Oriented Attachment Mechanism in Anisotropic Nanocrystals: A “Polymerization” Approach

DOI: 10.1002/cphc.200500508

Oriented Attachment Mechanism in AnisotropicNanocrystals: A “Polymerization” ApproachCaue Ribeiro, Eduardo J. H. Lee, Elson Longo, and Edson R. Leite*[a]

1. Introduction

Tailoring the shape and properties of nanostructures is one ofthe major challenges that must be overcome to allow thelarge-scale use of nanotechnology. The controlled preparationof nanocrystals with specific size and shape has been exten-sively investigated in studies involving the synthesis of nano-particles[1–9] and methods for growing anisotropic crystals suchas nanowires,[10,11] nanoribbons and nanorods,[12–14] andothers.[15] However, this is still a very active field strongly moti-vated by the preparation of nanostructures with tailored mor-phologies using nanoparticles as building blocks.

In this sense, investigations into growth and coarseningmechanisms in the nanometer range are fundamental to aclear understanding of how anisotropic nanostructures areformed. Recent efforts in this field have focused on anisotropicgrowth in various systems in which particle coarsening can bewell explained by the Ostwald ripening mechanism,[16–19] suchas ZnO,[20,21] TiO2,

[21,22] etc. These systems are characterized bysignificant solubility of the crystals in the liquid medium, whichallows for surface diffusion and generally results in sphericalparticles. However, some studies have demonstrated the possi-bility of fostering anisotropic growth in these systems by main-taining the solute concentration higher than the equilibriumconcentration.[5,13, 23] This prevents the smaller particles frombeing dissolved, and growth occurs by precipitation of dis-solved ions in high-energy facets instead of coarsening. On theother hand, oriented attachment (OA) has also proved to bean effective mechanism for particle coarsening, and may leadto anisotropic growth.[6,24–32] Despite its importance, the role ofOA coarsening has not featured in the majority of studies onthe anisotropic growth of nanostructures. As a matter of fact, arecent paper by Yu and collaborators[33] postulated that theOstwald ripening mechanism may act in anisotropic structuresin the smoothing of surfaces and edges by diffusion into thecrystal. However, the authors ascribe the actual formation ofthe structure to the OA mechanism.

This mechanism (also known as the grain coalescence mech-anism) can be described as the annihilation of the boundariesof two crystallographically aligned particles, thermodynamicallydriven by the reduction of interfacial energy.[34–37] Previous-

ly[38,39] we have proposed that the OA mechanism may occurin two ways for colloids: 1) nonaligned particles suffer crystal-lographic rotations until a favorable geometry is obtained andcoalescence takes place, and 2) coalescence may occur by ef-fective collision between particles with the same crystallo-graphic orientation. The former is possible when particle-to-particle contact occurs, as in agglomerates, for example, whilethe latter is the dominant mechanism in dispersed suspen-sions. The mechanism in dispersed conditions treats the nano-particle analogously to a molecule, and the first attachmentstep is the following molecular reaction [Eq. (1)]:

Aþ A ! B ð1Þ

that is, two particles coalescing and resulting in one, whichoccurs in consecutive steps. Intermediate steps may be relatedto the yield of effective collisions or to agglomeration and re-alignment processes. This interpretation has been used bysome researchers aiming to gain a more thorough understand-ing of the growth behavior in colloidal suspensions of severalsystems, such as ZnS,[40,41] SnO2,

[42,43] and TiO2[44] . Some authors

have reported that the formation of highly anisotropic struc-tures (e.g. nanowires) occurs by OA and depends on the exis-tence of a mechanism for organized agglomeration. Tang andco-workers[10] discussed the formation of CdTe nanowires frombuilding-block nanoparticles, whereby the shapes achievedwere brought about by a spontaneous orientation betweenthe nanoparticles in response to dipole interactions in theliquid medium. Cho and co-workers[32] used the same argu-ment to explain the formation of PbSe nanowires and nanor-ings in solution.

Here, we attribute the alignment explicitly to dipole interac-tions as a first step in the process, with oriented attachment as

[a] C. Ribeiro, E. J. H. Lee, Prof. Dr. E. Longo, Prof. Dr. E. R. LeiteUniversidade Federal de S¼o Carlos, Departamento de Qu!micaRod. Washington Luiz, km 235—13565-905, S¼o Carlos, SP (Brazil)Fax: (+55)16-3361-5215E-mail : [email protected]

The kinetic model of stepwise polymerization is revisited, withsome adaptations for its application to the kinetics of orientedattachment of nanoparticles in colloidal suspensions, which re-sults in the formation of anisotropic particles. A comparison with

experimental data reported in the literature shows good agree-ment with the model and supports comparisons with other sys-tems.

664 ? 2006 Wiley-VCH Verlag GmbH&Co. KGaA, Weinheim ChemPhysChem 2006, 7, 664 – 670

the subsequent step. Jun et al.[13] suggested that, under theseconditions (growth by OA), a linear or exponential kineticgrowth behavior should follow. This suggestion can be inter-preted as the formation of a “polymeric” particle, where the“monomer” is the primary particle. This interpretation is quiteplausible; in sol–gel methods, the formation of inorganic poly-mers by the polycondensation of metal hydroxides, which sub-sequently generate metal oxides by the elimination of watermolecules, is a widely accepted concept.[1] Also, the OA poly-merization is similar in some aspects to self-assemblygrowth,[45] except that, in this case, the particles are linked bystrong chemical bonds. Finally, we can understand this processas a non-classical growth process, concurrent or precedent tothe Ostwald ripening coarsening, which is present in severalgrowth processes in nature and in synthetic procedures.

In this article we discuss the formation of anisotropic nano-particles in analogy to a “polymer”, adapting the classicalmodel to the stepwise polymerization[46] described by Flory[47]

(reviewed in ref. [48]). Some experimental results reported inthe literature in hydrothermally treated SnO2 nanoparticles—amodel material, since the Ostwald ripening mechanism ishighly unfavored and hence OA is the only coarsening mecha-nism—are analyzed according to the OA polymerization con-cept, showing the applicability and the deviations of this prop-osition. In addition we discuss results, reported in the litera-ture, obtained from hydrothermally treated TiO2 and from thesynthesis of anisotropic anatase particles, which has been wellstudied and reported.

2. The “Polymeric” Attachment Approach

A small modification of the previously proposed OA coarseningmodel[39] is necessary to facilitate the interpretation of poly-merization attachment. In this case, the reaction may be inter-preted as the junction of two active surfaces. For the sake ofsimplicity, we consider that a primary particle in suspension[identified as A in Eq. (1)] behaves as an S�A�S molecule,where S represents an active surface and A corresponds to thebody of the particle, as shown schematically in Figure 1. Weassume that the coalescence events follow Equation (2):

S�A�Sþ S�A�S ! S�AA�S ! S�ðAÞx�S ð2Þ

The particle S�(A)x�S is defined as a single particle that hasundergone a coalescence sequence of x primary particles.Since the active center in all the chemical equations is S, andassuming k is constant throughout the process, the consump-tion rate of active surfaces is then expressed by Equation (3):

u ¼ dSdt

¼ �k½S�2 ð3Þ

The second-order rate law [Eq. (4)] can be solved easily byintegration:

½S� ¼ ½S�01þ k½S�0t

ð4Þ

where S0 is the initial concentration of active surfaces. Accord-ing to the initial assumptions, S0=2A0, where A0 is the initialconcentration of particles.

The probability P of a particular link occurring between twoparticles, is defined in terms of the total amount of active sur-faces reacting at time t, is given by Equation (5):

P ¼ ½S�0 � ½S�½S�0

ð5Þ

By this definition, the probability of the inexistence of linksbetween two particles is 1�P. In a system where x configura-tions are possible, the number of links in a single structure isx�1 and each structure will be left with two potential linkages(i.e. two active surfaces) (see Figure 1). Hence, the probabilityof the existence of a particular configuration is Px�1·(1�P)2.Therefore, the probability of the existence of any configurationx [Eq. (6)] is given by:

Px ¼ x Px�1 ð1� PÞ2 ð6Þ

Equation (6) can be interpreted as the probability distribu-tion for particle configurations. Figure 2 shows examples ofsuch distributions of configurations, obtained by assumingconsumptions of 50, 66, 75, 80 and 83% of the total amountof reactants. The configurations can be represented as a varia-ble, the coalescence degree, that is, the number of particles re-quired to form a single coalesced particle. The plotted curvesdepict an important feature: the average size is not an ade-quate parameter to represent linear growth, especially in theinitial stages, since the curves are not symmetrical. In thissense, the most probable configuration (i.e. the most probabledegree of coalescence) would represent the growth more ade-quately. Since the most probable configuration is given bydPx/dx=0, one has [Eq. (7)]:

dPx

dx¼ ð1� PÞ2 ðPx�1 þ xPx�1 ln PÞ ð7Þ

The solutions are xmin=1 and xmax=�1/lnP, where 0<P<1.Substituting P [Eq. (5)] into the second solution, one has [Eq.(8)]:Figure 1. Scheme of the coalescence events represented by Equation (2).

ChemPhysChem 2006, 7, 664 – 670 ? 2006 Wiley-VCH Verlag GmbH&Co. KGaA, Weinheim www.chemphyschem.org 665

Oriented Attachment in Nanocrystals

www.chemphyschem.org

xmax ¼ � 1

ln ½S�0�½S�½S�0

� � ð8Þ

Replacing the value of [S] as defined in Equation (4):

xmax ¼ � 1

ln k½S�0 t1þk½S�0t

� � ð9Þ

The solution xmax [Eq. (9)] gives the most probable configura-tion for a given t and represents a more consistent interpreta-tion of the growth behavior of anisotropic nanocrystals: the in-itial stages are determined by a strong logarithmic behaviorfollowed by a linear-type behavior for high consumption of re-actants, since �1/lnP�1/(1�P) for P near 1, that is, xmax�1+k[S]0t in this condition. At this limit, the most probable configu-ration is equal to the average configuration.

The existence probability, as described by Equation (6), canbe interpreted as the distribution of configurations, or as distri-butions of particle sizes or weights. Figure 3 presents themean degree of coalescence, which is closely related to theconfiguration distribution, for a SnO2 colloidal sample subject-ed to hydrothermal processing for 24 h at 200 8C, as well asthe fitting of the experimental data with Equation (6). Theresult shows a significant congruence between the experimen-tal data and the fitting, where the value of P obtained (0.76) isvery representative, since this value, when applied to Equa-tion (5), results in an average degree of coalescence of 4.2 (i.e.a particle in the system is composed, on average, of 4.2 pri-mary particles).

2.1. The Constant Rate k

Although an approximate calculation indicated that k assumesa constant value during the overall process [see Equations (3)and (4)] , this can be a reasonable assumption for extended re-action times. On the other hand, with shorter times, the mobi-

lity of coalesced particles is strongly affected in relation to un-coalesced particles. We can estimate the dependence of k onthe size of the particles involved in the reaction, using the ki-netic theory of gases. According to this approach, the collisiondensity of pairs in random systems is given by the collisioncross section of the two particles involved (defined asp(R1+R2)

2, where R1 and R2 are the radii of the two particles),and by the reduced mass of the two particles (1/m1+1/m2), asfollows [Eq. (10)]:

k ¼ Psteric pðR1 þ R2Þ2 1m1

þ 1m2

� �12

ð8kBTN2AÞ

12e�Ea=RT ð10Þ

where Psteric is the steric factor (the probability of a successfulcollision), kB is the Boltzmann constant, T is the temperature,NA is Avogadro’s constant, Ea is the activation energy and R isthe universal gas constant. Since it was also assumed that theparticles formed are linear, the probability p of a collision atthe extremities is determined by the gyration ratio of the ani-sotropic particle: [Eq. (11)] ,

p � 22RpnR

¼ 4pn

ð11Þ

where n is the number of “monomers” in the particle. Now,considering two particles, as illustrated in Figure 4, composedof n and m “monomers”, the probability of a successful colli-sion between two particles (Psteric) is given by Equation (12):

Psteric ¼42

p2

1nm

ð12Þ

Similarly, the collision cross-section will be determined bythe gyration ratio of the anisotropic particles. Finally, themasses of the particles are equal to nMinitial and mMinitial, whereMinitial is the mass of the primary particle. Substituting thiswhole expression into Equation (10), and regrouping all theconstant terms into a single constant, k0, one obtains Equa-

Figure 2. Examples of the distribution of configurations or coalescence de-grees for different values of P (ad.=adimensional).

Figure 3. Distribution of particle sizes in terms of the coalescence numberfor hydrothermally processed SnO2 sample, fitted using Equation (6).

666 www.chemphyschem.org ? 2006 Wiley-VCH Verlag GmbH&Co. KGaA, Weinheim ChemPhysChem 2006, 7, 664 – 670

E. R. Leite et al.


tion (13):

k ¼ k0 1nm

ðnþmÞ2 1nþ 1m

� �1=2

ð13Þ

Considering the specific case of n=m, one obtains Equa-tion (14):

k ¼ 4ffiffiffi2

pk0

1n

ð14Þ

This result [Eq. (14)] shows that the value k cannot be statedas constant in the entire range of the reaction. Because k be-haves as y=1/x (as illustrated in Figure 5), the value should beconsidered as constant only during extended reaction times.Considering that n is equal to [S]0/[S] and replacing this valuein Equations (14) and (3), one obtains a rate law [Eq. (15)] ex-pressed as:

d½S�dt

¼ k0

0½S�3 ð15Þ

where k0

0 ¼ 4ffiffiffi2

pk0=½S�0. The integration of the above expres-

sion results in Equation (16):

½S� ¼ ½S�0ð1þ 8

ffiffiffi2

p½S�0k0tÞ1=2

ð16Þ

Applying this result to the definition of P and xmax [Equa-tions (5) and (8)] , one obtains a general form of the time de-pendence of xmax. However, the exponent 1/2 in Equation (16)is obtained through the approximation n=m, that is, the colli-sion of two particles of equal coalescence number. Based onthe above, we suggest that a more coherent approach is toassume the existence of an exponent a, where 0<a<1, inwhich case the general expression takes on the form of Equa-tion (17):

xmax ¼ � 1

a ln 8ffiffi2

pk0 ½S�0t

1þ8ffiffi2

pk0 ½S�0t

� � ð17Þ

This final expression was fitted with good agreement toprocessing SnO2 hydrothermal colloid experiments reported byLee[43] , as illustrated in Figure 6. Three different conditionswere used—100 8C, 200 8C and a four-fold-diluted 200 8C sus-pension—and the fittings obtained proved congruent with theexperimental data. The higher deviation observed in the sam-ples subjected to 200 8C in more concentrated suspensionsmay have been related to growth effects, possibly occurringduring heating of the samples.

A comparison of the two sets of experiments carried out at200 8C reveals the effect of the number of collisions on the

Figure 4. Scheme of the collision between two particles formed by n and mprimary particles.

Figure 5. Behavior of k/k0 with the coalescence degree.

Figure 6. Fitting of experimental data in hydrothermallized SnO2 suspen-sions under three different conditions: 100 8C, 200 8C, and 200 8C in dilutedconditions.[43]




growth of the particles : since [S]0 is higher in more concentrat-ed suspensions, the frequency of collisions is higher than in di-luted conditions and the variation of the effective k is different.In fact, the values obtained for the initial kinetic parameters(the constant 8

ffiffiffi2

pk0=½S�0) and for the exponent a (Table 1) in-

dicate that the more aggressive conditions imposed at 200 8Cin concentrated suspensions led to a growth behavior domi-nated by the coalescence of larger particles (an exponent a of0.64 was obtained, which was close to 0.5, the value obtainedfor n=m conditions), while, in the other conditions, the expo-nent showed a higher dispersion of events.

An evaluation of the model over extended periods of time isperformed using the experimental data of Penn and Ban-field,[27] for the case of anatase growth in hydrothermal experi-ments. The authors observed the growth of highly aligned andcoalesced anatase nanoparticles, generating linear particlesalong the [001] plane. The growth was monitored over longperiods (up to 300 h) in the [001] plane and<101>direction,and the authors suggested a two-step growth mechanism,with linear behavior prevailing over extended periods of time.The results can be compared in terms of the degree of coales-cence by dividing the length in [001] by 5 nm, related to thesize of the initial crystals. The fitting of the experimental data(Figure 7) shows a good correlation, albeit impaired in the final

measurements. The deviation in this case can be attributed todiffusion effects, such as accommodation at the interfaces ofthe coalesced particles, or association with other growthmechanisms. Nevertheless, the model is reasonable underthese conditions and agrees with the conclusions regarding atwo-step growth behavior.

In the two conditions discussed above, one condition as-sumed in the model—the existence of only two active surfa-ces—cannot be guaranteed since, in the hydrothermal treat-ment of colloids, growth is determined statistically. In thissense, some of the systems reported in the literature, in whichgrowth occurs in the presence of surfactants adhering to spe-cific surfaces of the primary particle, are better suited to themodel, for example, the growth of TiO2–anatase nanorods re-ported by Jun and co-workers.[13] These authors measured thelength of nanorods formed by the surfactant-assisted elimina-tion of high-energy facets in a crystal seed, observing a bimo-dal distribution. The authors explained this growth behavior aspreferential growth in two distinct crystallographic directions.Based on this assumption, we can interpret each distributionas a different reaction route with distinct kinetic parameters.Assuming that the oriented attachment mechanism is predom-inant, we can apply the considerations used here to fit thegrowth behavior, as done in Figure 8. The fittings of both

curves are representative and coherent, the fitting parametersobtained, shown in Table 1, are closer to each other than ex-pected. The applicability of the model, in this case, can be con-firmed by the nanorod diameters observed during the sametime interval, (inset in Figure 8). The absence of diametralgrowth is a strong indication that the growth occurredthrough the attachment of previously formed nanoparticles insuspension, by a mechanism very similar to that describedhere. It is also important to note that the fast growth of nano-

Table 1. Values obtained from the fit to experimental data presented inFigure 6, 7 and 8.

8ffiffiffi2

pk0=½S�0 a

SnO2 hydrothermallized at 200 8C 0.55 0.64SnO2 hydrothermallized at 100 8C 0.56 0.30SnO2 hydrothermallized at 200 8C (diluted) 0.53 0.31TiO2 nanoribbons (1st distribution) 0.86 0.19TiO2 nanoribbons (2nd distribution) 0.92 0.12Anatase growth along [001] 6O10�4 0.055

Figure 7. Fitting of the experimental data obtained from Penn and Ban-field.[27]

Figure 8. Fitting of the experimental data obtained from Jun and co-work-ers.[13] Two kinetic behaviors obtained from the bimodal distribution are pre-sented in the main graph. The inset shows the evolution of the diameter ofthe nanorods at the same time interval.


E. R. Leite et al.


particles (the process was observed during 15 min) was consis-tent with the mechanism described here.

A few problems of the polymerization OA model must bediscussed in further detail. The approximation of the reactionconditions to gas phase conditions [used to evaluate thedecay of k in Eq. (14)] neglects the effects of viscosity imposedby the fluid. Undoubtedly the main failure of the proposedmodel lies in the assumption that each particle will coalesceonly twice, which is clearly unrealistic. High-resolution tunnel-

ing electron microscopy (HRTEM) images shown in Figure 9confirm the presence of particles with more than two points ofcoalescence. We can suppose that this deviation will occurmainly over extended reaction times, during which the largeparticles will have large contact areas. However, as mentionedearlier, in some cases the supposition may be very reasonable.Another example is the strategy developed for the synthesis ofCdTe nanowires, as proposed by Tang and co-workers.[10] Inthis route, the nanoparticles are covered with a surfactant,which is gradually eliminated from the surfaces during consec-utive coalescence events, finally resulting in a nanowire. Undersuch conditions, the active surfaces responsible for the OA arenot covered by surfactants. Finally, alternatives to the specifica-tion of only two coalescence events for each particle are im-plicit in several reports describing nanowire or nanorod forma-tion.

3. Conclusions

In summary, we describe a model to explain growth in colloidsof several crystalline structures, involving processes that rangefrom geological formations to synthetic materials, wherelinear-like shapes (e.g. nanorods and nanowires) are observed,and for which the oriented attachment mechanism is assumedto be the predominant process. The model uses a “polymeric”approach (i.e. it interprets particle coarsening analogously topolycondensation reactions) to describe the kinetics of thestructure formation since, in the oriented attachment mecha-nism, the particles’ surfaces may be interpreted in the samemanner as a termination in a monomer. The fitting of themodel to hydrothermally treated SnO2 nanoparticles showedvery good agreement. We believe that this model provides agreater insight into the growth mechanism of anisotropicnanocrystals as well as numerical information for controllingthe kinetics of the formation of these structures.

Acknowledgments

The authors gratefully acknowledge the financial support of theBrazilian research funding agencies FAPESP and CNPq.

Keywords: colloids · crystal growth · kinetics ·nanostructures · polymerization

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Received: September 8, 2005Revised: November 15, 2005Published online on February 13, 2006


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Oriented Attachment Mechanism in Anisotropic Nanocrystals: A “Polymerization” Approach