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A Kinetic Model to Describe Nanocrystal Growth by the Oriented Attachment Mechanism

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The classical model of particle coagulation on colloids is revisited to evaluate its applicability on the oriented attachment of nanoparticles. The proposed model describes well the growth behavior of dispersed nanoparticles during the initial stages
  A Kinetic Model to Describe NanocrystalGrowth by the Oriented Attachment Mechanism Caue Ribeiro, Eduardo J. H. Lee, Elson Longo, and Edson R. Leite* [a] Introduction The synthesis of different kinds of nanoscaled materials, frommetals to inorganic compounds, [1–9] has been reported in thepast decade. The increasing interest in nanotechnology hasbeen motivated by the possibility of tailor-making nanomateri-als with defined properties by controlling some of the charac-teristics of the nanostructures, such as their size and morphol-ogy. Hence, there has been a considerable effort to understandhow nucleation, growth, coarsening, and aggregation process-es affect these characteristics. It is well-known that the above-mentioned processes may occur simultaneously, and that theyare directly related to synthesis variables. A clear understand-ing of this relationship is necessary to enable the preparationof highly controlled nanostructures. [1,4,5,10–14] Supersatured solutions induce the precipitation of nuclei,due to thermodynamic stabilization. These nuclei grow addi-tionally as a result of the surface deposition of solvated ions. [15] After crystallization is complete, nanocrystal growth is an im-portant parameter to be controlled. Usually, nanoparticlegrowth is associated with coarsening, which is also known as Ostwald ripening . This mechanism can be described as a diffu-sion-limited growth of nanoparticles at the expense of smallerones. [16,17] A kinetical model for the Ostwald ripening mecha-nism was rigorously developed by Lifshitz, Slyozov, [18] andWagner, [19] and is also known as the LSW model. The LSWmodel predicts that the mean particle radius should evolve asa function of time according to Equation (1):  r  n   r  n 0 / t   ð 1 Þ Where  n  is dependent of the limiting step to the growth. Thisresult is obtained by combining the Gibbs–Thompson equa-tion—which describes the dependence of the particle solubili-ty as a function of its size—and Fick’s first law. The power lawcoefficient  n = 3 is obtained by considering dilute conditions,where diffusion of ions in solution is the limiting step. In con-centrated conditions (e.g., in solids, or in the case of nuclea-tion in melts), the coefficient may be  n = 2 or  n = 4 dependingon the limiting step involved in the interfacial reactions (i.e.,dissolution or reprecipitation). [20,21] The LSW equation [Eq. (1)] was extensively used to explainthe growth kinetics of dispersed colloids. However, recentworks have demonstrated that this mechanism cannot be con-sidered responsible for the growth process in some sys-tems, [6,22–24] since the main postulates of the theory are fre-quently neglected. The oriented-attachment mechanism wasproposed as another significant process that may occur duringnanocrystal growth. This mechanism leads to the formation of nanoparticles with irregular morphologies, which are not ex-pected in precipitation-based growth. Several studies indicatethat the oriented-attachment effect is very significant, even inthe early stages of nanocrystal growth, and may lead to theformation of anisotropic nanostructures in suspensions, suchas nanorods, by the consumption of nanoparticles as buildingblocks. [25,26] This mechanism has already been experimentallyobserved in micrometer-sized metallic systems for severalyears. [27–30] Recently, it was modeled by Moldovan et al. [31–34] and investigated by molecular dynamics studies. [35–38] In all of the above-mentioned theoretical works, the authors assumedthat the nanoparticles were in contact with each other. The ori-ented attachment occurred by means of relative rotations be-tween the particles, or by plastic deformation associated withdislocation motion, until a thermodynamically favorable inter-face configuration (i.e., crystallographic alignment) is reached.Considerable effort was expended in literature to obtain ad-equate models that describe the coalescence of nanoparticlesin suspension and on surfaces. [39] Penn and Banfield proposed,in a recent paper, that dispersed nanoparticles can be treatedas molecules or molecular clusters [23] in solution. This treatmenthas already been used by Huang et al. [20,40] in the developmentof a kinetic model used to explain the growth of ZnS nanopar-ticles induced by hydrothermal treatments. Penn has also de-veloped a kinetic model for oriented-attachment growth byconsidering the electrostatic interaction between particles in [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 235–13565–905S¼o Carlos, SP (Brazil)Fax: (    55)16-3361-5215E-mail:  The classical model of particle coagulation on colloids is revisited to evaluate its applicability on the oriented attachment of nano- particles. The proposed model describes well the growth behavior of dispersed nanoparticles during the initial stages of nanoparti-cle synthesis and during growth induced by hydrothermal treat-ments. Moreover, a general model, which combines coarsening(i.e., Ostwald ripening) and oriented attachment effects, is pro- posed as an alternative to explain deviations between experimen-tal results and existing theoretical models. 690   2005 Wiley-VCH Verlag GmbH&Co. KGaA, Weinheim  DOI: 10.1002/cphc.200400505  ChemPhysChem  2005  , 6, 690–696  solution. [41] In this model, particle collisions lead to the forma-tion of complexes (i.e., agglomerates), where coalescence maytake place. The transformation of a complex into a coalescedparticle may be described as an equilibrium equation.Ribeiro et al. [15] proposed another mechanism for oriented-attachment growth in dispersed nanoparticles. The authorsconsidered—to explain the growth behavior in SnO 2  colloidalsuspensions—that coalescence may also occur when particleswith similar crystallographic orientations (or with slight differ-ences) collide. This mechanism is based on the assumptionthat nanoparticles dispersed in a liquid medium present a veryhigh degree of freedom for rotation and translation motions.Hence, in suspensions where agglomeration does not takeplace, growth by means of oriented collisions should be moreeffective than that due to surface mechanisms (i.e., coales-cence induced by relative rotations between particles in con-tact). Dispersed nanoparticles should present a high velocity,due to the Brownian motion. Hence, it is expected that nano-particles in suspension present a high frequency of collisions.Therefore, growth by coalescence may be interpreted statisti-cally, since collisions may be considered effective (i.e., leadingto coalescence) or ineffective (i.e. elastic event). This mecha-nism is similar to the Smoluchowsky coagulation model, [42–44] which has been extensively used to explain polycrystalline col-loidal growth and aggregation mechanisms in suspension.If we assume that all of the above-mentioned considerationsare valid, the coalescence of two particles in suspension maybe interpreted as [Eq. (2)]:A þ A  k  ! B  ð 2 Þ where A is a primary nanoparticle and B is the product of thecoalescence of two nanoparticles. Figure 1a illustrates this re-action, and Figure 1b shows a high-resolution transmissionimage of two SnO 2  nanoparticles coalesced in this way.Herein, we propose a kinetic model, which is based on asimplified interpretation of the Smoluchowsky coagulationmechanism. This model was developed to describe the coales-cence process in the early stages of particle growth. Therefore,it is applied for the interpretation of the growth behavior insystems where particle-synthetic processes still take place (andespecially in diluted systems), which is a reasonable assump-tion for most of the bottom-up routes. [8] The proposed modelis compared qualitatively with experimental data and may pro-vide an insight into understanding the mechanisms that leadto the size control of nanocrystals during synthesis, and alsoduring subsequent processes such as hydrothermal treat-ments. Collision Frequency The collision frequency of dispersed nanoparticles may be eval-uated by assuming that the Brownian motion in dilute suspen-sions may be described by Maxwell–Boltzmann statistics. Inthis model, the frequency evaluation is done in analogy to thekinetics of gas molecules. The collision frequency for a singleparticle is given as a function of the mean velocity  u¯   by Equa-tion (3):  z   ¼  ffiffiffi 2 p   p  D 2  u N V  ð 3 Þ where  D  is the particle diameter,  N   is the total number of parti-cles, and  V   is the total volume occupied by the system. The vis-cous force is given by  mp  2 u¯  D 2 , where  m  is the viscosity, which isnegligible for systems composed of low-viscosity fluids, suchas in the case of nanoparticles dispersed in water. Hence, themean velocity of the dispersed particles may be estimated bythe equipartition theorem [Eq. (4)]:  u ¼  ffiffiffiffiffiffiffiffiffiffi 3 k  B T m r   ð 4 Þ The mass of a spherical nanoparticle with a radius of 2 to5 nm and a density of 3 to 10 gcm  3 is on the order of 110  8 g. At room temperature, this nanopaticle presents a meanvelocity of approximately 1.1 ms  1 , which is a very high valueif particle size is considered. In the study reported by Ribeiroet al., [15] the SnO 2  nanoparticle concentration (for dilute sus-pensions) was estimated to be on the order of 110 18 particlesper liter.If these parameters are inserted in Equation (3), the collisionfrequency is estimated to be of approximately 240 collisionss  1 for each particle. Although this value may not be consideredprecise, it still indicates that the number of (total) collisionsmay be indeed significant. It can be observed that the nano-particle collision frequency is much lower than the value ex-pected for gas molecules, which is obviously due to size andmass effects. Moreover, it is also important to note that colli-sions are only effective if particles with the same crystallo-graphic orientation collide. The Kinetic Model As proposed in Equation (2), the rate of reaction of the orientedattachment mechanism may be given by Equation (5): Figure 1.  a) Mechanism of oriented collision, which leads to the coalescence of two particles. b) HR-TEM image showing two coalesced primary SnO 2  nanopar-ticles. ChemPhysChem  2005  , 6, 690–696   2005 Wiley-VCH Verlag GmbH&Co. KGaA, Weinheim  691 Kinetic Model for Nanocrystal Growth  u ¼ 12d ½ A  d t   ¼ k  ½ A  2 ð 5 Þ where [  A ] is the concentration of primary (i.e., uncoalesced)particles (particles/volume) and  k   is the reaction constant. As-suming that the reaction occurs in a single step, [  A ] is definedin terms of the initial concentration [  A ] 0  by Equation (6): ½ A ¼ ½ A  0 1 þ 2 k  ½ A  0 t   ð 6 Þ However, this result is applicable only if particles are directlyin contact. In suspensions, particles need to achieve an equili-brium condition (which is provided by collisions) to form com-plexes—that is, two particles in contact—that may coalesce.This process may be described as a two-step reaction, as fol-lows [Eqs. (7) and (8)]:A þ A k  1 , k  0 1    !    AA  ð 7 Þ AA  k  2    ! B  ð 8 Þ Therefore, the kinetics of this process is described by threereaction rates: The first two rates,  u 1 ! and  u 1  [Eqs. (9) and(10)], correspond to the complex-formation equilibrium reac-tion, and the third one,  u 2 , is related to the coalescence event[Eq. (11)]: u 1 ! ¼ k  1 ½ A  2 ð 9 Þ u 1  ¼ k  0 1 ½ AA  ð 10 Þ u 2  ¼ k  2 ½ AA  ð 11 Þ Assuming that the complex concentration is in the steadystate, that is, d[AA]/d t  = 0 [Eq. (12)], it can be written that[Eq. (13)]:d ½ AA  d t   ¼ k  1 ½ A  2  k  0 1 ½ AA  k  2 ½ AA ¼ 0  ð 12 Þ½ AA ¼  k  1 ½ A  2 k  0 1 þ k  2 ð 13 Þ The rate of formation of coalesced particles  B  [Eq. (14)] is ob-tained from Equations (11) and (13):d ½ B  d t   ¼ 12d ½ A  d t    k  2  k  1 k  0 1 þ k  2  ½ A  2 ð 14 Þ This equation can be solved similarly to Equation (6), where( k  2 k  1 )/( k  ’ 1   k  2 ) can be interpreted as the rate constant  k  T  of thetotal reaction. According to the initial proposition, the orient-ed-collision-induced coalescence should be a very fast processwhen compared to mechanisms of coalescence induced byparticle rotation. Therefore, it is clear that the attachment willbe dominated by the first step (i.e., by the collision process),and it can be assumed that  k  1 @ k  2 , which implies that  k  T  k  2 .The total particle flux,  J  T , through a stationary spherical parti-cle is given by Equation (15):  J  T  ¼ 4 p  r  2   J   ð 15 Þ where  J   is the flux around a particle (A) with surface area 4 p  r  2 ,as defined by Fick’s 1st law [Eq. (16)]:  J  ¼ D A d ½ A  d  x  ð 16 Þ where  D A  is the diffusion coefficient of A and x is the distance.This term is obtained in analogy with atomic diffusion, and itactually describes the diffusion process of nanoparticles. Theconcentration of primary particles, [A], may be defined interms of the total particle flux [Eq. (17)] by integrating Equa-tion (16): ½ A   x   ¼½ A   J  T 4 p  D A  x   ð 17 Þ During the collisions, whenever a particle is within a dis-tance of 2 r   from the surface of another particle A, the forma-tion of the complex AA occurs, as shown in Figure 2. Hence,for this situation, the concentration of primary particles aroundthe collision site can be considered equal to zero (i.e. [A]  x  = 0),and  J  T  can be given as a function of [  A ] and  R = 3 r   by Equa-tion (18):  J  T  ¼ 12 p  D A r  ½ A  ð 18 Þ The number of particles in suspension is defined as [A] N  A  V  , where  N  A  is Avogadro’s number and  V   is the totalvolume of the suspension. Therefore, the overall flux of parti-cles can be defined as  J  T [A] N  A  V  , since particles are notreally stationary. Because the overall flux is time-dependent,the following approximation [Eq. (19)] can be used:d ½ AA  d t   ¼ð 6 p  rD A N  A Þ½ A  2 ð 19 Þ Figure 2.  Model of the contact of two particles A forming a complex AA. Thedashed line corresponds to the collision cross-section. 692   2005 Wiley-VCH Verlag GmbH&Co. KGaA, Weinheim  ChemPhysChem  2005  , 6, 690–696 E. R. Leite et al.  By comparison with Equation (9), the term in parenthesiscan be interpreted as the reaction rate constant  k  2 , as follows[Eq. (20]: k  2  ¼ 6 p  rD A N  A  ð 20 Þ This result indicates that the particle radius is affected bythe reaction rate. However, this dependence can be re-evaluat-ed if the diffusion constant is assumed to be equivalent to theStokes–Einstein equation definition [Eq. (21)]: D ¼  k  B T  6 ph r   ð 21 Þ where  h  is the liquid-medium viscosity. Hence, the rate con-stant  k  2  may be defined as [Eq. (22)]: k  2  ¼ N  A k  B T  h ð 22 Þ This very simple result shows that the viscosity of the liquidmedium plays an important role in the growth by particle coa-lescence, which is governed by an inverse proportional rela-tionship with respect to the rate constant. The temperaturedependence is not direct, since the viscosity is also tempera-ture-dependent in an Arrenhius form, that is,  h = h 0 e E  a / Nk  B T  . Thisdependence is observed in Figure 3. At lower temperatures(near the freezing point of the solvent), the viscosity effect ismore pronounced, which results in an inhibition of the in-crease of the rate constant  k  2 ; at higher temperatures, the vis-cosity stabilizes reaching a nearly constant value, and conse-quently  k  2  behaves linearly with respect to the temperature.Another interesting feature that can be observed in Equa-tion (22) is the dependence of the solid content on the viscosi-ty, as described by the Einstein equation [Eq. (23)]: h ¼  m ð 1 þ 52  Þ ð 23 Þ where  m  is the viscosity of the solvent and  f  is the solid volu-metric fraction. Equation (23) is obtained by considering thatthe particles behave as highly dispersed rigid spheres with lowinteractions, which is a reasonable assumption for dilute colloi-dal suspensions. Figure 4 shows that increments in the solidcontent may slightly interfere with the rate constant. However,this effect is not consistent with the model, which is applicablefor solid contents below 0.03. Growth Kinetics It was initially considered for the development of the modelthat particle growth occurs only by the oriented attachmentmechanism. Therefore, it is assumed that there are only twotypes of nanocrystals: 1) primary particles, A, which have notbeen exposed to any coalescence events and 2) coalesced par-ticles, B. Hence, it is easily observed that B particles have twicethe volume of A particles. The mean particle radius is an im-portant parameter for the evaluation of nanoparticle growth. Itis assumed that the mean particle radius of a coalesced parti-cle can be considered equivalent to the radius of a spherewith the same volume (i.e., equivalent radius), thus the totalparticle mass  M T  (invariant) can be described by Equations (24)and (25): M T  ¼½ A  0  N   4 = 3 p   r  3i  ð 24 Þ M T  ¼ð½ A þ½ B Þ N   4 = 3 p   r  3eq  ð 25 Þ where  r  i  is the initial mean particle radius and  r  eq  is the equiva-lent radius at a time  t  . From Equation (2), the total number of particles can be expressed as [A] 0 = [A]   2[B]. By comparingthis relation with the expressions above, it is possible to ob- Figure 3.  Temperature dependence of the reaction rate constant. Figure 4.  Dependence of the reaction rate constant on the solid content, ac-cording to the Einstein model. ChemPhysChem  2005  , 6, 690–696   2005 Wiley-VCH Verlag GmbH&Co. KGaA, Weinheim  693 Kinetic Model for Nanocrystal Growth  serve that the equivalent diameter depends on [A], as follows[Eq. (26)]: ½ A  0  r  3i  ¼½ A þ½ A  0 2   r  3eq  ð 26 Þ By inserting Equations (6) and (22) into Equation (26), it ispossible to write the following relation [Eq. (27)]: r  3eq  r  3i  ¼ Nk  B T  h  ½ A  0 t  1 þ Nk  B T  h  ½ A  0 t   r  3i  ð 27 Þ Since all the terms are constant, except  t  , we observe thatthis equation behaves as a function of the type  y  =  x  /(1    x  ).This behavior is slightly different to the one expected for theOstwald ripening mechanism. After very long periods of time(Figure 5), Equation (27) stabilizes and reaches a constant valuethat corresponds to the moment at which all primary particleshave undergone a coalescence process. This interpretationconsiders only the first stage of coalescence, where a single co-alescence event occurs for each particle. However, at longertime periods, events such as A   B ! C (attachment of previous-ly coalesced particles) may also occur. As a matter of fact, it ishighly improbable that each particle coalesces only once.All the terms in Equation (27) can be evaluated to predictthe particle growth behavior. Figure 5 presents a comparisonbetween an experimental curve, showing the mean particleradius as a function of time during the initial stages of SnO 2 nanoparticle synthesis, and a theoretical curve, which was cal-culated for  T  = 300 K,  h = 1 cP (water), and [A] 0 = 110  6 molL  1 (mol of particlesL  1 ) using the initial mean particleradius. The mean particle radius values from the experimentaldata were estimated by means of optical spectroscopy meas-urements and the effective-mass model (absorbance spectracan be seen on the inset). The synthesis of SnO 2  nanoparticleswas carried out by using the same procedure previously de-scribed by Leite et al. [45,46] It is possible to observe that the the-oretical curve fits the experimental data very well, althoughthere are slight differences in the values (the experimentaldata range from 1.15 to 1.28 nm 3 , while the theoretical curveranges from 1.15 to 1.7 nm 3 ). This discrepancy can be ascribedto several factors, such as the presence of other growth mech-anisms, inconsistencies in some of the model’s approximations,or deviations in the measurements obtained experimentally.On the other hand, the very good agreement between the ex-perimental data and the theoretical predictions is consistentwith the fact that, in tin oxide, oriented attachment is thedominant mechanism due to its extremely low solubility inwater or alcohol.A similar comparison with good correlation can be per-formed on SnO 2  nanoparticles submitted to hydrothermaltreatments, as performed by Lee. [47] Figure 6 shows the com-parison between experimental data obtained by direct meas-urements in high-resolution transmission electronic microscopy(HR-TEM) (at least 200 particles) and the fit to Equation (27).The average ratio was determined by using the same supposi-tions used in the present model. Lee discussed the growth interms of coalescence events by dividing the observed particlesin “primary” (rounded particles, without any visible defect inHR-TEM) and “secondary” particles (anisotropic particles withseveral shapes and random presence of characteristic defectsof oriented attachment). When observing only the “primary”particles, the author did not see any significant change in sizebetween the treatments, although the average size was in-creasing with time; this made him conclude that SnO 2  particlesunder those conditions did not grow significantly by means of the Ostwald ripening mechanism, and that the key process inthat case was oriented attachment (represented by the “secon-dary” particles). The calculated value for the initial particleradius (obtained by fitting) was 1.31 nm, a value extremelyclose to that reported by the author (1.2 nm). Figure 7 shows Figure 5.  Comparison between the theoretical curve predicted by Equation (27)and experimental data obtained during the synthesis of SnO 2  nanoparticles. Figure 6.  Fitting of the experimental data obtained for hydrothermally treated SnO 2  nanoparticles (from Lee [47]  ). The particle size was obtained by direct meas-urements using HR-TEM images. 694   2005 Wiley-VCH Verlag GmbH&Co. KGaA, Weinheim  ChemPhysChem  2005  , 6, 690–696 E. R. Leite et al.
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