The Dynamics of Nanoparticle Growth and Phase Change During

Article pubs.acs.org/JPCC

The Dynamics of Nanoparticle Growth and Phase Change During Synthesis of β‑NaYF4 Paul B. May, John D. Suter, II, P. Stanley May, and Mary T. Berry* Department of Chemistry, University of South Dakota, Vermillion, South Dakota 57069, United States

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S Supporting Information *

ABSTRACT: We implement a model for nanocrystal growth and crystallographic phase transition during the synthesis of βNaYF4. In this model, the size of the α-phase nanoparticles, formed during the heating of the precursor materials, grows slightly in mean diameter and broadens in distribution width until some particles reach a size at which the β phase is thermodynamically favored. Individual particles crossing this threshold convert to the β phase, and then, being less soluble than the α phase, grow at the expense of dissolving α-phase particles. Implementing a straightforward kinetic formalism for individual particle growth and a variable phase definition depending on particle size, the model reproduces, in a quantitative fashion, the experimentally observed growth dynamics of βNaYF4:Yb,Er. This work supports a hypothesis that the β-particle seeds arise from a phase transition in individual α-phase particles. The model also suggests that the great variability observed in the duration of the stage during which the α particle ripen, before β particles begin to appear, may be attributed to rather small differences in the size distribution of the α particles formed during the heating of precursor material.

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convincingly that the “size focusing” of the β-phase product particles occurs due to their growth in the presence of the αphase particles formed in the initial stages of synthesis.12,13 The higher solubility of the α-phase relative to the β-phase provides a condition of supersaturation for β-phase particle growth, which leads to the size focusing that is expected in the diffusionlimited growth regime. Van Veggel et al. have exploited this effect to produce size-focused, β-phase, core−shell nanocrystals by ripening β-phase cores in the presence of sacrificial nanoparticles of α-phase shell material.14 By contrast, β-phase particles growing in isolation, in the absence of sacrificial αphase particles, exhibit the size broadening expected for normal Ostwald ripening. Suter et al., by monitoring the NIR-to-visible UC emission signal from the “heat-up” synthesis of NaYF4:Yb,Er upconversion particles in real-time, obtained detailed kinetic information regarding the progression of the reaction through its various stages.15 A significant finding of this study was that there is a large synthesis-to-synthesis variation in total reaction time, and that the variation in reaction time is due almost entirely to the variation in the delay between the initial formation of the small α-phase nanoparticles and the start of the phase transition to form the larger β-phase product particles. Interestingly, the final size distribution of the β-phase nanoparticles was quite reproducible, and showed no correlation with reaction time.

INTRODUCTION Hexagonal-phase β-NaYF4 doped with lanthanides is one of the most commonly employed upconversion (UC) phosphors. With appropriate choice of dopants, the materials show exceptional quantum efficiency,1,2 excellent red, green, or blue color purity,3,4 and, in the form of nanoparticles, good dispersibility in a wide range of polar and nonpolar matrices.5,6 Many applications of UC nanoparticles require narrow particle-size distributions. Among the various coprecipitation methods of synthesis, the so-called “heat up” method reliably produces narrow size distributions of the β-phase material and has rapidly increased in popularity since its introduction by Zhang et al.7,8 Here, all precursors are combined in a highboiling-point solvent mixture prior to heat-up, as opposed to injecting the lanthanide precursors into an already hot solution containing the fluoride reactant. After heat-up, the mixture is maintained at a temperature close to 300 °C for approximately 1 h, and then allowed to cool. An initial amorphous precipitate is formed from the combination of precursors in the cold solvent mixture. During the heat-up, small cubic α-phase particles appear, followed by the later appearance of larger β-phase particles. For a given set of synthesis conditions, the resulting β-phase particles exhibit very reproducible sizes with remarkably narrow size distributions. There have been several revealing studies that shed light on the mechanism for this reaction including early work by Lifshitz, Slyozov, and Wagner, (LSW)9,10 and by Talapin et al.11 Two later studies by Voss et al. demonstrated rather © 2016 American Chemical Society

Received: February 8, 2016 Revised: April 12, 2016 Published: April 15, 2016 9482

DOI: 10.1021/acs.jpcc.6b01365 J. Phys. Chem. C 2016, 120, 9482−9489

Article

The Journal of Physical Chemistry C

Figure 1. (Left) Schematic illustration of the spectroscopic signature of green and red upconversion corresponding to the four stages identified in the synthesis of NaYF4 nanoparticles. (Right) TEM of a typical product distribution from synthesis as described in refs 15 and 16.

synthesis to synthesis (10 ± 2 min). The UC intensity plateaus after the phase change is complete (Stage IV) and the reaction is terminated shortly thereafter to avoid further Ostwald ripening of the phase-pure β product. Based on the work of Voss et al. discussed above, it is clear that the tightly focused size distribution for the final β-phase product produced by the heat-up method is a result of the βphase particles growing in the presence of sacrificial α-phase particles in Stage III. The more soluble α-phase particles maintain a condition of supersaturation in the dispersion, so that the β-phase particles grow in the diffusion limit, illustrated schematically in Figure 3.

Referring to Figures 1 and 2 and data from ref 15, weak UC luminescence is evident even prior to heat-up, but no crystalline

Figure 2. Particle size distribution during synthesis of β-NaYF4:Er,Yb. Aliquots were withdrawn and analyzed by TEM during real-time monitoring of upconversion luminescence (inset). The relative number of α particles as compared to the number of β particles is not represented, 100 particles from any phase present were used to calculate the distribution. The t = 0 on the time scale is set at the beginning of heating for Stage I (data from ref 16).

Figure 3. Schematic representation of β-particle growth during Stage III of the reaction. During Stage III, the β-particles grow at the expense of the more soluble α-phase by consuming the dissolved components, referred to as “dissolved monomer” for historical reasons.

material can be isolated. During heat-up (Stage I), small αphase particles begin to form and, by the time the temperature reaches 300 °C (end of Stage I), all detectable solid in the mixture is in the form of small (∼4−5 nm) α particles. Stage II is characterized by a period of relative stasis of widely variable duration (63 ± 22 min), during which the spectroscopic signature remains nearly constant, corresponding to only minor changes in the initial α-particle population (in the form of slight particle growth and modest increase in the width of the sizedistribution). Subsequently, a rapid α → β phase transition (Stage III) is signaled by a steep increase in UC intensity and corresponding increase in the green-to-red intensity ratio. The duration of this phase-change stage was fairly reproducible from

What is not clear is the reason for the high variability in the duration of the ripening of α-phase particles (Stage II) leading up to the phase change (Stage III). Moreover, the origin of the β-particle seeds that appear at the beginning of Stage III is not established. Also interesting, yet unexplained, is the observation that there appears to be no correlation between size distribution of the final product and the duration of the αparticle ripening (Stage II). In addition we make note of the observation that, although the duration of the β growth period (Period III) was very consistent for a given set of reaction conditions, it was quite sensitive to the solvent makeup, 9483


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consideration arises in that, under diffusion-limited growth, the concentration of precursor in solution at the surface of the nanoparticles (Cs) may be different that the concentration in the bulk solution (C∞). Taking these factors into consideration, we have

specifically, that is, to the relative amounts of oleic acid and octadecene.16 In this study, we present a kinetic model for the reaction mechanism for the synthesis of β-NaYF4 nanoparticles via the heat-up method. The model quantitatively accounts for the evolution of the reaction mixture, starting after the initial formation of the intermediate α-phase particles, through to completion of the α → β phase transformation that results in the final β-phase products. The model assumes that β-phase seed particles are formed through the phase transformation of individual α-phase particles that reach a critical size, where a phase transition from α to β is thermodynamically favored. The variation in observed reaction times is well explained in terms of slight differences in size distributions of the α-phase particles as they are initially formed during the heat up to reaction temperature. The model simulations are also consistent with the observation that consistent final β-particle sizes are obtained, independent of the total reaction time.

( Lr )

k1(SA)C∞ − k 2(SA) exp dm = aL dt exp r + (k1r /D)

( )

where k1 and k2 are rate constants for the bulk material, D is an effective diffusion coefficient for the precursor material, and 2γV L = RTm . The quantity, L, is often referred to as a “capillary length” and relates to a size below which the nanoparticle solubility is greatly accelerated. It is different than the transition radius, rT, which defines a size at which the α and β phases are in thermodynamic equilibrium. In the denominator of eq 3, the term,

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⎛ δμ‡ ⎞ ⎛ aL ⎞ ⎟ exp⎜ ⎟ = exp⎜ ⎝ r ⎠ ⎝ RT ⎠

METHODS Mathematical Model. The critical size for thermodynamically favored α → β phase transition, rT, is determined by the differences in the bulk lattice energies (lower for β-phase) and the surface energies (lower for α-phase) for the two phases. rTransition = rT =

−2ΔγVm Δμ lattice

describes the effect of any increase in the free energy of the activated complex for solution/dissolution in the nanomaterials as compared to the bulk. These quantities are illustrated in Figure 4.

(1)

where, Vm represents the molar volume of the crystal, Δγ is the change in surface energy, and Δμlattice is the change in molar lattice energy, using the approximation that the molar volume is similar for the two phases. (See Supporting Information (SI) for derivation of eq 1.) The surface energies may be quite sensitive to surfactant coverage and therefore the critical size for phase conversion may also depend on surfactant coverage. There is no direct experimental evidence to indicate whether the growing α particles do indeed become the seeds for the final β-phase crystals or whether, alternatively, the β seeds form de novo from solution. However, the assumption in our model is consistent with the relative size-dependent thermodynamic stability of the two phases. A similar size-dependent phase transformation has been previously observed for nanoparticulate zirconia.17 The purpose of this study then is to test the model as proposed here for agreement with the observations made during real-time monitoring of the growth dynamics of β-NaYF4 and, in future, to use the model to create testable predictions. Here we develop the model for nanoparticle growth in any single phase. The nanoparticles are approximated as spherical so that their size is defined by a single variable, r, and their mass by m = 4πr3ρ/3, where ρ is the density of the material. This treatment parallels that of Talapin, and details may be found in the SI.10 For a given nanoparticle, dm = k1np(SA)(Cs) − k 2np(SA) dt

(3)

Figure 4. Arrhenius diagrams for the growth and dissolution of bulk solid (left) and nanoparticles (right). The activation energy for growth in the nanoparticles, Δ t̵μ 1np is shown with an increase of 2γV δμ‡ = a r m relative to the bulk solid, resulting in a net decrease in the activation energy for dissolution, Δt̵μnp 2 by (1 − a) this work we have employed a = b = 1/2.

2γVm r

=b

2γVm . r

In

Equation 3 can describe the growth of either the α or β phase particles. Starting with an initial Gaussian distribution of approximately 6000 α particles, the particles grow according to eq 3 until those in the leading edge of the distribution reach the transition radius, rT given in eq 1. As particles cross this transition radius, the particles’ identity is reassigned as β and those β-particles continue to grow as described by eq 3, but now grow with the newly assigned rate constants, ki and capillary length, L, appropriate to their new β-phase identity. These β-particles, having a lower solubility than the α-particles, consume precursor material from solution, in response to which, the remaining α-particles begin to dissolve. A derivation of eq 3 is given in the SI. Fitting Methods. To compare the model growth dynamics to the experimental results from the real-time monitoring experiments, we have treated several of the characteristic physical constants for the materials as variable parameters and

(2)

where m is the mass of the nanoparticle, SA is the surface area, and Cs is the concentration of precursor material in solution. np The kinetic rate constants, knp 1 and k2 control nanoparticle growth and dissolution, respectively. The “np” designation is given to the rate constants because they are expected to be size dependent and differ from the bulk values. A further 9484


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The mean radius (μ0 = ⟨rα⟩0) and standard deviation (σ0) of the starting size distribution are also included among the parameters. The starting distribution is created by distributing 6000 particles, in appropriate proportion for a normal distribution, among 60 bins of equal width, in the range μ0 ± 3σ0. The particles within each bin are then evenly distributed in size across the bin, such that no particles have exactly the same size. Because we do not include particles that lie further than 3 standard deviations from the mean, the total number of initial particles in the simulation is 5951. An additional strategy for creating a skew-normal starting distribution is discussed toward the end of the next section.

determined the values, through a simplex algorithm, that give the best fit to specific experimental observables. Tables 1 and 2 Table 1. Physical Constants Serving as Free Parameters in a Simplex Fit of Simulation Results to Observables Given in Table 2 Fit Aa parameter values

Fit Ba parameter values

4.91 × 10−7 m/s 3.07 × 10−8 kg/m2s 4.79 × 10−6 m/s 1.03 × 10−8 kg/m2s 1.49 nm 4.87 nm 4.92 nm

17.3 × 10−7 m/s 5.51 × 10−8 kg/m2s 3.39 × 10−6 m/s 0.18 × 10−8 kg/m2s 0.39 nm 5.13 nm 5.18 nm

2.12 × 10−14 m2/s 2.25 nm

46.8 × 10−14 m2/s 2.12 nm

4.6 × 10−4 nm

0.30 nm

n/a

31 particles (0.5%)

parameters kα1 kα2 kβ1 kβ2 L(α) L(β) rT D μ0 σ0 ix

rate constant for deposition of bulk solid α-phase material rate constant for dissolution of bulk solid α-phase material rate constant for deposition of bulk solid β-phase material rate constant for dissolution of bulk solid β-phase material capillary length capillary length thermodynamic phase transition radius effective diffusion coefficient for precursor mean radius for initial α distribution standard deviation of the radius for initial α-particle distribution skew parameter

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RESULTS AND DISCUSSION Using a fourth-order Runge−Kutta method18 to propagate the growth of an initial α-particle distribution, and a simplex19 routine to choose the parameters which best reproduced the observables, we obtained the parameter values given under “Fit A” in Table 1. The values in the column under “Fit B” are derived from a skewed starting distribution and will be discussed later. Initial fits, under “A” included the observables for Stage II and III duration and for final β-particle size distribution as averages from the six syntheses reported in ref 15. Also included was the early Stage II, average size for the αparticle distribution, rαt. This latter data point was from a single measurement reported in ref 15 but is consistent with other samples we have previously measured. The relevant observables are given in Table 2. The growth of the nanoparticles is illustrated in Figure 5. The illustrations are shown for particle diameter rather than radius, as this is more typical for reports on experimental work. The histograms in the top panel reflect the changing particle size distribution of the α particles whose distribution spreads in width and increases in mean up until around 59 min when β particles begin to appear. By 73 min, the α-particle size distribution has receded to very few in number with a mean diameter near 2 nm. The middle panel of Figure 5 illustrates the growing β-particle size distribution after β particles first appear around 59 min. The bottom panel shows how the total mass is distributed between the two phases during the course of the reaction. From the derived parameters in Table 1 we may extract solubilities, surface energies (γ) and the molar change in lattice energy (Δμlattice) for the phase transition. These are given in Table 3, again under Fit A, along with comparisons to evaluate the extent to which they have physically reasonable magnitudes. The surface energy may be extracted from the capillary length,

The parameters for Fit A provide a best fit given a Gaussian starting distribution of α particles. The parameters for Fit B result when given a negatively skewed starting distribution. a

Table 2. Measured and Simulated Values of Relevant Observables in the Synthesis of NaYF4 Nanomaterials observables: average values from six syntheses (refs 15 and 16) Stage II duration (T2) Stage III duration (T3) μβ = rβfinal Final product radius σβ, std dev of the final product radius. α-particle radii measured from synthesis. early Stage II, rαt average αparticle size(t = 10% Stage II) early Stage II, σt std dev αparticle size distribution late Stage II, rαt average αparticle size. (t = 90% Period II) Late Stage II, σt std dev αparticle size distribution.

simulationa results (Fit A)

simulationa results (Fit B)

63 ± 22 min 10 ± 2 min 19 ± 1.5 nm

60.0 min 10.3 min 19.1 nm

61.7 10.2 19.0

0.9 ± 0.3 nm

0.90 nm

0.90

one individual 2.4 nm

simulated α-particle diameters 2.3 nm 2.1 nm

0.5 nm

0.001 nmb

3.3 nm

b

2.4 nm

0.47 nm 3.0 nmb

L= 1.2 nm

0.7 nmb

1.0 nmb

2γVm RT

For the α-phase, we find, γα = 0.088 J/m2, which is similar in magnitude to the surface tension of water in air, whereas the surface energy for the β-phase is three times higher with γβ = 0.255 J/m2, consistent with the relative greater stability of the α-particles, as compared to β-particles, at small radii. The change in the surface energy, Δγ, allows us to extract the change in bulk lattice energy from the transition radius,

a

The parameter values in Table 1 are used in the simulations along with molar volume, Vm = 4.62 × 10−5 m3/mol, for both α- and β-phase particles. The time step for the simulations was 0.2 s. bThese values were not included as criteria in the simplex fits, but are included for comparison.

give a summary of the parameters and observables, respectively, used in the fit. The observables include average values for Stage duration and particle size distributions. The parameters include bulk rate constants, capillary lengths, diffusion coefficients, and transition radius, as defined for eqs 1 and 3.

rT =

−2ΔγVm Δμ lattice

yielding, Δμlattice = −3.3 kJ/mol, which is physically reasonable for a solid−solid phase transition. Furthermore, the bulk 9485


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Table 3. Physical Constants Extracted from the Parameter Values of Table 1 physical constants γβ (310 °C) γα (310 °C) Δγ (310 °C) Δμlattice (310 °C) bulk solubility αphase (310 °C) bulk solubility βphase (310 °C) γwater (25 °C) γLiF, γCaF2, γBaF2 (25 °C) −Δμfusion water (25 °C) solubility LaF3 in water (25 °C)

magnitude Fit A

magnitude Fit B

0.255 0.088 0.17 −3.3

0.269 0.020 0.25 −4.4

J/m2 J/m2 J/m2 kJ/mol

from L(β) from L(α) γβ − γα from rT

3.3 × 10−4

1.7 × 10−4

mol/L

from k2/k1

1.1 × 10−5

2.8 × 10−6

mol/L

from k2/k1

units

Comparison Values 0.07 J/m2 0.34, 0.45, 0.28 J/m2

notes

ref 20 ref 21

−6.02

kJ/mol

ref 20

2.4 × 10−5

mol/L

ref 22 (from Ksp = 3.1 × 10−19)

Figure 6. Illustration of the skewed sized-distribution used as a starting point for α-particle growth. The mean, the standard deviation, and the extent of the skew (4.2 nm, 0.6 nm, and 2% in this illustration) were optimized in the fit as 4.2 nm, 0.6 nm, and 0.5%, but are shown here with the larger skew for ease of visualization. Figure 5. Evolution of particle size distribution and phase from simulation using parameters from Fit A. (a) The α-particle size distribution remains very narrow up to 29 min, but broadens significantly by 44 min. (b) By 59 min some particles have exceeded the thermodynamic phase transition diameter, 2rT = 9.8 nm, and grow as β particles. (c) Proportion of α material and β material during the course of the reaction. The resolution of the bin size for the histograms is 1 nm. For the simulations, the time, t = 0 is equivalent to the start of Stage II in the reaction, whereas for the experiments, t = 0 is defined as the start of Stage I, approximately 30 min earlier.

might be much smaller than the diffusion coefficient for a single molecule, as it must represent the simultaneous diffusion of six ions, Na+ + Y3+ + 4F−, with some unknown coordination environment, through a matrix of long chain organic molecules. Less easily explained is the very narrow width of the initial αparticle size distribution. The simulated size distribution is very narrow for all of the first half of Stage II. All of the particles in the histogram in the top panel of Figure 4 fall at d = 4 nm at 29 min and the fit thus shows a poor match to the observed width of the size distribution at early Stage II as shown in Table 2. A second fit was performed, fixing the value of σ0 at a more reasonable value of 0.25 nm, but we were unable to reproduce the size of the final β-particle distribution or the other observables very closely. We have considered several possibilities for elements that may have been left out of this simple model, which are manifested in the requirement for an unphysically narrow starting α distribution. The small σ0 requirement appears to arise from the need for a fairly steep leading slope for the α distribution as it crosses the phase transition size, rT. With a gentler slope, fewer particles cross the transition size before the

solubilities (k2/k1) of the α- and β-phase materials (in organic solvent at 310 °C) are comparable to the bulk solubility of LaF3 in water at 25 °C, with the α-phase more soluble than the βphase. These extracted values give some confidence that the proposed model for nanoparticle growth and phase transition can reproduce the observed course of the reactions with, for the most part, parameters with physically reasonable magnitudes. Two of the parameters which were directly determined from the fit bear some further scrutiny. Specifically, the effective diffusion coefficient, D, and the width of the initial α-particle size distribution, σ0, seem very small in Fit A. For the effective diffusion coefficient one might argue, that the model parameter, 9486


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Figure 8. Results of simulation of six different experimental runs reported in refs 15 and 16. (a) Upconversion luminescence intensity as a function of reaction time. (b) The simulated fraction of the mass in the β phase serving as a proxy for the relative upconversion intensity. (c) Comparison of the measured and simulated final product distribution. The height of the columns represents the average diameters, 2 μβ, and the error bars represent the width of the distributions, 2σβ.

Figure 7. Evolution of particle size distribution and phase from simulation using parameters from Fit B starting from a skew-normal distribution of α particles. (a) The α particle size distribution grows and broadens slightly over the first 58 min. (b) After 58 min, some particles have exceed the thermodynamic phase transition diameter, 2rT = 10 nm, and grow as β particles. (c) Proportion of α material and β material during the course of the reaction. The resolution of the bin size for the histograms is 1 nm.

motivates consideration of other factors that might cause the αdistribution to maintain a greater focus in its size distribution as it ripens. One possibility is that at the starting point for the simulation, which is equivalent to the start of Stage II, there is an additional phase present, one more soluble than the α phase, which would focus the α distribution in the same way that the presence of α material focuses the growing β size distribution. However, if this is the case, the material is being washed away in the workup of the extracts made during the early stage of Stage II as there is no evidence for it in the TEMs reported in ref 15. A second possibility is that there is a significant kinetic barrier to the α → β phase transition, such that α particles of a size where the phase transition is thermodynamically favored, “stack up” just beyond rT, growing slowly at the α-particle rate, before achieving the ultimate α → β phase transition, and then

β particles begin to consume the α particles and the α distribution size begins to recede away from rT. The number of particles that cross the transition size determines the final number of β particles and since the total mass is fixed by the initial α distribution, determination of the number of β particles also determines the final size of the β particles. In this model, starting with a wider initial α particle distribution results in final β particles that are larger than we observe experimentally. Since the simulated α particle distribution broadens through Oswald ripening throughout Stage II, the fit is choosing a very narrow starting distribution so that the width of the distribution near the end of Stage II is on the order of 1 nm similar to the experimental observation during late Stage II. This observation 9487


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The Journal of Physical Chemistry C Table 4. Effect of Starting Distribution Parameters on Stage Duration and Final Product Size Distributiona EXPT

simulation results

starting distribution

run number

T1 (min)

T2 (min)

T2 (min)

T3 (min)

2 μβ (nm)

2σβ (nm)

2 μ0 (nm)

2σ0 (nm)

skew

1 2 3 4 5 6 average std dev fit to average

33 33 29 23 27 33 29.7 4.1 n/a

30 46 56 79 81 85 62.8 22.3 63

30 46 56 79 80 85 62.7 22.2 62

10 10 10 11 10 10 10.2 0.4 10

38.3 39.3 38.5 39.0 38.0 37.4 38.4 0.7 37.9

2 1.8 1.8 1.8 1.8 1.8 1.8 0.1 1.8

4.83 4.91 4.58 3.85 4.31 3.46 4.3 0.6 4.24

1.33 0.87 0.70 0.40 0.34 0.38 0.67 0.39 0.60

0.54% 0.54% 0.54% 0.75% 0.82% 0.84% 0.67% 0.15% 0.52%

T1, T2, T3 refer to the duration of Stages I, II, and III; 2 μβ and 2σβ define the final particle size distribution in terms of the average and standard deviation of the diameter. a

growing rapidly toward the final size. Consistent with this assumption, Tolbert and Alivisatos have shown that a kinetic barrier in CdSe quantum dots leads to significant hysteresis in the pressure-dependent phase-transition from wurzite to rocksalt structures.23 The first scenario, inclusion of an additional more soluble phase in which starting materials is sequestered, is fairly easy to model. The second presents more of a problem in that stochastic behavior is incompatible with the simplex method. However, the barrier can be crudely represented by defining a delay time, Q, so that each α particle, as it crosses the phase transition size, is delayed by Q minutes before it undergoes the α → β phase transition. Inclusion of either of these components in the model does allow an increase in the width of the starting distribution, as expected, but only a modest increase. A third possibility is that the starting α distribution is negatively skewed rather than normal. A negative skew would, in fact, be the expected distribution. The 4 nm particles that are present at the start of Stage II, corresponding to the start of our simulation, likely result from Ostwald ripening of much smaller seeds that nucleated during heat-up. As reported by Talapin in the ripening of a Gaussian distribution of 1 nm particles, the particle distribution develops a negative skew by the time the mode of the distribution reaches 3 nm for both reaction- and diffusion-limited growth.11 To create a computationally efficient skewed distribution, we have adopted a strategy illustrated in Figure 6. In Figure 6 the blue curve represents a normal distribution of particles sizes with a mean and standard deviation typically observed in early stage α particle growth. For an illustration of the skewed distribution, the largest 2% of the particles from the normal distribution are reflected in size about the d98% position, defined such that 98% of the particles in the normal distribution have diameters smaller than d98%. A new simplex fit was performed in which the extent of the skew was treated as a free parameter. The experimentally observed width of the early Stage II size distribution was included as an additional criterion to the fit. The new parameter values and the simulated results for the observables are given in Tables 1 and 2, respectively, under the column heading Fit B. Note that a very small skew parameter, 0.5%, is required to provide a good fit to the observables, including the early Stage II width of the particle size distribution. The size distribution at late-stage Stage II is also well reproduced in the simulation, although these observables, arising from a measurement during

a single synthesis, and potentially showing some variability from run-to-run, were not included as criteria in the fit. The evolution of the nanoparticle size distribution for simulation using the parameters derived from Fit B is illustrated in Figure 7, where it can be seen that the width of the distribution at early times is now very similar to observed experimental distributions. To determine whether the model described above could account for the observed variation in Stage II duration while maintaining the consistency in Stage III duration and in the final product distribution, we tested the simulation results for sensitivity to each of the parameters. Not surprisingly perhaps, Stage II duration is very sensitive to the size of the starting αparticle distribution. Figure 8 and Table 4 illustrate the results of simulating the six different experiments separately as opposed to simulating an average outcome as done previously. In the simulations, only the parameters relating to the starting α-particle distribution were allowed to vary, and all other parameters were fixed at the values determined to give the best fit under Fit B. As can be seen in Table 4, physically reasonable variations in the starting α-particle distribution easily reproduce a wide variation, 63 ± 22 min, in the Stage II duration, while maintaining very similar Stage III durations (10 min) and final product size distributions, (2 μβ = 37−39 nm). Figure 8 illustrates the six experiments for real-time monitoring of upconversion luminescence compared to the simulation results. For the simulations, the fraction of the total mass in the β-phase serves as a proxy for the relative upconversion intensity. The time scale of the simulations is adjusted to match the time scale for the experiments by adding the length of the Stage I duration (not simulated). The lower panel of Figure 8 compares the measured and simulated final β-particle size distributions for each of the experiments. As can be seen, there is a very good correspondence between the experimental and simulated results for the six individual experiments, and the differences between the runs can be explained simply in terms of small differences in the starting α-particle size distribution. The purpose of this paper is not to suggest that the parameters derived here are accurate estimates of the relevant physical constants, but rather to show that the observed growth and phase transition behavior can be very closely reproduced by a simple model of Ostwald ripening and thermodynamically controlled phase transition, invoking physical constants well within a reasonable range. 9488


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CONCLUSION We have shown that a simple model of nanoparticle growth via Ostwald ripening and size-dependent α → β phase transition reproduces the experimentally observed changes in the size distribution and crystallographic phase of NaYF4 nanoparticles during synthesis of β-NaYF4 by the “heat-up” method. Using reasonable values for the relevant physical constants, and starting with a normal distribution or a slightly skewed-normal distribution of α particles, the model predicts growth and broadening of the α particle distribution until some particles exceed the diameter at which the β phase become thermodynamically stable. The newly formed β-phase particles, having a lower solubility than the α phase, grow at the expense of the α phase, focusing to a final diameter of 2 μβ = 38 nm. The model also reproduces the observed timing of the various stages of the reaction, reported from experiment as approximately 60 min for Ostwald ripening of the α particles and 10 min for subsequent completion of the α → β phase transition and, further, explains the observed variation in delay for onset of the phase transition as arising from small variations in the size distribution for the α particles formed during the heat up of the precursor material. Up to this point, the origin of the seeds for the β particles has been unclear. This work supports a hypothesis that the β-particle seeds arise from a phase transition in individual α-phase particles.

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.6b01365. Detailed derivation of Equations 1 and 3. Relationship of transition radius to capillary length. (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Computations were performed on the High Performance Computing Cluster at the University of South Dakota. The authors acknowledge support from the National Science Foundation (DGE-0903685) and (IIP-1414211).

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REFERENCES

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