Abstract
A molecular dynamics (MD) study of the lithium ultraphosphate glass series, xLi
Abstract
.5) was used to investigate the changes in the Li environment with increasing modiÐer concentration. The results from the MD simulations indicate that only gradual structural variations in the Li coordination environment occur as a function of modiÐer content. Changes in the type of oxygen coordinated to the Li are observed to correlate with the minimum in the glass transition temperature. Additionally, changes in the number of shared phosphorus vertices were seen with increasing modiÐer concentration, in support of recent models involving the role of the cation modiÐer in the extended range structure of phosphate glasses. Empirical calculations of the 6Li NMR chemical shifts directly from the MD simulation structures are also reported and compared to recent experimental solid-state NMR results.
Introduction
Phosphate glasses are technologically important materials, and are used in a variety of applications including glass-tometal seals,1 sealing frits,2 nuclear waste forms,3h5 lasers,6h11 and optical components.12 Understanding the relationship between the structure and the observed physical properties of phosphate glasses remains an area of active research. In many instances this structureÈproperty correlation has not been realized. For example, in binary alkali ultraphosphate glasses a minimum in the glass transition temperature is observed (T g ) near the 20 to 25 mole percent alkali modiÐer concentration. Fig. 1 While there have been numerous experimental investigations into this anomalous behavior in phosphate glasses, includ-T g ing NMR, Raman and XRD studies (see the recent reviews by Brow15 and Hoppe,16 plus references cited therein), an explanation of the structural changes responsible for the anomalous minimum has yet to be conÐrmed. T g Both theoretical ab initio and molecular dynamics (MD) investigations of phosphate glass structure have been reported. In contrast to the wealth of studies on silicate glass systems,17h25 the number of theoretical investigations in phosphates has been limited, with the anisotropy of the phosphate bonding being a major complicating factor. The phosphorus atom contains Ðve valence electrons, and yet maintains a tetrahedral hybridization such that the PÈO bonds are intrinsically anisotropic. For example, in the single component glass this hybridization is clearly evident with three of P 2 O 5 the PÈO bonds being bridging (with a PÈO bond distance of 1.581 along with a single terminal P2O bond (with a PÈO A ) distances of 1.432 The theoretical investigations to date A ). 26 show that for accurate simulations of phosphate materials the structural asymmetry of the P bonding must be reproduced, along with the ability to distinguish between P2O and PÈO bonds. Uchino and co-workers have reported a series of ab initio orbital calculations on the electronic, structure and vibrational properties of alkali metaphosphate glasses.27h29 MD simulations have been reported for the NaPO 3 ,30 and glasses,31,32 as well as
Introduction
) glass system were the Ðrst theoretical study to directly address the question of the anomalous behavior as T g a function of alkali modiÐer concentration in ultraphosphate glasses. In that initial report, the minimum in the was repli-T g cated, along with a preliminary analysis of the structural variations that were observed with changing concentraLi 2 O tion. It was demonstrated that the changes in the phosphate ring structure correlated nicely with the minimum, and that T g ring strain may play an important role in determining the glass properties.
Introduction
Hoppe and co-workers16,36,37 have recently presented evidence that the coordination and cross-linking of the alkali or alkaline earth cations used as backbone modiÐers also play a signiÐcant role in the observed physical properties of phosphate glasses. In general, the initial addition of the modifying cation results in the depolymerization of the phosphate crosslinked network with breaking of the PÈOÈP bond and conversion of Q3 to Q2 units (in the Qn terminology, n, denotes the number of PÈOÈP bonds in the phosphate tetrahedra). 38, 39 This depolymerization process results in the initial observed decrease of At low modiÐer concentrations the cation is T g . isolated and is coordinated by terminal oxygens (TO), both double bonded oxygens (DBO) and non-bridging oxygens (NBO). Above some critical concentration it is proposed that the TO are all coordinated to the metal cations, and that the metal coordination tetrahedra begin to link. 
Introduction
where l is the metal (Me) valency. For the Li modiÐer, with a CN D 4È5,40 the critical concentration is expected to occur for x D 0.2È0.25, corresponding nicely with the observed T g minimum.
Introduction
It is therefore important to investigate the role of the alkali environment on the phosphate glass structure as observed in MD simulations. In this manuscript we detail the variations in the Li cation environment observed in the MD simulation of the Li ultraphosphate series,
Introduction
Correlation of this environment with the T g minimum will be discussed. In addition, a comparison between recently reported 6Li NMR results40 and the Li chemical shifts empirically calculated from the simulated MD structures is presented.
Computational methods

MD simulations
The creation of a reliable force Ðeld to describe the interaction potentials involving phosphates is a very important step in the modeling of phosphate systems. The correct representation of the phosphate bonding anisotropy is important in the development of accurate models. For the MD simulations detailed here a forceÐeld model that explicitly incorporates both twoand three-body interactions was employed.34 This model allows the P bonding asymmetry to be realized while treating all oxygen atoms identically. Partial charges instead of formal charges were used to describe the Coulombic (electrostatic) component of the system. In addition to the standard Coulombic interactions, a Lennard-Jones (LJ) potential function was used to describe the short-range two-body interactions :
MD simulations
where r represents the interatomic distances, with and R 0 D 0 being adjustable parameters. For the LJ potential only cationÈanion and anionÈanion interactions make signiÐcant contributions to the overall energy, and were explicitly evaluated. The three-body energy terms were incorporated to describe both the OÈPÈO and PÈOÈP interactions, and are deÐned by
MD simulations
where h is either the OÈPÈO angle or the PÈOÈP angle, with adjustable parameters and The partial charges and the h 0 k 0 . forceÐeld potential parameters in eqn. (2) and (3) were determined by Ðtting the structure of crystalline lithium metaphosphate, and the two orthorhombic phases of LiPO 3 P 2 O 5 as previously described,34 and are summarized in Table 1 .
MD simulations
MD simulations for eight compositions in the solid solution series,
MD simulations
.4 and 0.5, were preformed using the CERIUS2 software and the OFF energy program. 41 The MD simulations of the glasses were performed using a constant volume and temperature (NVT) canonical ensemble. For any given composition in the series, a rectangular simulation box containing 300È400 atoms was generated.34 Details of the cell parameters are given in Table 1S of the supplementary material.¤ The compositions were equilibrated at 3000 K to achieve active melting for 50 ps, utilizing 1 fs time steps. Quenching of this equilibrated system from 3000 to 300 K was performed in 500 K intervals (the last step from 500 to 300 K), with each temperature step lasting 7 ps. This simulation sequence corresponds to a quench rate of D7.1 ] 1011 K s~1. At 300 K, an additional 50 ps simulation was used to equilibrate the system.
MD simulations
The glass-transition temperatures were determined by (T g ) monitoring the volumeÈtemperature relationship for a given composition in the MD simulations where the phosphate glass was treated as an NPT ensemble (constant pressure and temperature at each given temperature step) as previously described.34 This computational approach is analogous to the experimental determination of using dilatometry.
MD simulations
T g
Calculation of 6Li NMR chemical shifts
Koller et al. 42 has recently demonstrated that the observed NMR chemical shift of alkali species can be related to a chemical shift parameter A, where A is deÐned as the summation of the shift contributions for all the oxygens located within the Ðrst (and possibly the second) coordination sphere around the cation. For the present study of Li phosphate glasses, the chemical shift correlates directly to bond valence of the coordinating oxygen. An empirical bond valence (s ij ) between oxygen i and the cation j can be calculated from the cationÈoxygen bond length, using43h45 r ij ,
Calculation of 6Li NMR chemical shifts
where is the empirically derived oxygenÈcation bond length r 0 of unit valence, and B \ 0.37 is a constant previously a See text and eqn. (2) and (3) for deÐnitions of the symbols and parameters.
Calculation of 6Li NMR chemical shifts
deÐned.43h45 The total valence of the ith oxygen is simply (W i ) the summation over all oxygenÈcation bond valences for s ij each of the j cations bonded to the oxygens, including both lithium and phosphorous cations :
Calculation of 6Li NMR chemical shifts
The chemical shift parameter A will then be a summation of the oxygen shift contributions with a (where is the LiÈO 1/r j 3 r i bond distance) distance dependence :
Calculation of 6Li NMR chemical shifts
The dependence was empirically obtained previously by 1/r i 3 Koller et al. 42 from Ðtting of model compounds. The resulting A values obtained from the MD simulations are given in Table 2 . We recently reported an empirical linear relationship between the observed 6Li chemical shift (d) and the chemical shift parameter A for lithium phosphate systems,40
Calculation of 6Li NMR chemical shifts
allowing the 6Li chemical shift to be determined directly from the MD simulations.
Results
To address the performance of these MD forceÐeld potentials, the results of the simulations should be directly compared to the radial distribution functions, G(r), obtained experimentally from X-ray and neutron di †raction studies of these glass compositions. Unfortunately, to the best of our knowledge di †rac-tion studies for the lithium ultraphosphate glasses series has not been reported. The G(r) obtained from a neutron di †raction46 investigation of the lithium metaphosphate (x \ 0.5) glass and a X-ray study47 for a near metaphosphate glass composition (x \ 0.52) are available, and are compared to the MD simulation for the x \ 0.5 composition in Fig. 1S in the supplementary material.¤ The agreement between the MD simulation G(r) and the neutron di †raction results46 is very good, with the simulation slightly underestimating the PÈO distance and distribution. The experimental X-ray and neutron G(r) results are very di †erent (See Fig. 1S in the supplementary material¤), with the near metaphosphate glass composition showing a signiÐcantly broader distribution of radial distances. These variations may result from the di †erent experimental techniques used to obtain the G(r), as well as di †erences in the glass preparation and thermal history. The lack of experimental data for the entire lithium ultraphosphate composition series prevents a critical analysis of the MD potential performance, but the agreement at the metaphosphate composition is encouraging.
Li pair distribution functions
The variation of the Li environment for the phosphate glass series, (0. Tables 2 and 3 .
Li pair distribution functions
The LiÈO pair distribution functions (PDF) for di †erent concentrations are shown in Fig. 2S in the supplemenLi 2 O tary material.¤ The shape of the LiÈO PDF shows little variation with increasing modiÐer concentration. The Ðrst peak for LiÈO PDF ranges from 1.95 to 1.97 with a full width at A , half maximum (FWHM) ranging from 0.32 to 0. 35 The Ðrst A . peak is asymmetric, tailing towards longer coordination distances. The LiÈO PDF does not return to a null value after the Ðrst coordination sphere (see Fig. 2S ), demonstrating that there is overlap between the Ðrst and second coordination shells. A similar result has been reported for the NaÈO PDF Table 2 Average coordination numbers and chemical shift parameters derived form the pair distribution functions in the xLi 2 a CN \ Average coordination number. The cuto † distance (in used in the calculation of CN is shown in parentheses. b Chemical shift A ) parameter as deÐned in eqn. (6). c Chemical shift calculated using the empirical relationship in eqn. (7). Table 3 Structural distances derived from the pair distribution functions in the glass series in the MD simulation of glass.30 Because of this coorNaPO 3 dination sphere overlap, it is important to include the cut-o † distances used in the determination of structural parameters, including average bond lengths and CN. Variations in the cut-o † distance have a dramatic e †ect on the resultant values. Due to this asymmetry and coordination sphere overlap an average LiÈO distance using a 3.0 cuto † is also reported A ( Tables 2 and 3 . The average CN ranges between 4 and 5, and is consistent with experimental estimations of CN. 40, 47, 48 The LiÈP PDF as a function of concentration is Li 2 O shown in Fig. 3S of the supplementary material,¤ with the Ðrst coordination peak varying between 3.25 and 3. 27 and a A FWHM between 0.5 and 0.63
Li pair distribution functions
The second coordination A . peak is very broad and asymmetric, such that the peak position was not reliably determined. The shape of the LiÈP PDF is unchanged with increasing concentration. The average Li 2 O LiÈP bond length calculated using a 4.5 cuto † was found to A decrease gradually with increasing modiÐer concentration, and reÑects the change in the valency of the coordinated oxygen in the LiÈOÈP bond (see discussion below). The average LiÈP CN varied between 6.5 and 7.4 (see Table 2 ), but did not show any clear trends.
Li pair distribution functions
The LiÈLi PDF is shown in Fig. 4S of the supplementary material,¤ and reveals a very broad and asymmetric peak that varies between 2.45 and 2.93 having a FWHM peak width A , ranging between D0.5 to 1.5
Li pair distribution functions
The average LiÈLi distance A . using a 6 cuto † varied between 4.05 and 4.88 The A A . average LiÈLi CN steadily increases between 1.6 and 14.2 between x \ 0.1 and 0.5. For x [ 0.1 the LiÈLi PDF shows signiÐcant tailing for LiÈLi distances greater than 4 The A . x \ 0.1 PDF is shifted to slightly smaller LiÈLi distance, and does not appear to have as many long range LiÈLi distances present within the PDF. While these may suggest shorter LiÈLi interactions at very low concentrations, the very Li 2 O low average LiÈLi CN (Table 2) shows that initially the Li cations are relatively isolated. The broad nature of the LiÈLi PDF strongly suggests that the Li cations are randomly distributed throughout the glass network for the entire ultraphosphate concentration regime.
Coordination environment
To address the change in the oxygen type that is coordinated to the Li with variation of the concentration, selected Li 2 O PÈO PDF for x \ 0.1, 0.25 and 0.4 are shown in Fig. 2a , while Fig. 2b shows the average PÈO bond distance for oxygens that are within the Ðrst coordination sphere of the Li (cuto † distance 3.0
Coordination environment
The minimum PÈO distance for each coordi-A ). nating oxygen was used to determine the average shown in Fig. 2b , and only includes oxygens that are coordinating in the average. In Fig. 2a the di †erent PÈO bond distances are clearly evident in the simulated PÈO PDF. The TO PÈO bond distance (both NBO and DBO) is approximately 1.45 with A the BO PÈO distance being approximately 1.53 There is a A . very small increase in the TO and BO PÈO bond lengths with the addition of (1.47 and 1.57 respectively at x \ 0.5). with the BO PÈO distance being 1.6 A , Lengthening of the PÈO bond length with increasing modi-A . Ðer content has been observed experimentally in ultraphosphate glasses,26 and is of similar magnitude to that observed in the MD simulations (see Fig. 5 of ref. 26) . The relative amount of BO also decreases with increasing concentraLi 2 O tion as predicted due to depolymerization of the phosphate network.
Coordination environment
By determining the minimum PÈO bond distance for oxygens in the Li coordination sphere (Fig. 2b) , the type of coordinating oxygen can be addressed. In the MD simulations this average PÈO distance ranges from 1.455 to 1.480 A between x \ 0.1 and 0.5. This average is similar to the average TO PÈO bond distance observed for the entire glass series (see Fig. 2a ). This clearly demonstrates that BO oxygens do not play an important role in the coordination of the Li modiÐer, and that the Ðrst coordination sphere around Li is predominantly TO. Fig. 3 shows the variation of the average number of Li surrounding a phosphorus atom as a function of concentraLi 2 O tion. A cuto † distance of 4.5 was utilized, and only CN for A phosphorus that actually have Li within this coordination sphere were utilized in calculation of the average CN. At x \ 0.1 this PÈLi CN is D1.5 suggesting that on average the Li do not signiÐcantly share P vertices. This coordination increases steadily with higher concentration, with a disLi 2 O tinct change in the variation occurring near x \ 0.25.
6Li NMR chemical shifts
In order to compare the structural information about the Li environment derived from MD simulations to experimentally observed NMR results, the 6Li chemical shifts were calculated directly from the MD structures. The 6Li NMR chemical shift is expected to be sensitive to variations in the local Li coordination environment, and should reÑect any changes in the Li PDF. Using eqn. (4)È(7) the average 6Li chemical shift was calculated from the MD structures as a function of conLi 2 O centration and are given in Table 2 and Fig. 4 . The intercept of this line can be adjusted by variation in the cuto † distance used in the calculation of the A parameter. For example, increasing the cuto † distance to include values [3.5 will A increase A and similarly d through eqn. (6) and (7). It is possible to increase this cuto † distance to provide an improved Ðt between the experimental and the MD calculated chemical shifts. For the present discussion a 3.5 cuto † distance was A utilized, as this was the original cuto † distance used in the development of the empirical relationship in eqn. (7). 40 
Discussion
These MD simulations show that in general there are only small variations of the Li environment with increasing Li 2 O concentration. In particular, there are no major structural changes in the Li coordination environment that would correlate with the minimum observed in Fig. 1 . This is in con-T g trast to the observation of major changes in the phosphate ring structure as a function of modiÐer concentration previously reported for this,
Discussion
The MD simulations presented here do reveal subtle variations in the Li coordination environment that are consistent with the modiÐer restructuring model forwarded by Hoppe and co-workers. 16, 36, 37, 49 A schematic illustration of this model is shown in Fig. 5 , for low, medium and high modiÐer concentrations. As seen in Fig. 2b there is a step in the average minimum PÈO bond distance for concentrations x [ 0.25, signaling a change in the coordinating oxygen type. It is expected that the average TO PÈO bond would show a linear lengthening with increasing modiÐer content,26 such that a random distribution of DBO and NBO in the Li coordination sphere would result in a linear variation of the minimum PÈO bond distance. The step in the minimum PÈO bond distance of coordinating oxygens Ðts well with the pictorial representation of the Hoppe modiÐer restructuring model where the initial coordination involves both DBO and NBO in isolated polyhedra. At the critical concentration LiO n the number of TO available is equal to the CN eqn. [M TO , (1)], such that any additional increase in the concentraLi 2 O tion will require polyhedra to link together by corner LiO n and edges. This linking of polyhedra would be expected to manifest itself in the lengthening of the average coordinating PÈO bond length, as the electron density shifts along the PÈO bond to charge compensate the charge of the shared modiÐer sites. 26 The low number of Li coordinating to a given phosphorus atom (Fig. 3) , along with the low LiÈLi CN (Table 2) show that initially the Li cations are relatively isolated (CN D 1.5). Above the critical concentration of x D 0.25, the average coordination of Li around a phosphorus increases dramatically, consistent with edge and corner sharing. The increase in the average LiÈLi CN (Table 2) is consistent with the formation of OÈLiÈO cross-links between neighboring phosphorus tetrahedra, but an increase in the LiÈLi CN is also expected to result from increases in the Li number density.
Discussion
The MD simulations also provide insight into the surprising linear behavior of the 6Li MAS NMR chemical shifts. Table 3 we can see that there is a minimum in LiÈO CN at x D 0.2, which should result in a minimum in the magnitude of the shift parameter A and subsequently a minimum in the chemical shift [eqn. (7)]. Inspection of Table 2 shows that there is also a variation of the LiÈO bond distance with a minimum near x D 0.2, which maximizes the shift parameter A. The combination of these two non-linear e †ects tends to balance out to produce the observed linear behavior. This observation demonstrates the NMR analysis of alkali chemical shifts may not reveal subtle changes in the coordination environment, since the shift depends on the combination of several di †erent variables describing the coordination environment.
Discussion
The results of the MD simulations also agree with recent NMR 7LiÈ31P and 31PÈ7Li rotational-echo double resonance (REDOR) MAS experiments.50 REDOR experiments allow a measure of the dipolar coupling between di †erent nuclei, and thus a measure of spatial proximity. REDOR experiments on Li phosphate glasses have shown that the dipolar coupling between Li and P is rather independent of the concenLi 2 O tration, demonstrating that the P environment around a Li cation is unchanged. The MD simulations (Tables 2 and 3 ) predict a similar behavior since the average LiÈP distance is constant, and the average LiÈP CN shows only a small variation. The REDOR experiments also demonstrate that the Li environment around a P changes distinctly near x D 0.25. This experimental observation may correspond to the change in slope noted in the average PÈLi CN (Fig. 3 ) obtained for the MD simulations. A more future analysis of these REDOR results is warranted.
Discussion
The MD simulations demonstrate that while there are changes in the Li environment that do correlate with the observed minimum, the e †ects or changes are more gradual T g that one would predict from the simpliÐed cation restructuring model deÐned by eqn. (1) and schematically detailed in Fig. 5 . A gradual variation of the coordination environments with changes in modiÐer concentration would appear to be more realistic than a sudden and distinct environmental change. Whether the anomalous in alkali T g ultraphosphate glasses is produced solely by these Li environment changes, results from the formation of phosphate ring structures34 (still not experimentally veriÐed), or is a combination of these two e †ects remains to be determined. Further comparison between MD simulations and experimental results in phosphate glass systems are presently being pursued.

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