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A Shell Model for Atomistic Simulation of Lithium Diffusion in Mixed Mn/Ti Oxides Sebastien Kerisit, Anne M. Chaka, Timothy C Droubay, and Eugene S Ilton J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/jp506025k • Publication Date (Web): 30 Sep 2014 Downloaded from http://pubs.acs.org on October 5, 2014

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The Journal of Physical Chemistry

A Shell Model for Atomistic Simulation of Lithium Diffusion in Mixed Mn/Ti Oxides Sebastien Kerisit*, Anne M. Chaka, Timothy C. Droubay, and Eugene S. Ilton Physical Sciences Division, Pacific Northwest National Laboratory, Richland, WA 99352 USA *

Corresponding Author: [email protected]

Abstract Mixed Mn/Ti oxides present attractive physicochemical properties such as their ability to accommodate Li for application in Li-ion batteries. In this work, atomic parameters for Mn were developed to extend an existing shell model of the Li-Ti-O system and allow simulations of pure and lithiated Mn and mixed Mn/Ti oxide polymorphs. The shell model yielded good agreement with experimentally-derived structures (i.e. lattice parameters and inter-atomic distances) and represented an improvement over existing potential models. The shell model was employed in molecular dynamics (MD) simulations of Li diffusion in the 1×1 c-direction channels of LixMn1-yTiyO2 with the rutile structure, where 0 ≤ x ≤ 0.25 and 0 ≤ y ≤ 1. In the infinite dilution limit, the arrangement of Mn and Ti ions in the lattice was found to have a significant effect on the activation energy for Li diffusion in the c channels due to the destabilization of half of the interstitial octahedral sites. Anomalous diffusion was demonstrated for Li concentrations as low as x = 0.125, with a single Li ion positioned in every other c channel. Further increase in Li concentration showed not only the substantial effect of Li-Li repulsive interactions on Li mobility but also their influence on the time dependence of Li diffusion. The results of the MD simulations can inform intrinsic structure-property relationships for the rational design of improved electrode materials for Li-ion batteries. Keywords: rutile, pyrolusite, titanium manganese oxides, molecular dynamics, diffusion.

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Introduction Manganese dioxide polymorphs are an important class of materials with technological applications to energy storage, catalysis, sensors, and groundwater remediation.1 In manganese dioxides such as α-, β-, and γ-MnO2, MnO6 octahedra arrange in edge-sharing single and double chains linked through corner-shared oxygens to form channels of different sizes (2×2, 1×1, and 2×1, respectively). The formation of channels allows for the insertion of small cations and in particular Li, which makes MnO2 polymorphs of great interest for the development of Li-ion battery cathodes.2-5 Lithiated MnO2 can also adopt a spinel structure, which shows attractive capacity and has been studied extensively.6-11 Titanium dioxide polymorphs form similar structures as manganese dioxides and their ability to accommodate Li has also been the subject of extensive work as a result.12 In addition to the pure end-members, titanium-manganese mixtures are being considered for several applications. For example, manganese doping has been investigated to reduce the band gap energy of TiO2 in order to allow for a greater proportion of the solar spectrum to be absorbed in solar cell applications.13-15 Formation of lithiated Mn-Ti solid solutions with the spinel structure has been considered to minimize the capacity loss during cycling of pure lithiated Mn spinel oxide for Li-ion batteries.16-18 In our own work, epitaxial single-crystal thin films of MnxTi1-xO2 with the rutile structure have been synthesized19 to facilitate the derivation of intrinsic structureproperty relationships for energy materials, in particular as it relates to diffusion of inserted lithium ions. Application of computational modeling techniques to Li diffusion in titanium and manganese oxides has provided significant insights into the elementary, atomic-level processes that give rise to the macroscopic performance characteristics of these materials. In particular,


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electronic structure calculations have allowed calculations of Li insertion voltage, atomic structural changes upon Li insertion, and relative stabilities of lithiated polymorphs.20-24 However, there is a need for simulating large systems over extended periods of time to obtain reliable information on Li diffusion as a function of Li concentration, which means that classical molecular dynamics (MD) techniques are often required to extrapolate knowledge obtained from electronic structure calculations to scales that can be linked to material performance. Although Li diffusion in the pure end-members has been studied with MD simulations based on classical force fields (also referred to as potential models) for a number of polymorphs,25-30 classical MD techniques have not been applied yet to Li diffusion in mixed Mn/Ti oxides. This current limitation is partly due to the lack of a potential model that can accurately reproduce the structures of a range of Mn oxide and Ti oxide polymorphs as well as the insertion of Li in these materials. In previous publications,28,29,31 Kerisit and co-workers developed and applied a shell model for lithium diffusion in pure titanium oxides based on the model originally introduced by Matsui and Akaogi.32 In this work, this model is extended to include manganese to allow polymorphs of mixed Mn/Ti oxides to be considered. As an example of the application of this newly developed model, Li diffusion in mixed Mn/Ti oxides with the rutile structure is also investigated.

Computational Methods Atoms are represented as point-charge particles that interact via long-range Coulombic forces and short-range interactions. The latter are described by parametrized functions and represent the repulsion between electron charge clouds and van de Waals attraction forces. The short-range



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interactions are described using a Buckingham potential, and therefore, the pairwise interaction energy, Uij, takes the following form:

U ij =

1

qi q j

4πε 0 rij

 rij  Cij − + Aij exp −  ρ ij  r 6   ij

(1)

where ε0 is the permittivity of vacuum, rij is the interatomic distance between ions i and j, qi is the charge of ion i and Aij, ρij and Cij are the Buckingham pair parameters. The Buckingham pair parameters are derived from the atomic parameters, Ai, Bi, and Ci:  Ai + A j Aij = f Bi + B j exp  Bi + B j 

(

)

   

(3)

ρ ij = Bi + B j

(4)

Cij = Ci C j

(5)

where f is 4.184 kJ mol-1 Å-1. The atomic parameters (qi, Ai, Bi, and Ci) used to simulate Ti and O are those optimized by Matsui and Akaogi32 (MA) for the TiO2 polymorphs rutile, anatase, brookite, and TiO2-II. In a previous publication,31 Kerisit and co-workers derived a shell model version of the MA potential. In the shell model,33 a polarizable ion is composed of two particles, a core and a shell, which share the ion’s charge and are linked by a harmonic spring, k: U c − s = k × rc2− s

(2)

where rc-s is the core-shell separation distance. Only oxygen anions were considered to be polarizable in this model. Subsequently, the parameters employed to describe Li were fitted to the lattice parameters, elastic constants, and bulk modulus of Li2O while keeping the MA parameters for oxygen constant.22 This set of parameters was used on several occasions to simulate lithiated titanium oxides.28-30,34-38 As described in detail in the Results and Discussion section, the parameters for Mn are derived in this work and they are combined with the existing 4 ACS Paragon Plus Environment

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Ti, Li, and O parameters to simulate pure and lithiated Mn oxides and mixed Mn/Ti oxides. All of the parameters used in this work are reported in Table 1. In the first half of this work, which is concerned with the derivation of a potential model and its application to the calculation of the structures of pure and lithiated Mn oxides and mixed Mn/Ti oxides, constant-pressure energy minimizations were carried out with the computer code METADISE39 without space group symmetry constraints. In the second half of this work, which focuses on Li diffusion in mixed Mn/Ti oxides with the rutile structure, molecular dynamics simulations were performed with the computer code DL_POLY.40,41 TABLE 1: Model Parameters Used in This Worka

a

ion

qi-core (e)

qi-shell (e)

Ai (Å)

Bi (Å)

Ci (Å kJ mol-0.5)

Mn(IV)

2.196

-

1.1250

0.077

22.5

Mn(III)

1.647

-

1.1813

0.077

22.5

Ti(IV)

2.196

-

1.1823

0.077

22.5

Ti(III)

1.647

-

1.1823

0.077

22.5

Li(I)

0.549

-

0.8000

0.050

0.0

O(-II)

-1.598

0.500

1.6339

0.117

54.0

3

0.5

The oxygen core-shell spring constant, k, is set to 44.3 eV Å-1. All the molecular dynamics simulations were carried out at zero-applied pressure in the NPT

ensemble (constant number of particles, constant pressure, and constant temperature). In these simulations, the temperature and pressure were kept constant by use of the Nosé-Hoover thermostat42 and the Hoover barostat,43 respectively. The electrostatic forces were calculated by means of the Ewald summation method. A 9 Å cutoff was used for the short-range interactions and the real part of the Ewald sum. The Verlet leapfrog integration algorithm was used to integrate the equations of motion with a time step of 0.2 fs. The shells were given a mass of 0.2



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au, and their motion was treated as that of the cores following the adiabatic shell model first introduced by Mitchell and Fincham.44

Results and Discussion This section is divided into three subsections. In the first subsection, the derivation of model parameters for Mn is presented and the ability of the newly-derived model to reproduce the structure of various oxides containing Mn, Ti, and Li is evaluated. In the second subsection, the performance of the model is compared to that of other potential models previously published in the literature. Finally, in the third subsection, the model is employed to study Li diffusion in the 1×1 c-direction channels (referred to as “c channels” hereafter) of mixed Mn/Ti oxides with the rutile structure. Derivation of Mn parameters and evaluation of Mn-Ti-Li-O model. One of the goals of this work is to develop a potential model that yields good agreement with a range of experimentally-derived structures of pure and lithiated Mn and Ti oxide polymorphs as well as lithiated Mn/Ti mixed oxides. Therefore, all the parameters previously derived for the Ti-Li-O system were kept constant throughout. When comparing calculated and experimental lattice parameters and inter-atomic distances, percentage differences of 1% or less are considered as very good agreement, between 1 and 2% as good agreement, between 2 and 3% as marginal agreement, and greater than 3% as poor agreement. As noted in the Computational Methods section, the TiO2 model is a shell model version of the Matsui and Akaogi model.32 The MA model uses atomic parameters (Ai, Bi, and Ci) for Ti and O, from which pair interactions can be calculated in a Buckingham potential form.32 In a previous publication,22 the atomic parameters for Li were fitted to the lattice parameters, elastic constants, and bulk modulus of Li2O while


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keeping the O atomic parameters from the MA model constant. Here, the atomic parameters B and C for Mn are set to those of Ti; then, the atomic parameter A is varied to optimize the agreement between the experimental and calculated lattice parameters of pyrolusite (β-MnO2)45, which adopts the rutile structure. For Ti, it was shown in a previous publication34 that using the atomic parameters derived for Ti(IV) to represent Ti(III), with its partial charge set to 1.647 e (3/4 of 2.196 e) to reflect the change in oxidation state, led to very good agreement (within 1%) with the lattice parameters of Ti2O3, which adopts the corundum structure. When using the atomic parameters derived here for Mn(IV) to model the Mn(III) oxide bixbyite (α-Mn2O3)46, the calculated lattice parameter was 3% shorter than the experimental value. Therefore, the atomic parameter A for Mn(III) was modified to be slightly higher than that of Mn(IV) to optimize the agreement with the lattice parameter of Mn2O3. In simulations where Mn is on average in a fractional oxidation state, the charge and atomic parameter A of Mn are scaled linearly. The atomic parameters used in this work are shown in Table 1; the corresponding Buckingham pair parameters are given in Supporting Information. The newly derived parameters were evaluated against two groups of structures. The first group consisted of the tetravalent manganese oxide polymorphs hollandite (α-MnO2),47 ramsdellite (γ-MnO2),48 and spinel (λ-MnO2),49 in addition to the two structures employed in the parameter derivation. Table 2 compares the calculated and experimental lattice parameters and Mn-O distances of the four pure tetravalent Mn oxide polymorphs. The percentage deviations from the experimental lattice parameters are all within 2%. The individual Mn-O distances are also well reproduced by the model with a large majority of the distances within 2% of experiment. The second group consisted of LixMn2-yTiyO4 compounds with the spinel structure9,17 (Table 3). When simulating the LixMn2O4 spinels, Mn atoms were positioned in



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octahedral 16d sites only and Li atoms were placed in tetrahedral 8a sites for x ≤ 1 and in octahedral 16c sites for x > 1. When simulating the LiMn2-yTiyO4 spinels, Li atoms were placed in 8a sites only and Mn and Ti atoms were placed in 16d sites only; then, all possible combinations for arranging the Mn and Ti atoms in the 16d sites were generated and energy minimized and the configuration with the most negative potential energy was used for comparison with experimental data. This differs slightly from the best model used by Krins et al.,17 in which a small amount of Mn occupied the 8a sites and a small proportion of Li was found in the 16d sites in addition to Mn and Ti. Table 3 shows that the agreement with experimental data is also good for this group, with all lattice parameters reproduced within 2%. Additionally, good agreement is obtained for M-O distances (M = Mn, Ti, or Li) with all distances within 2% of experiment, except for the Li-O distances in the tetrahedral 8a sites for x ≤ 1, for which the calculations predicted a larger tetrahedral site than observed experimentally. Elastic data on these materials is rare; however, Lin et al.50 reported a bulk modulus of 119 ± 4 GPa for LiMn2O4, which is in good agreement with our calculated bulk modulus of 116 GPa. In a previous publication,29 it was shown that using the parameters derived for Ti, Li, and O gave good agreement with the experimental structure of Li2TiO3 without additional modifications of the parameter set. Table 4 shows that this is also true for the Mn analog Li2MnO3,51 with lattice parameters and Mn-O and Li-O distances that differ from experiment by 2% at most. Li2MnO3 adopts a monoclinic layered structure whereby octahedral Li layers with two distinguishable Li sites (Li1 and Li2) alternate with mixed Li (Li3) and Mn octahedral layers.


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TABLE 2: Calculated and Experimental Bulk Properties of a Series of MnO2 Polymorphs polymorph

hollandite

pyrolusite

ramsdellite

spinel

property

expt.

calc.

∆ (%)

expt.

calc.

∆ (%)

expt.

calc.

∆ (%)

expt.

calc.

∆ (%)

a (Å)

9.788

9.732

-0.6

4.398

4.373

-0.6

4.53

4.460

-1.6

8.045

8.175

1.6

9.27

9.098

-1.9 1.91×6

1.93×6

0.7

b (Å) c (Å)

2.865

2.903

1.3

2.873

2.886

0.5

2.87

2.905

1.2

Mn-O (Å)

1.89×1

1.84×1

-2.8

1.90×2

1.86×2

-2.1

1.87×1

1.83×1

-1.9

Mn-O (Å)

1.92×1

1.94×1

0.9

1.88×4

1.90×4

1.0

1.89×1

1.95×1

3.2

Mn-O (Å)

1.88×2

1.90×2

1.2

1.89×2

1.91×2

0.8

Mn-O (Å)

1.93×2

1.92×2

-0.5

1.92×2

1.91×2

-0.5



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TABLE 3: Calculated and Experimental Bulk Properties of Lithiated Pure Mn and Mixed Mn/Ti Spinels (LixMn2-yTiyO4) stoichio.

a (Å)

octahedral M-O (Å)

tetrahedral M-O (Å)

x

y

expt.

calc.

∆ (%)

expt.

calc.

∆ (%)

expt.

calc.

∆ (%)

0

0

8.045a

8.175

1.6

1.91

1.93

0.7

-

-

-

0.5

0

8.150b

8.247

1.2

1.94

1.95

0.6

1.94

1.99

3.0

1

0

8.245c

8.301

0.7

1.96

1.94

-0.7

1.97

2.04

3.2

2

0

8.380d

8.357

-0.3

2.10

2.09

-0.5

-

1.92

-

1

0

8.243e

8.301

0.7

1.95

1.94

-0.5

1.98

2.04

3.2

1

0.25

8.257e

8.337

1.0

1.96

1.98

0.9

1.98

2.00

1.0

1

0.5

8.282e

8.372

1.1

1.96

1.99

1.3

1.99

2.01

0.8

1

0.75

8.308e

8.409

1.2

1.97

2.00

1.3

2.00

2.02

1.1

1

1

8.334e

8.451

1.4

1.98

2.00

1.2

2.00

2.03

1.8

49

a

Thackeray et al.

b

Kanamura et al.9

c

Strobel et al.10

d

Mishra and Ceder20 (density functional theory). Octahedral M-O distances are averages of 16c and 16d sites.

e

Krins et al.17


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TABLE 4: Calculated and Experimental Lattice Parameters and Interatomic Distances of Li2MnO3 property

expt.a

calc.

∆ (%)

a (Å)

4.937

4.939

0.0

b (Å)

8.532

8.528

0.0

c (Å)

5.030

4.967

-1.2

β (°)

109.46

109.53

0.1

Mn-O (Å) ×2

1.91

1.90

-0.6

Mn-O (Å) ×2

1.92

1.91

-0.6

Mn-O (Å) ×2

1.90

1.90

-0.1

Li1-O (Å) ×2

2.07

2.05

-0.9

Li1-O (Å) ×4

2.05

2.05

-0.2

Li2-O (Å) ×2

2.04

2.08

2.0

Li2-O (Å) ×2

2.10

2.06

-1.8

Li2-O (Å) ×2

2.17

2.16

-0.2

Li3-O (Å) ×2

2.01

2.01

-0.1

2.15

2.13

-0.7

Li3-O (Å) ×4 a

Strobel and Lambert-Andron

51

Comparison with previous potential models. In 1995, Cygan et al.52 published a shell model (C-95) for simulating lithiated Mn spinel oxides. The model used formal charges and parameters taken from Lewis and Catlow53 with the exception of the parameters for the Mn4+-O2interactions, which were derived in that study to reproduce the structures of β-, γ-, and λ-MnO2. In 1997, Ammundsen et al.54 introduced a shell model (A-97) for simulating pure and lithiated Mn spinel oxides. This model used formal charges and was based on the refinement of previously derived parameters53,55 for Mn3+-O2- and Mn4+-O2- interactions and the O core-shell force constant, whereas the parameters for Li+-O2- interactions were taken from a previous publication56 without modifications. In 1998, and again with the aim to study lithiated Mn spinel 11 ACS Paragon Plus Environment


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oxides, Cygan et al.25 published a new model (C-98) that differed significantly from their 1995 model, in which all ions were represented as non-polarizable ions and short-range interactions were treated with Lennard-Jones potentials rather than Buckingham potentials. In 1999, Ammundsen et al.57 published a modified version of their 1997 model (A-99) with modified ρ values for the Mn3+/Mn4+-O2- interactions as well as modified Mn and O core-shell force constants. In 2000, Suzuki et al.58 reported a rigid-ion model (S-00) for simulating lithiated Mn spinel oxides. This model used partial charges for Mn(IV) and O(-II), +2.4 e and -1.2 e, respectively, but the formal charge for Li(I) and a partial charge for Mn(III) that is exactly 1 e less than that of Mn(IV), +1.4 e; as a result compounds such as Mn2O3 and Li2MnO3 cannot be simulated with this model. In 2005, Sayle et al.59 published a potential model (S-05) for simulating pyrolusite. The model is identical to the MA potential of TiO2, a rigid-ion model with partial charges, with the Mn(IV)-Mn(IV) and Mn(IV)-O(-II) pair parameters slightly modified to reproduce the structures of β- and γ-MnO2. In 2009, Sayle et al.60 added potential parameters for simulating Li insertion. To compare the ability of the different models to accurately reproduce known experimental structures, a figure of merit was calculated as the root mean square deviation (RMSD) of the differences between the experimental and calculated lattice parameters of the pure Mn(IV) and Mn(III) oxides considered in this work (i.e., α-, β-, γ-, and λ-MnO2 and α-Mn2O3). The RMSD obtained with the model derived in this work was the lowest of all the models introduced above (1.2 compared to 12.1, 4.3, 2.4, 3.0, 2.3, and 2.5 for C-95, A-97, C-98, A-99, S-00, and S-05, respectively). Similarly, the RMSD calculated for this model for lithiated Mn oxides (Li2MnO3 and LixMn2O4 spinels with x=0.5, 1.0, and 2.0) was again the lowest of all the models (0.7 compared to 2.6, 1.2, 2.0, 0.8, 1.7, and 1.9 for C-95, A-97, C-98, A-99, S-00, and S-05,


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respectively). Details of the two sets of comparison are presented in Supporting Information. In addition, all constant-pressure energy minimizations yielded a valid minimum with this model, whereas several of the previously published models resulted in the presence of imaginary frequencies for a small number of structures (Table S1). Finally, Fleming et al.61 published a charge-fluctuating model for MnO2 polymorphs that showed good agreement with the experimental structures of β-, γ-, and λ-MnO2; but parameters for simulating Li were not included. Li diffusion in LixMn1-yTiyO2. MD simulations were carried out to simulate Li diffusion in LixMn1-yTiyO2 with the rutile structure, where 0 ≤ x ≤ 0.25 and 0 ≤ y ≤ 1. The simulation cells consisted of a 5 × 5 × 8 supercell (400 TiyMn1-yO2 units). The Ti mole fraction y was varied by increments of 0.25 and for each value, four different Li concentrations and arrangements were considered: (1) a single Li ion in the supercell to model the infinite dilution limit (=1 Li per 400 Mn/Ti); (2) one Li ion in every other c channels (=1 Li per 16 Mn/Ti), as shown in Figure 1; (3) one Li ion in every c channel (=1 Li per 8 Mn/Ti), also shown in Figure 1; and (4) two Li ions in every other c channels (=1 Li per 8 Mn/Ti). Insertion of Li is accompanied by the addition of a charge-compensating negative charge distributed equally over all Ti/Mn sites. Although electron polarons can localize on Mn and Ti sites, previous work34 indicated that their diffusion is faster than Li hopping, suggesting that a distributed excess electron charge is more adequate than Mn(III) and Ti(III) sites fixed at their initial positions throughout the simulations, given the constraints of the method. Additionally, the driving force for the formation of nearest-neighbor electron polaron-Li+ pairs is expected to decrease with increasing Li concentration, leading to a more homogenously distributed excess electron charge at higher Li concentrations.



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In each case, MD simulations were run at five temperatures (300, 350, 400, 450, and 500 K) for 10 ns after an equilibration period of 20 ps where the velocities were scaled to the target temperature. For values of y other than 0 and 1, two different arrangements of Mn and Ti cations in the octahedral site were considered. In configuration #1, rows of cations in the [001] direction are occupied only by either Mn or Ti whereas, in configuration #2, these rows consist of alternating Mn and Ti, as illustrated in Figure 1. (a)

(b)

(c)

(d)

Figure 1. Illustrations of two Li concentrations (a) one Li in every other c channel and (b) one Li in every c channel as well as (c) configuration #1 (Mn- and Ti-only rows) and (d) configuration #2 (rows of alternating Mn and Ti). Li atoms are shown in yellow, Mn in purple, Ti in gray, and O in red. The self-diffusion coefficients of Li were obtained from its mean square displacement (MSD): MSD = ri (t ) − ri (0 )

2

(6)

where ri(t) is the position of atom i at time t. A configuration was recorded every 1 ps and the MSDs were calculated for a correlation period of 250 ps. In the case of “normal” diffusion, the diffusion coefficient, D, is given by the Einstein relation

D=

MSD 2nt

(7)


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where n is the number of dimensions in which the diffusion coefficient is considered. All the diffusion coefficients calculated in this work are those in the [001] direction; therefore, n equals 1. However, the MSD of Li ions in the c channels is only linear with time in the infinite dilution case, as shown in Figure 2 for Li diffusion in TiO2 at 400 K. A log-log plot of the same data is also presented in Figure 2 to establish the power law of the time dependence of the MSD MSD = Ct α

(8)

where C is a constant and α is the slope of the log-log plot. If α = 1, diffusion follows Einstein relation and Equation 7 can be used to obtain the diffusion coefficient. Values of α ≠ 1 are indicative of anomalous diffusion. In particular, single-file diffusion yields MSDs proportional to the square root of time (α = 0.5).62 In the case of anomalous diffusion, a generalization of Equation 7 is written instead F=

MSD 2t α

(9)

where F is termed the mobility factor.

Figure 2. (left) Normalized MSD as a function of time for the four Li concentrations in TiO2 at 400 K. (right) log-log plot of the same data showing the slopes of the linear regressions (α).



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For the infinite dilution case, the Li diffusion coefficients, DLi, were derived using Equation 7. The average and standard deviation of the Li diffusion coefficients were obtained from dividing each molecular dynamics trajectory into four equal blocks. The activation energies and pre-exponential factors were then obtained from Arrhenius plots of the Li diffusion coefficients, as shown in Figure 3. Uncertainties in the activation energies and pre-exponential factors were obtained from the standard errors on the gradients and intercepts of the Arrhenius plots, respectively. The activation energy (0.04 ± 0.01 eV) and pre-exponential factor (2.09 ± 0.06 × 1012 s-1) for Li diffusion in pure TiO2 compare well with those obtained in a previous study34 with the same model (0.05 eV and 2.85 × 1012 s-1), albeit for shorter MD simulations. In addition, the calculated activation energy is in excellent agreement with the energy barrier derived by Koudriachova et al.63-65 from density functional theory calculations (0.04 eV). The experimental activation energy derived by Johnson66 from macroscopic experiments on rutile single crystals was much larger (0.33 eV). As will be described later, there are strong concentrations effects at finite Li concentrations that could contribute to this difference. Additionally, the experimental value is likely to be an effective macroscopic activation energy rather than the energy barrier for an elementary hop between two nearest-neighbor sites along the c channel. Indeed, the presence of point or extended defects in the single crystals could constitute bottlenecks that control Li diffusion and prevent a 1-to-1 equivalence between the calculated and experimental results. The good agreement between the activation energy calculated in this work and the energy barrier obtained by Koudriachova et al. supports this hypothesis. The activation energy for Li diffusion in pure MnO2 is calculated to be twice as high as that in TiO2 (0.08 eV). This is approximately half the activation energy obtained by Tompsett et al.67 from DFT calculations using the generalized gradient approximation with Hubbard U corrections and a


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supercell containing 192 MnO2 stoichiometric units (0.17 eV). Wang et al.24 obtained an activation energy of 0.26 eV following a similar computational approach including the use of GGA+U; but with a much smaller supercell containing 16 MnO2 stoichiometric units. Therefore, the effect of cell size on the GGA+U calculations is a potential source of the difference between the DFT calculations and the MD simulations. An additional potential source of this difference is the possible localization of an electron polaron in the vicinity of the Li ion in the DFT calculations.

Figure 3. Arrhenius plots (left) and resulting activation energies (right) for Li diffusion in Mn1-yTiyO2 for the infinite dilution case. Although there is only a small decrease in activation energy with increasing Ti content, a large difference is observed between the two Mn/Ti arrangements, with the configuration where Mn and Ti alternate in each cation row along the [001] direction leading to much higher activation energies. Potential of mean force (PMF) calculations were carried out to elucidate this difference. For each configuration, a series of 48 energy minimizations was carried out whereby the Li ion was constrained in the [001] direction at intervals of 0.123 Å, which spans two unit cells. The constraint force along the reaction coordinate (i.e. the [001] direction) was integrated to obtain the potential of mean force. One Ti ion away from the Li ion was frozen to prevent the 17 ACS Paragon Plus Environment


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lattice from translating to compensate for the force introduced by moving the Li ion away from its energy minimum. The PMF profiles thus obtained are shown in Figure 4.

Figure 4. Potential of mean force for Li diffusion in Ti0.5Mn0.5O2 for two arrangements of the Mn/Ti cations in the infinite dilution case. The free energy barriers (0.06 and 0.18 eV for configurations #1 and #2, respectively) are in good agreement with the activation energies obtained from the MD simulations (0.07 and 0.16 eV). When the Ti and Mn cations are arranged in alternating rows containing only either Mn or Ti, the PMF resembles that of the pure TiO2 or pure MnO2 with two free energy minima per unit cell corresponding to interstitial octahedral sites separated by small free energy barriers on the order of 0.05-0.1 eV. However, when the Ti and Mn cations are arranged in rows of alternating Mn and Ti, the PMF shows a clear destabilization of one free energy minimum versus the other. The geometry of both energy minima is similar with the Li ion forming two short bonds with oxygen ions (~1.8 Å) and two pairs of longer bonds. In the more energetically-stable of the two minima, the two pairs of longer bonds have similar lengths (2.06 and 2.01 Å) whereas, for the other minimum, the octahedral coordination environment of Li is much more distorted (2.19 and 1.98 Å) as a result of the difference in size between Mn and Ti. Because of the large


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destabilization of one of the two free energy minima in the c channels, the effective activation energy for Li diffusion in the [001] direction is increased by a factor of approximately 2 to 3. PMF calculations for Li diffusion in the [110] direction (i.e. inter-channel diffusion) were also carried for pure MnO2 in the infinite dilution limit and yielded an activation energy of 1.94 eV, which compares well with that obtained by Tompsett et al.67 from DFT+U calculations (2.22 eV). This result confirms that Li diffusion is predominantly along the [001] direction and that Li diffusion across c channels is not expected within the timescale of the MD simulations. For the three finite Li concentrations, the Li mobility factors, FLi, were derived using Equation 9. As illustrated in Figure 2, values of α for the case where the supercell contained one Li in every other c channel were approximately 0.7; therefore, this value of α was used in Equation 9 to extract the Li mobility factors at this Li concentration. Similarly, values of α of approximately 0.5, i.e. as typically obtained for single-file diffusion, were calculated for the cases where the supercell contained 1 Li per 8 Mn/Ti, and therefore, a value of 0.5 was used in Equation 9 for these cases. It should be noted that the values of α were found generally to increase slightly with temperature for a given concentration and Mn/Ti content; however, good fits were obtained nonetheless with values of α fixed with temperature. As before for the Li diffusion coefficient in the infinite dilution limit, the average and standard deviation of the Li mobility factors were obtained from dividing each MD trajectory into four equal blocks and the activation energies and pre-exponential factors were then obtained from Arrhenius plots.



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Figure 5. Activation energies (left) and pre-exponential factors (right) for Li diffusion in the [001] direction as a function of Mn and Ti content for one Li ion in every other c channel (top), one Li ion in every c channel (middle), and two Li ions in every c channel (bottom). Figure 5 presents the activation energies and pre-exponential factors calculated from Arrhenius plots of the Li mobility factors. As the Li concentration is increased from the infinite dilution case to one Li ion in every other c channel, the activation energies increase to values between 0.22 and 0.27 eV. However, when the Li concentration is increased further to one Li per c channel, the activation energies diminish slightly and are found between 0.16 and 0.26 eV, but the pre-exponential factors decrease by a factor of 3 to 7 for most cases. As a result, the mobility factors are generally lower when the concentration is increased. When the distribution of Li ions is changed from one Li per c channel to two Li in every other c channel, the activation energies decrease further (i.e. in the range from 0.12 to 0.17 eV), but again, the pre-exponential factors also decrease by at least one order of magnitude for most Mn/Ti contents, which leads to lower 20 ACS Paragon Plus Environment

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mobility factors overall when more than one Li occupy an individual c channel. These results indicate that Li-Li repulsive interactions strongly affect Li diffusion along the [001] direction. If neighboring interstitial sites along the c channels are destabilized to the same extent by the presence of other Li ions in the same or neighboring c channels, an increase in concentration does not necessarily translates to an increase in activation energy; however, the pre-exponential factors, which can be conceptualized as the attempt frequencies for overcoming the free energy barriers between neighboring free energy minima in the c channels, decrease in magnitude with increasing Li concentration. The MD simulations also show that this decrease is more significant when adding Li ions in a single channel than when increasing the number of occupied channels.

Conclusions Atomic parameters for Mn consistent with an existing shell model of pure and lithiated Ti oxide polymorphs were derived in this work and applied to calculate the structures of a range of pure and lithiated Mn oxide polymorphs as well as lithiated mixed Mn/Ti spinel oxides. A figure of merit determined based on differences between calculated and experimental lattice parameters showed that the model derived in this work provides a more accurate representation of a number of pure and lithiated Mn oxide structures than previously published models. This finding indicates that the shell model is sufficiently transferable to be employed in simulations that cover a wide range of Li insertion concentrations. The shell model was then used to evaluate with MD simulations the effects of Li concentration and Mn/Ti content on Li diffusion in the c channels of LixMn1-yTiyO2 with the rutile structure, where 0 ≤ x ≤ 0.25 and 0 ≤ y ≤ 1. The MD simulations revealed that small, local structural changes due to different arrangements of Mn and Ti cations in the rutile lattice for a



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given concentration can result in large changes in activation energy for Li diffusion. In the infinite dilution limit in particular, distributing the cations in alternating Mn- or Ti-only [001] rows led to activation energies 2 to 3 times lower than for a configuration where [001] rows are composed of alternating Mn and Ti. As the Li concentration increases, Li-Li repulsive interactions begin to affect Li mobility significantly. Initially, they lead to an increase in activation energies but, as the Li concentration increases further, the reduction in Li mobility is primarily due to a decrease of the preexponential factors. In addition, time dependence of the Li displacement is no longer linear and reduces as the Li concentration increases. A square root dependence on time, typical of singlefile diffusion, is predicted even when a single Li ion occupies each c channel due to the strong interactions between Li ions in neighboring channels.

Acknowledgement This research was supported by the Laboratory Directed Research and Development Program at Pacific Northwest National Laboratory (PNNL), a multi-program national laboratory operated by Battelle Memorial Institute for the U.S. Department of Energy (DOE) under Contract DE-AC0576RL01830. The computer simulations were performed in part using the Molecular Science Computing (MSC) facilities in the William R. Wiley Environmental Molecular Sciences Laboratory (EMSL), a national scientific user facility sponsored by the DOE’s Office of Biological and Environmental Research (OBER) and located at PNNL.

Supporting Information Available. Buckingham pair potential parameters; comparison of lattice parameters of pure and lithiated Mn(IV) and Mn(III) oxide polymorphs obtained with


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several potential parameters available in the literature. This material is available free of charge via the Internet at http://pubs.acs.org.



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References (1) Post, J. E. Manganese Oxide Minerals: Crystal Structures and Economic and Environmental Significance. Proc. Natl. Acad. Sci. 1999, 96, 3447-3454. (2) Thackeray, M. M. Manganese Oxides for Lithium Batteries. Prog. Solid St. Chem. 1997, 25, 1-71. (3) Ling, C.; Mizuno, F. Capture Lithium in αMnO2: Insights from First Principles. Chem. Mater. 2012, 24, 3943-3951. (4) Jiao, F.; Bruce, P. G. Mesoporous Crystalline β-MnO2 - A Reversible Positive Electrode for Rechargeable Lithium Batteries. Adv. Mater. 2007, 19, 657-660. (5) Zachau-Christiansen, B.; West, K.; Jacobsen, T.; Skaarup, S. Insertion of Lithium into the Manganese Dioxides: Pyrolusite and Ramsdellite. Solid State Ionics 1994, 70/71, 401-406. (6) Thackeray, M. M.; David, W. I. F.; Bruce, P. G.; Goodenough, J. B. Lithium Insertion into Manganese Spinels. Mat. Res. Bull. 1983, 18, 461-472. (7) Tarascon, J. M.; Wang, E.; Shokoohi, F. K.; McKinnon, W. R.; Colson, S. The Spinel Phase of LiMn2O4 as a Cathode in Secondary Lithium Cells. J. Electrochem. Soc. 1991, 138, 28592864. (8) Gummow, R. J.; de Kock, A.; Thackeray, M. M. Improved Capacity Retention in Rechargeable 4 V Lithium/Lithium-Manganese Oxide (Spinel) Cells. Solid State Ionics 1994, 69, 59-67. (9) Kanamura, K.; Naito, H.; Yao, T.; Takehara, Z. Structural Change of the LiMn2O4 Spinel Structure Induced by Extraction of Lithium. J. Mater. Chem. 1996, 6, 33-36.


Page 25 of 32

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


(10) Strobel, P.; Le Cras, F.; Seguin, L.; Anne, M.; Tarascon, J. M. Oxygen Nonstoichiometry in Li-Mn-O Spinel Oxides: A Powder Neutron Diffraction Study. J. Solid State Chem. 1998, 135, 132-139. (11) Ouyang, C.; Du, Y.; Shi, S.; Lei, M. Small Polaron Migration in LixMn2O4: From First Principles Calculations. Phys. Lett. A 2009, 373, 2796-2799. (12) Yang, Z.; Choi, D.; Kerisit, S.; Rosso, K. M.; Wang, D.; Zhang, J.; Graff, G. L.; Liu, J. Nanostructures and Lithium Electrochemical Reactivity of Lithium Titanites and Titanium Oxides: A Review. J. Power Sources 2009, 192, 588-598. (13) Shao, G. Electronic Structures of Manganese-Doped Rutile TiO2 from First Principles. J. Phys. Chem. C 2008, 112, 18677-18685. (14) Lu, L.; Xia, X.; Luo, J. K.; Shao, G. Mn-Doped TiO2 Thin Films with Significantly Improved Optical and Electrical Properties. J. Phys. D: Appl. Phys. 2012, 45, 485102. (15) Brus, V. V.; Pidkamin, L. J.; Abashin, S. L.; Kovalyuk, Z. D.; Maryanchuk, P. D.; Chugai, O. M. Optical Constants and Polarimetric Properties of TiO2-MnO2 Thin Films. Opt. Mater. 2012, 34, 1940-1945. (16) Arillo, M. A.; Cuello, G.; López, M. L.; Martín, P.; Pico, C.; Veiga, M. L. Structural Characterisation and Physical Properties of LiMMnO4 (M = Cr, Ti) Spinels. Solid State Sci. 2005, 7, 25-32. (17) Krins, N.; Hatert, F.; Traina, K.; Dusoulier, L.; Molenberg, I.; Fagnard, J. F.; Vanderbemden, P.; Rulmont, A.; Cloots, R.; Vertruyen, B. LiMn2-xTixO4 Spinel-Type Compounds (x ≤ 1): Structural, Electrical and Magnetic Properties. Solid State Ionics 2006, 177, 1033-1040.



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

Page 26 of 32

(18) Avdeev, G.; Amarilla, J. M.; Rojo, J. M.; Petrov, K.; Rojas, R. M. Composition and Structure of Acid Leached LiMn2-yTiyO4 (0.2 ≤ y ≤ 1.5). J. Solid State Chem. 2009, 182, 32263231. (19) Ilton, E. S.; Droubay, T. C.; Chaka, A. M.; Kovarik, L.; Varga, T.; Arey, B. W.; Kerisit, S. N. Epitaxial Single-Crystal Thin Films of MnxTi1-xO2-δ Grown on (Rutile) TiO2 Substrates with Pulsed Laser Deposition: Experiment and Theory. Surf. Sci., (in press). (20) Mishra, S. K.; Ceder, G. Structural Stability of Lithium Manganese Oxides. Phys. Rev. B 1999, 59, 6120-6130. (21) Koudriachova, M. V.; de Leeuw, S. W.; Harrison, N. M. Orthorhombic Distortion on Li Intercalation in Anatase. Phys. Rev. B 2004, 69, 054106. (22) Kerisit, S.; Rosso, K. M.; Yang, Z.; Liu, J. Computer Simulation of the Phase Stabilities of Lithiated TiO2 Polymorphs. J. Phys. Chem. C 2010, 114, 19096-19107. (23) Tompsett, D. A.; Islam, M. S. Electrochemistry of Hollandite α-MnO2: Li-ion and Na-ion Insertion and Li2O Incorporation. Chem. Mater. 2013, 25, 2515-2526. (24) Wang, D.; Liu, L.-M.; Zhao, S.-J.; Li, B.-H.; Liu, H.; Lang, X.-F. β-MnO2 as a Cathode Material for Lithium Ion Batteries from First Principles Calculations. Phys. Chem. Chem. Phys. 2013, 15, 9075-9083. (25) Cygan, R. T.; Westrich, H. R.; Doughty, D. H. Molecular Dynamics Study of Lithium Diffusion in Lithium-Manganese Spinel Cathode Materials. Mater. Res. Soc. Symp. Proc. 1998, 496, 109-114. (26) Gotlib, I. Y.; Murin, I. V.; Piotrovskaya, E. M. Molecular Dynamics Simulations of LixMn2O4 Spinel Solid Solutions with Simple Potential Models. Inorg. Mater. 2003, 39, 404408.


Page 27 of 32

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


(27) Tateishi, K.; du Boulay, D.; Ishizawa, N.; Kawamura, K. Structural Disorder along the Lithium Diffusion Pathway in Cubically Stabilized Lithium Manganese Spinel II. Molecular Dynamics Calculation. J. Solid State Chem. 2003, 174, 175-181. (28) Vijayakumar, M.; Kerisit, S.; Rosso, K. M.; Burton, S. D.; Sears, J. A.; Yang, Z.; Graff, G. L.; Liu, J.; Hu, J. Lithium Diffusion in Li4Ti5O12 at High Temperatures. J. Power Sources 2011, 196, 2211-2220. (29) Vijayakumar, M.; Kerisit, S.; Yang, Z.; Graff, G. L.; Liu, J.; Sears, J. A.; Burton, S. D.; Rosso, K. M.; Hu, J. Combined

6,7

Li NMR and Molecular Dynamics Study of Li Diffusion in

Li2TiO3. J. Phys. Chem. C 2009, 113, 20108-20116. (30) Yildirim, H.; Greeley, J. P.; Sankaranarayanan, S. K. R. S. The Effect of Concentration on Li Diffusivity and Conductivity in Rutile TiO2. Phys. Chem. Chem. Phys. 2012, 14, 4565-4576. (31) Kerisit, S.; Deskins, N. A.; Rosso, K. M.; Dupuis, M. A Shell Model for Atomistic Simulation of Charge Transfer in Titania. J. Phys. Chem. C 2008, 112, 7678-7688. (32) Matsui, M.; Akaogi, M. Molecular Dynamics Simulation of the Structural and Physical Properties of the Four Polymorphs of TiO2. Mol. Simul. 1991, 6, 239-244. (33) Dick, B. G.; Overhauser, A. W. Theory of the Dielectric Constants of Alkali Halide Crystals. Phys. Rev. 1958, 112, 90-103. (34) Kerisit, S.; Rosso, K. M.; Yang, Z.; Liu, J. Dynamics of Coupled Lithium/Electron Transfer Diffusion in TiO2 Polymorphs. J. Phys. Chem. C 2009, 113, 20998-21007. (35) Yildirim, H.; Greeley, J. P.; Sankaranarayanan, S. K. R. S. Effect of Concentration on the Energetics and Dynamics of Li Ion Transport in Anatase and Amorphous TiO2. J. Phys. Chem. C 2011, 115, 15661-15673.



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

Page 28 of 32

(36) Yildirim, H.; Greeley, J. P.; Sankaranarayanan, S. K. R. S. Concentration-Dependent Ordering of Lithiated Amorphous TiO2. J. Phys. Chem. C 2013, 117, 3834-3845. (37) Murphy, S. T.; Zeller, P.; Chartier, A.; Van Brutzel, L. Atomistic Simulation of the Structural, Thermodynamic, and Elastic Properties of Li2TiO3. J. Phys. Chem. C 2011, 115, 21874-21881. (38) Azuma, K.; Dover, C.; Grinter, D. C.; Grau-Crespo, R.; Almora-Barrios, N.; Thornton, G.; Oda, T.; Tanaka, S. Scanning Tunneling Microscopy and Molecular Dynamics Study of the Li2TiO3(001) Surface. J. Phys. Chem. C 2013, 117, 5126-5131. (39) Watson, G. W.; Kelsey, E. T.; de Leeuw, N. H.; Harris, D. J.; Parker, S. C. Atomistic Simulation of Dislocations, Surfaces and Interfaces in MgO. J. Chem. Soc., Faraday Trans. 1996, 92, 433-438. (40) Smith, W.; Forester, T. R. DL_POLY_2.0: A General Purpose Parallel Molecular Dynamics Simulation Package. J. Mol. Graphics 1996, 14, 136-141. (41) Smith, W.; Forester, T. R.; Todorov, I. T. The DL_POLY Classic User Manual. STFC Daresbury Laboratory, United Kingdom. (42) Hoover, W. G. Canonical Dynamics - Equilibrium Phase-Space Distributions. Phys. Rev. A 1985, 31, 1695-1697. (43) Melchionna, S.; Ciccotti, G.; Holian, B. L. Hoover NPT Dynamics for Systems Varying in Shape and Size. Mol. Phys. 1993, 78, 533-544. (44) Mitchell, P. J.; Fincham, D. Shell Model Simulations by Adiabatic Dynamics. J. Phys.: Condens. Matter 1993, 5, 1031-1038. (45) Baur, W. H. Rutile-Type Compounds. V. Refinement of MnO2 and MgF2. Acta Cryst. B 1976, 32, 2200-2204.


Page 29 of 32

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


(46) Geller, S. Structures of α-Mn2O3 (Mn0.983Fe0.017)2O3 and (Mn0.37Fe0.63)2O3 and Relation to Magnetic Ordering. Acta Cryst. B 1971, 27, 821-828. (47) Rossouw, M. H.; Liles, D. C.; Thackeray, M. M. Alpha Manganese Dioxide for Lithium Batteries: A Structural and Electrochemical Study. Mat. Res. Bull. 1992, 27, 221-230. (48) Byström, A. M. The Crystal Structure of Ramsdellite, an Orthorhombic Modification of MnO2. Acta Chem. Scand. 1949, 3, 163-173. (49) Thackeray, M. M.; de Kock, A. Synthesis and Structural Characterization of Defect Spinels in the Lithium-Manganese-Oxide System. Mat. Res. Bull. 1993, 28, 1041-1049. (50) Lin, Y.; Yang, Y.; Ma, H.; Cui, Y.; Mao, W. L. Compressional behavior of bulk and nanorod LiMn2O4 under nonhydrostatic stress. J. Phys. Chem. C 2011, 115, 9844-9849. (51) Strobel, P.; Lambert-Andron, B. Crystallographic and Magnetic Structure of Li2MnO3. J. Solid State Chem. 1988, 75, 90-98. (52) Cygan, R. T.; Westrich, H. R.; Doughty, D. H. Ionic Modeling of Lithium Manganese Spinel Materials for Use in Rechargeable Batteries. Mater. Res. Soc. Symp. Proc. 1995, 393, 113-118. (53) Lewis, G. V.; Catlow, C. R. A. Potential Models for Ionic Oxides. J. Phys. C: Solid State Phys. 1985, 18, 1149-1161. (54) Ammundsen, B.; Rozière, J.; Islam, M. S. Atomistic Simulation Studies of Lithium and Proton Insertion in Spinel Lithium Manganates. J. Phys. Chem. B 1997, 101, 8156-8163. (55) Cherry, M.; Islam, M. S.; Catlow, C. R. A. Oxygen Ion Migration in Perovskite-Type Oxides. J. Solid State Chem. 1995, 118, 125-132. (56) Bush, T. S.; Gale, J. D.; Catlow, C. R. A.; Battle, P. D. Self-Consistent Interatomic Potentials for the Simulation of Binary and Ternary Oxides. J. Mater Chem. 1994, 4, 831-837.



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

Page 30 of 32

(57) Ammundsen, B.; Burns, G. R.; Islam, M. S.; Kanoh, H.; Rozière, J. Lattice Dynamics and Vibrational Spectra of Lithium Manganese Oxides: A Computer Simulation and Spectroscopic Study. J. Phys. Chem. B 1999, 103, 5175-5180. (58) Suzuki, K.; Takami, S.; Kubo, M.; Miyamoto, A.; Kikuchi, M.; Yamazaki, N.; Mita, M. Structural Properties of LixMn2O4 as Investigated by Molecular Dynamics and Density Functional Theory. Jpn. J. Appl. Phys. 2000, 39, 4318-4322. (59) Sayle, T. X. T.; Catlow, C. R. A.; Maphanga, R.; Ngoepe, P. E.; Sayle, D. C. Generating MnO2 Nanoparticles Using Simulated Amorphization and Recrystallization. J. Am. Chem. Soc. 2005, 127, 12828-12837. (60) Sayle, T. X. T.; Ngoepe, P. E.; Sayle, D. C. Simulating Mechanical Deformation in Nanomaterials with Application for Energy Storage in Nanoporous Architectures. ACS Nano 2009, 3, 3308-3314. (61) Fleming, S. D.; Morton, J. R.; Rohl, A. L.; Ward, C. B. Interatomic Potentials for Simulating MnO2 Polymorphs. Mol. Sim. 2005, 31, 25-32. (62) Hahn, K.; Kärger, J. Molecular Dynamics Simulation of Single-File Systems. J. Phys. Chem. 1996, 100, 316-326. (63) Koudriachova, M. V.; Harrison, N. M.; de Leeuw, S. W. Effect of Diffusion on Lithium Intercalation in Titanium Dioxide. Phys. Rev. Lett. 2001, 86, 1275-1278. (64) Koudriachova, M. V.; Harrison, N. M.; de Leeuw, S. W. Density-Functional Simulations of Lithium Intercalation in Rutile. Phys. Rev. B 2002, 65, 235423. (65) Koudriachova, M. V.; Harrison, N. M.; de Leeuw, S. W. Diffusion of Li-Ions in Rutile. An Ab Initio Study. Solid State Ionics 2003, 157, 35-38. (66) Johnson, O. W. One-Dimensional Diffusion of Li in Rutile. Phys. Rev. 1964, 136, 284-290.


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(67) Tompsett, D. A.; Parker, S. C.; Bruce, P. G.; Islam, M. S. Nanostructuring of β-MnO2: The Important Role of Surface to Bulk Ion Migration. Chem. Mater. 2013, 25, 536-541.



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