Ripening of Semiconductor Nanoplatelets - ACS Publications

Letter Cite This: Nano Lett. XXXX, XXX, XXX-XXX

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Ripening of Semiconductor Nanoplatelets Florian D. Ott,† Andreas Riedinger,† David R. Ochsenbein,‡ Philippe N. Knüsel,† Steven C. Erwin,§ Marco Mazzotti,*,‡ and David J. Norris*,† †

Optical Materials Engineering Laboratory, ‡Separation Processes Laboratory, Department of Mechanical and Process Engineering, ETH Zurich, 8092 Zurich, Switzerland § Center for Computational Materials Science, Naval Research Laboratory, Washington, DC 20375, United States S Supporting Information *

ABSTRACT: Ostwald ripening describes how the size distribution of colloidal particles evolves with time due to thermodynamic driving forces. Typically, small particles shrink and provide material to larger particles, which leads to size defocusing. Semiconductor nanoplatelets, thin quasi-two-dimensional (2D) particles with thicknesses of only a few atomic layers but larger lateral dimensions, offer a unique system to investigate this phenomenon. Experiments show that the distribution of nanoplatelet thicknesses does not defocus during ripening, but instead jumps sequentially from m to (m + 1) monolayers, allowing precise thickness control. We investigate how this counterintuitive process occurs in CdSe nanoplatelets. We develop a microscopic model that treats the kinetics and thermodynamics of attachment and detachment of monomers as a function of their concentration. We then simulate the growth process from nucleation through ripening. For a given thickness, we observe Ostwald ripening in the lateral direction, but none perpendicular. Thicker populations arise instead from nuclei that capture material from thinner nanoplatelets as they dissolve laterally. Optical experiments that attempt to track the thickness and lateral extent of nanoplatelets during ripening appear consistent with these conclusions. Understanding such effects can lead to better synthetic control, enabling further exploration of quasi-2D nanomaterials. KEYWORDS: Colloidal semiconductor nanoplatelets, Ostwald ripening, growth kinetics, nucleation

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We recently introduced a simple model that can explain the formation of this highly anisotropic shape.26 We considered the standard 2D nucleation and growth mechanism,27 which describes how a new monolayer of material nucleates as a 2D island and expands to cover an exposed crystal facet. We showed that the nucleation barrier for this process can be strongly reduced on atomically narrow facets compared to wide facets. This can lead to a kinetic instability in which growth in the lateral directions of a platelet is highly accelerated while its thickness remains constant. This then provides a straightforward explanation for the formation of semiconductor nanoplatelets. However, a more comprehensive model must also be consistent with experiments that examine how the size distribution of nanoplatelets evolves over longer time scales. During prolonged growth reactions, the nanoplatelet shape can be affected by thermodynamically driven phenomena. One such process is Ostwald ripening.28,29 In general, it causes smaller crystals within a colloidal size distribution to dissolve while larger ones grow. Typically, this broadens (or defocuses) the

emiconductor nanocrystals have physical properties that strongly depend on their size and shape.1 Due to extensive research on these materials, syntheses have been developed to control both of these attributes. Available shapes now include spheres,2 rods,3,4 tetrapods,5,6 and quasi-two-dimensional (quasi-2D) platelets.7−16 In general, nonspherical particles can be prepared by exploiting the underlying crystallographic symmetry of the material. For example, growth along a unique crystal axis can be induced to create rods or wires. By modifying the conditions (e.g., by adding surfactants that selectively bind to specific exposed facets), various shapes can be obtained even from the same material.17 Alternatively, asymmetric shapes can be created by constraining growth within templates. In particular, molecular lamellae have been invoked to explain the formation of nanosheets and nanobelts.9,10,12,18 However, absent such templates, equivalent exposed facets of a crystallite are expected to exhibit the same stability and growth behavior.17,19−24 Thus, it has been challenging to understand the formation of zinc blende CdSe nanoplatelets. These rectangular, atomically flat particles with thicknesses of only a few atomic layers grow without templates. The underlying crystal structure is isotropic, and all of the exposed facets are identical {001} surfaces.25,26 © XXXX American Chemical Society

Received: July 26, 2017 Revised: September 25, 2017 Published: October 9, 2017 A

DOI: 10.1021/acs.nanolett.7b03191 Nano Lett. XXXX, XXX, XXX−XXX

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Figure 1. Size dependence of the solubility and kinetic barriers for CdSe nanoplatelets at T = 450 K. (a) Solubility (eq 3) for 1- to 5-monolayer-thick CdSe nanoplatelets (colored lines) plotted versus their lateral length L (see inset) in units of angstroms (Å; bottom axis) and monolayers (top axis). (b) Island nucleation barriers on 1- to 5-monolayer-thick facets (colored lines) as a function of the monomer supersaturation c/c*∞. For comparison, the thick black curve presents the nucleation barrier in the wide-facet limit. It shows that at increased concentrations, where the size of the critical (i.e., thermodynamically stable) wide-facet surface island is reduced, fewer narrow facets exhibit a reduced kinetic barrier. Indeed, at sufficiently high concentrations, far above the plotted range, no kinetic instabilities will exist because a single monomer added to a crystal surface already corresponds to a stable surface island. The inset shows how surface islands grow in the wide- and narrow-facet regime. The additional crystal surface area and step length that emerge during island growth are marked in blue and red, respectively. See Section 1 in the Supporting Information for further discussion.

are set by the material system, which includes the semiconductor and the surface ligands (here CdSe with carboxylate ligands). Using these three parameters, we previously showed that the nucleation barrier of a 2D surface island decreases with the thickness of the facet on which it grows.26 (For a summary, see Section 1 and Figure S1 in the Supporting Information). This decrease then affects the growth kinetics. In addition, the model predicts that the thermodynamic stability of a crystallite increases with its thickness. This means that the thinnest platelets grow the fastest (due to kinetics) but are also the least stable (due to thermodynamics). To include both effects, we now consider the formation energy (describing the thermodynamics) of an m-monolayerthick square-shaped nanoplatelet with L × L lateral dimensions:

size distribution, which is detrimental for exploiting the sizedependent properties of semiconductor nanocrystals. For spherical particles (e.g., colloidal quantum dots), this phenomenon is well-understood. In contrast, ripening in nanoplatelets has not been studied despite several unique attributes. The size distribution of nanoplatelets can be essentially discrete in one dimension due to their atomicscale thickness of a few monolayers.11,30 Moreover, each thickness population can be experimentally tracked due to its specific optical characteristics. For example, the primary absorption peak from CdSe nanoplatelets of 2, 3, 4, and 5 monolayers each occurs at a distinct wavelength.11 Using such spectral features, experiments have shown that the thickness distribution for nanoplatelet samples does not defocus with time, as would be expected from conventional Ostwald ripening. We previously argued26 that this suggests that thicker nanoplatelets do not arise by adding another layer to a thinner platelet. Instead, thinner platelets dissolve laterally, while thicker nuclei expand laterally. In this case, the stability of a nanoplatelet would not necessarily scale with particle volume, as in conventional Ostwald ripening. Rather, material could transfer from large-volume thin nanoplatelets to small-volume thick ones. To clarify these issues, here we investigate in detail how ripening occurs in semiconductor nanoplatelets. Such a study provides an additional test for our kinetic model26 of nanoplatelet growth. More importantly, it allows an exploration of the ripening process in this unique colloidal system. As thermodynamic driving forces are critical for this process, we must incorporate these effects to obtain a complete description of nucleation, growth, and ripening in semiconductor nanoplatelets. We develop such a model and then demonstrate how competition between kinetics and thermodynamics leads to counterintuitive shape evolution in these materials during ripening. Our model depends on three energy parameters: (i) the difference in chemical potential between a monomer in the crystal and in the growth solution, Δμ, (ii) the crystal surface energy, σ, and (iii) the step energy of a 2D surface island, κ. Δμ is a function of the monomer concentration in solution. σ and κ

ΔGmcryst(L) = L2mh

Δμ + (2L2 + 4Lmh)σ V0

(1)

where h is the monolayer height and V0 is the volume per CdSe monomer in the crystalline phase. In the zinc blende lattice these are related as V0 = 2h3. Under conditions where crystallization is thermodynamically favorable, Δμ is negative. Thus, the first term in eq 1 represents the energy gain due to formation of crystallite volume. The second term is the energy cost due to creation of crystallite surface (σ > 0). As expected, the formation energy is minimized for the smallest surface-tovolume ratio.19 Moreover, the thermodynamic driving force for a CdSe nanoplatelet to increase in thickness is much higher than in the lateral direction. However, this driving force is hindered by the high kinetic barriers for growth on wide facets of nanoplatelets. Thus, a transition from an m- to an (m + 1)monolayer-thick platelet via direct addition of another layer is unlikely. Nevertheless, experiments show that, during prolonged heating (i.e., ripening), nanoplatelets with m layers slowly disappear, while platelets with m + 1 layers slowly appear.26,31,32 Our aim is to determine the mechanism behind this process. Because the thermodynamic driving forces depend on the magnitude of Δμ, it is a key parameter for understanding ripening in nanoplatelets. In our previous work,26 we could not fully explore ripening because our treatment assumed Δμ was B


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Nano Letters constant. In reality, |Δμ| decreases as monomers from the solution are consumed during growth. As |Δμ| decreases, the thermodynamic driving force for crystallization is also reduced. Thus, thin nanoplatelets, which have a very high surface-tovolume ratio, eventually become unstable. This effect can be quantified by starting with the standard relation for Δμ in terms of the solute concentration: * Δμ(c) = −kBT ln c /c∞

(2)

where kB is the Boltzmann constant, T is the reaction temperature, c is the free-monomer concentration in the growth solution, and c∞ * is the monomer solubility for a bulk crystal. If the monomer concentration c is equal to the c*∞, Δμ is exactly zero, and no growth or dissolution occurs for a bulk crystal. If c is greater than c∞ * (i.e., the solute is supersaturated), growth can occur. However, the concentration for which growth and dissolution are in equilibrium (i.e., the solubility) depends on the size of the crystal. The solubility for a finitesized particle is higher than c∞ * because it is less stable due to the positive surface energy σ. In other words, the concentration of free monomers must be higher to grow on a small particle than on a bulk crystal. This dependence of the solubility on the crystal size is the basis of Ostwald ripening. For nanoplatelets, the solubility is primarily governed by the narrow facets (see Section 2 in the Supporting Information).29 Because growth on the wide facets is hindered, it can (for the moment) be neglected. The solubility concentration for an mmonolayer-thick nanoplatelet with L × L lateral dimensions is then well-approximated as ⎡ 2σV0 ⎛ 1 1 ⎞⎤ * exp⎢ ⎜ cm*(L) = c∞ + ⎟⎥ ⎝ L ⎠⎦ ⎣ kBT mh

(3)

At c = cm*(L), such a nanoplatelet is in equilibrium with respect to lateral growth, implying that growth and dissolution are equally fast on the narrow facet. Figure 1a plots c*m (L)/c*∞ versus lateral side length L. The solubility of a nanoplatelet increases both with decreasing thickness and decreasing lateral dimension. To model the ripening process, we now combine the kinetic and thermodynamic effects and develop a model consisting of a system of rate equations (see Sections 3 and 4 in the Supporting Information). The rates describe the addition or removal of monolayers according to 2D nucleation and growth. Thus, we consider the “attachment” and “detachment” rates for monolayers (instead of monomers). For addition of material on a narrow facet, new surface layers are assumed to have an attachment rate equal to that of 2D-surface-island nucleation, given by (see ref 27): ⎡ ΔG barrier(c) ⎤⎛ c ⎞2 m ⎥⎜ ⎟ Im(c) = I0 exp⎢ − *⎠ ⎢⎣ ⎥⎦⎝ c∞ kBT

Figure 2. Attachment and detachment rates on surface facets of CdSe nanoplatelets at T = 450 K. (a) Monolayer attachment rates for 1- to 5-monolayer-thick facets (colored lines) and on wide facets (thick * . The black curve) as a function of the supersaturation c/c∞ disappearance of reduced kinetic barriers on narrow facets with increasing supersaturation can be observed as the colored lines merge into the wide-facet result. (b) Monolayer detachment rates for 1- to 5monolayer-thick facets (colored lines) as a function of lateral length L in units of angstroms (Å; bottom axis) and monolayers (top axis). The rates depend on the dimensions of the surface facet, here given by the nanoplatelet thickness in monolayers (m) and lateral side length L. The detachment rates are plotted for L ≥ mh, and the points correspond to square-shaped facets with L = mh.

For the removal of material from a narrow facet, dissolution of layers must be included. By imposing the equilibrium condition at c = c*m (L), where the rate of removal equals the rate of addition, we can immediately find a monolayer detachment rate:

(4)

where I0 is a constant prefactor (with an estimated value of ∼1 Hz), and ΔGbarrier (c) is the concentration-dependent nucleation m barrier of a surface island on an m-monolayer-thick facet (see Section 3 in the Supporting Information). Figures 1b and 2a plot the barrier ΔGbarrier (c) and the monolayer attachment rate m Im(c)/I0, respectively, as a function of free-monomer concentration for different facet thicknesses. Curves for the wide-facet limit are shown for comparison. The decrease of the barrier and increase of the attachment rate for narrower facets is the origin of the intrinsic instability.

Dm(L) = Im(cm*(L))

(5)

This condition then incorporates the thermodynamic constraints into the model. (See Section 4 in the Supporting Information for a detailed discussion of this important relationship.) Figure 2b shows the detachment rate versus lateral size for various nanoplatelet thicknesses. C


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Nano Letters Equations 4 and 5 provide the rates at which a monolayer detaches or attaches from a mh × L narrow surface facet. Moreover, we show in Section 4 of the Supporting Information that the same equations can provide these rates for wider facets, given its dimensions and the concentration of monomers in solution. Thus, they can also be applied to an L × L wide facet. In general, the unique behavior of nanoplatelets originates in their atomic-scale thickness, which leads to facile growth in the lateral directions and hindered growth in the thickness direction. Above, to estimate c*m (L), we assumed that the latter is zero. Now, we relax this condition and allow growth (no matter how slow) on wide facets. We simply apply eqs 4 and 5, which describe growth and dissolution as a function of facet dimensions and monomer concentration. This allows us to study the evolution of a population of nanoplatelets during ripening. For simplicity, we describe all crystals as thin rectangular parallelepipeds that are m monolayers thick with L × L lateral dimensions, where L = nh. Each platelet population has a concentration Zm,n. In addition, we consider platelets with thicknesses only up to cubes. This means that, by definition, the thickness of any crystallite must be equal to or smaller than its lateral size, a condition fully consistent with the kinetic instability. For each nanoplatelet population characterized by m and n, the change in concentration dZm,n/dt is described by a kinetic rate equation which accounts for all surface reactions either forming or consuming the corresponding nanoplatelet33−35 (see Section 4 and Figure S2 in the Supporting Information). Finally, we need to address the role of the growth unit, assumed to be CdSe monomers. This highly reactive species forms from precursor molecules upon heating with rapid mixing.36 For a fast (burst) nucleation process, high monomer concentrations are required at the early stages of the reaction. This can be achieved by high precursor conversion rates.37 For simplicity, we assume here that the precursors react extremely fast and have completely transformed to monomers at the very beginning of the crystal-growth reaction (see Section 5 in the Supporting Information for calculations with slower monomerformation rates). Numerical solutions of the rate equations for nanoplatelets growing at T = 450 K are shown in Figures 3 and 4 (see Section 5 in the Supporting Information for calculations at different temperatures). Based on experimental knowledge that 1monolayer-thick nanoplatelets do not form, we assumed that the initial concentration of monomers c0 is smaller than the solubility of any 1-monolayer-thick platelet with a finite lateral size. We thus set c1 = c*1 (∞) = 284c*∞ (see Section 5 in the Supporting Information for calculations with different initial conditions). We included nanoplatelets with thicknesses up to 6 monolayers. The material that exceeded this thickness during our calculations (0.015%) was regarded as irreversibly lost and removed. Figure 3a plots the time dependence of the fraction of the initial monomers that remain free (black curve) or that are incorporated into different nanoplatelet populations with thicknesses ranging from 1 to 5 monolayers (colored curves). It shows clearly the successive appearance of increasingly thicker nanoplatelet populations. For each additional layer in thickness, the reaction time increases by several orders of magnitude, explaining why the selective synthesis of one specific thickness can be well-controlled (usually in practice by adapting the temperature instead of the reaction time). In the profiles in Figure 3a, the plateaus represent the solubility where

Figure 3. Calculated evolution of thickness populations for CdSe nanoplatelets growing at T = 450 K. (a) Fraction of initial monomers remaining as free monomers (thick black curve) or incorporated into nanoplatelet populations with thicknesses ranging from m = 1 to 5 monolayers (colored curves) plotted versus dimensionless time τ = I0t. The successive formation of nanoplatelet populations with increasing thicknesses can be observed. (b) Corresponding concentrations, Zm, of the same populations as in panel a, relative to the initial monomer concentration c0. All populations nucleate at very early stages in the synthesis. Note that 1-monolayer-thick crystals with negligible monomer fraction do exist. As can be seen by comparing panels a and b, these crystals are in direct equilibrium with free monomers.

free monomers are at equilibrium with the platelet surface for each thickness. These results clearly show that different nanoplatelet thicknesses form sequentially. We might conclude that another layer simply adds onto the wide facet of an existing nanoplatelet to increase its thickness by one monolayer. However, Figure 3 exhibits features that are very different from what one would obtain if this were the underlying mechanism. Rather, our calculations indicate that direct growth in the thickness direction only occurs at very early stages of the synthesis when the nanoplatelets are small “nuclei” and the monomer concentration is high. This is clear from Figure 3b where the concentration of nanoplateletsirrespective of their lateral D


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Figure 4. Calculated evolution of the lateral size distribution at T = 450 K for CdSe nanoplatelet thickness populations of m = 2−4 monolayers (in columns left to right). In each plot, the concentration Zm,n for nanoplatelets with thickness m and lateral side length L = nh is plotted versus L. The concentrations are relative to the initial monomer concentration c0. In each column, dimensionless time τ = I0t proceeds downward. Note that different ranges for τ and L are used for each thickness. The red vertical line represents the average side length for the plotted distribution. For each thickness population, the initial growth leads to a relatively narrow lateral size distribution. Eventually smaller lateral sizes within each thickness population become unstable, leading to Ostwald ripening in the lateral direction. The smaller (larger) nanoplatelets shrink (grow), broadening the lateral size distribution. Eventually (bottom plot in each column), the average lateral size decreases as all the nanoplatelets of a given thickness become unstable and laterally dissolve.

sizeis plotted versus time. The amount of platelets in each thickness population quickly reaches a maximum value with small delays for thicker populations. In comparison with the monomer fractions shown in Figure 3a, the nanoplatelet concentrations increase very uniformly. Therefore, the concentration ratios between different thickness populations are determined at the very beginning of the synthesis during the nucleation phase.37 After that, during the growth phase, the population concentrations slightly decay and then quickly drop to zero as each thickness population becomes thermodynamically unstable. This behavior can be well-understood from the time evolution of the lateral size distribution, as shown in Figure 4 for 2- to 4-monolayer-thick nanoplatelets (see Figure S3 in the Supporting Information for additional results). For each

thickness, the growth sequence evolves as follows: (i) Nanoplatelets nucleate and start to grow. In this phase, the distribution of lateral sizes for each thickness is relatively narrow. (ii) In a second stage, as the free-monomer concentration is reduced, growth slows down, and within each thickness population a fraction of the laterally small nanoplatelets becomes thermodynamically unstable. This means that lateral detachment is faster than attachment for these laterally small crystals. They then shrink laterally while the larger ones of the same thickness grow laterally. This phenomenon within an individual thickness population strongly resembles conventional Ostwald ripening, but in 2D. (iii) At even lower free-monomer concentrations, all lateral sizes of a given thickness population become unstable. All nanoplatelets for this thickness dissolve via lateral detachment and eventually E


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Figure 5. Experimental results for CdSe nanoplatelet growth and ripening at 453 K (180 °C). (a) The evolution of the absorption spectrum in time (listed in minutes). The three colored vertical lines at 395, 462, and 512 nm correspond to the heavy-hole absorption peak of 2-, 3-, and 4monolayer-thick nanoplatelets, respectively. In each case, this peak is accompanied by a second feature at shorter wavelengths due to the light−hole transition (see ref 11). (b) The maximum absorbance from the three peaks in panel a plotted versus time. (c−e) The wavelength of the absorption peak over time for the 2-, 3-, and 4-monolayer thick nanoplatelets, respectively. The blue horizontal lines represent the absorption wavelengths for laterally large nanoplatelets. The shifts in panel c imply that 2-monolayer-thick nanoplatelets shrink laterally while their absorption reduces at longer times. Panels d and e are consistent with our predictions that thicker populations grow from laterally small nuclei instead of forming directly from thinner nanoplatelets that increase in thickness.

disappear. Our calculations show that under these conditions the lateral detachment rates are at least 6 orders of magnitude faster than either attachment or detachment on the wide facets. The results described above are already consistent with the experimental observation that nanoplatelets of a given thickness slowly disappear during prolonged heating, while those with one additional layer appear.26,31,32 However, to seek additional evidence we performed experiments in which CdSe nanoplatelets were heated over an extended time span (3 days) at 453 K (180 °C) (see Section 6 in the Supporting Information for experimental methods). Nanoplatelets exhibit spectroscopic characteristics that are extremely valuable for tracking the details of their growth. As stated above, each thickness population exhibits a primary optical absorption peak at a distinct wavelength.11,38,39 However, this peak shifts slightly to longer or shorter wavelengths as the lateral side length of each thickness population expands or shrinks.40,41 These shifts occur due to quantum confinement of the exciton in the lateral direction. Therefore, by simply measuring the absorption spectra as a function of time, the evolution over different thicknesses and lateral dimensions can be followed. In Figure 5a, absorption spectra taken during the 3-day reaction are shown. As predicted by our model, the thinnest 2monolayer-thick nanoplatelets grow faster than any other population leading to the immediate build-up of the absorption feature at 395 nm. At much slower rates (over several hours), the 3-monolayer population develops, as can be seen from the peak rising at 462 nm. Then, at the very end of our experiments, a small absorption feature due to 4-monolayerthick nanoplatelets appears at 512 nm. Figure 5b plots the

absorption maximum for each population at its respective wavelength versus time. This result confirms the subsequent evolution of different populations as predicted by our model. In addition, this experiment can potentially reveal information about the evolution of the lateral size within a thickness population. If nanoplatelets dissolve laterally instead of growing in thickness, we would expect to observe a blue shift of the absorption peak for a given thickness. Indeed, as seen in Figure 5c, the absorption peak for 2-monolayer-thick platelets first shifts red and then blue as a function of time. This is consistent with lateral growth and then dissolution, as expected from our model. For 3- and 4-monolayer-thick nanoplatelets, only a red shift is observed (Figures 5d,e). This is expected if these populations continue to expand laterally, implying that they did not start to dissolve during the time frame of the experiment. While these experimental results agree well with our theoretical predictions for the evolution in thickness and lateral size of nanoplatelets, such small spectral shifts in absorption features could be explained by other effects, such as changes in the chemical environment, surfactant coverage, or nanoplatelet aspect ratio.40,42 However, we note that, in our experiments, the blue (red) spectral shifts occur simultaneously with a reduction (increase) in absorbance. This combination is consistent with lateral dissolution (growth). We also note that these same trends were reproduced in several synthetic runs. To provide further experimental support, we also used transmission electron microscopy (TEM) to investigate the size evolution of nanoplatelets during growth (see Section 7 in the Supporting Information). Unfortunately, it was not possible to extract both F


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the lateral size and thickness for the same nanoplatelet by TEM, which would be required to track the evolution of the thickness populations. In summary, we have combined the previously proposed microscopic mechanism of facet-size-dependent growth with the underlying thermodynamic constraints given by shapedependent crystal stabilities to obtain a complete model describing the processes during nanoplatelet nucleation, growth, and ripening. The predicted behavior is consistent with experimental results on CdSe nanoplatelets. Within each thickness population the evolution of the lateral size distribution can be described by two steps: concerted growth of the distribution followed by a dissolution of the laterally smaller platelets at lower monomer concentrations. This second step would broaden the lateral size distribution for a given thickness and is consistent with conventional Ostwald ripening, but in 2D. As the monomer concentration decreases further, we predict that a given thickness will eventually dissolve completely and disappear. Instead of direct growth in the thickness direction, we expect that thicker nanoplatelets arise from nuclei that are present early in the synthesis. These would then expand laterally by capturing material as thinner platelets dissolve. Altogether, these processes explain how thicknessdependent growth rates and stabilities allow different nanoplatelet thicknesses to be selectively obtained at well-separated time or temperature scales. We believe that our general model is applicable to a wide range of platelet materials and can enable better control over their size and shape. Finally, we note the resemblance between the nanoplatelet ripening process and polymorph transformations, in which the evolution among different crystal structures of the same material is described.43 As stated by Ostwald’s rule of stages, unstable solid phases typically form faster and transform into more stable crystal structures at slower rates.44 However, these transformations are driven by the reduction of the energy per volume in different crystal phases, whereas the driving force to increase the thickness of nanoplatelets arises from the reduction of the total surface area (the energy per unit volume and that per unit area remain the same).

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F.D.O., M.M., and D.J.N. conceived the study. F.D.O. developed the model with inputs from all authors. F.D.O. and D.R.O. performed the calculations. A.R. and P.N.K. developed and performed the experiments. A.R. analyzed the experimental results. F.D.O., A.R., D.R.O., M.M., and D.J.N. wrote the manuscript with input from S.C.E. and P.N.K. Funding

This work was supported by ETH Research Grant ETH-38 141, by the Swiss National Science Foundation under grant nos. 200021-140617 and 200020-159228 and by the U.S. Office of Naval Research (ONR) through the Naval Research Laboratory’s Basic Research Program. F.D.O. benefitted from an ONR Global travel grant. Computations were performed at the ETH High-Performance Computing Cluster Euler. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge P. Kumar, S. Mazzotti, S. Meyer, F. Rabouw, and A. Rossinelli for useful discussions.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.7b03191.

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REFERENCES

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. ORCID

Marco Mazzotti: 0000-0002-4948-6705 David J. Norris: 0000-0002-3765-0678 Present Addresses

A.R.: Max Planck Institute for Polymer Research, Ackermannweg 10, 55128 Mainz, Germany. D.R.O: Janssen: Pharmaceutical Companies of Johnson & Johnson, Hochstrasse 201, 8200 Schaffhausen, Switzerland. G


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