Pair distribution function analysis driven by atomistic simulations Application to microwave radiation synthesized TiO 2and ZrO 2 Shuyan Zhang1 Jie Gong1 Daniel Xiao2 B. Reeja Jayan1 and Alan J. H. McGaughey1

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Pair distribution function analysis driven by atomistic simulations: Application to

microwave radiation synthesized TiO2and ZrO2

Shuyan Zhang1, Jie Gong1, Daniel Xiao2, B. Reeja Jayan1, and Alan J. H. McGaughey1∗

1Department of Mechanical Engineering, Carnegie Mellon University,

Pittsburgh, Pennsylvania 15213, USA and

2Department of Materials Science and Engineering, Carnegie Mellon University,

Pittsburgh, Pennsylvania 15213, USA

A workﬂow is presented for performing pair distribution function (PDF) analysis of defected ma-

terials using structures generated from atomistic simulations. A large collection of structures, which

diﬀer in the types and concentrations of defects present, are obtained through energy minimization

with an empirical interatomic potential. Each of the structures is reﬁned against an experimental

PDF. The structures with the lowest goodness of ﬁt Rwvalues are taken as being representative

of the experimental structure. The workﬂow is applied to anatase titanium dioxide (a-TiO2) and

tetragonal zirconium dioxide (t-ZrO2) synthesized in the presence of microwave radiation, a low

temperature process that generates disorder. The results suggest that titanium vacancies and inter-

stitials are the dominant defects in a-TiO2, while oxygen vacancies dominate in t-ZrO2. Analysis

of the atomic displacement parameters extracted from the PDF reﬁnement and mean squared dis-

placements calculated from molecular dynamics simulations indicate that while these two quantities

are closely related, it is challenging to make quantitative comparisons between them. The workﬂow

can be applied to other materials systems, including nanoparticles.

I. INTRODUCTION

The atomic structure of good quality crystalline materials can be obtained from X-ray crystallography, which

only measures the Bragg peaks that result from the atomic periodicity. Crystallography alone is not suﬃcient for

specifying the atomic structure of nanocrystalline and/or highly-defected samples [1–4]. In pair distribution function

(PDF) experiments, on the other hand, X-rays or neutrons are used and a wider angular range is detected so that

both Bragg scattering and diﬀuse scattering are collected, the latter of which results from disorder, allowing for

structural characterization without assuming periodicity [4–6, 8]. The PDF is thus a powerful tool for quantitatively

characterizing short-range and long-range atomic structure [6].

The PDF, denoted by G(r), provides the scaled probability of ﬁnding two atoms a distance of rapart [2]. An

experimentally-measured PDF can be analyzed by adjusting the parameters of an assumed structure model, such as

the lattice constants, atomic positions, and grain/particle size. The structure is reﬁned in real-space by minimizing

the diﬀerence between its PDF and the experimental PDF [1, 2]. The reﬁnement process to perform this analysis has

been implemented in PDFgui [9], DiﬀPy-cmi [10], and TOPAS [11].

A major challenge in PDF modeling is the selection of the starting atomic structure [1, 2, 12]. Signiﬁcant information

about the sample (e.g., the crystal phase) is required to achieve satisfactory results. Typically, PDF modeling involves

manual trial-and-error reﬁnement of multiple structure models [12, 13]. There have been attempts to automate and

accelerate this process, where a large number of structures are either pulled from a materials database or generated

automatically [2, 13–15]. Such approaches can be eﬃcient for the identiﬁcation of the crystal structure of an unknown

material. They may not be suﬃcient, however, when the material and phase are already known and detailed structural

information is required. For example, if the sample is disordered and/or a nanoparticle, where there can be atomic

displacements away from the perfect bulk structure due to defects and/or surfaces [7]. Since the crystal structures

available in most databases are for perfect bulk phases [16–18], a method for generating candidate structures that

takes into account the atomic displacements induced by disorder and/or surfaces is needed.

Atomistic simulations provide a solution to this challenge. While density functional theory (DFT) can perform ﬁrst

principles-based energy calculations, it is limited to small systems with minimal complexity by its large computational

cost. Empirical interatomic potentials, on the other hand, can eﬃciently provide the total energy of a large, complex

system as an explicit function of its atomic coordinates [19]. A well-parameterized potential can maintain high

accuracy when compared to a DFT calculation.

In minimizing the energy of a system, the atomic positions will change to account for perturbations to the perfect

structure (e.g., defects). Structure models generated via energy minimization with an interatomic potential can

∗mcgaughey@cmu.edu.

arXiv:2210.05890v1 [cond-mat.mtrl-sci] 12 Oct 2022

therefore be used to represent complex systems that are not available in materials databases. Furthermore, an

energy minimization-based approach is advantageous compared to the reverse Monte Carlo (RMC) method. In RMC,

atoms are allowed to move under certain constraints (e.g., bond angles, bond lengths) until the diﬀerence between the

calculated and experimental structural characteristic of interest (e.g, the PDF) is minimized [20]. Incorrect structures,

however, can be generated if the appropriate combinations of such constraints are not applied [21].

Defect-induced changes in the local atomic environment can be described by atomistic calculations if the crystal

phase and defect information (e.g., types, concentrations) are speciﬁed. Herein, we describe a PDF analysis workﬂow

for modeling defected crystals. Structures with diﬀerent types and concentrations of defects are ﬁrst created and then

relaxed using energy minimization. The PDF of each relaxed structure is then reﬁned against the experimental PDF.

The defect types and concentrations are then estimated from the structures that yield reﬁned PDFs with the smallest

diﬀerences compared to the experimental PDF.

Measured PDFs of anatase titanium dioxide (TiO2) and tetragonal zirconium dioxide (ZrO2) grown using low

temperature solution synthesis in the presence of microwave radiation (MWR) are used to demonstrate this method.

External ﬁelds can induce defects that alter the diﬀusion of space charges across grain boundaries and ultimately lead

to eﬃcient processing [22–25]. Our goal is to identify the types and concentrations of point defects that are present

in MWR-grown anatase TiO2and tetragonal ZrO2. While we focus on structures with local disorder, this workﬂow

can also beneﬁt PDF analysis of nanoparticles [26] and perovskites [27], which are challenging to characterize using

conventional methods.

The rest of the paper is organized as follows. In Secs. II A and II B, the atomistic calculations, structure generation,

and PDF analysis are described. In Sec. III A, the interatomic potential selection process is presented, which is

guided by predictions of lattice constants and elastic constants. In Sec. III B, defect formation energy predictions are

compared with literature values. The PDF analysis is presented in Sec. IV. We ﬁnd that Ti vacancy and Ti interstitial

are the dominant defects in MWR-grown anatase TiO2and that O vacancy is the dominant defect in MWR-grown

tetragonal ZrO2.

II. METHODS

A. Atomistic calculations and structure model generation

The nature of the bonding in TiO2and ZrO2, which have relatively strong covalent characteristics, makes it

challenging to model the atomic interactions using an interatomic potential. We also require that the potential be able

to handle heterogeneous, defected systems. Atomistic calculations of materials with electrostatic interactions typically

maintain ﬁxed charges on the ions. Charge transfer may happen around a defect site, however, causing deviations

of the electrostatic energies and particle positions [28, 29]. A variable-charge potential allows for the redistribution

of charges based on the local environment [30–32]. We identiﬁed two candidate variable charge potentials: second

moment tight-binding charge equilibrium (SMTB-Q) [33, 34] and reactive force ﬁeld (ReaxFF) [35–37], which are

evaluated in Sec. III A.

All atomistic calculations are performed using the Large-scale Atomic/Molecular Massively Parallel Simulator

(LAMMPS) [38]. The space groups, supercell sizes, and number of atoms in the simulation boxes of anatase and

rutile TiO2, and tetragonal, cubic, and monoclinic ZrO2are listed in Table I. While we are focused on anatase TiO2

and tetragonal ZrO2, the other phases were included to best assess the potentials. Periodic boundary conditions

are applied in all three directions. The long-range electrostatic interaction is calculated using the Wolf summation

method [33, 39]. The time step for the molecular dynamics (MD) simulations is 0.2 fs. The charge on each atom is

updated every time step using the charge equilibrium (Qeq) scheme developed by Rapp´e and Goddard [40].

We built structures with diﬀerent types and concentrations of randomly-placed point defects and relaxed them at

zero temperature by energy minimization. To investigate the sensitivity of the reﬁned PDF to the defect locations,

ten structures were created for each conﬁguration. We consider common point defects for both the cation (Ti and

Zr) and the anion (O): vacancies, interstitials, Frenkel pairs (a cation vacancy and interstitial pair), and anti-Frenkel

pairs (an anion vacancy and interstitial pair), and combinations of these defect types.

B. Pair distribution function (PDF) analysis

We use measured PDFs for MWR-grown anatase TiO2and tetragonal ZrO2thin ﬁlms from previous studies [6, 25].

The data acquisition was performed at the X-ray Powder Diﬀraction Beamline, 28-ID-2, at the National Synchrotron

Light Source II at Brookhaven National Laboratory.

TABLE I. Space groups, supercell sizes, and number of atoms in the TiO2and ZrO2simulation boxes. a,b, and care the

lattice constants.

Phase Space Group Supercell Number of Atoms

Anatase TiO2I41/amd 7a×7a×3c1764

Rutile TiO2P42/mnm 6a×6a×9c1944

Tetragonal ZrO2P42/nmc 7a×7a×5c1470

Cubic ZrO2F m3m5a×5a×5a1500

Monoclinic ZrO2P21/c 5a×5b×5c1500

The PDF reﬁnement is performed using the DiﬀPy-cmi package [10]. The structure models are ﬁt against the

measured PDFs by reﬁning global and phase-speciﬁc scaling factors, lattice parameters, isotropic atomic displacement

parameters (ADPs, denoted by Uiso), a low-rpeak sharpening coeﬃcient for correlated motion of nearby atoms, and

a PDF peak envelope function, which dampens the signal as a function of separation to account for grain/particle

size. The quality of the ﬁt is quantiﬁed by a goodness-of-ﬁt value Rw, which is calculated as [4]

Rw=sPn(Gobs,n −Gcalc,n)2

PnG2

obs,n

,(1)

where Gobs,n is the nth point on the experimentally measured PDF and Gcalc,n is the nth point on the reﬁned PDF.

The PDF reﬁnement is done from 1.5 ˚

A to 30 ˚

A in an increment of 0.01 ˚

A. Periodic boundary conditions are applied

to the structure models [42]. This step is important because if the supercell is not periodic, the PDF peaks will

dampen as the atomic separation approaches the simulation box size.

III. POTENTIAL SELECTION AND ASSESSMENT

A. Lattice constants and elastic constants

We performed atomistic calculations using SMTB-Q and ReaxFF for both TiO2and ZrO2. The SMTB-Q po-

tential [34] and two ReaxFF parametrizations, i.e., ReaxFF – TiO2/H2O [36] and ReaxFF – defect [37] for TiO2

all produce stable structures in MD simulations and will be further considered in the lattice constant and elastic

constant calculations. The ReaxFF parametrized for yttria-stabilized zirconia [35], however, does not generate stable

structures in MD simualtions for the tetragonal, cubic, and monoclinic phases of ZrO2. The SMTB-Q potential for

ZrO2[33] does generate stable structures for perfect and defected phases and will be used in the subsequent analysis.

The lattice constants and elastic constants of anatase TiO2calculated using the SMTB-Q and ReaxFF are presented

in Table II along with experimental values from the literature [43, 44]. The calculation details are provided in

Appendix A. The lattice parameters predicted by all three potentials are close to the experimental values with a

largest deviation of 1.7%. The accuracy of the elastic constants is quantiﬁed by the root-mean-square error (RMSE)

of the unique elements of the elastic constant tensor compared to the experimental values. The elastic constant RMSE

from the SMTB-Q potential is 17.9 GPa, thirteen and ﬁve times smaller than those predicted by ReaxFF – TiO2/H2O

and ReaxFF – defect. In addition, the elastic constants calculated by SMTB-Q maintain the crystal symmetry (i.e.,

C11 =C22,C13 =C23, and C44 =C55), which is not the case with those calculated by ReaxFF – TiO2/H2O and

ReaxFF – defect. Based on the lattice constant and elastic constant results, we decided to use the SMTB-Q potential

in our further calculations of TiO2. The lattice constants and elastic constants of rutile TiO2are provided in Table

S1 of the Supplemental Material [45].

The lattice constants and elastic constants of tetragonal ZrO2calculated using the SMTB-Q potential are also

reported in Table II. Although the ZrO2elastic constants have a larger RMSE compared to that for TiO2, the lattice

constants are well predicted with an average error of 2.4% compared to the experimental values [46, 47]. The elastic

constants that represent the reaction to a normal stress (i.e., C11,C22, and C33) are reasonably accurate, with an

average error of 18%. The lattice constants and elastic constants of monoclinic and cubic ZrO2are provided in Table

S2 of the Supplemental Material [45].

TABLE II. Lattice constants (a,c) (˚

A), elastic constants (cij ) (GPa), bulk modulus (BV) (GPa), and shear modulus (GV)

(GPa) calculated using the SMTB-Q and ReaxFF potentials, and literature experimental values.

Anatase TiO2Tetragonal ZrO2

SMTB-Q ReaxFF-TiO2/H2O ReaxFF-defect Exp. [43, 44] SMTB-Q Exp. [46, 47]

a3.809 3.798 3.832 3.784 3.681 3.591

c9.597 9.550 9.673 9.515 5.285 5.169

c11 341 401 402 337 294 327

c22 341 401 412 337 294 327

c33 198 211 245 192 178 264

c12 104 87 101 139 191 100

c13 103 115 145 136 16.5 62

c23 103 113 155 136 16.5 62

c44 56 -675 -238 54 41.7 59

c55 56 -58 90 54 41.7 59

c66 63 35 53 60 181 64

cij RMSE 17.9 249 106 - 190 -

a167 182 207 178 135 152

b58 -32 -32 57 89 83

aEq. (A3)

bEq. (A4)

B. Defect formation energy

The relaxed point defect structures predicted by SMTB-Q in anatase TiO2and tetragonal ZrO2, the phases that are

grown under MWR, are shown in Figs. 1(a)–1(e) and 2(a)–2(e). The calculation details are presented in Appendix A.

The resulting atomic displacements around the point defects in TiO2(denoted by arrows) are similar to those observed

in previous DFT calculations [48] and range from 0.1 ˚

A (O interstitial) to 0.4 ˚

A (Ti vacancy). For ZrO2, the atomic

displacements around a point defect are less than 0.0001 ˚

A and are not shown.

FIG. 1. Relaxed atomic structures of anatase TiO2from the SMTB-Q potential: (a) perfect, (b) Ti interstitial, (c) Ti

vacancy, (d) O interstitial, and (e) O vacancy. The dashed circles indicate the defect sites. The arrows indicate the direction

of displacements after relaxation. The structures are qualitatively similar to DFT-predicted structures [48].

The anatase TiO2defect formation energies predicted by SMTB-Q and from previous DFT calculations are provided

in Table III. Also provided are the tetragonal ZrO2defect formation energies predicted by SMTB-Q, previous DFT

calculations, and previous calculations for the polarizable Born-Mayer (BM) core-shell potential [49]. In the DFT

studies, the defect formation energies depend on the charge states, the elemental chemical potentials, the Fermi level,

and the valence-band maximum of the perfect crystal. As such, several discrete values are often reported for a given

FIG. 2. Relaxed atomic structures of tetragonal ZrO2from the SMTB-Q potential: (a) perfect, (b) Zr interstitial, (c) Zr

vacancy, (d) O interstitial, and (e) O vacancy. The dashed circles indicate the defect sites. The atomic displacements around

a point defect are less than 0.0001 ˚

A and are not shown.

TABLE III. Defect formation energies in eV for anatase TiO2and tetragonal ZrO2from the SMTB-Q potential, and previous

DFT and BM potential calculations.

Anatase TiO2Tetragonal ZrO2

SMTB-Q DFT SMTB-Q DFT [50] BM [49]

O Vacancy 1.85 0.70–4.23 [51] 4.42 -0.76 – 6.10 15.62

Cation Vacancy 2.93 -0.7 – 5.80 [48] 9.63 6.11 85.04

O Interstitial 4.37 0.60 – 4.30 [48] 7.74 1.79 -10.42

Cation Interstitial 1.07 0.70 – 7.74 [51] 1.85 1.96 -67.41

O Frenkel Pair 3.86 – 11.39 8.07 – 18.57 17.63

Cation Frenkel Pair 5.38 – 12.12 4.11 – 7.41 5.20

defect type. The full range is provided in Table III. The defect formation energies for other phases of TiO2and ZrO2

are provided in Tables S3 and S4 of the Supplemental Material [45].

For anatase TiO2, the SMTB-Q defect formation energies fall within the DFT energy ranges except for the O

interstitial (0.07 eV above the maximum DFT value). SMTB-Q predicts that the smallest formation energy is for a

Zr interstitial. For tetragonal ZrO2, the SMTB-Q defect formation energies fall close to the DFT results except for the

O interstitial. All SMTB-Q results are in better agreement with the DFT results than the BM potential. SMTB-Q

predicts that the smallest formation energy is for a Ti interstitial.

IV. PAIR DISTRIBUTION FUNCTION (PDF) ANALYSIS

A. Anatase TiO2

We now perform PDF analysis with starting structures obtained as described in Sec. II A to determine the defect

types (vacancies, interstitials, Frenkel pairs, anti-Frenkel pairs) and concentrations present in MWR-grown anatase

TiO2(this section) and tetragonal ZrO2(Sec. IV B). The variation of Rwwith defect concentration for diﬀerent defects

in anatase TiO2is plotted in Fig. 3. The concentration of a defect type is speciﬁed by dividing the number of that

type of defect by the total number of atoms before defects are inserted. The mean values and standard deviations (as

shown by the error bars) are obtained from ten randomly generated structures for each defect type and concentration.

For all six defect types, Rwﬁrst decreases starting from 0.325 (the perfect structure, plotted as a dashed horizontal

line), reaches a minimum, and then increases as the defect concentration varies from 0 to 4.5%. Introducing Ti

defects improves the PDF reﬁnement more than O defects as indicated by the lower Rwacross all concentrations.

The small error bars indicate that Rwis insensitive to the defect conﬁguration except at the higher concentrations,

when clustering is possible. The lowest Rw(0.274) comes from the structure with 2.3% Ti Frenkel pairs.

Based on this result, we conducted a more comprehensive study. The concentrations of the four types of point

defect (Ti vacancy, Ti interstitial, O vacancy, and O interstitial) were simultaneously varied and the PDFs of the

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Pairdistributionfunctionanalysisdrivenbyatomisticsimulations:ApplicationtomicrowaveradiationsynthesizedTiO2andZrO2ShuyanZhang1,JieGong1,DanielXiao2,B.ReejaJayan1,andAlanJ.H.McGaughey11DepartmentofMechanicalEngineering,CarnegieMellonUniversity,Pittsburgh,Pennsylvania15213,USAand2DepartmentofMaterial...

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Pair distribution function analysis driven by atomistic simulations Application to microwave radiation synthesized TiO 2and ZrO 2 Shuyan Zhang1 Jie Gong1 Daniel Xiao2 B. Reeja Jayan1 and Alan J. H. McGaughey1.pdf

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Pair distribution function analysis driven by atomistic simulations Application to microwave radiation synthesized TiO 2and ZrO 2 Shuyan Zhang1 Jie Gong1 Daniel Xiao2 B. Reeja Jayan1 and Alan J. H. McGaughey1

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