THE JOURNAL OF CHEMICAL PHYSICS 129, 084504 共2008兲
Short- and medium-range structure of multicomponent bioactive glasses
and melts: An assessment of the performances of shell-model
and rigid-ion potentials
Antonio Tiloccaa兲
Department of Chemistry and Materials Simulation Laboratory, University College London,
London WC 1H 0AJ, United Kingdom
共Received 19 June 2008; accepted 25 July 2008; published online 25 August 2008兲
Classical and ab initio molecular dynamics 共MD兲 simulations have been carried out to investigate
the effect of a different treatment of interatomic forces in modeling the structural properties of
multicomponent glasses and melts. The simulated system is a soda-lime phosphosilicate
composition with bioactive properties. Because the bioactivity of these materials depends on their
medium-range structural features, such as the network connectivity and the Qn distribution 共where
Qn is a tetrahedral species bonded to n bridging oxygens兲 of silicon and phosphorus network
formers, it is essential to assess whether, and up to what extent, classical potentials can reproduce
these properties. The results indicate that the inclusion of the oxide ion polarization through a
shell-model 共SM兲 approach provides a more accurate representation of the medium-range structure
compared to rigid-ion 共RI兲 potentials. Insight into the causes of these improvements has been
obtained by comparing the melt-and-quench transformation of a small sample of the same system,
modeled using Car–Parrinello MD 共CPMD兲, to the classical MD runs with SM and RI potentials.
Both classical potentials show some limitations in reproducing the highly distorted structure of the
melt denoted by the CPMD runs; however, the inclusion of polarization in the SM potential results
in a better and qualitatively correct dynamical balance between the interconversion of Qn species
during the cooling of the melt. This effect seems to reflect the slower decay of the fraction of
structural defects during the cooling with the SM potential. Because these transient defects have a
central role in mediating the Qn transformations, as previously proposed and confirmed by the
current simulations, their presence in the melt is essential to produce an accurate final distribution
of Qn species in the glass. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2972146兴
I. INTRODUCTION
The special bioactive properties of some surface-active
phosphosilicate glasses depend on the presence of large
amounts of sodium and calcium modifier cations, whose
charge balance is achieved by breaking Si–O–Si bridges and
creating a corresponding high concentration of nonbridging
oxygen 共NBO兲 atoms, and thus a significant fragmentation of
the silicate network.1–3 Indeed, experimental and simulation
data show that the bulk structure of common, melt-derived
bioactive compositions such as 45S5® 共Ref. 2兲 is dominated
by interconnected chains of Q2 silicate tetrahedra 关where the
Qn notation denotes a tetrahedron with n bridging oxygens
共BOs兲 and 4 − n NBOs兴 and predominantly isolated phosphate groups.4–6 The partial dissolution of this highly fragmented structure in a physiological 共aqueous兲 environment is
thus relatively fast and leads to a series of reaction steps at
the glass surface, which ends with the formation of a strong
interfacial bond between the glass and living tissues, such as
bone or muscle.3 This unique tissue-bonding ability is exploited in many clinical applications of implants made or
derived from Na2O : CaO: SiO2 : P2O5 bioactive glasses,
which therefore represent a key reference in the field of
a兲
Electronic mail: [email protected].
0021-9606/2008/129共8兲/084504/9/$23.00
biomaterials.1,3 Even though significant scientific and technological advances have been achieved since their discovery
in the early 1970s, investigations on the atomistic structure
of bioactive glasses have only started to appear very recently,
despite the obvious importance of these highly detailed data
to improve our understanding of their properties. The disordered and multicomponent nature of bioglass systems represents a challenge to experimental and computational approaches to unveil their structure. For instance, while binary
glasses have often been modeled using classical molecular
dynamics 共MD兲 simulations,9–13 not as many studies have
focused on amorphous oxides containing three or four base
components, and in particular bioactive glasses have been
seldom modeled by MD.5–7 The complex nature of bioactive
glasses complicates the use of standard MD force fields commonly used for ionic oxides. Si–O and P–O bonds show a
mixture of covalent and ionic character, whose specific balance depends on the local coordination environment, such as
the concentration and geometry of Na/Ca cations surrounding the individual SiO4 or PO4 tetrahedra or the local intertetrahedral connectivity. Therefore, in principle, the interaction terms of an accurate force field to model
multicomponent amorphous silicates should take into account the specific environment surrounding each ion and especially the highly polarizable oxide ions.14,15 An implicit,
129, 084504-1
© 2008 American Institute of Physics
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084504-2
J. Chem. Phys. 129, 084504 共2008兲
Antonio Tilocca
mean-field inclusion of polarization as in rigid-ion 共RI兲 models using fixed partial charges may not fully reproduce the
diverse, often distorted arrangements of silicate and phosphate bonding environments in disordered biomaterials, neither can it account for the dynamical fluctuations in the
charge distributions resulting from even small rearrangements in these coordination environments. On the other
hand, the explicit inclusion of electronic polarization enables
the model to respond to local electric field fluctuations
brought about by the ionic motion,14,15 as shown by the success of shell-model 共SM兲 approaches16 to model a large variety of condensed-phase systems, ranging from ionic solids
and liquids and their interfaces, to aqueous solutions,17–21
and should then yield a more flexible and accurate representation of the structure of bioactive glasses. Some indications
in this sense came from our previous SM investigations of
binary and ternary silicate glasses, for which the explicit inclusion of the oxide ion polarizability improved the intertetrahedral structure, compared to an RI potential.22 Following
these indications, we recently developed and applied an SM
potential to model bioactive glasses.6–8 Even though these
studies confirm the general suitability of a SM approach to
model bioactive compositions, several important issues remain open. The dissolution and bioactivity of these glasses
are closely linked to their structure in the medium range; for
instance, the network connectivity reflects the structural arrangement beyond the short range and shows interesting correlations with the glass bioactivity.23,24 As ab initio models
of medium-range structural properties would require too
large computational resources, it is essential to thoroughly
assess the performance of empirical potentials from this perspective. To this purpose, in this work, we compare the structure of large samples of 45S5® bioactive glass obtained using classical MD with SM and RI potentials, and highlight
the advantages of an SM approach in reproducing the key
structural features of the glass, such as the Qn distributions of
Si and P network formers. The Qn distribution is a convenient measure of the ability of a potential to describe the
medium-range structure of the 45S5 Bioglass®, both because
it reflects the glass network connectivity8 and because of the
availability of NMR and Raman spectroscopy data, which
can be used as direct reference.4,25,26
Thereafter, we examine the possible effects that determine the better performance of the SM approach by focusing
on the glass formation process from the melt. Because of the
lack of experimental data on these effects, the corresponding
reference data are obtained from Car–Parrinello 共CP兲
ab initio MD simulations27,28 of a smaller sample, whose
structural evolution during the melt-and-quench process is
compared to the results of classical MD simulations using
the SM and RI potentials. This detailed comparison on two
different scales allows us to highlight specific features in the
short and medium ranges whose accurate description needs
an explicit treatment of polarization. From a different perspective, the present calculations provide new insight into
whether and up to what extent polarization effects influence
the structural properties of complex amorphous systems such
as multicomponent glasses.
The twofold goal of this work can be reached by com-
bining two different levels of analysis. First, large-scale
models, obtained using classical MD, are used in conjunction
with experimental data to show the superiority of an SM
approach to reproduce the medium-range structure of bioactive glasses. Then, smaller models are created with the specific purpose to analyze the dynamics of the melt in greater
detail. By comparing the small classical models to the corresponding ab initio reference, one can investigate local effects
共short-range structure and exchange dynamics in the Si and P
coordination shell兲, which are at least partly responsible of
the different performances of SM and RI potentials on a
larger scale. While a representative Qn distribution can only
be achieved using large classical models 共provided that the
interatomic potential is accurate兲, the small models are employed here to show that the inclusion of polarization effects
is necessary to reach an accurate dynamic balance between
the transformations occurring in the melt. Only when this
favorable balance 共obtained with SM potentials兲 is coupled
with a sufficiently large system size and a lower quenching
rate will the result be an accurate Qn distribution for Si and P,
which in turn is a prerequisite for large-scale simulations of
the bioactivity.
II. COMPUTATIONAL METHODS
The composition of the system was 46.2SiO2 24.4Na2O
26.9CaO 2.6P2O5 mol %, roughly corresponding to the standard 45S5 Bioglass®.3 A large sample, described in detail
below, was modeled using only the empirical RI and SM
potentials, whereas a smaller sample of the same composition, also described below, was also simulated using the
ab initio CP approach, in addition to the empirical potentials.
The experimental density of 45S5 glass29 was used for both
small and large samples.
A. Large sample
A large glass sample, including 10 017 atoms in a
51.14 Å side cubic supercell, was obtained with both RI and
SM empirical potentials and used to investigate the mediumrange structure of the glass and highlight the different performances of the classical potentials in relation to the available experimental data. The melt-and-quench procedure was
similar to the one employed in our recent classical MD studies of different bioactive glasses.6,8 A random initial configuration 共the same for the RI and SM samples兲 was heated to
3200 K for 100 ps and then cooled to 300 K at a nominal
cooling rate of 10 K/ps, before a final room-temperature production run of 200 ps, from which the Qn distributions of Si
and P were calculated.30
B. Small sample
A smaller system containing 116 atoms in total, in a
cubic periodic supercell of 11.63 Å side, was modeled using
MD with the empirical SM and RI potentials, and compared
to recent ab initio data obtained with the CP method.31 Because the purpose of these smaller samples was to closely
compare the local environments and short-range structure in
the melt and the glass to the reference represented by the
ab initio data, it was important to remove any possible bias
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J. Chem. Phys. 129, 084504 共2008兲
Structure of multicomponent bioactive glasses and melts
C. Car–Parrinello simulations
The details of the CPMD simulations can be found in
Ref. 31. In summary, the CP code of the QUANTUM-ESPRESSO
package32 was employed, with the Perdew-Burke-Ernzerhof
共PBE兲 exchange-correlation functional33 to treat the electronic structure within density functional theory and ultrasoft
pseudopotentials,34 explicitly including semicore shells for
Na and Ca to represent core-valence electron interactions.
The MD time step ␦t and fictitious electronic mass ␮ were 7
and 700 a.u., respectively.
D. SM potential
Classical MD simulations with the SM were carried out
within the adiabatic SM scheme implemented in the DL_POLY
code.35,36 The SM interatomic potential used here, based on a
potential originally developed for quartz,37 was recently
adapted by us to model silicate glasses incorporating Na, Ca,
and PO4 ions.6,22 As in standard charge-on-spring models,14
the polarizable oxide ions are represented by core-shell units
of opposite charge, connected by harmonic springs. Formal,
full ionic charges and short-range Buckingham potentials are
used to describe the interaction of the oxygen shells with
each other and with Si, Na, Ca, and P cations, and three-body
screened harmonic potentials are used to control the intratetrahedral O–Si–O and O–P–O angles. In the adiabatic SM,36
Si
50
SM
RI
40
30
20
10
n
related to the computational setup. Therefore, exactly the
same melt/quench procedure and control parameters used in
the CPMD simulations were used to generate the small RI
and SM samples. In this way any deviation of the small SM
and RI samples from the reference CP reveals an intrinsic
lack of accuracy of the specific treatment of interatomic
forces, whereas the effect of the small size and fast quench
rate, being the same in all the samples, will be cancelled out.
As done for the large sample, the initial configuration
was created by randomly placing the appropriate number of
atoms in the simulation box using shortest distance cutoffs to
avoid significant overlaps. In each case, following the protocol used to obtain the CPMD reference sample, the same
initial random configuration was heated up and thermalized
to around 2300 K, and a microcanonical trajectory of
⬃14 ps was then carried out, where the temperature T oscillated around 2300 K. The melt was then rapidly cooled down
to room temperature through nine subsequent NVT runs of
10 ps each, with the target temperature decreased in 200 K
steps. This corresponds to a nominal cooling rate around
20 K ps−1. Even though slower cooling rates can be easily
afforded in glass simulations with empirical potentials, as
explained before, this choice is dictated by the need to
closely match the procedure previously used to obtain the
CPMD reference.31 After the cooling phase, the system was
further equilibrated at 300 K for 10 ps, before a final microcanonical production run of 12 ps. An additional MD trajectory at T = 3000 K was also started from the 2300 K liquid,
and continued for 14 ps, the last 10 ps of which were used in
the analysis, after checking the thermal equilibration of
sample using the ionic mean square displacement, as discussed in Ref. 31.
%Q
084504-3
0
0
80
1
2
3
P
4
5
SM
RI
60
40
20
0
0
1
2
n
3
4
5
FIG. 1. 共Color online兲 Qn distributions of Si 共top panel兲 and P 共bottom
panel兲 in the large glass samples obtained using the SM 共square symbols兲
and RI 共circle symbols兲 interatomic potentials. Error bars were estimated
from the standard deviation of the two different 45S5 bioglass samples
modeled in Ref. 6.
the shells are given a small fraction of the core mass 共0.2 a.u.
for oxide ions in this work兲. Provided that the MD time step
is chosen small enough to follow the high vibrational frequency of the core-shell units, the integration of conventional equations of motion should allow the shells to follow
the ionic motion adiabatically, i.e., the total force acting on
shells will constantly remain close to zero in the MD trajectory. In order to enforce this condition over relatively long
time scales, it is necessary to add a damping term to the
core-shell harmonic forces.22,36 The SM potential parameters
are listed in Refs. 6 and 7.
E. RI potential
As a standard RI potential, we employed a revised version of the Teter forcefield,38,39 with partial ionic charges and
short-range Buckingham terms for the pair interactions between oxygen and Si, P, Na, Ca, and O ions. Previous
simulations38,39 showed that the quality of its reproduction of
the structure of pure silica and sodium silicate glasses is
close or slightly improved with respect to the common van
Best-Kramer-van Santen 共BKS兲 potential.40
III. RESULTS AND DISCUSSION
A. Qn structure
The distributions of Qn silicate and phosphate species in
the large glass samples obtained using classical MD are
shown in Fig. 1. For Si, both potentials correctly lead to a
predominantly Q2 structure, in agreement with experimental
data,4,25 together with significant amounts of both Q1 and Q3
species. Even though a binary model with only Q2 and Q3
species is sometimes assumed,4 a three-component structure
with Q1, Q2, and Q3 silicates turned out to provide a better fit
of very recent NMR data on the 45S5 Bioglass®,25 and the
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J. Chem. Phys. 129, 084504 共2008兲
Antonio Tilocca
presence of more than two Qn共Si兲 species in this glass was
also inferred from Raman spectra.26 Previous NMR experiments also highlighted the coexistence of several different
Qn species in depolymerized alkali silicate glasses.41–44 As
the presence of other species besides Q1, Q2, and Q3 is less
likely, the narrower Qn共Si兲 distribution predicted by the SM
model, with a 10% higher Q2 silicate content, and correspondingly lower amounts of Q0, Q1, and Q4, represents a
closer match to the experimental results compared to the RI
potential, which predicts a broader Qn共Si兲 distribution, similar to the one obtained using a different RI, partial-charge
model.25 A clear improvement arising from the SM potential
is also observed in the Qn distribution of phosphate species.
The SM glass is dominated by isolated orthophosphate 共Q0兲
species, with a Q0 abundancy of 82%, and the rest of the
phosphate species represented by Q1 pyrophosphates, where
one of the four P–O bonds forms a P–O–Si bridge with an
adjacent silicate. On the other hand, the Qn共P兲 distribution of
the RI glass is again much broader, with similar amounts
共47% and 42%, respectively兲 of orthophosphates and pyrophosphates, and a significant fraction of chain metaphosphates 共Q2兲, which are not present in the SM glass. Even
though the determination of the phosphate speciation in these
compositions using one-dimensional magic angle spinning
NMR techniques is not straightforward, recent data do suggest that a small percentage of nonorthophosphate species
coexist with the majority of isolated Q0 in the 45S5
Bioglasss®.4,25 Rather than P–O–P, it is more likely that
these species contain P–O–Si bonds, on statistical grounds.8
As a matter of fact, the preference of for P–O–Si rather than
P–O–P linkages was recently revealed in mesoporous bioactive glasses, using cross-polarization NMR techniques.45
Therefore, even if the match with the experiments can still be
improved 共the 17% fraction of nonorthophosphate species is
presumably still too high兲, the SM potential provides a more
accurate model of the phosphate speciation in the 45S5
Bioglass® compared to RI potentials. Together with the results mentioned above for the silicate Qn structure, there are
clear evidences that the inclusion of polarization in the SM
potential leads to consistent improvements in the mediumrange structure of the glasses. The residual discrepancies are
most likely related to the intrinsic limitations of the computational procedure used to make the glass: namely, the limited size of the periodic supercell, which imposes artificial
constraints due to the periodic boundaries, and the much
faster cooling rate used in the simulations than in the experimental melt-and-quench procedure.46 For the purpose of the
present paper, in the following, we will focus on understanding the microscopic origin of the improvements to the final
glass structure brought about by the SM potential. We will
investigate how the inclusion of polarization in the model
affects the structural evolution from the melt to the glass. As
explained before, the reference system is represented in this
case by a small periodic sample where the interatomic forces
are treated with ab initio accuracy. Even though the absolute
Qn amounts of these systems cannot be representative of a
macroscopic sample, important insight is provided by
the short-range structure and the dynamics of the exchanges
40
Si-O
30
O-O
5
4
CP
SM
RI
20
3
2
10
1
P-O
40
g(r)
084504-4
Si-Si
5
4
30
3
20
2
10
1
Na-O
8
Ca-O
8
6
6
4
4
2
2
0
1
2
3
4
2
3
4
5
r (Å)
FIG. 2. 共Color online兲 rdfs of the small melt samples 共3000 K兲 modeled
using the classical SM and RI interatomic potentials and the CPMD
approach.
in the Si/P coordination environment during the hightemperature trajectory.
B. Local „short-range… structure
First, we examine the short-range atomic arrangements
of the amorphous system and its melt precursor in order to
highlight even small differences which can play a role in the
process of glass formation from the melt. The radial distribution functions 共rdfs兲 of the melt modeled using SM, RI,
and CP approaches are shown in Fig. 2. In the melt, the rdfs
obtained using the two interatomic potentials are rather similar to each other. The most significant difference with respect
to the CP rdfs is the Si–Si distribution, which in the ab initio
sample has a short-distance tail extending to 2.2 Å, whereas
no such short Si–Si contacts are formed in the MD runs
using the classical potentials. Moreover, while the peak positions of the classical and ab initio rdfs are rather close, the
latter tend to be broader than the corresponding SM and RI
ones, denoting a larger degree of structural disorder in the
ab initio melt. In fact, Table I shows that a larger fraction of
Si3c and Si5c ions are found on average in the CPMD melt,
compared to both melt samples obtained with the classical
potentials. We have previously observed that some of these
TABLE I. Average Si coordination 共%兲 in 3000 K melt. Standard errors are
reported between parentheses.
SM
RI
CP
Si2c
Si3c
Si4c
Si5c
Si6c
0
0
0.1 共0.01兲
0.7 共0.07兲
1.7 共0.05兲
5.5 共0.08兲
97.2 共0.1兲
96.5 共0.1兲
88.8 共0.1兲
2.3 共0.06兲
1.8 共0.05兲
5.5 共0.08兲
0
0
0.1 共0.01兲
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084504-5
J. Chem. Phys. 129, 084504 共2008兲
Structure of multicomponent bioactive glasses and melts
40
p(θ)
0.02
CP
SM
RI
0.01
Si-O
30
O-Si-O
O-O
4
CP
SM
RI
20
Si-O-Si
3
2
10
90
120
θ (deg)
150
FIG. 3. BADs of intratetrahedral 共OSiO兲 and intertetrahedral 共Si–O–Si兲
angles of the 3000 K melt samples obtained with the classical SM and RI
interatomic potentials and with CPMD.
defects are found in or are associated with small rings containing two or three silicon ions.31 Even though the small
size of the samples does not allow us to draw more quantitative conclusions, the CP melt at 3000 K is certainly characterized by a more distorted local structure. This is further
highlighted in the bond angle distributions 共BADs兲 in Fig. 3.
The O–Si–O angle distribution of the CP melt is broader and
with a tail extending to lower angles. The intertetrahedral
Si–O–Si angle in the CP distribution is also significantly
broader, due to a shoulder around 90°, which corresponds to
the short-distance tail in the Si–Si rdf mentioned above31 and
is absent from the SM and RI BADs. These additional features in the CP BADs are mostly associated with the higher
fraction of structural defects.31 It is important to remark that
in agreement with previous results for soda-lime silicate
glasses,22 the SM potential seems to provide a better description of the intertetrahedral geometry of the silicate network
in the melt, with its main intertetrahedral peak shifted significantly closer to the CP reference than the RI potential.
Except for the latter observation, both classical potentials
show some inadequacy in the description of the considerably
distorted short-range structure of the melt, highlighted in the
CP simulations. This reflects the fact that the RI and SM
potentials are fitted to the structural properties of crystalline
oxides.6 Even though they are transferable to amorphous
phases, the classical potentials are not completely able to
reproduce local environments which are not commonly
found in the low-temperature solid because by construction
they are biased against large distortions in bond distances
and angles. On the other hand, the rdfs of the glass 共Fig. 4兲
confirm the superiority of a potential including polarization
to model glass structural features, not only in the medium
range, as shown in the previous section, but also in the short
range. The CP rdfs for the glass can be used again as a direct
reference, unbiased by system-size effects, which were carefully screened out thanks to the procedure discussed before.
The SM provides a closer match to the ab initio data for
essentially all the atomic pairs whose rdfs are shown in
Fig. 4. This trend is summarized in the total distribution
functions T共r兲 共Ref. 22兲 shown in Fig. 5, which denote a
rather poor performance of the RI potential in the 2 – 4 Å
range, where the nonbonded nearest neighbor interactions
are represented. The Rx factors,48 calculated with respect to
the CP T共r兲, are 9.6% and 17.2% for the SM and RI poten-
P-O
40
180
Si-Si
5
4
30
3
20
2
10
1
Na-O
8
Ca-O
8
6
6
4
4
2
2
0
1
2
3
4
2
3
4
5
r (Å)
FIG. 4. 共Color online兲 rdfs of the small glass samples obtained with the
classical SM and RI interatomic potentials and with CPMD.
tials, respectively, confirming the better performance of the
SM potential also in reproducing the short-range structure of
the glass. Some of the characteristic distances and angles are
collected in Table II. The SM potential provides a more accurate representation of the different strength of T-NBO and
T-BO bonds, where T is a network-former atom, i.e., Si or P.
In particular, the ab initio and SM models yield T-BO bonds
0.07– 0.08 Å longer than T-NBO, whereas the difference is
only 0.04 Å for the RI potential. Both classical potentials
predict an average Si–O distance slightly shorter, but close to
the experimental estimate of 1.61⫾ 0.02 Å 共Refs. 4 and 49兲
共RI and SM potential yield 1.59 and 1.60 Å, respectively兲,
whereas the ab initio Si–O mean distance is 1.625 Å; therefore both classical potentials appear to provide an accurate
average description of the Si–O bond. The inclusion of polarization does not improve the average Si–O distance, which
is already accurate without polarization, but seems to be im2
1.5
T(r)
60
1
1
CP
SM
RI
0.5
0
∆T(r)
30
g(r)
0
0
5
0
1
2
3
4
5
6
r (Å)
FIG. 5. 共Color online兲 共Top panel兲 Total distribution function T共r兲 of the
small glass samples obtained with the classical SM and RI interatomic
potentials and with CPMD. 共Bottom panel兲 Corresponding error TI共r兲
− TCP共r兲, where I = SM or RI.
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J. Chem. Phys. 129, 084504 共2008兲
Antonio Tilocca
TABLE II. Glass short-range structure. Selected average distances 共Å兲 and
angles 共deg兲 involving BO and NBO atoms. Values within parentheses are
coordination numbers, calculated by integrating the corresponding rdf up to
the first minimum cutoff 共3.1 and 3.2 Å for Na–O and Ca–O, respectively兲.
Estimated uncertainties for all reported values are ⱕ1% 共ⱕ0.01 Å and
ⱕ0.5° for distances and angles, respectively兲.
1.58
1.66
1.54
1.61
2.63
2.67
2.63
113
110
106
2.33 共4.0兲
2.45 共1.6兲
2.27 共5.5兲
2.59 共0.4兲
59–91
62–85
1.58
1.62
1.48
1.52
2.42
2.61
2.60
110
108
107
2.38 共4.6兲
2.52 共1.0兲
2.38 共5.4兲
2.73 共0.8兲
57–87
58–85
20
CP
1.60
1.67
1.55
1.63
2.67
2.70
2.63
112
110
105
2.28 共4.0兲
2.46 共1.4兲
2.30 共5.8兲
2.42 共0.4兲
60–91
65–81
10
C. Temperature-dependence of the Qn structure
We have seen in the previous section that the better performance of the SM potential in reproducing both the shortand medium-range structure of the melt-derived glass does
not reflect a 共significantly兲 more accurate description of the
melt precursor compared to the RI potential. As the accuracy
of the structural model of the melt precursor itself does not
appear crucial for the quality of the final glass structure, we
examined the time evolution of the system during the cooling
in order to understand where the inclusion of polarization
positively impacts the model of the glass formation. The
5
4
Q
1
50
Q
3
2
2
Q
4
Q
2
Q
30
20
1
6
SM
5
4
3
Q
10
Si5c
Si3c
40
3
Si5c
1
2
1
Si3c
Q
50
2
Q
30
20
2000
1000
5
Si5c
4
3
Q
10
6
CP
40
3000
portant in determining the correct balance between Si-BO
and Si-NBO distances within the SiO4 tetrahedra. The better
intratetrahedral geometry of the SM is also evident from the
共N兲BO-共N兲BO distances and 共N兲BO-Si-共N兲BO angles in
Table II, which are significantly closer to the corresponding
CP values than those yielded by the RI potential. In particular, the SM produces a more marked difference between
O–Si–O angles formed by two NBOs and by two BOs, again
in close agreement with the CP data. When we examine the
coordination environment of modifier Na and Ca cations in
Table II, the description provided by both classical potential
appears fairly accurate, and slightly closer to the CP data in
the case of the SM potential. For the purpose of the present
work, the present comparison shows that a SM provides a
realistic description of the distorted tetrahedral shell of Si
and P, with an accurate representation of the different bonding patterns involving BOs and NBOs. Despite the slightly
less accurate performance in this context, RI potentials can
be considered sufficiently adequate in the case where the
interest is primarily in modeling the short-range structure of
these materials; however, the SM should be the method of
choice when an accurate description of the medium-range
structure and network connectivity is sought.
6
RI
3
30
n
rSi-NBO
rSi-BO
rP-NBO
rP-BO
rNBO-NBO
rNBO-BO
rBO-BO
␪共NBO-Si-NBO兲
␪共BO-Si-NBO兲
␪共BO-Si-BO兲
rNa-NBO
rNa-BO
rCa-NBO
rCa-BO
␪共O − Na− O兲
␪共O − Ca− O兲
RI
Q
40
% Q (Si)
SM
50
% Sin
084504-6
3
1
2
Si3c
0 3000
T (K)
2000
1
1000
0
0
FIG. 6. 共Color online兲 共left panels兲 Qn distributions and 共right panels兲 average amounts of three- and five-fold coordinated Si of the RI, SM, and CP
systems as a function of the temperature. The vertical dashed lines qualitatively mark the onset of the glass transition in each case.
changes in the Si coordination, as well as in the Qn distributions, as the melt is gradually cooled to room temperature are
highlighted in Fig. 6. At this stage, it is worth noting again
that a more significant test of the medium-range structure is
provided by the large glass samples discussed before, and the
absolute Qn abundancies of the small samples should not be
considered representative of the specific performances of the
SM and RI potentials from this perspective.50 However, the
comparison of relative trends in the modeled melts does provide relevant insight. In particular, the small models incorporate an accurate representation of local effects that can
influence the extended network connectivity in larger
samples, such as the exchange dynamics in the coordination
shell of Si and P.31 The advantage of performing three additional sets of calculations with a smaller sample is that we
can now directly compare the dynamical trends emerging
from the melt-quench trajectory as the system is cooled, with
the accurate reference represented by the CP system. The left
panels in Fig. 6 show that, starting from the melt at 3000 K,
during the cooling the CP and SM systems evolve in a similar and consistent way, with a decrease in Q1 and corresponding increase in Q2 and Q3 as the melt is cooled down.
On the other hand, in the RI system, the opposite Q2 → Q1
transformation is taking place instead, while at the same
time, the initial small amount of Q4 silicates is not further
transformed during the cooling phase and is still found in the
glass, whereas all Q4 are converted into lower-n Q species in
the CP and SM runs. The main message that can be gathered
from Fig. 6 is that the dynamical behavior of the high-
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084504-7
J. Chem. Phys. 129, 084504 共2008兲
Structure of multicomponent bioactive glasses and melts
% (Si+P) defects
5
4
SM
RI
3
2
1
0
3000
2500
2000
1500
T (K)
FIG. 8. Percentage of nontetrahedral T atoms: Si3c + Si5c + P3c + P5c in the
large melt sample as a function of the temperature.
FIG. 7. 共Color online兲 A series of snapshots extracted from the CP-MD
trajectory at 3000 K, showing the mechanism of the changes to the Qn
speciation of a selected Si. The atoms involved in the transformations are
highlighted as ball and stick. Color codes areas follows: Si 共turquoise兲, O
共red兲, Na 共gray兲, Ca 共green兲, and P 共yellow兲.
temperature RI model is rather different from the reference
CP, whereas the SM model appears to follow more closely
the ab initio data. This seems to indicate that the less accurate Qn distribution of the large RI samples might be a consequence of some inadequacy of the RI potential to describe
the melt-quench process.
Figure 7 shows several snapshots extracted from the
CPMD trajectory at 3000 K, highligthing the changes in the
Qn connectivity of a selected Si ion. The initial Q1 coordination of the tagged Si 共panel a兲 changes to Q2 in 共b兲 by forming a Si–O–Si bridge with an adjacent Si, which keeps its
tetrahedral coordination by losing a bonded BO to another
adjacent Si. When the latter bond is reformed as in 共c兲, the
result is a 4-M ring containing a fivefold-coordinated Si adjacent to the tagged Q2 Si. Internal rearrangement of this
complex, with the formation of a Si–O–Si bond between the
Q2 Si and the opposite Si in the ring, reduces the size of the
ring to a 3-M ring still containing an Si5c 共panel d兲. The latter
species then recovers fourfold coordination when one of its
oxygens is captured by a neighboring Si 共this O can still be
spotted in the bottom right of panel e兲, and a fully saturated
3-M ring is formed 共panel e兲. This ring is then further transformed due to the interaction of the tagged Q2 Si with another adjacent Si, which becomes fivefold coordinated while
the tagged Si becomes Q3 共f兲. Again, the tetrahedral coordination is restored by removal of an extra oxygen 共top right of
panel g兲, and the final result is a Q3 Si still incorporated in
the 3-M ring 共g兲. Finally, the 3-M ring is opened due to the
action of an orthophosphate group, which donates one of its
NBOs to form a new P–O–Si link 共h兲, which then replaces
the Si–O–Si bridge in the ring, thus converting the tagged Si
back to Q2 共i兲. These complex rearrangements take place in
less than ⬃10 ps and show how the transient formation of
coordinative defects 共especially Si5c兲 and small rings bring
about frequent changes in the Qn speciation. The active participation of structural defects to the interconversion between
Qn species in silicate melts had been previously proposed.51
Due to their established role in the Qn transformation, it is
important to see how the fraction of undercoordinated and
overcoordinated Si also changes during the cooling phase,
especially because, as noted before, the overall fraction of
these defects found in the 3000 K melts modeled with both
classical potentials was reduced with respect to the CP reference. The data shown in the right panels of Fig. 6, while
confirming that a higher fraction of defects is always present
in the CP melt, also show that most defects have been healed
in the RI sample already at 1800 K, whereas a non-negligible
percentage of fivefold-coordinated Si defects is still present
in the SM sample at lower temperatures, until no residual
defects are left below 1200 K. This interesting trend was
confirmed in the larger system. Figure 8 shows that the decay
of the initial total fraction of miscoordinated Si and P is
faster in the RI than in the SM sample, in which a higher
fraction of defects is always present during the cooling.
These observations seem to denote a correlation between a
more accurate description of the medium-range structure of
the glass and the persistence of miscoordinated atoms during
the cooling, which can assist the interconversion process between the Qn species.
D. Dynamics of Qn interchanges
The interchange between the various Qn species can be
further analyzed by focusing on the dynamics of Qn Qn+1
transformations in the melt. The characteristic time for the
Qn Qn+1 transformations, with n = 1 , . . . , 4 was averaged
over all such processes completed in the MD trajectories at
T = 3000 K. The statistics are summarized in Table III. An
important point highlighted in the table is the more facile
nature of Qn Qn+1 processes in the CP melt, where all
transformations occur faster, and therefore a larger number
of interconversions is completed in the same simulation time,
compared to the SM and especially to the RI run. The slower
dynamics of Qn interchanges with the RI potential might
denote an important role of polarization effects in accounting
for these transformations in a more realistic way, even
though their rate in the SM system is still lower than in the
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084504-8
J. Chem. Phys. 129, 084504 共2008兲
Antonio Tilocca
TABLE III. Mean Qn Qn+1 interchange times 共femtosecond兲, averaged over the MD trajectory at 3000 K performed using SM and RI potentials and CPMD.
The first value is the forward n → n + 1 reaction time, followed by the corresponding backward time and the tb / t f ratio. Standard errors are reported in
parentheses.
n
1
2
3
SM
tf
221共40兲
297共35兲
223共30兲
tb
401共45兲
199共27兲
48共3兲
RI
Kn,n+1
1.81共0.39兲
0.67共0.12兲
0.22共0.03兲
tf
325共70兲
277共41兲
447共83兲
reference CP system, as to indicate that additional effects
would need to be incorporated in the classical potentials for
further improvement.
For each transformation, the Kn,n+1 ratio between the
measured Qn+1 → Qn and Qn → Qn+1 mean times is also reported. These ratios are used to assess the relative stability of
two Qn species in the melt based on the time it takes on
average to be converted into one another. The data in the
Table III show that the formation of Q2 from Q1, as well as
from Q3, is strongly favored in the CP and SM cases,
whereas with the RI potential the Q2 → Q3 transformation is
on average faster than the inverse process. The same analysis
was then repeated in the 2000–1400 K range, in which Fig. 6
denoted a significantly different behavior of the RI potential
with respect to the SM one and the CP simulations. In this
range, the K1,2 and K2,3 values are 3.2 and 0.54 for CP, 7.5
and 0.66 for SM, and 0.98 and 3.1 for RI, respectively. Despite the different absolute values, the SM shows the correct
trend, favoring the formation of Q2 from both Q1 and Q3,
whereas in the same temperature range, which is crucial for
the glass formation, the RI potential is biased toward a depletion of the Q2 abundancies in the system, in favor of Q1 and
Q3. Even though the equilibrium Qn speciation does not necessarily reflect the Kn,n+1 ratios,52 there is some evidence that
the dynamical and energetic interaction between the Qn species in the melt is better described when polarization is included.
IV. SUMMARY AND FINAL REMARKS
The present simulations show that the approximate inclusion of polarization in the interatomic potential results in
significant improvements in the intertetrahedral structure of
melt-derived multicomponent glasses, and is therefore essential to model processes where the glass network connectivity
plays a crucial role, such as the dissolution and reactivity in
physiological environments and technological applications.
The physical reasons for these improvements are not related
to a better description of the melt precursor, for which the
classical potentials, with and without including polarization,
do not fully reproduce the highly distorted short-range structure highlighted by CP simulations. Despite this discrepancy,
which reflects the intrinsic bias of the classical interatomic
potentials toward “regular” local arrangements, the structure
of the glass obtained upon cooling the melt with the SM
potential is significantly improved, not only in the mediumrange as mentioned before, but also in the short-range. In
tb
539共58兲
376共47兲
134共38兲
CP
Kn,n+1
1.66共0.40兲
1.36共0.26兲
0.40共0.10兲
tf
73共6兲
123共11兲
81共7兲
tb
146共11兲
88共5兲
52共5兲
Kn,n+1
2.00共0.22兲
0.71共0.08兲
0.64共0.08兲
particular, the inclusion of polarization improves the intratetrahedral geometry as a consequence of a more realistic treatment of T-BO and T-NBO bonds.
Because the structure of the melt is not improved with
the SM, whereas the final glass structure is, including the
polarization of oxide ions seems to be essential to steer the
melt → glass transformation upon cooling in a more reliable
way. This hypothesis was confirmed by examining smaller
samples of this system, whose atomistic properties could be
directly compared to a reference ab initio model of the same
size. The analysis of interchanges between the Qn species as
a function of the temperature denotes a higher stability of Q2
species during the cooling with the SM potential, in agreement with the CP data, whereas the dynamical evolution of
the melt with RI potential reveals a lower stability of Q2
species already in this precursor phase. In particular, despite
the faster rates measured in the CP dynamics, the inclusion
of polarization in the SM potential determines sufficiently
accurate rates of interconversion between the different Qn
species.
In order to model the glass medium-range properties
such as the Qn distribution, periodic samples of large size
must be obtained, which rule out ab initio MD and impose
the use of classical MD. The ab initio MD sample was used
in this work mainly to provide an unbiased reference for the
local interactions occurring at high temperature. This comparison highlighted the more realistic behavior of the SM
melt. It is only when this accurate local balance is combined
with a large system size that an accurate Qn distribution for
both Si and P can be obtained. There are some indications
that the better performances of the SM potential reflect its
ability to allow the persistence also at lower temperatures of
a small amount of miscoordinated Si defects, which are involved in the interconversion process of Qn. In addition,
larger-scale CPMD simulations would be needed to further
test and confirm this hypothesis.
In conclusion, in this work we have shown that the inclusion of polarization effects at the level of a SM potential
is essential to model multicomponent silicate glasses,
whereas more sophisticated force fields might be needed to
accurately model the corresponding melts, especially if a
very accurate description of structural and coordinative distortions at high temperature is sought. Possible reasons behind the improved glass structure provided by the SM were
discussed, such as the persistence of fivefold-coordinated Si
at lower temperatures, which in turn might lead to the quali-
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084504-9
Structure of multicomponent bioactive glasses and melts
tatively accurate balance between the dynamical processes of
interconversion between the Qn species during the cooling
phase.
ACKNOWLEDGMENTS
The author would like to thank the U.K.’s Royal Society
for the award of a University Research Fellowship, and Professor A. N. Cormack 共Alfred University, USA兲 for fruitful
comments and discussions. Computer resources on the HPC
x service were provided via the U.K.’s HPC Materials Chemistry Consortium and funded by EPSRC 共portfolio Grant No.
EP/D504872兲.
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