Chapter 4: Hypothesis of Diffusion-Limited Growth

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1 Suary This section derives a useful equation to predict quantu dot size evolution under typical organoetallic synthesis conditions that are used to achieve narrow size distributions. Assuing diffusion-controlled growth before Ostwald ripening, organoetallic synthesis kinetics should be characterized by an activation energy, Q. Traditionally, Q would be extracted fro the slope in an Arrhenius plot of ln(rate) versus 1/T, where T is the absolute teperature. Recently, a study of dse quantu dots grown in stearic acid (SA) used Arrhenius analysis to estiate Q fro olar growth rates. 68 It has also been deonstrated that Q can be estiated fro photoluinescence (PL) redshift rates (i.e. the changes in PL peak wavelength per change in reaction tie between sequential saples). 69 The Proble Peng clearly stated that the growth of dse quantu dots fored via organoetallic synthesis should be liited by the diffusion of etal precursors, rather than by the surface reaction rate. 58 Alost all literature in this field offers evidence that supports this idea. Despite detailed theoretical treatents, we still lack reasonable analytical expressions for the ean radius versus tie that are consistent with diffusionliited growth in a batch process. Although diffusion generally exhibits Arrhenius 58

2 teperature dependence, diffusion activation energies have not been experientally easured and published for nanocrystal synthesis in various solvents. Therefore, the goal of this chapter is to develop analytical expressions, describing the evolution of nanocrystal radii as a function of tie and teperature, in order to help estiate activation energies fro experiental easureents. Assuptions Soe siplifying assuptions ay facilitate achieving a anageable expression for R versus t. I specifically hypothesize that the diffusion of precursors into a dse nanocrystal is reasonably approxiated by a linear concentration gradient across a diffusion length, L D, that should correspond to the ective ligand length when synthesis includes strong coplexes of d with TOPO and with TDPA. To visualize how a linear concentration gradient ight surround a quantu dot that was synthesized in stearic acid, Figure 4-1 uses a decreasing intensity of color to represent how the concentration of cadiu stearate olecules ay decrease towards the surface, due to stearic hindrance fro ligands attached to the dse nanocrystal. To be explicit, the atheatical odel is based on the following assuptions: 1. A fast reaction rate reduces the surface concentration of d coplexes, inter (in oles c -3 ), to a level that is orders of agnitude lower than in the bulk fluid, so that inter ~ During the growth stage, the diffusion of d coplexes is significantly slower than the diffusion of Se, so that the reduced supply of d is the rate-liiting step for dse nanocrystal growth. 59

3 3. The bulk reaction fluid outside of the diffusion boundary layer is well stirred to aintain a hoogeneous concentration of d coplexes, bulk (in oles c -3 ), which are gradually consued with reaction tie. 4. A diffusion barrier region of constant thickness, L D, exists around the surface of each nanocrystal. If this barrier is assued to be caused by ligands pacifying dangling bonds on the surface of the nanocrystal, then L D ay correspond to the hydrodynaic diaeter or the ective length of the ligand. However, L D could also represent a wider reactant depletion length in the reaction solvent. 5. The radial concentration gradient fro x = R to x = L D is approxiately linear fro the nanocrystal surface to the outside of the diffusion barrier; so the gradient is given by (4-1), where x is the radial distance fro the center of the nanocrystal. d dx x= R ~ bulk L D int er ~ L bulk D (4-1) 6. Growth conditions are controlled to insure roughly spherical quantu dots, not rods, with a radius of R (t) (in c), which increases with reaction tie, so that the surface area, A, is given by (4-2). A = 4πR 2 (4-2) 7. Mobility of the d coplex is liited by standard diffusion with Arrhenius teperature behavior, Equation (4-3). Then the diffusivity, D (in c 2 s -1 ), is characterized by a diffusion activation energy, Q (in ev), and by a pre-exponential diffusivity coicient, D o (in c 2 s -1 ), where k b is Boltzan s constant, and T is the absolute teperature (in K). D o does not depend on the d coplex concentration. D = D exp { Q k T} (4-3) o b 60

4 8. The odel only siulates quantu dot growth after nucleation and before Ostwald ripening, and it assues that the nuber of nuclei, N, reain constant during this period. 59 Se d stearate L D Figure 4-1. Diagra of a dse quantu dot growing in stearic acid. The illustration shows how the diffusion length, L, ight represent the thickness of a ligand barrier. Using the assuptions of linear concentration gradients (4-1), spherical nanocrystals (4-2), and Arrhenius diffusion (4-3), Fick's first law (4-4) yields a governing Equation (4-5) for the olar growth rate, v, where J is the flux of d into the nanocrystal (in oles c -2 s -1 ). d v = A J = A D (4-4) dx 61

5 Q 2 bulk ( t) v( t ) 4π R( t) Do exp (4-5) LD kbt Teporal Evolution of Nanocrystal Radius Positive and negative feedback on the growth rate can be seen by expressing v (t), bulk, and R in ters of, the total nuber of oles of precursor (4-6), where V org (in c 3 ) is the volue of all the organic aterial in the reaction. The olar volue of dse (V = c 3 ole -1 ) is its olecular weight (W = g ole -1 ) divided by its density (ρ = 5.74 g c -3 ), based on unit cell volue. 64 Others 59 estiate V to be c 3 ole -1. N is the ective nuber of spherical nanocrystals that would contain the nuber of oles of d that have been consued. 1 d v =, N dt bulk =, and V org R = 1 / 3 3V 4πN 1 / 3 (4-6) The governing Equation (4-5) becoes a siple differential Equation (4-7), with a coicient, Θ (Τ), which does not vary with tie, but does depend on teperature. Integration fro an interediate tie after nucleation, t i, (when the reaining oles of precursor is i ) to the reaction tie, t, yields an expression (4-8) describing how changes during the reaction. d dt = Θ T, where 1/ 3 ( ) Θ ( T ) 4πD o 3V V L org 4πN 2 / 3 exp Q k T b (4-7) ( t) ( 2 / 3 i ( 2 / 3) Θ( T ){ t ti 3 / 2 = } ) (4-8) It is helpful to express the differential Equation (4-7) and its solution (4-8) in unitless fors (4-10) and (4-11), respectively, by defining a reaining precursor fraction 62

6 f, an interediate precursor fraction f i (at t i ), and a reaction copletion tie t c (4-9). Integrating (4-10) fro an interediate tie t i after nucleation to a given synthesis tie t (in s) yields an expression (4-11) that describes how reactants are gradually depleted. The interediate precursor fraction f i is less than 1.0 because soe reactants are consued during nucleation. 58 bulk ( t) f =, o bulk ( to) bulk ( ti), i f i = o bulk ( to) 3( f ) 2 / 3 i i τ c (4-9) 2Θ( T ) df dt = Θ (4-10) 1 / 3 ( T ) f 3 / 2 { t t } i f ( ) = 1 t fi (4-11) tc Growth of the average quantu dot radius (4-12) can be derived geoetrically fro (4-11) and (4-6), where o is the original olar concentration of d in the batch (in oles c -3 ). Using Equation (2-6), the wavelength of the first absorbance peak λ abs can be estiated fro the average radius according to (4-13), where X = 0.82 x 10-7 ev-c for dse. R ( t) (1- f(t) ) o3v = 4πN 1 / 3 and R c o3v 4πN 1/3 (4-12) hc λ abs = (4-13) E + X / R g If the peak photoluinescence photon eission energy, E PL, is always lower than the 1s 1s absorbance peak energy E abs by an energy shift ( E AP ~ ev x E abs or E AP ~0.025 ev for dse quantu dots when 2 ev < E abs < 2.5 ev based on initial experiental observations), then (4-13) can be odified to calculate the wavelength of 63

7 the PL peak λ PL (in c) (4-14). Right when all d precursors have been consued, the average nanocrystal radius reaches the copletion radius R c (4-12), and PL eission occurs at a copletion wavelength λ c (4-14), which arks the knee in a plot of λ PL versus synthesis tie. hc λ PL = and c PL( R= Rc ) E g + X / R E AP λ λ (4-14) Teperature Dependence of Growth Rate It would be convenient to onitor the olar growth rate and the nanocrystal radius using photoluinescence spectra, because PL peaks for well-defined Gaussian distributions in photon energy. To estiate R, Equation (4-14) is solved for R to obtain (4-15). Theoretically, Arrhenius analysis of redshifts is possible because v is proportional to dλ PL /dt, as will be shown. The bulk d concentration decreases fro its original level according to (4-16), due to the growth of spherical nanocrystal, if N is constant, and if d is not consued by any other echanis. However, if the reaction yield, y, is not 100%, then the actual nuber of quantu dots N is less than N, or N = y*n. The olar growth rate of each individual quantu dot is proportional to the rate at which bulk is depleted (4-17). Substituting (4-15) and (4-16) into (4-17) and then differentiating with respect to tie yields Equation (4-18). In this case, the experiental 1/R photon energy dependence was used instead of the 1/R 2 dependence fro the ective ass approxiation; therefore R 4 appears in (4-18) instead of R 5, as published earlier. 69 R = hc λ PL X + E AP E g (4-15) 64

8 bulk ( t) 3 N 4πR( t) = o (4-16) V 3V org Vorg dbulk v = (4-17) N dt 4 dλpl R 4πhc v = (4-18) 2 dt λ V X PL To the degree that the above assuptions hold, Equation (4-18) is a convenient way to track the QD growth rates by onitoring PL redshift rates. To estiate Q fro dλ/dt, redshift rates at different teperatures could be copared at a coon test wavelength. Equation (4-18) shows how fixing λ PL iniizes sensitivity to R, in order to focus on the teperature dependence of v. If the olar growth rate, v, has Arrhenius teperature dependence, then an Arrhenius plot of ln{v} versus 1/T, would follow a straight line, with the slope, -Q/k b. It ay be possible to plot ln{dλ PL /dt} versus 1000/T, or to plot ln{1/t c } versus 1000/T, and get the sae activation energy fro the slope, -Q/1000T. 65