Machine-Learning Approach for the Development of StructureâEnergy

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C: Physical Processes in Nanomaterials and Nanostructures

Machine-Learning Approach for the Development of Structure-Energy Relationships of ZnO Nanoparticles Mingyang Chen, and David A Dixon J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b01667 • Publication Date (Web): 20 Jul 2018 Downloaded from http://pubs.acs.org on July 25, 2018

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The Journal of Physical Chemistry

Machine-Learning Approach for the Development of Structure-Energy Relationships of ZnO Nanoparticles Mingyang Chen,1,*,† and David A. Dixon2,*,† 1

2

Beijing Computational Science Research Center, Beijing, 100193, China. Department of Chemistry, The University of Alabama, Shelby Hall, Tuscaloosa, AL,

35487-0336, USA Abstract The structure-energy relationships for the zinc oxide morphologies were investigated using a newly developed fragment-based energy decomposition approach. In this approach, the local chemical compositions of a material are abstracted as fragment types that serve as the material’s genes with respect to its thermodynamic properties. A machine learning-based fragment recognition scheme was developed to learn about the fragment-related knowledge from a relatively small training set consisting of computationally viable ultra-small nanoparticles. The knowledge gained including the fragment geometries and fragment energy parameters can be used the classification and energy expression of the test sets consisting of different polymorphs and morphologies of that material at various scales. The stabilities of ZnO nanoparticles with different morphologies were expressed explicitly as functions of the particle size. The size-related phase transitions among various morphologies including wurtzite prisms, wurtzite octahedrons, body centered tetragonal particles, sodalite-like particles, single-layered cages, multi-layered cages and nonpolar hexagonal prisms were predicted. The multi-layered cages with

†

Emails: Mingyang Chen: [email protected]. David A. Dixon: [email protected] 1 ACS Paragon Plus Environment

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nonpolar surfaces exhibit superior stability among the low energy morphologies, but wurtzite nanoparticles are more favorable under practical synthesis and growth conditions under the control of the kinetics. The growth mechanism for ZnO clusters, ultra-small nanoparticles, nanocrystals, and bulk-sized particles are proposed based on the synergy between the size-related phase transitions and external factors that affect the surface energies of the particles. Our interpretation of the Wulff theorem at a fragment-sized resolution provides new chemical insight for understanding the structural phase transition and particle growth for ZnO at various scales.

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Introduction Zinc oxide (ZnO) is a wide band gap semiconductor that has been utilized in many areas. ZnO can be prepared as a solid material, thin film, or nanoparticle. ZnO nanoparticles (NPs) in various morphologies and shapes can be synthesized including hexagonal prisms,1 triangular prisms, 2 regular octahedrons, 3 tetrapods, 4 needles, 5 flowers, 6 nanodisks, 7 nanowire, 8 nanobelts,8 nanoporous,8 etc. The physical and chemical properties of ZnO nanoparticles depend on the different particle morphologies, shapes, and surface structures.9 The versatility in the properties of ZnO have more to do with the surface structures of the morphologies rather than the bulk structures, as the nanoparticles and solid materials almost always possess a wurtzite structure. The low index 0001 and 0001 surfaces of wurtzite are known to be polar with an exterior Zn- (on 0001) or O- (on 0001) sublayer. These surfaces remain polar after cleavage without significant reconstruction.10,11 Because of the relatively high surface energies of the 0001 and 0001 surfaces, the 0001 growth direction is the most common primary growth direction for ZnO. The semi-polar 1010 and 1120 surfaces were also found as the dominating exterior surfaces for various ZnO morphologies due to the low surface energies. Non-wurtzite polymorphs were also found but were less common. The thin films that exhibit similar structure to GaN have honeycomb-like nonpolar surfaces, and are only found to be stable with thickness of a few monolayers. The body centered tetrahedral (BCT) polymorph can be formed from the nonpolar thin film as the film thickness increases12,13 and the BCT polymorph contains several types of semi-polar surfaces. 3 ACS Paragon Plus Environment


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The intrinsic thermodynamics of the ZnO surfaces determines that the thermodynamic favorability of the morphologies is dependent on the particle size, as particle size will affect the surface energy density as well as other interrelated factors that influence particle’s stability. The thermodynamic

stability

of

different

ZnO

morphology classes

have

been

studied

theoretically.14,15,16 Among these studies, Viñes et al.14 used a fitting approach to determine the stability of ZnO morphologies as polynomials of the inverse of the particle sizes for a various range of ZnO structural motifs. In our prior study, we performed a comprehensive global minimum search for (ZnO)n clusters and ultra-small nanoparticles (n up to 168), and explicitly showed the thermodynamic favorability for different ZnO structural motifs at different particle sizes < 2nm.17 In addition, the relative stabilities of different ZnO surfaces can also be affected by various external factors including temperature, pH, solvent, etc., and the morphology of ZnO can be controlled by these factors. 7,8,9,18 The particle size plays even more important roles when the size is typically under 10 nm, where the surface-to-volume ratio is high and the surfaces effects are not negligible. Such particles are referred as ultra-small nanoparticles (USNPs). Most of the size-related phase transitions of metal oxide nanoparticles occur in the USNP region. Therefore, the thermodynamic favorability of the bulk sized polymorphs and morphologies are normally “predetermined” at the smaller USNP scale. USNPs of metal oxides can contain structural information about the clusters, NPs, and bulk phases, and, thus, is important to understand the structural and thermodynamic properties of metal oxides at various scales. One significant feature of the USNP is that the size 4 ACS Paragon Plus Environment

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plays an important role in determining many physical and chemical properties. The effects of changing particle size on the particle properties can be complex. Changes in size can give rise to a different surface energy and chemical potential that may induce the reconstruction of the old particle into a different shape/morphology. On the other hand, due to the molecule nature of the USNP, altering the size can change the local chemical environment (e.g. bonding and electronic density) of the atoms in the USNP, and thereby induce the local geometry rearrangements that could lead to a different polymorph. The synergistic roles of particle size effects must be considered when attempting to understand size-property relationships for USNPs. One approach that we have taken is the use of fragment-based models.19,20 In the fragment-based model, a USNP is partitioned into fragments that are associated to the local contributions of the particle’s properties. Because the size-related information can be implied from the fragment types and fragment counts, the fragment composition-property relationship can be used to approximate the computationally inconvenient size-property relationship. We have previously applied a fragment-based energy decomposition scheme (FBED) to elucidate the structure-energy relationships for metal oxide and metal hydroxide USNPs.19,20,21 The FBED interpretation has proven to yield reasonable descriptions for the thermodynamics of these inorganic particles ranging from small clusters to bulk-sized NPs. Thermodynamics-related properties of the inorganic particles can be derived based on the fragment energy parameters obtained from the FBED, including the ideal aspect ratio of the particle and the size-related structural phase transition between morphologies. 5 ACS Paragon Plus Environment


In our previous studies, breaking the USNPs into fragments, a crucial step of the FBED scheme, is done mainly based on the local translational symmetry and point group symmetry found in the USNPs and thus relies on chemical intuition. In the present work, we have generalized the fragmentation procedure and have developed a computer-aided fragmentation technique based on fragment fingerprint recognition. The new technique allows the FBED to be applied extensively to various kinds of USNPs including amorphous USNPs, in contrast to the initial energy decomposition scheme which could only be used for USNPs with specific symmetries. One crucial difference between the new FBED scheme and the original one is that the new scheme adopted methodology similar to machine learning so that the knowledge gained from the training set (i.e. fragment types learnt from calculated USNP structures) can be applied to unclassified test sets, whereas the original FBED scheme is essentially a regression analysis. With carefully benchmarked density functional approaches (functionals and basis sets), the new fragmentation and energy decomposition scheme can predict the thermodynamic properties with CCSD(T)-level of accuracy for large NPs (>3nm) for which explicit CCSD(T) or DFT calculations are not feasible. In this study, we describe and then use the new FBED method to extract universal fragment energetic parameters from the USNP structure pools that are shared by various low energy ZnO polymorphs and morphologies. “Universal” implies these parameters are morphology- and size-independent. Such universal fragment energetic parameters and the related fragment types can be used as “genes” for a variety of ZnO polymorphs and morphologies. The 6 ACS Paragon Plus Environment

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present study demonstrates, with the molecular genes recognized from limited sized structure pools (with less than 200 (ZnO)n USNP structures) by the FBED approach, that the structure-energy relationships for a versatile range of ZnO polymorphs and morphologies at various scales can be elucidated. Our new approach allows us to predict the thermodynamics of the ZnO particles with sizes up to the bulk without performing explicit ab initio calculations. We can use the resulting FBED predictions to explain several phenomena during the growth of ZnO particles leading to crystal formation. Computational Approaches Fragment-based energy decomposition For a metal oxide USNP with an integer number of formula units, denoted as (MxOy)n, the total energy can be partitioned into on-site energies of the atoms. Since we define the O atom to always be formally O2-, we can approximately treat the on-site energies of all the O atoms as a constant. In practice, the fluctuation of the on-site energy of O will be reflected in the fragment energy by including the coordination number of O in the fragmentation settings. If we fragment the USNP into nx MOy/x fragments, the total energy can then be written as a sum of fragment energies that are composed of the onsite energy for 1 M and y/x O, i.e. y/x E(O) + E(M), where E(M) is dependent on the chemical environment of the M center, such as the number of O atoms about M, the spatial positions of O atoms and the chemical environment of the O atoms. The definition of a fragment in our FBED approach is therefore distinguished from its usual definitions. In this work, a fragment has a dual definition from both a geometric and energetic point of view. From the geometric point-of-view, a fragment 7 ACS Paragon Plus Environment


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includes an M atom and all the O atoms in the M atom first coordination sphere, and from the energetic point of view, a fragment is always equivalent to a MOy/x molecule where its energy is determined by the geometry. Although the multiple fragment geometries can share some O atoms from the geometric point of view, their fragment energies are not overlapping, as the shared O’s only contribute partly to the fragment energy. It is noted that although we only focus on the applications of FBED for the metal oxides in the current work, the FBED method tends to work for other inorganic systems such as metal hydroxide, as shown in our prior work.20 The total electronic energy of a (MxOy)n particle can be written as the sum of the fragment energies (Equation (1)): = ∑ ∗

(1)

where the fragment energy is the total energy for a fragment type frag, and mfrag is the number of the frag fragment type in (MxOy)n. The constraint of Equation 1 is given by Equation (2): = ∑

(2)

The basic idea of the FBED is that the energy of a local geometry (fragment) could be solved via the method of multivariate least squares (according to Equation (1)) when the electronic energies of the molecules that share the fragment types are known (Equation (3a)): ̂ =

(3a)

where M is a matrix whose matrix element is the count of i-th fragment types in the j-th particle of the structure pool (Equation (3b)), 8 ACS Paragon Plus Environment

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… $ … # = ⋮ # ! ! ! … "

(3b)

̂ is the vector containing the fragment energies with vector element ̂ = for the i-th fragment type, and is the vector containing the total electronic energies of the molecules in which the vector element is the total electronic energy for the j-th particle. Since the energetics of the fragment is not directly calculated from the short-range local environment of a single fragment, but instead fit from the DFT calculations, if the long-range interactions are included in the DFT energies, also contains the long-range interactions. This can be done by choose ultra-small nanoparticles with sufficiently large sizes (greater than 1 nm3) for the fitting, so the long-range interactions are well included in the fitting data. The values of ̂ can be used to derive thermodynamic related energy parameters for the fragment types. The fragment normalized clustering energy NCEfrag and surface energy density per fragment SEDfrag can be evaluated by using Equations (4a) and (4b) %& = '()('* −

(4a)

,- = − ./01

(4b)

where '()('* is the total electronic energy of the gas phase monomer MxOy, and ./01 is the fragment energy of the bulk fragment type for the solid crystal (wurtzite for ZnO). The NCEfrag and SEDfrag thermodynamic parameters not only can be used to indicate the stabilities of the fragment types, but also to predict a range of thermodynamics-related properties of the NP series, including the structural phase diagram, ideal aspect ratio (morphology), and surface 9 ACS Paragon Plus Environment


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reactivities.20 For the sake of simplicity, for the remainder of this work, we will set x = 1, i.e. for (MOy)n, as the corresponding equations for (MxOy)n only vary by a factor of x as compared to the equations for (MOy)n. The normalized clustering energy for the metal oxide particle %& can be obtained from Equation (5a), %& = '()('* − /

(5a)

or from Equation (5b) in terms of ’s,

%& = '()('* − ∑ ∗ )

(5b)

or from Equation (5c) in terms of %&./01 34 ,- ’s, %& = %&-./01 − ∑

'5678 ∗(:;10 nm) are dominated by the face-to-total (F/T) ratio (indicated by the linear region of the %&F:G vs. n-1/3 plot). This can be seen by rewriting Equation (5c) to obtain Equation (6a), which requires one to be able to partition a particle into the bulk, face, edge and corner regions and evaluate the region-averaged fragment energy parameters for each region: I ; %& ≈ %&-./01 − ∗ J, K* − ,--./01 L − ∗ J,-*M * − ,--./01 L − N

J,K()* − ,--./01 L

(6a)

where ,O is the m-weighted average for all of the fragment types in region R (R = face, edge, or corner) given by Equation (6b) ∑P∈R 'P :;

Machine-Learning Approach for the Development of StructureâEnergy

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