www.elsevier.com/locate/jnoncrysol

Journal of Non-Crystalline Solids 349 (2004) 16–21

Molecular dynamics simulation of vitreous state by ionic pairpotentials

Hiroyuki Inoue *, Futoshi Utsuno, Itaru Yasui

Institute of Industrial Science, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan

Available online 28 October 2004

Abstract

Structural models of a single component vitreous state were systematically developed using molecular dynamics simulations

incorporating ionic pair potentials. The M–X (M is a cation and X is an anion) pair bond distance was constrained between

1.4A and 2.7A. The charge of the M ion ranged from 2 to 5. The structural models can be classified into two groups based on

the atomic arrangement. In the structural models the coordination number around the M ion is nearly constant up to some max-

imumM–X distance whose value depends upon the coordination numbers. Conversely, the coordination number around the M ions

increases with an increase of the M–X distance when the M–X distance exceeds this maximum value. It was found that the atomic

arrangement in which the coordination number is almost constant, is more stable than the other types of atomic arrangements.

These structural models correspond to several glassformers which can form the vitreous state with a single component.

� 2004 Elsevier B.V. All rights reserved.

PACS: 61.43.Bn

1. Introduction

Woodcock, Cheeseman and Angell have reported

structural models for ZnCl2 and SiO2 glasses on the

basis of molecular dynamics (MD) simulations [1]. They

obtained a tetrahedral network in a model for vitreous

silica by use of the ionic pair potentials of Born–Mayertype. Since then, other numerical studies of simulation

for vitreous states have been reported as well. Inter-

atomic potentials for MD simulations have been devel-

oped from empirical classical potentials to theoretical

ones. Furthermore, a first-principle MD simulation of

amorphous SiO2 has been reported [2]. During the past

quarter-century, MD simulations have become an

important tool to study not only glass structure but alsothe properties of glass.

0022-3093/$ - see front matter � 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.jnoncrysol.2004.08.256

* Corresponding author. Tel.: +81 3 5452 6315; fax: +81 3 5452

6316.

E-mail address: inoue@iis.u-tokyo.ac.jp (H. Inoue).

The Zachariasen�s criteria [3] and random network

model are of great importance for glass-formation.

However, some fluoride glasses with atomic arrange-

ment that are an exception to the Zachariasen�s criteriahave been reported [4]. It was found that the network

former of these glasses had not satisfied the second

and third rules of Zachariasen�s [5,6]. Recently theseglasses have been developed for practical application

[7]. Therefore, it is necessary to extend the rules for

glass-formation. In this paper, the structural models

for the single component vitreous state were systemati-

cally prepared using molecular dynamics simulations

with ionic pair potentials. The structural rules of the vit-

reous state were reexamined.

2. Experimental procedure

Structural models for the single component of MOn

(n = 3/2, 2 and 5/2) and MFn (n = 2, 3 and 4) (M is a cat-

ion) were prepared by MD simulations. Two hundred

-20

0

20

60

40

Coo

rdin

atio

n N

umbe

r

Mol

ar V

olum

e (c

m3 /

mol

)

2

4

6

8

10

12

M-O Distance (Å)1.5 2.0 2.5

Fig. 1. Calculated molar volumes (�) and coordination numbers (n)

for the MO2 system as a function of the M–O distance. The broken

lines are drawn as guides for the eyes.

H. Inoue et al. / Journal of Non-Crystalline Solids 349 (2004) 16–21 17

cations and the corresponding number of anions were

placed randomly in a cubic cell with periodic boundary

conditions. The simulations were carried out at a con-

stant volume. The potential of the Born–Mayer type

was used with formal ionic charges.

/ijðrÞ ¼e2

4pe0

ZiZj

rþ Bij exp � r

qij

!; ð1Þ

where Z is the electronic charge, B is a parameter of the

ionic radius and q is a softness parameter for an ionicpair. The Coulomb force was evaluated by the Ewald

summation. MD simulation was performed for 10 differ-

ent sets of random initial coordinates. After 10000 time

steps of 1 fs at 3000K, the temperature of the simulation

was lowered from 3000K to 293K for 10000 time steps.

After 5000 time steps at 293K, the coordinates of the

last step were used for further calculation. The values

of the potential parameters, qMO and qOO, of 0.17Aand 0.31A were used for the M–O and O–O pairs,

respectively. The value of BO-O was fixed at

5.69 · 10�16J. For the fluoride, the value of 0.3 A was

used. The value of BFF was fixed at 8.43 · 10�17J. The

values of the potential parameters used were determined

from the radial distribution curves of ZBLAN glass [8].

The distance of the M–X (X is an anion) pair was con-

trolled from 1.4A to 2.7A by the value of BMX The pres-sure at individual potential parameters and cell sizes

at 293K was obtained. The molar volume at ordinary

temperature and pressure was estimated for each M–X

distance. The bulk modulus was estimated from the rela-

tion between volume and pressure of the cell.

In order to evaluate the stability of the structural

models, an electric field was applied to the structural

models at 293K for 4000 time steps. It was assumed thatthe magnitude of force by the applied electric field was

proportional to the charges of ions. The magnitude of

the applied electric field was given from 0.1 to 10GV/

m. After 4000 time steps, the average deviation from

the coordinates in the structural models at 293K and

the number of recombinations of the M–X bond were

obtained.

3. Results

3.1. Structural models for the MO2 system

The calculated molar volume for the MO2 system as a

function of the M–O distance is shown in Fig. 1. The

molar volume increased monotonously with an increasein the M–O distance. The maximum molar volume at an

M–O distance of 1.85A was 43.5cm3/mol. Then, the

molar volume decreased to 24.1cm3/mol at an M–O dis-

tance of 2.04A. The molar volume increased again with

an increase in the M–O distance. There was a large dis-

continuity of the molar volume at the M–O distance be-

tween 1.85A and 2.04A. It is possible to classify two

kinds of structural models at this discontinuity. The

oxygen coordination number around the M4+ ions ob-

tained from the area of the first peak of the M–O pair

is shown in Fig. 1. The coordination number in the

model, in which the M–O distance was shorter than thediscontinuity of the molar volume, was almost 4. On

the other hand, the oxygen coordination number around

the M4+ ions in the model, in which the M–O distance

was longer than the discontinuity, increased with an

increase in the M–O distances. The structural model in

which the M–O distance is shorter than the discontinu-

ous point is called a CCN (constant coordination num-

ber) model, and the structural model in which M–Odistance longer than the discontinuous point is called

the VCN (variable coordination number). Fig. 2 shows

the bulk modulus obtained from the volume and pres-

sure of unit cells. The bulk modulus for the CCN model

decreased from 90GPa to 9GPa with an increase in the

M–O distance. On the other hand, that for the VCN

model was about 110GPa and increased slightly with

an increase in the M–O distance. The cation coordina-tion number around the oxygen ion is listed in Table 1.

In the CCNmodel, which consists of tetrahedra, the oxy-

gen ion is coordinated by two cations. The coordination

number of the oxygen ion in the VCN model rapidly

increased to 2.8 and increased with an increase in the

M–O distance.

An electric field was applied to the structural models,

and the structural change was investigated. Fig. 3 showsthe average deviation of ions from the coordinates in the

models after 4000 steps under the electric field as a func-

tion of M–O distance. As can be seen from the figure,

the average deviation of the VCN model was larger than

1.5 2.0 2.5

50

100

M-O Distance (Å)

Bul

k M

odul

us (

Gpa

)

Fig. 2. Calculated bulk moduli of the structural models for the MO2

system as a function of the M–O distance. The broken lines are drawn

as guides for the eyes.

Table 1

The distribution of the cation coordination number around O ions in

the structural models for the MO2 system

RM–O (A) Relative percent (%) Average

1 2 3 4 5

1.47 5 95 1.95

1.53 3 96 1.97

1.61 2 98 1.98

1.68 1 99 2.00

1.76 98 2 2.01

1.85 89 11 2.11

2.04 18 80 2 2.85

2.13 6 85 9 3.04

2.22 2 77 21 3.19

2.32 57 43 3.43

2.42 40 59 1 3.60

2.51 31 68 1 3.70

2.62 22 76 2 3.80

1.5 2.0 2.50

1

2

M-O Distance (Å)

Ave

rage

Dev

iatio

n (Å

)

Fig. 3. Average deviation from the coordinates of ions in the

structural model after 4000 time steps. The broken lines are drawn

as guides for the eyes. The magnitude of the applied electric field were

(�) 1, (n) 3, (h) 4 and (,) 5GV/m.

0

20

40

60

Coo

rdin

atio

n N

umbe

r

Mol

ar V

olum

e (c

m3

mol

)

2

4

6

8

10

12

M-O Distance (Å)

1.5 2.0 2.5

0

20

40

60

Mol

ar V

olum

e (c

m3 /

/m

ol)

Coo

rdin

atio

n N

umbe

r

2

4

6

8

10

(a)

(b)

Fig. 4. Calculated molar volumes and coordination numbers for (a)

MO3/2 and (b) MO5/2 systems as a function of the M–O distance. The

broken lines are drawn as guides for the eyes.

18 H. Inoue et al. / Journal of Non-Crystalline Solids 349 (2004) 16–21

that of the CCN model. The recombination of the M–O

bond shows a similar tendency of deviation. Though the

CCN model shows the large volume change for the

stress, it is difficult for the M–O bond to be broken.

The VCN model shows reverse characteristics compared

with the CCN model.

3.2. Structural models for MO3/2 and MO5/2 systems

The calculated molar volumes and coordination

number around cations for the MO3/2 and MO5/2 sys-

tems as a function of the M–O distance are shown in

Fig. 4. The molar volumes of the MO3/2 system show

the large discontinuous change between 1.55A and

1.74A of the M–O distances. The oxygen coordination

0

20

40

60

4

6

8

10

12

M-O Distance (Å)

Mol

ar V

olum

e (c

m3 /

mol

)

Coo

rdin

atio

n N

umbe

r

0

20

40

60

4

6

8

10

12

1.5 2.0 2.5

0

20

40

60

2

4

6

8

10

12

Mol

ar V

olum

e (c

m3 /

mol

)

Coo

rdin

atio

n N

umbe

r

Mol

ar V

olum

e (c

m3 /

mol

)

Coo

rdin

atio

n N

umbe

r

(a)

(b)

(c)

Fig. 5. Calculated molar volumes and coordination numbers for (a)

MF2, (b) MF3 and (c) MF4 systems as a function of the M–F distance.

The broken lines are drawn as guides for the eyes.

H. Inoue et al. / Journal of Non-Crystalline Solids 349 (2004) 16–21 19

numbers around the M3+ ion are almost constant at 3

within the range of the M–O distance from 1.34A to

1.55A. Therefore, the structural models within the range

to 1.55A correspond to the CCN model for the MO2

system, and the structural models, in which the M–O

distances were over 1.74A, are attributed to the VCNmodel. In the MO5/2 system, the coordination number

around the M5+ ions is 4 and is within 1.54A of the

M–O distance, and the coordination number is 5 to

1.89A of the M–O distance. The discontinuity of the

molar volume was not clear at these M–O distances.

The increase of the molar volume over 2.2 A of the

M–O distance corresponds to the increase of the molar

volume of the VCN model of the MO2 and MO3/2 sys-tems. Thus, the structural models, in which the M–O

distances are less than 1.89A, seem to be attributed

to the CCN model. It was found that the magnitude

of the discontinuity of the molar volume between

CCN and VCN models decreased, and the M–O dis-

tance for the discontinuity of the molar volume length-

ened with an increase in the valence charge of the M

ions.In the CCN model, the atomic arrangement is con-

structed by the coordinated polyhedra. As in the VCN

model, the amount of oxygen per unit volume increased

slightly with an increase of the valence of the M ion,

thus the atomic arrangement of the VCN models seems

to be constructed by the random packing of the oxygen

ions.

3.3. Structural models for MF2, MF3 and MF4 systems

The calculated molar volumes for the MF2, MF3 and

MF4 systems as a function of M–F distance are shown

in Fig. 5. The molar volumes of the MF2, MF3 and

MF4 systems showed the discontinuous changes around

the M–F distances of 1.81A, 2.21A and 2.51A,

respectively.The coordination numbers of the M ions for the

MF2, MF3 and MF4 systems were 4, 6 and 8, respec-

tively, in the range where the coordination number

was constant. As can be seen from the figure, the curves

of the molar volume and the coordination number of

MF2 system are very close to those of the MO2 system.

It was shown that these curves are determined by the

ratio of the charges of cation and anion. The magnitudeof the bulk modulus of the MF2 system was about 1/4

that of the MO2 system. The deviation of the MF2 sys-

tem under an electric field became equivalent to that of

the MO2 system under an electric field of about 1/4.

The changes of the bulk modulus and structural stabil-

ity under the electric field were similar to the case of the

oxides. The changes of the bulk modulus and the effect

of an applied electric field were similar to those of theoxides, though the absolute values were greatly

different.

4. Discussion

In all calculated oxide systems, it was found that

there was a range where the coordination number

around the cation was determined independently on

the M–O distance. The coordination numbers for the

MO3/2, MO2 and MO5/2 systems were 3, 4 and 5, respec-

tively. The coordination polyhedra were stable, and theM–O bond did not recombine easily under the applied

electric field. At a longer M–O distance, the coordina-

tion number increased with an increase in the M–O dis-

tance. In spite of the high bulk modulus and high

20 H. Inoue et al. / Journal of Non-Crystalline Solids 349 (2004) 16–21

packing density, the M–O bond recombined under the

applied electric field. As the CCN model of the vitreous

state consists of the coordinated polyhedra, the slope of

the molar volume as a function of the M–O distance is

dependent on the coordination number. The VCNmodel

depends on the packing of the oxygen ion. Therefore, themolar volume was determined by the oxygen number of

the composition, and the slope of the molar volume is

almost the same in the oxides.

In the MO2 composition, the 4-fold coordinated pol-

yhedron is stable over the range of the M–O distance

from 1.47A to 1.85A. In the MO3/2 composition, the

3-fold coordinated polyhedron is stable over the range

of the M–O distance from 1.34A to 1.55A. SiO2 (Si–O = 1.61A), GeO2 (Ge–O = 1.74A) and B2O3 (B–

O = 1.37A), which can form vitreous states, are located

in these ranges. In the MO5/2 composition, the 4-fold

coordinated polyhedron is stable to M–O = 1.58A. This

4-fold coordination corresponds to P2O5. The limitation

of the M–O bond length for the coordinated polyhe-

dron, 1.55A for MO3 and 1.85A for MO4, can be linked

to the ratio of the radii of M and O ions. When the ra-dius of the O ion, RO, is assumed to be 1.3 A, RM/

RO = 0.19 for MO3 and RM/RO = 0.42 for MO4, where

RM is the radius of the M ion. These values are close

to the values of RM/RO for geometrically stable coordi-

nation polyhedra, such as RM/RO = 0.225 for MO3 and

RM/RO = 0.414 for MO4. The polyhedra with a larger

coordination number are stabilized, when the M–O dis-

tances exceed these values. The structural change seemsto occur when the coordination number increases.

It is possible to apply these relations to the model of

the fluoride. The limitations of 4-, 6- and 8-fold coordi-

nated polyhedra were at 1.81A, 2.21A and 2.51A of the

M–F distance, respectively. When the radius of a fluo-

rine ion, RF is assumed to be 1.28A, RM/RF = 0.41 for

MF4, RM/RF = 0.73 for MF6, and RM/RF = 0.96 for

MF8, respectively. It was found that the coordinationnumber of the cation was determined by the radius ratio

of cation and anion in the structural model obtained by

use of the ionic pair potential. This relation is Pauling�sfirst rule [9] itself. On the other hand, the coordination

number of the anion is estimated on the basis of Paul-

ing�s second rule [9]. As the strength of the electrostatic

valence bond of the 4-fold coordinated M4+ ion is 1.0 in

the MO2 system, the O2� ion is coordinated by two M4+

ions. In the cases of 3-fold coordinated M3+ and 5-fold

coordinated M5+ ions, the O2� ion is coordinated by

two M ions. The coordination numbers of the M ion

in the CCN model in the oxide systems corresponded

to the O2� ion coordinated by two M ions. In the fluo-

ride systems the charge of the F ion is �1. The required

strength of the electrostatic valence of the M–F bond is

1/2 for a 2-fold coordinated F� ion. In the MF2, MF3

and MF4 systems, the coordination numbers of the M

ions are estimated to be 4, 6 and 8, respectively. These

coordination numbers also agreed well with those in

the CCN model for the fluoride systems. It was found

that the atomic arrangement in the CCN model obeyed

Zachariasen�s first rule �(1) an oxygen atom is linked to

not more than two atoms A (A is a cation)�.The number of 3-fold coordinated anions increased

rapidly between the CCN and VCN models. The num-

ber of edge-sharing linkage also increased with an in-

crease in the number of 3-fold coordinated anions. In

gathering 3 polyhedra, the 3-fold coordinated anion is

formed. Therefore, the atomic arrangement is densified

with an increase in the number of 3-fold coordinated

anions. On account of the steric interactions among

the polyhedra and/or decrease of the repulsion betweenthe anions, the proportion of the edge-sharing linkage

increased with an increase of 3-fold coordinated anions.

According to Pauling�s third rule [9], the presence of

shared edges in a coordinated structure decreases its sta-

bility. The VCN models with the edge-sharing linkages

were unstable. It was found that the structural model

for the vitreous state by use of ionic pair potentials

obeyed Pauling�s third rule.Phillips [10] has proposed the formation rule of the vit-

reous state with the covalent bond, and Gupta and Coo-

per [11] have built topologically disordered infinite

networks of regular, congruent and rigid polyhedra and

they have also proposed the glass formation rule. In the

structural model by means of molecular dynamic simula-

tions, both the distributions of the coordination number

of polyhedra and the distorted polyhedra are allowed. Byconsidering Zachariasen�s first rule, it will be possible todiscuss the structure rule of the ionic glass on the basis

of the structural models. Though it is necessary to evalu-

ate the stability of the structural models quantitatively,

we propose a way to understand the structure of the vit-

reous state of a single component of oxide and fluoride.

5. Conclusion

The structural models for single component oxide

and fluoride glasses were obtained by means of molecu-

lar dynamics simulations with ionic pair potentials. The

characteristic changes of the molar volume and coordi-

nation number were obtained as a function of the

M–O or M–F bond distance. The curve of the molarvolume as a function of bond distance has at least one

maximum. The coordination number of the cations in

the models is almost constant until the distance increases

beyond this maximum. The coordination number of the

cation corresponds to the anion coordinated by 2 cati-

ons. In these structural models, the structural change

under the applied electric field is suppressed. The chem-

ical compounds, which can be vitrified in a single com-ponent, such as B2O3, SiO2, GeO2, P2O5 and BeF2, are

located in the range. Therefore, the structural model in

H. Inoue et al. / Journal of Non-Crystalline Solids 349 (2004) 16–21 21

the range is suitable for an ionic glass structure. Paul-

ing�s rules for the ionic crystals and Zachariasen�s first

rule are important for our structural models.

Acknowledgment

This study was financially supported by the Semi-

conductor Technology Academic Research Center

(STARC).

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