Monodisperse Nanocrystals and Their Superstructures
FEATURE ARTICLE
Organization of Matter on Different Size Scales: Monodisperse Nanocrystals and Their Superstructures** By Andrey L. Rogach,* Dmitri V. Talapin,* Elena V. Shevchenko, Andreas Kornowski, Markus Haase, and Horst Weller Advanced colloidal syntheses enable the preparation of monodisperse semiconductors and magnetic alloy nanocrystals. They can be further used as building blocks for the fabrication of ordered assemblies: two-dimensional and three-dimensional arrays and colloidal supercrystals. This article reviews our recent activities in these fields. A theoretical description of the evolution of an ensemble of nanoparticles in a colloidal solution is applied to the problem of control over the nanocrystal monodispersity.
1. Introduction Chemistry and physics on the nanometer scale have experienced an enormous development in the last decade leading to the appearance of the new interdisciplinary field of ªnanoscienceº. The interest in nanoscale materials arises from the possibility to manipulate them in one (quantum wells), two (quantum wires), and three (quantum dots) dimensions. In the last structures, atomic-like electronic energy levels are formed due to the charge-carrier confinement, so that the properties of semiconductors and metals become governed by size.[1,2] There are two distinct routes to produce quantum dots: in the ªphysicalº approach they are grown by lithographic or molecular beam techniques. In the ªbottom upº, or ªchemicalº approach, they are synthesized by methods of colloidal chemistry in a solvent medium, and the term nanocrystals is commonly used to denote them. A famous demonstration of the size-dependent
properties of semiconductor nanocrystals is the continuous change of their emission color with decreasing particle size (Fig. 1).
±
Fig. 1. ªTraffic lightsº and ªrainbowsº from luminescent colloidal nanoparticles. a) Thiol-capped CdTe nanocrystals synthesized in aqueous solution emit green, yellow, or red light depending on size (2.5, 3.0, and 4.0 nm, correspondingly) with room temperature quantum yield of up to 40 %. b) Hexadecylamine-trioctylphosphine oxide-trioctylphosphine capped CdSe/ZnS core±shell nanocrystals are soluble in non-polar organic solvents and emit in the whole visible spectral region depending on size with a quantum efficiency of 40±70 %. c) and d) CdSe/ZnS nanocrystals embedded into a polylaurylmethacrylate matrix retain their superior luminescent properties.
[**] The authors gratefully acknowledge all colleagues who have contributed to this work, especially mentioning Dr. S. Haubold (Nanosolutions GmbH) for his contribution to the synthesis of the InAs nanocrystals. The financial support was provided by the SFB 508, research projects BMBF-Philips and BIOAND, and by the DFG Schwerpunktprogramm ªPhotonic Crystalsº. A more complete treatment on the topic of semiconductor nanoparticles and their superstructures can be found in the chapter on ªSemiconductor Nanoparticlesº, which is to appear in the book Colloids and Colloidal Interfaces (Ed: F. Caruso), Wiley-VCH, Weinheim.
Because the strong bandgap luminescence of colloidally synthesized semiconductor quantum dots is tunable by size due to the quantum confinement effect, they are currently intensively investigated as emitting materials for thin film light-emitting devices,[3±5] optical amplifier media for telecommunication networks,[6,7] and biological labels.[8±10] The incorporation of luminescent semiconductor nanocrystals into 3D photonic crystals[11] and microcavities[12] has attracted considerable attention recently as a promising pathway to novel light sources with controllable emission. The preparation of ordered
[*] Dr. A. L. Rogach, Dr. D. V. Talapin, E. V. Shevchenko, A. Kornowski, Dr. M. Haase, Prof. H. Weller Institute of Physical Chemistry, University of Hamburg D-20146 Hamburg (Germany) E-mail: [email protected]; [email protected]; [email protected]
Adv. Funct. Mater. 2002, 12, No. 10, October
Ó 2002 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1616-301X/02/1010-0653 $ 17.50+.50/0
653
FEATURE ARTICLE
A. L. Rogach et al./Monodisperse Nanocrystals and Their Superstructures arrays of magnetic nanocrystals is attracting growing interest because of potential magnetic data-storage applications.[13±15] The synthesis of monodisperse nanocrystals of desired sizes is the first and very important step being a pre-requisite of any further investigation and their use in practice. Monodispersity is strongly required, e.g., for the purity of emission color in the case of luminescent semiconductor nanocrystals, as well as for many other potential applications. Colloidal chemistry routes to several semiconductor[16±20] and metal[13±15,21,22] nanocrystals with narrow particle size distributions, high crystallinity, and controllable surface properties have been developed providing nanoparticles in gram amounts just like ordinary chemical substances. The synthetic efforts are gradually being concentrated on both the improvement and simplification of existing syntheses[23,24] and the development of reliable approaches to more and more nanometer-sized compounds.[25±27] Despite of this, there is still a lack of theoretical understanding of the processes occurring during the growth of nanoparticles in colloidal solutions and the problem of keeping the particle size distribution narrow. Only a few reports deal with this important problem.[28±30] Further synthetic progress will definitely depend on our ability to understand and control the parameters governing the properties (e.g., size distribution and photoluminescence quantum efficiency) of colloidally grown nanocrystals.[31,32] A new field of research has recently emerged, which focuses on the use of individual monodisperse nanocrystals as building blocks for the fabrication of superstructures and the investigation of collective properties of these artificial quantum dot solids. Reviews on this topic have appeared[22,33±35] providing both a summary of the literature at the time of publication and the
developments done by the respective groups. This article covers the very recent activities of our group on the colloidal synthesis of monodisperse semiconductor and magnetic alloy nanocrystals and their use for the creation of two-dimensional (2D) and three-dimensional (3D) arrays and colloidal crystals by means of self-assembly. In colloidal crystals, individual nanocrystals play the role of building blocksÐartificial atoms in the next level of hierarchy. A recently developed theoretical description of the evolution of an ensemble of nanoparticles in a colloidal solution[30] is applied to the problem of control over the nanocrystal monodispersity.
2. Synthesis of Monodisperse Nanocrystals The availability of reliable colloidal syntheses leading to nanocrystals being uniform in composition, size, shape, and surface chemistry is crucial for the fabrication of superstructures. The use of thiols as capping agents in the syntheses of II±VI semiconductor nanocrystals allowed the preparation of extremely small molecular-like clusters of exact composition, e.g., [Cd17S4(SC6H5)28]2±,[36] [Cd32S14(SC6H5)36]´(DMF)4,[37] [Cd17S4(SCH2CH2OH)26],[38] [Cd32S14(SCH2CH(OH)CH3)36]´ (H2O)4,[39] and [Cd54Te32(SCH2CH2OH)52]8±[40] (the last formula was suggested based on extended X-ray absorption fine structure (EXAFS) data). The stable clusters correspond to pronounced minima of the free energy vs. particle-size dependence owing to their closed structural shells (the concept of so-called clusters of magic size in the earlier literature[41]) and are naturally ª100 % monodisperseº. These nanoparticles
Andrey L. Rogach received his Ph.D. degree in Physical Chemistry from the Belarussian State University in Minsk in 1995 for his work on the formation and properties of silver nanoparticles in different media. He was a DAAD Postdoctoral Fellow at the Institute of Physical Chemistry, University of Hamburg (Germany) in the group of Prof. Horst Weller, whereafter he returned to Belarus to a position of Senior Research Scientist at the Physico-Chemical Research Institute (Minsk). He was a guest scientist at the British Telecom Laboratories (Ipswich, UK) and the Oklahoma State University (USA) from 1998±1999. He revisited the University of Hamburg as an Alexander von Humboldt Research Fellow in 2000, followed by a position in the group of Prof. Weller in 2001. In September 2002 he joined the Photonics & Optoelectronics group of Prof. J. Feldmann at the Ludwig-Maximilians-University of Munich. His research focuses on different aspects of chemistry, physics, and applications of semiconductor and metal nanocrystals and on colloidal photonic crystals.
Dmitri V. Talapin studied chemistry at the Belarussian State University in Minsk, Belarus. Since 2000 he has been working at the Institute of Physical Chemistry, University of Hamburg (Germany), in the group of Prof. Horst Weller. He received his Doctor of Natural Sciences degree in 2002 for his work on experimental and theoretical studies on the formation of highly luminescent II±VI, III±V, and core±shell semiconductor nanocrystals. His current area of research focuses on the organometallic colloidal synthesis of semiconductor nanocrystals and the theoretical modeling of the evolution dynamics of nanometer-sized particles in a colloidal solution.
654
Ó 2002 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1616-301X/02/1010-0654 $ 17.50+.50/0
Adv. Funct. Mater. 2002, 12, No. 10, October
can be crystallized in macroscopically large single crystals allowing their investigation by single-crystal X-ray analysis.[36±39] Moving from molecular-like clusters of exact composition to larger particles whose crystalline core consists of ~ 102±104 atoms, nearly continuous tunability of the particle size becomes possible, as an addition or removal of a unit cell requires only a small variation of the nanocrystal free energy. The colloidal synthesis of nanocrystalline particles generally involves several consecutive stages: nucleation from initially homogeneous solution, growth of the pre-formed nuclei, isolation of particles from the reaction mixture after they reached the desired size, post-preparative size fractionation, etc. As a rule, temporal separation of the nucleation event from the growth of the nuclei is required for narrow size distribution.[35,42] The so-called hot-injection technique, where the precursors are rapidly injected into a hot solvent with subsequent temperature drop, satisfies this requirement.[16,35] During further growth of the nanocrystals, several regimes can be observed depending on the system and experimental conditions. They are discussed below.
dr S e1=r ds r Kea=r
with r
2.1. Ostwald Ripening Ostwald ripening (OR), the growth mechanism where the smaller particles dissolve and the monomer released thereby is consumed by the large particles,[43,44] takes place in most colloidal syntheses of both II±VI and III±V semiconductor nanocrystals. As a result, the average nanocrystal size increases with time, and the particle concentration decreases. The driving force of OR is a decrease of the particle solubility with increasing size as expressed by the Gibbs±Thompson equation:
2 γVm 2 γVm 0 0 C ( r ) = Cbulk exp ≈ Cbulk 1 + rRT rRT
(1)
where, C(r) and C0bulk are the solubilities of a particle with radius r and of the bulk material, respectively, c is the surface tension, and Vm is the molar volume of the solid. Validity of the Gibbs±Thompson equation was proven for very small (r ~ 1±2 nm) colloidal particles.[45,46] In case of ªlargeº (r > 25 nm) particles the kinetics of OR can be satisfactorily described analytically in the framework of the Lifshitz±Slyozov±Wagner (LSW) theory.[47,48] However, the LSW approach takes into account only two terms of the expansion of the Gibbs±Thompson equation and fails in the description of ensembles of particles smaller than ~ 50 nm in radius due to the large error arising from the truncation of the expansion of Equation 1. The coefficient 2cVm/(RT) called ªcapillary lengthº is of the order of 1±3 nm for most solid±liquid interfaces[49,50] and in the case of nanoparticles with r = 1±5 nm the particle solubility is strongly nonlinearly dependent on r±1. Moreover, the surface tension c of the nanoparticles can be considerably larger than that of the corresponding bulk material, as was reported for CdS,[51] Pt, and Au.[52] This results in a value of capillary length of ~ 33 nm for thiophenol-capped CdS nanocrystals under the particle growth conditions.[51] Additionally, for nanoscale particles the activation energies of the
Adv. Funct. Mater. 2002, 12, No. 10, October
growth and dissolution processes are also size-dependent.[30] In this case a general analytical solution describing all processes occurring during the evolution of the particle ensemble could not be obtained. In our recent study we applied the MonteCarlo simulation technique to describe the evolution of an ensemble of nanoparticles in a colloidal solution[30] and showed that OR in this case is characterized by some features that are not observed for large (sub-micrometer- and micrometer-sized) particles. For convenience, we will use the term ªnano-ORº in further discussion to distinct this particular case from OR of (sub-)micrometer-sized particles adhering to the LSW theory. OR implies that the largest particles in the ensemble have positive and the smallest ones have negative growth rates. The growth/dissolution rate of a single particle of radius r is given by the following equation, which seems to be valid for both nano- and (sub-)micrometer-sized particles:[30]
and τ =
(2)
RT r 2cVm
0 R 2T 2 DCbulk
4γ 2Vm
(3)
t
(4)
where r* and s are the dimensionless particle radius and time, respectively, and K is a dimensionless parameter describing the impacts of diffusion and surface reaction as kinetics-limiting factors (K > 1 to the surface-reaction-controlled one).[30] The dimensionless parameter S = [M]/C0bulk describes the oversaturation of the monomer in a solution, with the monomer concentration [M]. D is the diffusion coefficient for the monomer, and a is the transfer coefficient of the activated complex (0
Organization of Matter on Different Size Scales: Monodisperse Nanocrystals and Their Superstructures** By Andrey L. Rogach,* Dmitri V. Talapin,* Elena V. Shevchenko, Andreas Kornowski, Markus Haase, and Horst Weller Advanced colloidal syntheses enable the preparation of monodisperse semiconductors and magnetic alloy nanocrystals. They can be further used as building blocks for the fabrication of ordered assemblies: two-dimensional and three-dimensional arrays and colloidal supercrystals. This article reviews our recent activities in these fields. A theoretical description of the evolution of an ensemble of nanoparticles in a colloidal solution is applied to the problem of control over the nanocrystal monodispersity.
1. Introduction Chemistry and physics on the nanometer scale have experienced an enormous development in the last decade leading to the appearance of the new interdisciplinary field of ªnanoscienceº. The interest in nanoscale materials arises from the possibility to manipulate them in one (quantum wells), two (quantum wires), and three (quantum dots) dimensions. In the last structures, atomic-like electronic energy levels are formed due to the charge-carrier confinement, so that the properties of semiconductors and metals become governed by size.[1,2] There are two distinct routes to produce quantum dots: in the ªphysicalº approach they are grown by lithographic or molecular beam techniques. In the ªbottom upº, or ªchemicalº approach, they are synthesized by methods of colloidal chemistry in a solvent medium, and the term nanocrystals is commonly used to denote them. A famous demonstration of the size-dependent
properties of semiconductor nanocrystals is the continuous change of their emission color with decreasing particle size (Fig. 1).
±
Fig. 1. ªTraffic lightsº and ªrainbowsº from luminescent colloidal nanoparticles. a) Thiol-capped CdTe nanocrystals synthesized in aqueous solution emit green, yellow, or red light depending on size (2.5, 3.0, and 4.0 nm, correspondingly) with room temperature quantum yield of up to 40 %. b) Hexadecylamine-trioctylphosphine oxide-trioctylphosphine capped CdSe/ZnS core±shell nanocrystals are soluble in non-polar organic solvents and emit in the whole visible spectral region depending on size with a quantum efficiency of 40±70 %. c) and d) CdSe/ZnS nanocrystals embedded into a polylaurylmethacrylate matrix retain their superior luminescent properties.
[**] The authors gratefully acknowledge all colleagues who have contributed to this work, especially mentioning Dr. S. Haubold (Nanosolutions GmbH) for his contribution to the synthesis of the InAs nanocrystals. The financial support was provided by the SFB 508, research projects BMBF-Philips and BIOAND, and by the DFG Schwerpunktprogramm ªPhotonic Crystalsº. A more complete treatment on the topic of semiconductor nanoparticles and their superstructures can be found in the chapter on ªSemiconductor Nanoparticlesº, which is to appear in the book Colloids and Colloidal Interfaces (Ed: F. Caruso), Wiley-VCH, Weinheim.
Because the strong bandgap luminescence of colloidally synthesized semiconductor quantum dots is tunable by size due to the quantum confinement effect, they are currently intensively investigated as emitting materials for thin film light-emitting devices,[3±5] optical amplifier media for telecommunication networks,[6,7] and biological labels.[8±10] The incorporation of luminescent semiconductor nanocrystals into 3D photonic crystals[11] and microcavities[12] has attracted considerable attention recently as a promising pathway to novel light sources with controllable emission. The preparation of ordered
[*] Dr. A. L. Rogach, Dr. D. V. Talapin, E. V. Shevchenko, A. Kornowski, Dr. M. Haase, Prof. H. Weller Institute of Physical Chemistry, University of Hamburg D-20146 Hamburg (Germany) E-mail: [email protected]; [email protected]; [email protected]
Adv. Funct. Mater. 2002, 12, No. 10, October
Ó 2002 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1616-301X/02/1010-0653 $ 17.50+.50/0
653
FEATURE ARTICLE
A. L. Rogach et al./Monodisperse Nanocrystals and Their Superstructures arrays of magnetic nanocrystals is attracting growing interest because of potential magnetic data-storage applications.[13±15] The synthesis of monodisperse nanocrystals of desired sizes is the first and very important step being a pre-requisite of any further investigation and their use in practice. Monodispersity is strongly required, e.g., for the purity of emission color in the case of luminescent semiconductor nanocrystals, as well as for many other potential applications. Colloidal chemistry routes to several semiconductor[16±20] and metal[13±15,21,22] nanocrystals with narrow particle size distributions, high crystallinity, and controllable surface properties have been developed providing nanoparticles in gram amounts just like ordinary chemical substances. The synthetic efforts are gradually being concentrated on both the improvement and simplification of existing syntheses[23,24] and the development of reliable approaches to more and more nanometer-sized compounds.[25±27] Despite of this, there is still a lack of theoretical understanding of the processes occurring during the growth of nanoparticles in colloidal solutions and the problem of keeping the particle size distribution narrow. Only a few reports deal with this important problem.[28±30] Further synthetic progress will definitely depend on our ability to understand and control the parameters governing the properties (e.g., size distribution and photoluminescence quantum efficiency) of colloidally grown nanocrystals.[31,32] A new field of research has recently emerged, which focuses on the use of individual monodisperse nanocrystals as building blocks for the fabrication of superstructures and the investigation of collective properties of these artificial quantum dot solids. Reviews on this topic have appeared[22,33±35] providing both a summary of the literature at the time of publication and the
developments done by the respective groups. This article covers the very recent activities of our group on the colloidal synthesis of monodisperse semiconductor and magnetic alloy nanocrystals and their use for the creation of two-dimensional (2D) and three-dimensional (3D) arrays and colloidal crystals by means of self-assembly. In colloidal crystals, individual nanocrystals play the role of building blocksÐartificial atoms in the next level of hierarchy. A recently developed theoretical description of the evolution of an ensemble of nanoparticles in a colloidal solution[30] is applied to the problem of control over the nanocrystal monodispersity.
2. Synthesis of Monodisperse Nanocrystals The availability of reliable colloidal syntheses leading to nanocrystals being uniform in composition, size, shape, and surface chemistry is crucial for the fabrication of superstructures. The use of thiols as capping agents in the syntheses of II±VI semiconductor nanocrystals allowed the preparation of extremely small molecular-like clusters of exact composition, e.g., [Cd17S4(SC6H5)28]2±,[36] [Cd32S14(SC6H5)36]´(DMF)4,[37] [Cd17S4(SCH2CH2OH)26],[38] [Cd32S14(SCH2CH(OH)CH3)36]´ (H2O)4,[39] and [Cd54Te32(SCH2CH2OH)52]8±[40] (the last formula was suggested based on extended X-ray absorption fine structure (EXAFS) data). The stable clusters correspond to pronounced minima of the free energy vs. particle-size dependence owing to their closed structural shells (the concept of so-called clusters of magic size in the earlier literature[41]) and are naturally ª100 % monodisperseº. These nanoparticles
Andrey L. Rogach received his Ph.D. degree in Physical Chemistry from the Belarussian State University in Minsk in 1995 for his work on the formation and properties of silver nanoparticles in different media. He was a DAAD Postdoctoral Fellow at the Institute of Physical Chemistry, University of Hamburg (Germany) in the group of Prof. Horst Weller, whereafter he returned to Belarus to a position of Senior Research Scientist at the Physico-Chemical Research Institute (Minsk). He was a guest scientist at the British Telecom Laboratories (Ipswich, UK) and the Oklahoma State University (USA) from 1998±1999. He revisited the University of Hamburg as an Alexander von Humboldt Research Fellow in 2000, followed by a position in the group of Prof. Weller in 2001. In September 2002 he joined the Photonics & Optoelectronics group of Prof. J. Feldmann at the Ludwig-Maximilians-University of Munich. His research focuses on different aspects of chemistry, physics, and applications of semiconductor and metal nanocrystals and on colloidal photonic crystals.
Dmitri V. Talapin studied chemistry at the Belarussian State University in Minsk, Belarus. Since 2000 he has been working at the Institute of Physical Chemistry, University of Hamburg (Germany), in the group of Prof. Horst Weller. He received his Doctor of Natural Sciences degree in 2002 for his work on experimental and theoretical studies on the formation of highly luminescent II±VI, III±V, and core±shell semiconductor nanocrystals. His current area of research focuses on the organometallic colloidal synthesis of semiconductor nanocrystals and the theoretical modeling of the evolution dynamics of nanometer-sized particles in a colloidal solution.
654
Ó 2002 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1616-301X/02/1010-0654 $ 17.50+.50/0
Adv. Funct. Mater. 2002, 12, No. 10, October
can be crystallized in macroscopically large single crystals allowing their investigation by single-crystal X-ray analysis.[36±39] Moving from molecular-like clusters of exact composition to larger particles whose crystalline core consists of ~ 102±104 atoms, nearly continuous tunability of the particle size becomes possible, as an addition or removal of a unit cell requires only a small variation of the nanocrystal free energy. The colloidal synthesis of nanocrystalline particles generally involves several consecutive stages: nucleation from initially homogeneous solution, growth of the pre-formed nuclei, isolation of particles from the reaction mixture after they reached the desired size, post-preparative size fractionation, etc. As a rule, temporal separation of the nucleation event from the growth of the nuclei is required for narrow size distribution.[35,42] The so-called hot-injection technique, where the precursors are rapidly injected into a hot solvent with subsequent temperature drop, satisfies this requirement.[16,35] During further growth of the nanocrystals, several regimes can be observed depending on the system and experimental conditions. They are discussed below.
dr S e1=r ds r Kea=r
with r
2.1. Ostwald Ripening Ostwald ripening (OR), the growth mechanism where the smaller particles dissolve and the monomer released thereby is consumed by the large particles,[43,44] takes place in most colloidal syntheses of both II±VI and III±V semiconductor nanocrystals. As a result, the average nanocrystal size increases with time, and the particle concentration decreases. The driving force of OR is a decrease of the particle solubility with increasing size as expressed by the Gibbs±Thompson equation:
2 γVm 2 γVm 0 0 C ( r ) = Cbulk exp ≈ Cbulk 1 + rRT rRT
(1)
where, C(r) and C0bulk are the solubilities of a particle with radius r and of the bulk material, respectively, c is the surface tension, and Vm is the molar volume of the solid. Validity of the Gibbs±Thompson equation was proven for very small (r ~ 1±2 nm) colloidal particles.[45,46] In case of ªlargeº (r > 25 nm) particles the kinetics of OR can be satisfactorily described analytically in the framework of the Lifshitz±Slyozov±Wagner (LSW) theory.[47,48] However, the LSW approach takes into account only two terms of the expansion of the Gibbs±Thompson equation and fails in the description of ensembles of particles smaller than ~ 50 nm in radius due to the large error arising from the truncation of the expansion of Equation 1. The coefficient 2cVm/(RT) called ªcapillary lengthº is of the order of 1±3 nm for most solid±liquid interfaces[49,50] and in the case of nanoparticles with r = 1±5 nm the particle solubility is strongly nonlinearly dependent on r±1. Moreover, the surface tension c of the nanoparticles can be considerably larger than that of the corresponding bulk material, as was reported for CdS,[51] Pt, and Au.[52] This results in a value of capillary length of ~ 33 nm for thiophenol-capped CdS nanocrystals under the particle growth conditions.[51] Additionally, for nanoscale particles the activation energies of the
Adv. Funct. Mater. 2002, 12, No. 10, October
growth and dissolution processes are also size-dependent.[30] In this case a general analytical solution describing all processes occurring during the evolution of the particle ensemble could not be obtained. In our recent study we applied the MonteCarlo simulation technique to describe the evolution of an ensemble of nanoparticles in a colloidal solution[30] and showed that OR in this case is characterized by some features that are not observed for large (sub-micrometer- and micrometer-sized) particles. For convenience, we will use the term ªnano-ORº in further discussion to distinct this particular case from OR of (sub-)micrometer-sized particles adhering to the LSW theory. OR implies that the largest particles in the ensemble have positive and the smallest ones have negative growth rates. The growth/dissolution rate of a single particle of radius r is given by the following equation, which seems to be valid for both nano- and (sub-)micrometer-sized particles:[30]
and τ =
(2)
RT r 2cVm
0 R 2T 2 DCbulk
4γ 2Vm
(3)
t
(4)
where r* and s are the dimensionless particle radius and time, respectively, and K is a dimensionless parameter describing the impacts of diffusion and surface reaction as kinetics-limiting factors (K > 1 to the surface-reaction-controlled one).[30] The dimensionless parameter S = [M]/C0bulk describes the oversaturation of the monomer in a solution, with the monomer concentration [M]. D is the diffusion coefficient for the monomer, and a is the transfer coefficient of the activated complex (0
