the usual balance between
nucleation and growth that determines the particle size,it is the balance between rate of nucleation and
degradation of dicarboxy acetone that determines the particle size in the citrate process.It is this feature that
is able to explain the unusual dependence of the mean particle size on the ratio of citrate to gold salt
concentration.It is also found that coagulation plays an important role in determining the particle size at high
concentrations of citrate.
1.Introduction
Nanoparticles have potential for many applications,and their synthesis has assumed importance.Size obviously is their most important property,and many applications depend on it.Poly-dispersity in size is often an undesirable feature in the appli-cations.Thus,prediction of both size and dispersity for a given system is of concern in systematizing the manufacture of nano-particles.A variety of methods to synthesize nanoparticles are reported in the literature.A detailed account of the methods is provided by Cushing et al.2and Schmid.3Broadly,the methods can be classified into gas-pha and liquid-pha-bad methods.In gas-pha-bad methods,bulk material is evapo-
rated using high-energy sources such as resistive heating and lars to obtain a supersaturated gas pha,which,under con-trolled conditions,produces nuclei that grow to become nano-particles.In liquid-pha methods,also known as wet methods, precursors react to form a supersaturated solution,which nuc-leates and gives ri to particles ranging from1to100nm in size with stability ranging from a couple of hours to years.Wet-synthesis methods are attractive at least for two reasons:they are more energy efficient and they can be ud to produce nano-particles using the standard apparatus available in a laboratory. So far,two strategies have been followed for wet synthesis. In the first strategy,two reactants,usually both of them in (rever)micellized form,are mixed and nanoparticles form inside them by precipitation.The size of particles is controlled by the rates of nucleation and growth,and stabilization is provided by adsorption of surfactant.Nucleation is correlated to the size of micelles,and growth is determined by the rate of exchange of material between micelles.Both the features can be altered through chemical nature and concentration of surfactant.This strategy offers scope for good control over particle size and has been modeled by veral investigators. Results have been obtained by kinetic Monte Carlo simulations using the method of interval of quiescence in veral articles.4-6 Equivalently,mean-field population balance models have been developed for nanoparticle synthesis7-9and to study the effect of mixing on the synthesis process.10A large amount of surfactant is,however,ud in this method,which makes it exp
ensive.Recovery of ,by using nanofiltration, has been propod as a remedy,but it needs to be evaluated rigorously for determining the economics.
In the cond strategy,precipitation is carried out in bulk in the prence of stabilizers that adsorb on nanoparticles and prevent coagulation of particles.Most of the methods were first demonstrated for their ability to make gold nanoparticles and then extended to the synthesis of other types of particles, including tho of miconductors.Three widely ud bulk-precipitation-bad techniques for the synthesis of gold nano-particles are as follows:(i)citrate method of Turkevich et al.,1 (ii)citrate-tannic acid method of Muhlpfordt,11and(iii)Brust-Schiffrin method of Brust et al.12The first two methods yield particles which are stable against coagulation,whereas the last one produces particles which are also capped and cannot be grown bigger.Additionally,only the product of the Brust-Schiffrin method can be dried and redisperd.
In the simplest procedure for preparation of gold particles (Turkevich method),tetrachloroauric acid is reduced with trisodium citrate.Here,citrate is both the reducing agent and the stabilizer.Similarly,in the Brust-Schiffrin method,alkane thiols that provide long-term stability also take part in reduction of gold along with sodium borohydride,which is the main reducing agent.In all the methods,it is expected that the relative amounts of gold salt to reducing agent influence the particle size.The rate
of adsorption of stabilizer also plays a role in controlling particle size.Thus,a single reagent can play
*To whom correspondence should be addresd.E-mail:sanjeev@
in(Tel.,+91-080-22933110),gandhi@
in(Tel.,+91-080-22932320).Fax:+91-080-2360
8121.
†The authors are happy to contribute this investigation to the special
issue honoring Prof.M.M.Sharma.A part of this work formed the
basis for a paper prented at CHEMCON04held in Mumbai in
December2004.
3128Ind.Eng.Chem.Res.2007,46,3128-3136
10.1021/ie060672j CCC:$37.00©2007American Chemical Society
Published on Web10/07/2006
multiple roles,which creates more complexity.This feature combined with the fact that reactions that usually occur in the systems are fast makes the particle size more nsitively dependent on the concentrations of precursors,rate of addition of precursors,state of mixing in the reactor,and other variables.While this nsitivity requires strict adherence to protocols that work,it also permits particle size and polydispersity to be changed substantially merely by changing the concentrations of the precursors.All the three synthesis protocols listed above have been subquently modified by changing the ratio of common precursor tetrachloroauric acid to other reagents to synthesize nanoparticles of significantly different sizes.13,14Although direct precipitation using the citrate method is in wide u becau it produces uncapped but stable particles,a quantitative model to predict particle size and the associated distribution is not available.The objective of this paper is to develop a quantitative model for the citrate method of Turkevich et al.1Becau the chemistry involved and the process of nucleation and growth of particles are different from one another in the three methods,the model developed here is for the experimental method of Turkevich et al.1only.2.Features of the Citrate Process
优惠价We consider the formation of gold nanoparticles by reduction of tetrachloroauric acid by trisodium cit
rate.The method has been pioneered by Turkevich et al.,1and variants of it are still widely
ud 15to produce gold nanoparticles.Becau of its simplicity,various facets of the process have been investigated by Frens,13Freund and Spiro,16Chow and Zukoski,17Abid,18and others.In this process,an aqueous solution of tetrachloro-auric acid is brought up to boiling and a small volume of trisodium citrate is then added to it.In ∼10s,bluish color appears,indicating formation of gold nuclei.In a few minutes or less,the solution turns brilliant red becau of the formation of nanoparticles.The completion of reaction may take tens of minutes depending upon the amount of citrate taken.
A plot of mean particle size against the ratio of initial concentrations of citrate to gold species is prented in Figure 1.It summarizes data from Turkevich et al.,1Ferns,13Freund and Spiro,16Chow and Zukoski,17and Abid.18All the data shown in this figure,except tho of Chow and Zukoski,17are obtained for approximately the same concentration of auric chloride and by varying concentration of sodium citrate.It is interesting to note that the data spanning a 50-year period show a good correlated trend.
Figure 1reveals veral interesting features.First is the widely varying dependence of particle size on the ratio of citrate to gold.When this ratio increas by a factor of 5,from ∼0.4to ∼2,the particle size decreas by a factor of 7,which
corresponds to a change of nearly 3orders of magnitude in volume.Yet,when the ratio increas from about 2to 7,there is very little change in the particle size.On the other hand,when the concentration of sodium citrate is kept fixed and that of auric chloride is varied,Chow and Zukoski 17found that the particle size varies nsitively over a wide range of auric chloride concentrations investigated.With a decrea in initial concentra-tion of auric chloride,particle size first decreas,pass through a minimum,and then increas (the experimental data are shown in Figure 5).
Another interesting feature is that complete conversion of auric chloride has been reported at all the ratios of citrate to gold.Data with small ratios of citrate to gold were reported by Frens,13and the least of them corresponds to a ratio of ∼0.43.A model is prented below that attempts to explain the unusual features quantitatively.3.Model
3.1.Chemical Reactions.Gold nanoparticles have been synthesized using the citrate method for a long time now,and although most reactions occurring in the method and the intermediates formed in the process are known,some of the steps are still not fully understood.In the following discussion,we point out the and state how they are modeled in this work.The initial step of this multiple-step process,with reactions occurring in ries and parallel,is the oxidation of citrate,which yields dicarboxy acetone:
The cond step is the reduction of auric salt to aurous salt:
从句讲解The next step is the disproportionation of aurous species to gold atoms:
The disproportionation step requires three aurous chloride molecules to combine.This is facilitated by dicarboxy acetone,which,according to Turkevich et al.1and Abid,18plays the role of organizer through the formation of a complex.An illustration of the complex as visualized in this work is prented in Figure 2.In the chain-like structure pictured here,three Au +can be tethered by a minimum of two dicarboxy acetone molecules.
Figure 1.Variation of particle size with ratio of initial concentrations of citrate to gold salt.
Figure 2.Illustration of complex of aurous species and dicarboxy acetone.
AuCl 3+2e -f AuCl +2Cl -
(2)
3AuCl f 2Au o +AuCl 3
(3)
Ind.Eng.Chem.Res.,Vol.46,No.10,20073129
Once gold particles are formed,disproportionation can occur 19on the particle surface,too.
The overall stoichiometry of the reduction reaction can then be reprented as
If the were the only reactions that led to the formation of gold atoms,the overall stoichiometry would require three citrate molecules to reduce two auric chloride molecules.Since at high temperatures dicarboxy acetone is lost in side reactions 1to form acetone,
the stoichiometric ratio of citrate to gold
required for complete conversion of auric chloride should be even larger than 1.5.In contrast,Frens 13reports complete conversion of auric chloride at
a stoichiometric ratio as low as 0.43.He,however,does not discuss the chemistry involved.It appears reasonable to conclude that some products of degradation of dicarboxy acetone reduce trivalent gold,though the chemistry leading to the formation of gold is not clear.Turkevich et al.1and Davies 20have shown that acetone can reduce auric chloride to produce gold particles,though the conditions in the two experiments are different.In view of the foregoing,we assume that it is acetone formed by degradation of dicarboxy acetone that additionally reduces auric chloride for its complete conversion.To explain the obrvations of Frens,13we assign a stoichiometric ratio of 4for this reaction.
This allows for complete reduction of ∼3mol of auric chloride by 1mol of citrate.This assumption influences the model results only for ratios of citrate to gold <1.5.Becau the particle size obtained at the ratios is large,the practical significance of this regime may be less.
3.2.Overall Mechanism of Particle Formation.On the basis of the above t of chemical reactions,the following mechanism of particle formation and growth is propod.Au 3+is reduced in solution by citr
ate to Au +,which forms a multimolecular complex with dicarboxy acetone.The complex disproportion-ates,and gold atoms are formed.The atoms adsorb Au +and,by complexation with dicarboxy acetone,form large aggregates.Further disproportionation leads to formation of still larger aggregates of gold atoms.When the size of the aggregate reaches a critical value,a nucleus of gold atoms is formed.The gold atoms produced by disproportionation are absorbed by the particle,resulting in its growth.
3.3.Nucleation.Turkevich et al.1have investigated the nucleation process in the citrate method in detail.They considered nucleation to be possibly due to the prence of an impurity,statistical fluctuations,or some other mechanism.On the basis of their experimental findings,they ruled out the role of impurities and statistical fluctuations in nucleation and concluded that dicarboxy acetone plays a critical role in nucleation through an organizer mechanism by forming a complex with gold ions.Becau no other mechanism for nucleation of gold particles in the citrate process has been
propod,in this work,we have attempted quantification of the rate of nucleation by modeling the organizer mechanism through a chemical reaction step.
The number of gold atoms needed to form a nucleus is an open question.The so-called magic numb
ers determine the size of stable clusters,but in the abnce of stabilizing agents.A number of Au n clusters with n values of 10,15,18,etc.protected 21by glutathione are also reported,but they are not relevant here.A fundamental model is needed to predict the number of atoms that lead to the formation of a stable nucleus in the actual system.In the prent work,we have taken the size of the nucleus to be 2nm,as reported by Turkevich et al.1and Turkevich.22Becau the mass contained in a nanoparticle of typical size of tens of nanometers is much greater than that contributed by a nucleus (of 2nm diameter),we keep the model simple by using an overall rate expression for the nucleation process to capture the formation of nuclei.(Sensitivity of model predictions to this parameter is discusd later.)
3.4.Coagulation of Particles.Yet another issue to be addresd in the model is coagulation.There is some controversy regarding the stability of particles at small ratios of citrate to gold.For example,Turkevich et al.1reported that particles are unstable for a citrate-to-gold ratio <0.5.Frens,13however,reported formation of stable particles at a ratio of ∼0.4as well.Since neither citrate nor gold salt is left unreacted at the ratios,identification of the stabilizing agent is an issue.Li et al.23have recently reported that acetone can stabilize gold nanoparticles.Becau acetone is produced in the prent system,it can be expected to play this role.pte多少钱
Turkevich and co-workers 1obrved coagulation or ripening of nanoparticles of gold at high citrate concentrations and investigated the effect of addition of salts on coagulation in a ries of subquent papers.24They identified that coagulation is promoted becau of the compression of double layers by the high sodium ion concentration.Chow and Zukoski 17investigated this aspect systematically.Their experimental data (prented in Figures 1and 5)shows an increa in particle size at high counterion concentration,which suggests that coagulation of particles has to be considered in the model.are amorphous and can coalesce.The agglomerate becomes structured as it grows.Coalescence may be expected,since the melting point of clusters of atoms decreas with their size 25and surface atoms are quite mobile at temperatures well below the melting points.26It is perhaps also related to the lack of a sufficient number of adsorption sites on small particles,and hence,it is possible to stabilize only large particles.In this connection,it is interesting to note that particles of size smaller than ∼2.5nm have not been synthesized by citrate and citrate -tannic acid methods.This size corresponds to a particle formed by coalescence of two particles with a magic number of 55(∼2nm,the size of the nuclei chon in this work).Becau the fine structure of the particles is not of direct interest here,we do not assign any enhanced rate of coagulation to nuclei in the current investiga-tion.It may be mentioned in passing that Ostwald ripening does not occur at the temperatures of synthesis and need not be accounted for in the model.
ticket是什么意思
(CH 3)2C d O +4AuCl 3f 4AuCl +products (6)
3130Ind.Eng.Chem.Res.,Vol.46,No.10,2007
3.5.Model Equations.For the purpos of prediction of particle size,a model should quantitatively account for (i)reduction of Au 3+to Au +in the bulk,(ii)growth of particles by absorption of gold atoms formed by disproportionation on the particle surface,(iii)rate of nucleation,and (iv)growth due to coagulation.We consider the in detail below.
3.5.1.Scheme of Reactions.Let T and C reprent the auric and citrate species,respectively.Let M and S reprent the aurous and dicarboxy acetone species,respectively.Let D reprent acetone.The model can then be reprented by the following equations.
Becau the nucleation and growth of particles can occur
simultaneously,gold particles will not be monodisperd.Let P (V )d V reprent the number concentration of gold particles in the system in the size range V to V +d V .All the particles including tho formed by coagulation are considered to be spherical in shape.
3.5.2.Kinetics of Reductions in Bulk.Let us first consider the reactions involving trivalent auric specie
s.It is consumed by the homogeneous reduction reaction and produced in the disproportionation reaction.The reduction is assumed to be first order with respect to citric and auric species.It is also consumed in the reaction with acetone.We assume the rate of this reaction also to be first order with respect to citrate and acetone.
3.5.3.Kinetics of Disproportionation.Disproportionation of monochloride at various temperatures has been reported by Gammons et al.19At 100°C,equilibrium is heavily in favor of trichloride.This reaction can occur in the bulk but is catalyzed 19by the gold surface since Au +adsorbs on it.The surface of gold particles thus acts as a catalyst,and it is this mechanism that is dominant in the citrate process.Becau the stabilizer also competes for the surface of gold particles,and the concentration of aurous chloride is expected to be small in view of the unfavorable equilibrium,it is likely that the rate of the disproportionation reaction is controlled by the rate of adsorption of aurous chloride.This might also be expected since diffusional resistances are negligible at the length scales.Thus,the rate of this reaction on each particle is assumed to be proportional to the surface area of the particle and the concentration of aurous chloride [M].Simultaneous with the disproportionation of 3mol of aurous species,1mol of auric species is formed.
3.5.
4.Rate of Nucleation.As per the discussion in earlier ctions on the nucleation process,three molecules of aurous species are required for disproportionation to occur and at least two molecules of dicarboxy acetone are required for complexing aurous species in solution .In view of this,it was assumed that the order of the nucleation reaction is 3with respect to aurous and 2with respect to dicarboxy acetone.Nucleation occurs when a sufficient number of gold atoms agglomerate in the solution to reach critical size.
3.5.5.Mass Balance of Species.The mass balance of auric species in a batch system is,therefore,given by
where V o is the size of the nucleus.The last term on the right-hand side of the above equation accounts for the production of
auric chloride due to disproportionation leading to nucleation events.
Aurous chloride is formed through the reaction of auric chloride with citrate and acetone and is consumed when nuclei are born.Thus,the mass balance of aurous species is identical to the above with changed stoichiometric coefficients:
The mass balances for citrate and dicarboxy acetone can be written along the same lines:
The initial conditions for the above equations are given by
3.5.6.Population Balance.The above t of equations can
be solved once we find the number concentration of particles.A population balance equation for the number density of particles is ud to determine it.The number of particles in a size range changes when smaller particles grow into it and when particles in it grow out of it through the convective process of surface reaction.The Brownian coagulation of particles also results in birth and death of particles in the size range considered above.All the process are accounted by the following number or population balance equation:
The first term on the right-hand side accounts for the convective growth process due to surface reaction.The Brownian collision frequency,q (V ,V ′),is given by
Homogeneous reduction:T +C 98k c
M +S (7)变量英文
Heterogeneous disproportionation:
3M 98k h
particle
T +particle mass (8)Nucleation:3M 9
8k n ,S
nucleus +T (9)Degradation of dicarboxy acetone:S 98k s
D
(10)
Reduction by acetone:D +4T 9
8k d
4M +products (11)d[N]
d t
)2k n [M]3[S]2(12)
d[T]
d t
)-k c [C][T]-k d [D][T]+k h [M]
∫V o
∞V 2/3P (V )d V +k n FV o [M]3[S]2
(13)
d[M]
d t
)k c [C][T]+k d [D][T]-3k h [M]
英译汉∫V o
∞V 2/3P (V )d V -3k n FV o [M]3[S]2
(14)
d[C]
d t
)-k c [C][T](15)d[S]
d t
)k c [C][T]-k s [S ](16)d[D]d t )k s [S]-1
4
k d [D][T](17)
fag[T])[T]o ,[C])[C]o ,[M])[S])[D])0(18)∂P
∂t )-2F k h [M]∂∂V (V 2/3P )-P (V )∫
V o ∞q (V ,V ′)W
P (V ′)d V ′+
12∫
V o V q (V -V ′,V ′)
W
P (V -V ′)P (V ′)d V ′+
δ(V -V o )2k n [M]3[S]2(19)
q (V ,V ′))
2k u 3µ
[V -1/3
+V ′-1/3][V 1/3+V ′1/3]Ind.Eng.Chem.Res.,Vol.46,No.10,20073131
Further,W is the ratio of the number of collisions to tho that result in coagulation.The cond and t
hird terms on the right-hand side thus account for loss and gain due to Brownian coagulation,respectively.The last term accounts for the rate of creation of particles at size V o by nucleation.Here,δ(V-V o)is the Dirac delta function.Initially,the particle concentration is zero.One boundary condition needed to solve the equation for P(V)is provided as P(V o,t))0.
3.5.7.Coagulation Efficiency.Coagulation occurs when two particles colliding with each other becau of their Brownian motion overcome the repulsive force between them and reach primary minimum irreversibly.Repulsive force that resists approach of particles before they reach primary minimum is caud by double-layer interaction between the particles and steric repulsion due to the adsorption of acetone on the particle surface.Only a fraction of the collisions leads to coagulation. Thus,the stability of particles against coagulation is reprented by the stability factor W defined as28
In this paper,we incorporate all the interactions between the particles through repulsive forces of ionic origin and consider only irreversible coagulation of particles.Chow and Zukoski17 have shown that particles also reversibly aggregate in an open cluster,which breaks apart in later stages of synthesis.Since the particles are found to grow in the clusters unhindered (evidenced by monolith product),we ignore this aspect in the prent model.
Reerink and Overbeek29have developed a simplified expres-sion for W,applicable when interaction potential posss a relatively sharp maximum.The expression is given by Here,C e is the concentration of the counterions.Constant k1 varies with surface potential,and k2depends on the physical system only.The surface potential depends on coverage of the surface by the potential determining ions.Auric and aurous chlorides,dicarboxy acetone,acetone,and citrate ion are all possible candidates for adsorption on gold particles.Further-more,as the concentrations of the species change in the cour of particle formation,W also changes.To ascertain the basic features,veral simplifications need to be made.As a first step, we assume that adsorption of potential-determining ions is at equilibrium with the bulk.Since equilibrium data are not available,surface concentrations of ions can still not be estimated.The following approximate but simpler approach is pursued in the prent model.
Chow and Zukoski17examined simultaneous adsorption onto gold particles from solutions of citrate and tetrachloroauric acid (HAuCl4).They propod that the auric chloride adsorbs strongly on gold particles in comparison with citrate.Thus,the fraction of area occupied(f)is biad toward gold species,which we incorporate in the prent work using the following expression:
Surface potential,which depends on the adsorbed ions,is dif-ficult to measure or estimate from theor
y.17The data of Chow and Zukoski17indicate that the surface potential is about -90mV under some conditions.The charge on aurous is equal to that of auric but is three times smaller than that on citrate. To allow for variation in surface potential with the changing ionic environment in the bulk,the following empirical expres-sion is ud:
The somewhat higher value of surface potential ud here compensates for the steric stabilization that is not incorporated in the model.Coagulation efficiencies,theoretically calculated and reported by Chow and Zukoski,17were fitted approximately to the expression of Reerink and Overbeek29to obtain
The initial counterion concentration is given by the sum of3×the citrate concentration and the concentration of HAuCl4. 4.Results and Discussion
The model equations were converted to nondimensional form before solving them.All the concentrations were nondimen-sionalized using[T]o.Particle populations were scaled using N max,the maximum number of particles per unit volume that can be generated in the ,
Time is scaled using(k c[T]o)-1,the time scale for reaction between citrate and auric ions.The final t of nondimensional equations has the following nondimensional groups in it. The initial conditions
for the nondimensionalized[T]and[C] are1and R()[C]o/[T]o),respectively.All the other variables have zero value at the initial time.Model predictions require values for rate constants k c,k d,k h,k n,and k s.The nondimensional groups identified above indicate that,if the particles are stable against aggregation,only the ratios of the five rate constants need to be fixed to predict particle size distribution.K d should be smaller than unity becau the time taken for process completion with a limited amount of citrate is significantly larger than when citrate is in excess.The best-fit values of K h,K n, and K s required to predict the experimental data shown in Figure 1for a[T]o value of3×10-4M,the concentration ud by Frens,13were obtained.The value of F is taken to be0.1M/cm3 and V o is4.18×10-21cm3,corresponding to a nucleus of2 nm in size.The value of rate constant k c was fixed so as to
W)
number of collisions between particles
number of collisions that result in coagulation
log W)-k
1
log C
e
+k
2
(20)
f)
1
1+0.1
[C]
[M]+[T]
φ)-90[f+1.5(1-f)]
log W)
560
φ
log(3[C]
o
+[T]
o
)×10-4+27.5
N
max
)
[T]
o
FV
o
K
h
)
k
h
k
c
FV
o
1/3
(21)
K
n
)
k
n
k
c
FV
o
俄语学习[T]
o
3(22)
K
惊羡s
)
k
s
k
c
他妈的英文
[T]
o
(23)
K
d
)
k
d
k
c
(24)
K
q
)
1×10-3
k
c
FV
o
·
2k
B
T
3µ
(25)
3132Ind.Eng.Chem.Res.,Vol.46,No.10,2007
