Subscriber access provided by LAURENTIAN UNIV
Communication
Ripening of Semiconductor Nanoplatelets Florian Ott, Andreas Riedinger, David Ochsenbein, Philippe Knüsel, Steven C Erwin, Marco Mazzotti, and David J. Norris Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.7b03191 • Publication Date (Web): 09 Oct 2017 Downloaded from http://pubs.acs.org on October 9, 2017
Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.
Nano Letters is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.
Page 1 of 21
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters
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, Department of Mechanical and Process Engineering, ETH Zurich, 8092 Zurich, Switzerland ‡
Separation Processes Laboratory, Department of Mechanical and Process Engineering, ETH Zurich, 8092 Zurich, Switzerland #
Center for Computational Materials Science, Naval Research Laboratory, Washington, D.C. 20375, United States
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 quasitwo-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 to ( + 1) monolayers, allowing precise thickness control. We investigate how this counter-intuitive 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
ACS Paragon Plus Environment
Nano Letters
Page 2 of 21
Semiconductor nanocrystals have physical properties that strongly depend on their size and 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
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-twodimensional (quasi-2D) platelets.7-16 In general, non-spherical 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 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 size
ACS Paragon Plus Environment
1
Page 3 of 21
Nano Letters
distribution, which is detrimental for exploiting the size-dependent properties of semiconductor 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
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 atomic-scale 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 largevolume 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 counter-intuitive 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, ∆, (ii) the crystal surface energy, , and (iii) the step energy of a 2D surface island, . ∆ is a function of the monomer concentration in solution. and 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
ACS Paragon Plus Environment
2
Nano Letters
Page 4 of 21
grows.26 (For a summary, see Section 1 and Figure S1 in the Supporting Information). This decrease 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
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 -monolayer-thick square-shaped nanoplatelet with × lateral dimensions:
∆
∆
( ) = ℎ + (2 + 4 ℎ),
(1)
where ℎ is the monolayer height and is the volume per CdSe monomer in the crystalline phase. In the zinc blende lattice these are related as = 2ℎ . Under conditions where crystallization is thermodynamically favorable, ∆ 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 ( > 0). As expected, the formation energy is minimized for the smallest surfaceto-volume 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 - to an ( + 1)-monolayer-thick platelet via direct addition of another layer is unlikely. Nevertheless, experiments show that, during prolonged heating (i.e. ripening), nanoplatelets with layers slowly disappear while platelets with + 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 ∆, 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 ∆ was constant. In reality, |∆| decreases as monomers from the solution are consumed during growth. As |∆| 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 ∆ in terms of the solute concentration: ACS Paragon Plus Environment
3
Page 5 of 21
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters ∗ ∆(") = −$B & ln " ⁄"∞ ,
(2)
where $B is the Boltzmann constant, & is the reaction temperature, " is the free-monomer ∗ concentration in the growth solution, and "∞ is the monomer solubility for a bulk crystal. If the ∗ monomer concentration " is equal to the "∞ , ∆ is exactly zero and no growth or dissolution occurs ∗ for a bulk crystal. If " is greater than "∞ (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 finite-sized particle is higher than "∞ because
it is less stable due to the positive surface energy . 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 -monolayer-thick nanoplatelet with × lateral dimensions is then well approximated as: 0
∗ ( ) ∗ " = "∞ exp / 1
B2
3
4
4
+ 678.
5
(3)
∗ ( ), At " = " 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 " /"∞ versus lateral
side length . 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): : (") = : exp /−
barrier (A) ∆;
Communication
Ripening of Semiconductor Nanoplatelets Florian Ott, Andreas Riedinger, David Ochsenbein, Philippe Knüsel, Steven C Erwin, Marco Mazzotti, and David J. Norris Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.7b03191 • Publication Date (Web): 09 Oct 2017 Downloaded from http://pubs.acs.org on October 9, 2017
Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.
Nano Letters is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.
Page 1 of 21
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters
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, Department of Mechanical and Process Engineering, ETH Zurich, 8092 Zurich, Switzerland ‡
Separation Processes Laboratory, Department of Mechanical and Process Engineering, ETH Zurich, 8092 Zurich, Switzerland #
Center for Computational Materials Science, Naval Research Laboratory, Washington, D.C. 20375, United States
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 quasitwo-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 to ( + 1) monolayers, allowing precise thickness control. We investigate how this counter-intuitive 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
ACS Paragon Plus Environment
Nano Letters
Page 2 of 21
Semiconductor nanocrystals have physical properties that strongly depend on their size and 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
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-twodimensional (quasi-2D) platelets.7-16 In general, non-spherical 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 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 size
ACS Paragon Plus Environment
1
Page 3 of 21
Nano Letters
distribution, which is detrimental for exploiting the size-dependent properties of semiconductor 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
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 atomic-scale 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 largevolume 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 counter-intuitive 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, ∆, (ii) the crystal surface energy, , and (iii) the step energy of a 2D surface island, . ∆ is a function of the monomer concentration in solution. and 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
ACS Paragon Plus Environment
2
Nano Letters
Page 4 of 21
grows.26 (For a summary, see Section 1 and Figure S1 in the Supporting Information). This decrease 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
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 -monolayer-thick square-shaped nanoplatelet with × lateral dimensions:
∆
∆
( ) = ℎ + (2 + 4 ℎ),
(1)
where ℎ is the monolayer height and is the volume per CdSe monomer in the crystalline phase. In the zinc blende lattice these are related as = 2ℎ . Under conditions where crystallization is thermodynamically favorable, ∆ 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 ( > 0). As expected, the formation energy is minimized for the smallest surfaceto-volume 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 - to an ( + 1)-monolayer-thick platelet via direct addition of another layer is unlikely. Nevertheless, experiments show that, during prolonged heating (i.e. ripening), nanoplatelets with layers slowly disappear while platelets with + 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 ∆, 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 ∆ was constant. In reality, |∆| decreases as monomers from the solution are consumed during growth. As |∆| 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 ∆ in terms of the solute concentration: ACS Paragon Plus Environment
3
Page 5 of 21
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters ∗ ∆(") = −$B & ln " ⁄"∞ ,
(2)
where $B is the Boltzmann constant, & is the reaction temperature, " is the free-monomer ∗ concentration in the growth solution, and "∞ is the monomer solubility for a bulk crystal. If the ∗ monomer concentration " is equal to the "∞ , ∆ is exactly zero and no growth or dissolution occurs ∗ for a bulk crystal. If " is greater than "∞ (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 finite-sized particle is higher than "∞ because
it is less stable due to the positive surface energy . 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 -monolayer-thick nanoplatelet with × lateral dimensions is then well approximated as: 0
∗ ( ) ∗ " = "∞ exp / 1
B2
3
4
4
+ 678.
5
(3)
∗ ( ), At " = " 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 " /"∞ versus lateral
side length . 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): : (") = : exp /−
barrier (A) ∆;
