Role of Magnesium in Soda-Lime Glasses: Insight into Structural

Jun 28, 2008 - ... with compositions 15Na2O·(10 − x)CaO·xMgO·75SiO2 (x = 0, 5, and 10 mol) ... on the elastic properties (Young's modulus, shear ...
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J. Phys. Chem. C 2008, 112, 11034–11041

Role of Magnesium in Soda-Lime Glasses: Insight into Structural, Transport, and Mechanical Properties through Computer Simulations Alfonso Pedone,† Gianluca Malavasi,† M. Cristina Menziani,*,† Ulderico Segre,† and Alastair N. Cormack‡ Dipartimento di Chimica, UniVersita` di Modena e Reggio Emilia, Via G. Campi 183, 41100 Modena, Italy, and Kazuo Inamori School of Engineering, New York State College of Ceramics, Alfred UniVersity, Alfred, New York 14802 ReceiVed: February 26, 2008; ReVised Manuscript ReceiVed: April 20, 2008

The role of Mg in soda-lime glasses was elucidated by classical molecular dynamics (MD) simulations. The effect of the replacement of CaO for MgO on the structure, transport, and elastic properties of a series of glasses with compositions 15Na2O · (10 - x)CaO · xMgO · 75SiO2 (x ) 0, 5, and 10 mol) was studied. Different structural roles were found for the Ca and Mg ions. The former, coordinated by six oxygen atoms, acts as a network modifier, while the latter, four-fold coordinated, participates in the silica network. Consequently, Na ion diffusion is favored by the replacement of MgO for CaO in these glasses, as shown by variation in the computed diffusion coefficients and activation energy of the process in the series of glasses studied. Moreover, the consequences of these structural modifications on the elastic properties (Young’s modulus, shear modulus, bulk modulus, Poisson’s ratio, and compressibility) of the glasses were evaluated by means of energy minimization techniques carried out on the structures obtained by MD simulations. Introduction The design of new technologically important glasses requires a deep understanding as to how key properties of the final product change with composition. Soda-lime silicate glasses are the prototypes for a wide range of commercially significant glass products as well as bioactive glasses.1–4 As such, they have received a fair amount of experimental investigation, and a number of composition-property relationships have been developed. However, in commercial or technological glasses, sodium is often supplemented by other alkali cations and calcium by magnesium, for reasons related both to processing and to property improvement. It is now largely recognized that substitutions of CaO by MgO in the composition of silica glasses modifies their chemical durability5 and improves the mechanical properties of the glass.6 In fact, the replacement of CaO for MgO in soda-lime glasses leads to an increase of the fracture toughness and fracture surface energy with a simultaneous decrease of the Young’s modulus.7 This behavior is of particular interest for bioactive glasses since their poor mechanical properties as compared to human cortical bone limits their use as bone restoration with load bearing functions.6,8,9 It also is known that the partial substitution of CaO by MgO and Na2O by K2O is required to match the thermal expansion of bioactive glass coatings to that of Ti-based alloys.10 Most of the typical compositions of glasses and glassceramics for biomedical applications belongs to complex multicomponent systems. Thus, the understanding of the individual role of each component is not always straightforward. The role of MgO on the surface behavior of bioglasses has led to contradictory explanations, depending in part on whether it is considered as a network modifier or as a network former. Some in vitro results indicate that MgO inhibits mineralization,11 * Corresponding author. E-mail: [email protected]. † Universita ` di Modena e Reggio Emilia. ‡ Alfred University.

and others suggest that it does not affect apatite formation.12,13 However, significant amounts of MgO are present in some Bioverit glass-ceramics, whose bioactivity has been clinically confirmed for years.14 Notwithstanding these appealing characteristics, the consequences of the structural role of magnesium in the silica network have not been elucidated completely. Therefore, it is important to investigate the environment of magnesium in silicate glasses, which has been variously described as either tetrahedral or octahedral by NMR studies.15 Further investigation of key structural factors and their effect on transport and elastic properties can be greatly improved with the atomistic resolution of modern computational techniques. In fact, several molecular dynamics (MD) studies of the structure of bioactive glasses have been carried out so far,16–20 but few works have been devoted to the simulation of elastic21 and transport properties,22–26 except for binary silicate glasses. In this study, the effect of CaO/MgO ratio on the structure, ionic transport, and mechanical properties of a series of soda-lime glasses, taken as a model system, was investigated by means of computational simulation techniques with the aim of furnishing a microscopic-based interpretation to the macroscopic properties of interest. Computational Methods Simulation Procedure. The structures of soda-lime silicate glasses of composition 15Na2O · (10 - x)CaO · xMgO · 75SiO2 (x ) 0, 5, and 10 mol) and the sodium silicate glass with composition 15Na2O · 85SiO2 were modeled by means of NVT MD simulations. The initial configurations were generated by placing randomly 3480 atoms in a cubic box. Atomic compositions and size length of the simulation boxes are reported in Table 1, together with the glass densities at room temperature calculated according to Priven’s empirical method27 implemented in SciGlass software.28 The DLPOLY29 package was employed for MD simulations. Integration of the equations of motion was performed using the

10.1021/jp8016776 CCC: $40.75  2008 American Chemical Society Published on Web 06/28/2008

Role of Magnesium in Soda-Lime Glasses

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TABLE 1: Modeling Systems, Compositions, Densities, and Dimensions density atom cell size (g/cm3) number (Å)

composition 15Na2O · 10CaO · 75SiO2 (CA10) 15Na2O · 5CaO · 5MgO · 75SiO2(CA5MG5) 15Na2O · 10MgO · 75SiO2 (MG10) 15Na2O · 85SiO2 (NS15)

2.471 2.441 2.418 2.340

3480 3480 3480 3072

36.4255 36.4130 36.3651 35.1428

TABLE 2: Potential Parameters for Eq 1 -O Mg+1.2-O-1.2 Ca+1.2-O-1.2 Si+2.4-O-1.2 O-1.2-O-1.2 Na+0.6

-1.2

Dij (eV)

aij (Å)-1

jrij (Å)

Cij (eV Å12)

0.023363 0.038908 0.030211 0.340554 0.042395

1.763867 2.281000 2.241334 2.006700 1.379316

3.006315 2.586153 2.923245 2.100000 3.618701

5.0 5.0 5.0 1.0 22.0

Verlet Leapfrog algorithm with a time step of 2 fs. Coulombic interactions were calculated by the Ewald summation method30 with a cutoff of 12 Å and an accuracy of 10-4. The short-range interaction cutoff was set to 5.5 Å. The force-field developed by Pedone et al.31 was used in this work. This is based on a rigid ionic model, with partial charges to handle the partial covalency of silicate systems. The energy is given by the sum of three terms: (i) the long-range Coulombic potential; (ii) the short-range forces, which are represented by a Morse function; and (iii) an additional repulsive term C/r,12 which is added to model the repulsive contribution at high temperature and pressure. The expression for the model potential is therefore

Uij(r) )

zizje2 Cij + Dij[{1 - exp[-aij(r - r¯ ij)]}2 - 1] + 12 r r (1)

where zi, zj, Dij, aij, jrij, and Cij are parameters, and the indices i and j refer to the different atom species. The atomic charge for an alkali ion is assumed to be +0.6 e. The values of the parameters in eq 1 are reported in Table 2. They were derived by fitting both structural and mechanical properties of inorganic oxides according to the procedure implemented in the GULP code.32 It is noteworthy to discuss briefly a practical aspect of MD simulations. Periodic boundary conditions (PBCs) are used, so that the simulated system is, in fact, an infinite solid without external surfaces. As a consequence, structural coherence is created across the faces of the simulation box, and the system is more susceptible to superheating and supercooling because the structural coherence limits the ability of the small (relative, say, to Avogadro’s number) number of atoms in the simulation box to relax sufficiently. Because of these effects, effective temperatures in MD simulations are usually higher with respect to the ones used experimentally. The difference may well be ascribed to differences between thermodynamic melting and mechanical melting processes that occur in the case of MD simulations with periodic boundary conditions.33 The cooling procedure used in MD simulations is not directly related to the experimental laboratory procedure, but its efficacy is assessed by the comparison of the model structure with the known structural data. As in previous works that employed a rigid ionic model with partial charges,21,34–36 the system was heated at 6000 K, a temperature more than adequate to bring the system to its liquid state in the framework of the adopted force-field. The melt was

then equilibrated for 100 ps and subsequently cooled continuously from 6000 to 300 K in 1140 ps with a nominal cooling rate of 5 K/ps. The temperature was decreased by 0.01 K every time step using a Berendsen thermostat37 with the time constant parameter for the frictional coefficient set to 0.4 ps. Another 100 ps of equilibration at constant energy and 50 ps of data production were performed at 300 K. Configurations at every 0.1 ps were recorded for structural analysis. Transport Properties. Experimentally, transport properties such as diffusion constants and activation energies usually are obtained by tracer diffusivity experiments in the temperature range between 450 to 750 K with annealing times varying from 30 min at high temperatures to 90 days at low temperatures.38 In computer experiments, the phase space can be sampled for annealing time ranging from only hundreds of picoseconds to hundreds of nanoseconds depending on the CPU resources available.23,39,40 Higher temperatures must be employed to observe an appreciable ionic diffusion because of the reduced sampling time available. In the study of transport properties, as in the cooling procedure, the correspondence between the experimental temperature and the computational one is not straightforward. Therefore, it is anticipated that the absolute values of the transport coefficients will be difficult to reproduce, and we are more interested in reproducing their relative trends with respect to the sample compositions. In this case, it is the Arrhenius or activation energies that are more important. In this work, dynamical properties of the systems were determined at four different temperatures (1000, 1200, 1400, and 1600 K). For each system, the configuration obtained by MD simulations at 300 K was reheated using the Berendsen barostat37 with frictional constants set to 0.4 ps. At each temperature, the systems relaxed for 40 ps to allow them to reach the desired density. Subsequently, 4 ns of data production was run by using the Nose´-Hoover NVT ensemble. Configurations were saved at intervals of 0.2 ps and used subsequently to calculate dynamical properties. We note that the computer glass transition as observed on the time scale of 4 ns is ∼1600 K; in fact, at the end of the production run, silicon ions visited two different sites. No network diffusion was detected below this temperature. The mean square displacement σj(t) measures the average distance an atom of the j-th species travels in time t, and it is defined according to the following equation: Nj

Nj

∑

∑

1 1 σj(t) ) 〈∆ri(t)2 〉 ) 〈(r (t) - ri(0))2 〉 Nj i)1 Nj i)1 i

(2)

where ri(t) - ri(0) is the (vector) distance traveled by atom i over some time intervals of length t, and the squared magnitude of this vector is averaged over many such time intervals and over all atoms of the j-th species in the system. The limiting slope of σj(t), considered for time intervals sufficiently long for it to be in the linear regime, is related to the self-diffusion constant Dj by the Einstein relation

Dj )

d 1 lim σ (t) 6 tf∞ dt j

(3)

Fractional Free Volume (FFV). The FFV of different glasses was analyzed in terms of the Delaunay description of the void space in a disordered system.35 A set of four atoms contiguous to each other forms a tetrahedron, known as a Delaunay tetrahedron (DT). The whole set of the DTs constitutes the Delaunay tessellation since they fill the space without gaps and without overlaps. The FV is calculated as the summation of all void spaces inside the DTs that cover the entire box of the

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simulation.35 Therefore, the fractional free volume (FFV) is defined as the fraction of FV over the total volume of the box. This analysis requires that atoms be considered as spheres. Two sets of atom radii were used: these are the Pauling ionic radii41 (0.95, 0.99, 0.65, 0.41, and 1.40 Å for sodium, calcium, magnesium, silicon, and oxygen, respectively) and the Shannon’s radii42 (1.14, 1.14, 0.71, 0.40, and 1.21 Å for sodium, calcium, magnesium, silicon, and oxygen, respectively). Elastic Properties. The stiffness matrix elements for a crystalline system are defined as the second derivative of the energy (U) with respect to the strains ()43

Cij )

(

1 ∂U V ∂εi ∂ εj

)

(4) Results and Discussion

Once the stiffness matrix C is obtained, several related mechanical properties of anisotropic materials can be derived from their matrix elements or from the matrix elements of the compliance matrix S

S ) C-1

(5)

The bulk modulus B in the Reuss notations is given by

BReuss ) (S11 + S22 + S33 + 2(S31 + S21 + S32))-1

(6)

while the shear modulus is given by

GReuss ) 15 (7) 4(S11 + S22 + S33 - S12 - S13 - S23) + 3(S44 + S55 + S66) Poisson’s ratios are calculated from the compliance matrix for six components

νxy ) -

S21 S31 S32 , νxz ) - , ..., νzy ) S11 S11 S33

(8)

whereas the Young’s modulus values along three principal directions are given by

E-1 k ) Skk (k ) 1, 3)

(9)

For isotropic materials, the three principal values must be equal. Moreover, E, G, B, and ν must satisfy two relations

G)

E E and B ) 2(1 + ν) 3(1 - 2ν)

(10)

therefore, only two independent constants are needed to specify the elastic behavior of an isotropic material. The acoustic velocities in a solid can be derived from the density, F, and bulk and shear moduli of the material and can be expressed as transverse wave, VS, and longitudinal wave, VP, which are given by:

VS )

 GF and V )  4G3F+ 3B P

strictly depends on the second derivative of the energy in the local minimum reached. Therefore, a greater accuracy of the calculation is required, and this is achieved by using the same conditions used for force-field parametrization.31 Moreover, during minimization of the energy, the algorithm implemented in GULP truncates the interatomic potential at the cutoff without a switching process as is performed in MD simulations. Several tests were performed with different cut-offs during minimization and properties calculation, and the results are cutoff independent beyond 10 Å. To handle the amorphous character of glass, a cubic cell with no symmetry (space group P1) was used, which is the MD simulation cell.

(11)

The elastic properties were computed for glasses obtained from MD simulations by means of the GULP code.32 A Newton-Raphson44 energy minimization was performed at constant pressure (1 atm). The Coulomb term was evaluated by means of the Ewald method in which the real cutoff was determined according to an accuracy of 10-8. The minimum image convention was turned off, and the short-range potentials cutoff was set to 15 Å. In static energy minimization techniques, a different setting of the energy evaluation is necessary than is required for MD simulations. This is due to the fact that in the minimization technique, the calculation of elastic properties

Structure. To our knowledge, no experimental data concerning the structural effects of MgO substitution for CaO in silicate glasses are available in the literature. The analysis of the MD trajectories obtained here allows the assignment of different structural roles for the two alkaline-earth ions. As shown by Cormack and Du,45 the Ca ion is six-fold coordinated, by 4.5 nonbridging oxygens (NBO) and 1.5 bridging oxygens (BO), respectively, forming a reasonably regular octahedron. Therefore, Ca seems to play a similar, modifying, role as sodium in the structure, as depicted in Figure 1a. In contrast, 90% of Mg ions are four-fold coordinated and participate in the network, forming Si-O-Mg (24%) and Mg-O-Mg (2.4%) chains of tetrahedra (see Figure 1b), therefore acting as a network former. A small amount of Mg (10%) was found to be five-fold coordinated. The calculated total distribution functions T(r) of the three soda-lime silicate glasses and of the sodium silicate glass with composition 15Na2O · 85SiO2 (NS15) are shown in Figure 2. NS15 glass was taken as a reference to compare the network forming role of Mg with that of Si. Differences in the region ranging from 1.9 to 2.5 Å reflect a different short-range order around the Mg and Ca ions. In fact, the shoulder at 2.4 Å decreases upon replacing CaO with MgO, while a new peak appears at 2.0 Å. Moreover, differences in the medium range order with respect to NS15 are imputable to the Ca-Ca, Ca-Na, Mg-Na, and Mg-Mg distributions. Detailed analysis of the variation of the main short-range structural characteristics as a function of Mg substitution (Table 3) reveals that only very small differences are found, except for the total percentage of bonding oxygen atoms (% BO), which was computed by considering Si and Mg as network formers. The results obtained compare very well with the short-range order of soda-lime silicate glasses characterized experimentally, reported also in Table 3. Figure 3 reports the O-M-O bond angle distribution functions for Na, Ca, and Mg of CA5MG5 glass. Both O-Na-O and O-Ca-O distributions show a peak close to 90°, which generally results from Na or Ca atoms connecting two NBOs belonging to different chains of tetrahedra in the network. The second peak at 60° (which is more pronounced for Na) results from modifiers coordinated to two NBOs (or one NBO and one BO) belonging to the same tetrahedron.17 The broad distributions of Figure 3 also denote a high flexibility of the geometries of the coordination shell of these cations. Conversely, the O-Mg-O bond angle distribution is narrower and shows only a major peak at 101°, which reveals a tetrahedrally distorted environment of MgO4 units. Na-Na, Ca-Ca, and Mg-Mg pair distribution functions for CA5MG5 glass are shown in Figure 4. Na-Na presents only two broad peaks, at 3.0 and 6.5 Å, the former covering a range

Role of Magnesium in Soda-Lime Glasses

J. Phys. Chem. C, Vol. 112, No. 29, 2008 11037 distribution of sodium inside the MD box. No changes occur in the Na-Na distribution when MgO replaces CaO. The Ca-Ca distribution function is similar to the Na-Na one but with the narrower first peak (half-width of 1.0 Å) shifted to 3.5 Å. Both sodium and calcium lie in the percolation channels surrounded by the silicon network, as shown in Figure 1. Conversely, in the Mg-Mg pair distribution function, a doublet is obtained with maxima around 2.8 and 3.2 Å. The former is due to two-membered rings of Mg, which have a Mg-O-Mg angle of 80° and Mg-O bond distances ranging from 2.01 to 2.05 Å, while the latter is made up of Mg-O-Mg bridges in the network of silica with an Mg-O-Mg angle of ∼100° and Mg-O bond lengths ranging from 2.10 to 2.20 Å, as shown in Figure 5. Finally, Figure 6 shows that the substitution of Ca by Mg changes the ring size distribution, leading, for MG10 glass, to the same ring size distribution found for the reference 15Na2O · 85SiO2 glass. Transport Properties. The diffusion coefficients of Na ions (DNa) for the systems studied, obtained by fitting the linear region of the mean square displacements (σ) of sodium ions, are listed in Table 4. For an accurate computation of the diffusion constant, a mean square displacement of Na ions larger than 300 Å2 was ensured for all temperatures studied. Figure 7 shows the temperature dependence of diffusion constants of sodium for the four glasses modeled, which can be approximated by the Arrhenius function

ln D(T) ) ln D0 -

Figure 1. (a) Image of channels created by Ca (light cyan) and Na (light green) of CA10 glass. (b) Image of a percolation channel in MG10 glass. Mg ions (light yellow) enter into the Si-O (Si, light blue and O, red) network acting as a network former. Na ions (light green) are free to diffuse hopping among affinity sites. Ions were visualized with equal radii.

Figure 2. Total radial distribution function of the modeled glasses: CA10 ) 15Na2O · 10CaO · 75SiO2, CA5MG5 ) 15Na2O · 5CaO · 5MgO · 75SiO2, MG10 ) 15Na2O · 10MgO · 75SiO2, and NS15 ) 15Na2O · 85SiO2.

of 1.9 Å, which resembles free ions able to cluster at distances between the sum of the ionic radius and the 5.1 Å homogeneous

Ea kBT

(12)

The pre-exponential factor D0 and the activation energies (Ea) deduced from the linear regressions are listed in Table 5. The diffusivity of Na ions increases by increasing the MgO/ CaO ratio; in fact, the activation energy decreases from 71.9 to 59.2 kJ/mol for CA10 and MG10 glasses, respectively. These results are in agreement with the outcome of the experimental work carried out by O’Connell,46 who studied the K-Na ion exchange depth in float glass compositions. In fact, the penetration depth of alkali ions increases with an increasing MgO/CaO ratio, with other components held constant. Unfortunately, no experimental data are reported in the literature regarding Na diffusion as a function of the MgO/CaO ratio in our compositional ranges. However, some experimental data are available for compositions similar to the ones studied in this work. Terai and Kitaoka47,48 reported an activation energy of 86.1 kJ/mol for a 20Na2O · 10CaO · 70SiO2 glass, which decreased to 79.4 kJ/mol when CaO was replaced by MgO. Kolitsch and Richter49 reported an activation energy of 84 kJ/ mol in a 14.0Na2O · 12CaO · 74SiO2 glass in the range of 620-868 K. On the basis of these measurements, our results underestimate the activation energy by ∼15-20%. This may be ascribed to both the fact that we actually model self-diffusion, in the absence of a chemical potential gradient, and to differences between nominal and effective temperatures of the simulated structures. It is worth noting that Horbach et al.,50 using constant volume simulations in the range of 4000-2100 K with the box size kept fixed at a volume corresponding to the density at room temperature, overestimated the activation energy of soda-silicate glasses by ∼20-30%. The results obtained in this study clearly show that the AE of Na ions decreases upon increasing the MgO/CaO ratio. The explanation for this behavior is not immediately obvious since structural analysis showed that the environment around the Na ions does not change when MgO replaces CaO. Moreover, the average site potential for Na, computed for each of the three

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TABLE 3: Bond Lengths, Coordination Numbers, and Bond Anglesa this work parameter set

CA10 1.62

CA5MG5

MG10

NS15

obsd

Bond lengths (Å) 1.62 2.01

1.62

2.36

2.31

1.6357 1.90-2.0858–63 2.37-2.4960–62,64–67 2.30-2.6268–71

2.62 3.15

2.6657 3.1457

4.0

3.957 4.058,59,61–63 6.060–62,64–67 5.0-6.068–71

Si-O Mg-O Ca-O Na-O

2.36 2.34

1.62 2.01 2.37 2.35

O-O Si-Si

2.63 3.15

2.63 3.16

Si Mg Ca Na % BO

4.0 5.8 5.4 71.4

4.0 4.1 5.9 5.4 80.6

O-Si-O Si-O-Si

109 (14) 152 (35)

Bond angle (deg) (fwhm) 109 (14) 109 (13) 152 (34) 152 (34)

Interatomic distances (Å) 2.63 3.16 Coordination numbers 4.0 4.0 5.4 86.5

4.8 83.7 109 (12) 153 (34)

10957 14957

a The cation-oxygen bond lengths were estimated by the peak position of their pair distributions, and the CNs were calculated based on cutoffs obtained from the first minimum. They are 1.9 Å for Si-O, 2.2 Å for Mg-O, and 3.1 Å for Na-O and Ca-O. % BO was calculated considering Mg as a network former.

Figure 3. Bond angle distribution for O-Na-O, O-Ca-O, and O-Mg-O of CA5MG5 glass.

Figure 5. (a) Two-membered rings and (b) Mg-O-Mg bridges formed by MgO4 tetrahedra in the silicon network. Silicon, magnesium, oxygen, and sodium ions are represented as blue, yellow, red, and green spheres.

Figure 4. Na-Na, Ca-Ca, and Mg-Mg distributions of CA5MG5 glass.

Figure 6. Ring size distribution of modeled systems in which Mg is marketed as a network former: CA10 ) 15Na2O · 10CaO · 75SiO2, CA5MG5 ) 15Na2O · 5CaO · 5MgO · 75SiO2, MG10 ) 15Na2O · 10MgO · 75SiO2, and NS15 ) 15Na2O · 85SiO2.

soda-lime glasses studied, yields a constant value of ca. -7.05 eV/atom. However, the substitution of CaO with MgO increases the FFV of the glasses (Table 7) using either the Pauling ionic radii or the Shannon radii.42 Several reasons contribute to an increase in FFV, such as the change in CN of the exchanging ions (six for Ca but four for Mg) and the smaller size of Mg atoms as compared to Ca atoms. More importantly, this effect can be attributed to the network former role of Mg ions. Graphic visualization of glass structures outline different behaviors of Ca and Mg. Figure 1a shows that Ca ions lie in percolation

channels created by sodium ions. This agrees with findings reported by Cormack and Du,45 who studied as to how the replacement of sodium with calcium changes the structure of soda-lime silicate glasses. Calcium is seen to play a similar, modifying, role as sodium in the structure. Upon substitution, Ca replaces Na, entering modifier-rich regions in the glass. Having diffusivities several orders of magnitude smaller, Ca ions obstruct the channels and hamper Na diffusion. Mg, acting as a network former (see Figure 1b), does not obstruct the percolation channels, and Na ions are freer to diffuse in the

Role of Magnesium in Soda-Lime Glasses

J. Phys. Chem. C, Vol. 112, No. 29, 2008 11039

TABLE 4: Diffusion Coefficients (m2 s-1) of Na at Different Temperatures T (K)

CA10

1000 1200 1400 1600

1.33 × 10 5.33 × 10-10 1.43 × 10-9 3.53 × 10-9 -10

CA5MG5

MG10

1.38 × 5.05 × 10-10 1.33 × 10-9 2.64 × 10-9

1.67 × 10-10 5.33 × 10-10 1.25 × 10-9 2.41 × 10-9

10-10

structure, hopping among preferential sites inside channels with a vacancy-like mechanism that previously was described.22 Elastic Properties. Table 6 lists the computed values of E, G, B, and ν of soda-lime glasses studied, together with values predicted by Priven’s empirical model27 available in the SciGlass package.28 Experimental data measured by Wilantewicz7 for the Young’s modulus on glasses of slightly different compositions (17Na2O · (10 - x)CaO · xMgO · 73SiO2 with x ) 0, 5, and 10 mol) also are reported in Table 6. Overall, good agreement with experimental data is obtained. The experimentally observed trend for the elastic properties (i.e., a decrease in magnitude upon replacing CaO with MgO) is not reproduced by the Priven empirical method, which shows an anomalous value for MG10. Recently, we showed that three concurrent factors, with different consequences for the glass network, are responsible for the compositional dependence of elastic properties of silicate glasses.21 These factors are (i) the degree of polymerization of the silica network; (ii) the nature of the chemical bond and concentration of the modifiers added, and (iii) the packing of the overall structure. The effect of the first two factors (degree of polymerization of silica network and nature of chemical bond and concentration of the modifiers added) can be summarized by taking into account the bond density of the glass weighted by the strength of the different bonds present in the network. In this view, the only contribution to the macroscopic effect is given by the Ca and Mg cations. In fact, the Si-O bond density remains constant when MgO replaces CaO, as well as the Si-O

Figure 7. Temperature dependence of Na diffusion constants of modeled glasses: CA10 ) 15Na2O · 10CaO · 75SiO2, CA5MG5 ) 15Na2O · 5CaO · 5MgO · 75SiO2, MG10 ) 15Na2O · 10MgO · 75SiO2, and NS15 ) 15Na2O · 85SiO2.

TABLE 5: FFV Calculated Using Pauling’s Radii, Activation Energy, and Prefactor D0 and Average Site Potentials (Es) Acting on Na Ions for the Modelled Glassesa AE (kJ/mol)

FFV FFV Es D0 (m2 s-1) (Pauling) (Shannon) (eV/atom)

CA10 71.9 ( 2.1 7.40 × 10-7 CA5MG5 65.6 ( 0.5 3.67 × 10-7 MG10 59.2 ( 0.5 2.04 × 10-7

0.469 0.472 0.474

0.611 0.616 0.620

-7.05 -7.05 -7.05

a Linear regression data for CA10: y ) -8652.9x - 14.12, R ) -0.9992, SD ) 0.071; CA5MG5: y ) -7889.3x - 14.82, R ) -0.9999, SD ) 0.017; and MG10: y ) - 7118.5x - 15.41, R ) -0.9999, SD ) 0.016.

bond strength since no relevant shifts are observed in the projection of calculated vibrational density of states (VDOS) for Si. A semiquantitative estimate of the difference in the strength of chemical bonds between Ca and Mg with the surrounding oxygen atoms in the simulated glasses is obtained by the evaluation of force constants via vibrational frequencies. The projection of VDOS for Ca and Mg was computed in this work by direct diagonalization of the dynamic matrix51 by using the GULP code with the same energy setting as for elastic property calculations. The results obtained, shown in Figure 8, are consistent with vibrational frequencies deduced from infrared reflectivity spectroscopy measurements52,53 and Raman studies.54 VDOS of Ca and Mg are similar to the frequencies of the latter being shifted to higher values. In fact, Ca VDOS shows a peak at 200 cm-1, while Mg VDOS shows a major peak at 280 cm-1. However, this result, by itself, does not indicate that the Mg-O bond is stronger than the Ca-O bond since the lighter Mg inevitably gives rise to higher frequencies as compared to the heavier Ca. Therefore, isotopic substitution of 40Mg (same mass of Ca) for 24Mg in the computation of VDOS was carried out to allow the comparison of frequencies of Ca-O and Mg-O bonds. The results, reported in Figure 8, show equal force constants for Mg-O and Ca-O bonds, thus attributing the chemical strength of the cations in the glass to their coordination number. Finally, another determinant effect in the softening of the glass structure upon substitution of MgO for CaO is the increase in the FFV (Table 5). In fact, being six-fold coordinated and having a larger ionic radius than Mg, Ca ions produce a more packed structure (smaller FFV). The coordination of atoms plays an important role in the elastic behavior of materials because when atoms are highly coordinated, the deformations of M-O-M and O-M-O angles are hampered by steric effects. These results are in excellent agreement with those recently reported by Rouxel55 who claimed that both interatomic energies and atomic packing density (Cg), which is defined as the ratio between the minimum theoretical volume occupied by the ions and the corresponding effective volume of the glass, have to be taken into account to interpret elasticity data. In fact, the substitution of Ca by Mg in soda-lime glass with 78 mol % SiO2 and 15 mol % Na2O does not lead to a stiffness increase as would be expected from values of the bonding energies (U0,Mg-O > U0,Ca-O). Both the glass transition temperature and the Young’s modulus decrease from 785 to 773 K and 69 to 66 GPa, respectively. This behavior seems to be due to a significant decrease of the atomic glass density, as a consequence of an increase of the fractional free volume. The same explanation holds for the decrease of the stiffness observed in the same glass system when Mg is replaced by Si. Although U0,Si-O > U0,Mg-O, the glass atomic packing density decreases from 0.491 to 0.484 as the SiO2 content increases from 73 to 80 mol %. Comparing the results obtained in the present study with those reported by Rouxel,55 two important points must be highlighted: (i) the relative FFV values (FFVRouxel ) 1 - Cg) computed by the atomic packing density reported by Rouxel55 well correlate with the FFV reported in this paper and (ii) the assumption that Mg is six-fold coordinated in soda-lime glasses is in contrast with the results of the present simulations. However, the interatomic potential used in the simulations was fitted on magnesium oxide and successfully tested on magnesium-silicate crystals in which Mg is always six-fold coordinated; therefore, the results obtained of a four-fold coordination of Mg in the silicate glass cannot be considered as an artifact of the potential.

11040 J. Phys. Chem. C, Vol. 112, No. 29, 2008

Pedone et al.

TABLE 6: Elastic Properties of the Modelled Glassesa E (GPa) CA10 CA5MG5 MG10 NS15b

G (GPa)

ν

B (GPa)

calcd

Priven

exptl

calcd

Priven

calcd

Priven

calcd

Priven

67.4 65.4 62.3 61.1

69.7 67.4 68.4 62.5

68.4 66.5 63.2 62.9

26.6 26.1 25.8 25.1

28.7 27.9 28.1 26.2

45.7 43.4 40.2 35.6

40.8 39.6 40.0 33.8

0.260 0.250 0.243 0.216

0.215 0.215 0.215 0.192

a Av errors for calculated data are (1.5 GPa, (1.0 GPa, (0.6 GPa, and (0.01 for E, B, G, and ν, respectively. b Results taken from ref 21. FFV was found to be 0.484 and 0.631 by using Pauling’s and Shannon’s ionic radii.

Figure 8. Projection of VDOS on Ca, Mg, and 40Mg (same mass as Ca) for CA5MG5 glass.

Rather, the small size of Mg ions together with the release of the crystal constraints in the glass matrix might be determinants for the different coordination preferences of Mg in the crystal and in the glass. This is in agreement with a combined neutron and X-ray diffraction study performed by Wilding et al.56 on magnesiumsilicate glasses ranging in composition from estatite (MgSiO3) to forsterite (Mg2SiO4). Structural changes from a glass characterized by corner-shared SiO4 tetrahedra and an approximately equal mixture of MgO4 and MgO5 polyhedra to one in which the average coordination of Mg by oxygens is 5.0 were observed. Both these environments are very different from that of their crystalline counterparts in which Mg is sixfold coordinated. The decrease of the bulk modulus from CA10 to NS15 glass can be predicted from the behavior of the FFV (Table 6). In fact, the greater the FFV (going from CA10 to NS15), the greater the compressibility of the material. Similar behavior was observed for Poisson’s ratio, which is inversely correlated to the FFV and to the covalency of the bonds present in the system: the ionic Ca-O bond, the Mg-O bond (ionic but with a higher field strength), and the Si-O bond, which is more covalent. This behavior is not observed in values computed by using the empirical Priven method since Ca and Mg are both treated as modifiers. Conclusion The effect of the replacement of CaO for MgO on the structural, transport, and elastic properties of soda-lime glasses was studied. It was found that Ca and Mg ions exhibit different roles in the glass structure. The former is six-fold coordinated and shows a similar modifying role as Na ions, contributing to the formation of percolation channels together with Na ions. However, because of the strength of the Ca-O bond with respect to that of the Na-O bond, the calcium ions are less mobile in the structure and obstruct Na diffusion. Conversely, Mg ions are four-fold coordinated and exhibit a network former role, MgO4 tetrahedra being interconnected to the SiO4 network. The different behavior shown by Mg leads to an increase in the FFV of the glass, and the Na ions are freer to diffuse among similar sites along Na diffusion pathways.

The activation energy for diffusion of Na ions seems to be directly correlated to the FFV since the short-range order and average site potential of Na are independent of the CaO/MgO ratio. Elastic property calculations revealed that the replacement of CaO by MgO yields to a softening of the glass, which was explained by two concurrent factors: (ii) the bond density of the glass weighted by the strength of the different bonds present in the network and (ii) the FFV and packing of the structures. It was shown that the Mg–O and Ca–O force constants are of equal magnitude. However, the Ca-O bond density is higher as compared to the Mg-O bond density because of the higher coordination number of Ca. This factor contributes to the decrease of the elastic properties of glasses when MgO substitutes for CaO. Moreover, Ca ions produce a more packed structure (small FFV), leading to a more difficult deformation when an external stress is applied. Acknowledgment. This work was supported by the Ministero dell’Istruzione,Universita` eRicerca(MIUR,Grant2003032158_005). A.P. acknowledges financial support for his visit at Alfred University provided by the International Materials Institute for New Functionality in Glass (IMI-NFG, Grant DMR-0409588), Lehigh University, Bethlehem, PA. References and Notes (1) Kim, H. M.; Miyaji, F.; Kokubo, C.; Otsuki, C.; Nakamura, T. J. Am. Ceram. Soc. 1995, 78, 2405. (2) Kokubo, T.; Kim, H.-M.; Kawashita, M. Biomaterials 2003, 24, 2161. (3) Hench, L. L.; Splinter, R. J.; Alen, W. C.; Greenlee, T. K. J. Biomed. Mater. Res. 1971, 2, 117. (4) Hench, L. L.; Polak, J. M. Science (Washington, DC, U.S.) 2002, 295, 1014. (5) Barrere, F.; Van Blitterswijk, C. A.; De Groot, K.; Layrolle, P. Biomaterials 2002, 23, 1921. (6) Vallet-Regi, M. J. Chem. Soc., Dalton Trans. 2001, 97. (7) Wilantewicz, T. The effects of lithium, boron, and magnesium oxides on the mechanical properties of silicate glasses. Ph.D. Thesis, Alfred University, 1998. (8) Hench, L. L. Biomaterials 1998, 19, 1419. (9) Hench, L. L. J. Am. Ceram. Soc. 1998, 81, 1705. (10) Lopez-Esteban, S.; Saiz, E.; Fujino, S.; Oku, T.; Suganuma, K.; Tomsia, A. P. J. Eur. Ceram. Soc. 2003, 23, 2921. (11) Ebisawa, Y.; Kokubo, T.; Ohura, K.; Yamamuro, T. J. Mater. Sci., Mater. Med. 1990, 1, 239. (12) Moya, J. S.; Tomsia, A. P.; Pazo, A.; Santos, C.; Guitian, F. J. Mater. Sci., Mater. Med. 1994, 5, 529. (13) Pereira, D.; Cachinho, S.; Ferro, M. C.; Fernandes, M. H. V. J. Eur. Ceram. Soc. 2004, 24, 3693. (14) Vogel, W.; Holand, W. J. Non-Cryst. Solids 1990, 123, 349. (15) Fiske, P.; Stebbins, J. F. Am. Mineral. 1994, 79, 848. (16) Linati, L.; Lusvardi, G.; Malavasi, G.; Menabue, L.; Menziani, M. C.; Mustarelli, L. P.; Segre, U. J. Phys. Chem. B 2005, 109, 4989. (17) Tilocca, A.; de Leeuw, N. H. J. Phys. Chem. B 2006, 110, 51. (18) Tilocca, A.; Cormack, A. N.; De Leeuw, N. H. Chem. Mater. 2007, 19, 95. (19) Mead, R. N.; Mountjoy, G. Chem. Mater. 2006, 18, 3956. (20) Mead, R. N.; Mountjoy, G. J. Phys. Chem. B 2006, 110, 14273. (21) Pedone, A.; Malavasi, G.; Cormack, A. N.; Segre, U.; Menziani, M. C. Chem. Mater. 2007, 19 (13), 3144.

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