Direct Calculation of the Physical Properties of Sodium Borosilicate Glassfrom its Chemical Composition Using the Concept of Structural Units

Hiroyuki Inoue,‡ Atsunobu Masuno,‡,† Yasuhiro Watanabe,‡ Keiichi Suzuki,§ and Toru Iseda§

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

§New Glass Forum, 3-21-16 Hyakunin-cho, Shinjyuku-ku Tokyo 169-0073, Japan

The physical properties of sodium borosilicate glasses were

calculated directly from their chemical compositions in the

whole glass-forming region. These properties included molar

volume, refractive index, thermal expansion coefficient, elasticmoduli, and glass transition temperature. The physical property

values corresponding to the structural unit were estimated and

then the calculation was carried out by accumulating thephysical property values depending on the chemical composition

of the glass. In the case of sodium borosilicate glass, the struc-

tural units are BO3/2, BO4/2Na, BO2/2ONa, BO1/2O2Na2,

SiO4/2, SiO3/2ONa, SiO2/2O2Na2, and SiO1/2O3Na3. Toconduct the calculation, the structural units’ physical property

values were estimated by analyzing experimental data

extracted from the database of glass properties, INTER-

GLAD. The calculated values were similar to the experimentaldata in the whole glass-forming region. Furthermore, the calcu-

lation method provided, quantitatively, the structural units’

contributions to the glass’ physical properties. This study dem-onstrates that using the concept of structural units to calculate

the physical properties of sodium borosilicate glass is a simple

and powerful predictive tool.

I. Introduction

FOR over a hundred years, many attempts have been madeto calculate physical properties of glass directly from its

chemical composition. Several well-known methods can befound in a monograph by Volf published in 1988.1 Thesimplest method uses the following first-order mixture model:

P ¼Xi

bixi; (1)

where P is the property of the glass, xi is the mole fractionof the ith component, and bi is the coefficient of the first-order blending term involving xi. The value of bi is estimatedfrom a data set using least-squares regression, which can besatisfied within a narrow compositional range.

The following second-order mixture model is used whenthe interactions between the chemical components in theglass are considered:

P ¼Xi

bixi þXi

Xj

bijxixj (2)

where bij is the coefficient of the second-order blending terminvolving xi and xj. This model can be used to estimatephysical properties for a wider compositional range than the

first-order model. Nevertheless, it is possible to reproduceexperimental data from the chemical composition of glasswith the first-order mixture model1 by setting a structuralunit in the glass as the component in Eq. (1).2 Structuralunits are local structures with various atomic arrangements,including next-nearest-neighbor atoms. The structural unitscan be determined through a variety of methods, such as athermodynamics approach,3–6 hypothetical chemical equilib-ria between structural groups,7–11 or structural analyses byNMR.12–17 For example, the structural units in Na2O–B2O3

binary glass system are BO3/2 (BO3 triangle without nonbrid-ging oxygen), BO4/2Na (BO4 tetrahedron and one Na+),BO2/2ONa (BO3 triangle with one nonbridging oxygen andone Na+), BO1/2O2Na2 (BO3 triangle with two nonbridgingoxygens and two Na+). For Na2O–SiO2 binary glass system,SiO4/2 (Q4: SiO4 tetrahedron without nonbridging oxygen),SiO3/2ONa (Q3: SiO4 tetrahedron with one nonbridging oxy-gen and one Na+), SiO2/2O2Na2 (Q2: SiO4 tetrahedron withtwo nonbridging oxygens and two Na+) and SiO1/2O3Na3(Q1: SiO4 tetrahedron with three nonbridging oxygens andthree Na+) are well-known structural units. Budhwani andFeller2 built the relationship between a structural unit’schemical composition and proportions based on structuralstudies by NMR analysis for Na2O–B2O3–SiO2 systems.18,19

They showed that the model could predict densities.By using17O 3QMAS NMR analysis, it is possible to dis-

tinguish the cations which have bonds with the oxygen atom,such as Si–O–[4]B, Si–O–[3]B, [4]B–O–[3]B and [3]B–O–[3]B.20

The parentheses indicate the oxygen coordination numberaround Si or B atom. Furthermore, the number of the B orSi atoms around tetrahedral B atoms has been distin-guished.21 Therefore, if one can incorporate the additionalstructural information into the model, the atomic arrange-ment can be described in more detail. Furthermore, a topo-logical constraint22,23 and molecular dynamics simulation24

have provided the models, which have predicted values ofseveral physical properties well. Therefore, in the near futureit will be possible to describe firm linkage between the atomicarrangement and the physical properties of the glass.

In this study, we extended the first-order mixture modelusing the concept of structural units to calculate variousphysical properties of a Na2O–B2O3–SiO2 system. Theseproperties included molar volume, refractive index, thermalexpansion coefficient, elastic moduli, and glass transitiontemperature. We estimated the physical property values ofeach structural unit, which were expressed as coefficient bi inEq. (1), by using experimental data extracted from the data-base of glass properties, INTERGLAD.25 We evaluated thevalidity of the method by comparing the calculated physicalproperties to experimental data.

II. Calculation Method

The calculations are based on the structural units of Na2O–B2O3 and Na2O–SiO2 binary glass systems, which are

T. Rouxel—contributing editor

Manuscript No. 29733. Received May 13, 2011; approved October 18, 2011.†Author to whom correspondence should be addressed. e-mail: masuno@iis.

u-tokyo.ac.jp

211

J. Am. Ceram. Soc., 95 [1] 211–216 (2012)

DOI: 10.1111/j.1551-2916.2011.04964.x

© 2011 The American Ceramic Society

Journal

assumed to be similar to those in a Na2O–B2O3–SiO2 system.The proportions of the structural units, BO3/2, BO4/2Na,BO2/2ONa, BO1/2O2Na2, SiO4/2 (Q4), SiO3/2ONa (Q3),SiO2/2O2Na2 (Q2), and SiO1/2O3Na3 (Q1), are estimated fromthe glass composition by using Budhwani and Feller’smethod.2 They used the borate species’ proportions ofK = [SiO2]/[B2O3] and R = [Na2O]/[B2O3] to classify fourglass-forming regions as shown in Fig. 1. Region I:0 < R < Rmax = 0.5 + 0.0625 K. Region II: Rmax < R < R1 =0.5 + 0.25 K. Region III: R1 < R < R2 = 1.5 + 0.75 K.Region IV: R2 < R < R3 = 2 + K. For each region, the molefraction xi of the structural unit, such as f(BO3/2), f(BO4/

2Na), f(BO2/2ONa), and f(BO1/2O2Na2) and the number ofnonbridging oxygen per SiO4 tetrahedron f(SiNBO) can becalculated. The equations for each calculation are summa-rized in Table I. In the Na2O–SiO2 binary glass system,f(SiNBO) is equal to the ratio of the concentrations of Na andSi ions, [Na]/[Si]. The proportions of Q species, Q4, Q3, Q2,and Q1, in the Na2O–SiO2 binary system can be obtained forthe values of f(SiNBO) based on the NMR analysis.26 Here, itis assumed that the proportion of Q species at f(SiNBO) inthe Na2O–B2O3–SiO2 system is equal to the Q species in thebinary system. Maximum position of the distribution of Q3

units on the basis of this method was located at 0.8 bigger inR than that reported by means of 29Si MAS NMR.27

The experimental values of the physical properties for var-ious glass compositions all over the glass-forming regionwere extracted from the INTERGLAD. These propertiesincluded density ρ (g/cm3), molar volume Vm (cm3/mol),thermal expansion coefficient a (K�1), refractive index nD,mean dispersion nF � nC, Young’s modulus E (GPa), Shearmodulus G (GPa), Bulk modulus K (GPa), Poisson’s ratio c,and glass transition temperature Tg (K). The molar volumeVm and Young’s modulus E were measured at room temper-ature. The thermal expansion coefficient a was measured in

the range from 20°C to 300°C and the glass transition tem-perature Tg was obtained by means of thermal expansion.

The structural units’ coefficients, bis, were determined bymeans of multiple regression analysis. To evaluate the analy-sis’ validity, we estimated the coefficient of determination(R2), absolute (DP) and relative (dP) root mean square dis-crepancy (RMSD), and t-ratio (ti) as follows:

R2 ¼ 1�

Pj

yj � fj� �2

Pj

yj � �y� �2 (3)

DP ¼ 1

N

Xj

yj � fj� �2( )1=2

(4)

dP ¼ 1

N

Xj

yjfj� 1

� �2( )1=2

(5)

ti ¼ biffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPjðyj � fjÞ2

� �ðN� 2Þ� �r

1 P

jðxi � �xÞ2� �� �

(6)

where N is the number of experimental data points, y and fare the properties’ experimental and calculated values,respectively, and xi is the mole fraction of the ith structuralunit in a unit’s composition. The average of a variable isexpressed by a horizontal bar over the variable. In general, at-value with an absolute value � 2 is considered to be signifi-cant, with a statistical confidence level of approximately95%.

III. Results and Discussion

Figure 2 shows the mole fractions of Na2O–B2O3–SiO2 sys-tem’s structural units at K = 1 where [SiO2] equals [B2O3].The structural units’ mole fractions were determined as a + b +c = 1 in the composition of a Na2O·b B2O3·c SiO2. The struc-tural units can be classified into borate-related [Fig. 2(a)] andsilicate-related [Fig. 2(b)]. According to Budhwani and Feller,2

in Region I (0 < R < 0.5 + 0.0625 K) where the Na2O contentis <22 mol% at K = 1, all of the Na2O works to convert BO3/2

units into BO4/2Na units. Entering Region II from Region I,where the Na2O content exceeds 22 mol% at K = 1, theBO4/2Na units begin to decrease. In Region III (0.5 +0.25 K < R < 1.5 + 0.75 K), the Na2O content exceeds 27 mol% at K = 1, and the BO4/2Na units are converted to BO2/2ONaand BO1/2O2Na2 units. On the other hand, in the case of sili-cate-related structural units, all SiO4 tetrahedra are Q4 units inthe silica network in Region I. In Region II, Q4 units are con-verted to Q3 units, and the Q2 unit increases with an increasein Na2O content.

The structural units’ coefficients, bis, were estimated fromthe mole fractions of structural units and from the experimentaldata of several physical properties through multiple regression

Fig. 1. Four divided regions in the phase diagram of the Na2O3–B2O3–SiO2 glass system. I: 0 < R < Rmax = 0.5 + 0.0625 K. II:Rmax < R < R1 = 0.5 + 0.25 K. III: R1 < R < R2 = 1.5 + 0.75 K. IV:R2 < R < R3 = 2 + K. K = [SiO2]/[B2O3] and R = [Na2O]/[B2O3].

Table I. Mole Fractions of the Structural Units and the Number of Nonbridging Oxygen per SiO4

Tetrahedron. K = [SiO2]/[B2O3] and R = [Na2O]/[B2O3]

f(BO3/2) f(BO4/2Na) f(BO2/2Na) f(BO1/2Na) f(SiNBO)

I 1 � R R 0 0 0

II 1 � Rmax Rmax 0 0 2 R�Rmaxð ÞK

III 1� 0:125Kð Þ 0:75� R2þK

� �8þ Kð Þ 1

12 � R24þ12K

� �R�R1ð Þ 2�0:25Kð Þ

6þ3KR�R1ð Þ 2�0:25Kð Þ

4þ2K þ 23

KR8þ4K � K

16

� �9þ52

R�R12þKð Þ

24

IV 0 8þ Kð Þ 112 � R

24þ12K

� �43 � K

6

� �1� R

2þK

� �0:5� K

16

� �þ R�R2ð Þ 2�0:25Kð Þ2þKð Þ þ 2

3KR

8þ4K � K16

� �9þ52

R�R12þKð Þ

24

212 Journal of the American Ceramic Society—Inoue et al. Vol. 95, No. 1

analysis. Table II lists the number of experimental datapoints, the coefficients of determination, and absolute andrelative RMSD of the calculated properties for the experi-mental data in INTERGLAD. Although there is variation inthe number of data points between these properties, there areenough to estimate bis. The coefficients of determination R2

for molar volume Vm, refractive index nD, mean dispersionnF � nC, thermal expansion coefficient a, shear modulus G,and bulk modulus K were higher than 0.9. Furthermore, therelative RMSDs of nD and nF � nC were <1%. Though therelative RMSDs dP of the thermal expansion coefficient aand Young’s modulus E were 7.6% and 7.7%, respectively,those of other properties varied from 1.9% to 6.1%. Privenet al. calculated the physical properties of oxide glass andtheir relative RMSDs of the density, thermal expansion coef-ficient, and refractive index were 4.0%, 14.8%, and 1.0%,respectively.10 In their paper, the relative RMSD of elasticmoduli varied from 12% to 18% and the absolute RMSDsof characteristic temperatures of the glass ranged from 32 to45 K. According to Priven10, the physical properties valuesof the glass hardly changed, depending on B2O3 content. Ourrelative RMSDs values were superior to those reported byPriven, because our calculation was specific to the Na2O–B2O3–SiO2 system. Therefore, our calculation method repro-duces the physical properties more precisely.

Figure 3 compares the experimental data and the calcu-lated ones for (a) mean dispersion nF � nC. The coefficientof determination R2 for nF � nC was the highest among nineproperties. Therefore, the calculated and experimental valuesare almost the same.

Table III lists the estimated coefficients bis for variousproperties; coefficients with t-values <2 are in parentheses,and all other coefficients are statistically significant atapproximately the 95% confidence level. As shown, the coef-

ficients’ absolute values for the BO2/2ONa, BO1/2O2Na2, Q2

and Q1 units are considerably large, except for molar vol-ume. Considering that the mole fractions of these structuralunits averaged all over the regions were just more than 0.01,the coefficient values did not affect the estimation of physicalproperties, although t-values were >2.

To investigate the calculation validity, we focus on compo-sition dependence of density ρ, Young’s modulus E, andglass transition temperature Tg, among nine properties. TheNa2O content dependence is clearly seen under the conditionof [SiO2] = [B2O3]. Thus, the value of K is set as 1.0. As themolar volume of the BO3/2 unit was 19.2 cm3/mol and itsmolecular weight was 34.81 g/mol, the density of the BO3/2

Fig. 2. Composition dependence of the mole fraction of (a) borate-related and (b) silicate-related units when [B2O3] is equal to [SiO2] in theNa2O3–B2O3–SiO2 glass system. Vertical dotted lines are the boundaries between Regions I, II, III, and IV.

Table II. Numbers of Samples N, Coefficient of Determination (R2), Absolute (DP) and Relative (dP) Root Mean Square

Discrepancy (RMSD), and Remarks Column for Various Materials Properties

N R2 DP dP (%)

Molar volume, Vm (cm3/mol) 556 0.957 0.54 1.9 RTRefractive index, nD 137 0.951 3.56 9 10�3 0.24Mean dispersion, nF � nC 206 0.980 6.35 9 10�5 0.81Thermal expansion coefficient, a (10�7 K�1) 204 0.976 4.1 7.6 20–300 (°C)Young’s modulus, E (GPa) 296 0.875 4.6 7.7 RTShear modulus, G (GPa) 114 0.952 1.1 4.3Bulk modulus, K (GPa) 96 0.938 1.8 5.1Poisson’s ratio, c 37 0.718 1.38 9 10�2 6.1Glass transition temperature, Tg (K) 309 0.890 24.3 3.2 by thermal expansion

Fig. 3. The experimental data versus the calculated data of meandispersion nF � nC.

January 2012 Calculation of the Physical Properties of Sodium Borosilicate Glass 213

unit was estimated to be 1.81 g/cm3. In the same way, thedensities of BO4/2Na, BO2/2ONa, BO1/2O2Na2, Q4, Q3, Q2,were estimated to be 3.13, 2.27, 2.32, 2.28, 2.59, and 2.49 g/cm3, respectively.

Figure 4(a) shows the calculated density of K = 1.0. InRegion I, the mole fractions of BO3/2 and Q4 unitsdecreased and the mole fraction of the BO4/2Na unitincreased, as shown in Fig. 2. Due to the large value of thedensity of the BO4/2Na unit, the density of the glassincreased with the increase of the BO4/2Na unit in Region I.In Region II, the mole fraction of BO4/2Na unit began todecrease and the mole fraction of Q3 unit began to increase.As the density of Q3 was estimated to be smaller than thatof the BO4/2Na unit, density increased gradually. In RegionIII, the BO3 unit and the BO4/2Na units decreased rapidlyand the mole fractions of Q2, BO2/2ONa and BO1/2O2/2Na2units increased. These values of the density of the structuralunits were similar to those of the average of BO3 andBO4/2Na units. Therefore, the change in density was small.Figure 4(a) also shows the experimental data aroundK = 1.0 ± 0.5. As the coefficients were determined by multi-ple regression analysis using all the experimental data, ourestimated densities at K = 1 are similar to the experimentaldata reported by Budhwani and Feller2, even though thecompositional dependence was complicated and exhibitednonlinearity. It is acceptable and understandable that thecalculation method to accumulate molar volume of eachstructural unit was successfully conducted because the glassstructure can be considered as an ensemble of structuralunits.

In the case of Young’s modulus E, the coefficient bi ofBO3 was as small as 0.7 GPa. The coefficients bis of BO4/2Na,BO2/2ONa, BO1/2O2/2Na2, Q

4, and Q3 were 117, 609, �300,64.7, and 88.2 GPa, respectively.

Figure 4(b) shows that in Region I, the calculated Eincreased with an increase of Na2O content due to the largevalue of the coefficient of BO4/2Na. In Region III, Edecreased due to the rapid decreases of the BO3 and BO4/

2Na units and a gradual increase of the Q3 unit. Similar todensity, the calculated value of Young’s modulus are similar

to the experimental values, even though the Na2O contentdependence of E was complicated.

These results shown in Fig. 4(b) present that the calcula-tion method using the structural units is an effective way totrace the complicated change. The same complicated behav-ior was seen not only in compositional dependence of Tg,but also in the refractive index, and shear modulus. The cal-culated values of these properties are also similar to theexperimental values. By analyzing the compositional depen-dence of the physical properties, the quantitative distributionof the physical value of the structural units to the physicalproperties of the glass can be investigated.

Finally, we show all the calculated data for density ρ,Young’s modulus E, and glass transition temperature Tg inthe ternary phase diagram. The following are some commonfeatures in Fig. 5.

1. The minimum values are located at the B2O3 rich areain Region I.

2. The isolines are linear and parallel to each other inRegion I, indicating that the decrease of [B2O3] has alinear effect on physical properties, regardless ofwhether SiO2 or Na2O is substituted for B2O3.

3. The other minimum point is in the Na2O rich region.4. The maximum values are in the middle of the ternary

system. In the case of Young’s modulus and glass tran-sition temperature, there is lack of experimental datain the Na2O rich region and the calculation predictedthat the maxima would be in Region II. In the case ofthe density, however, the tendency of the composi-tional dependence on density to be different isobserved, because density is obtained from molarweight divided by molar volume.

As our calculation method is based on multiple regressionanalysis, the precision of the calculation, that is, the values ofthe coefficient of determination R2 and the RMSDs, isstrongly affected by deviations in the experimental values.The experimental values in the database vary by differentmeasurements and under different conditions. In the regionof phase separation, there are sometimes two values for one

Table III. The Coefficient bis of the Structural Units for Various Materials Properties

BO3/2 BO4/2Na BO2/2ONa BO1/2O2Na2 Q4 Q3 Q2 Q1

Molar volume, Vm (cm3/mol) 19.2 21.0 29.0 41.7 26.3 35.2 50.3 20.8Refractive index, nD 0.6961 1.636 2.098 1.739 1.471 2.238 5.545 �104.1Mean dispersion, nF � nC 3.348 9.546 16.03 8.188 7.069 12.81 73.63 �20.63Thermal expansion coefficient, a (10�7 K�1) 22.3 169 747 �141 25.0 138 4070 �145 000Young’s modulus, E (GPa) (0.70) 117 609 �300 64.7 88.2 1430 �64 400Shear modulus, G (GPa) �4.24 50.0 164 �95.2 26.4 30.0 559 �23 900Bulk modulus, K (GPa) (�0.57) 81.2 (�73.9) 307 32.8 78.8 �34.3 �220 000Poisson’s ratio, c 0.154 0.203 �1.48 4.22 0.221 0.355 �3.90 52.3Glass transition temperature, Tg (K) 224 950 2100 �522 818 1230 (82.7) 40 500

Fig. 4. Composition dependence of the experimental data of K = 1 ± 0.5 and the calculated data of K = 1. (a) density, (b) Young’s modulus,and (c) glass transition temperature. The circles indicate the experimental data and the lines indicate calculated data.

214 Journal of the American Ceramic Society—Inoue et al. Vol. 95, No. 1

chemical component. Here, the glass transition temperature isthe example. As a measurement method of the thermalexpansion was selected, the higher Tg was excluded in theregion of the phase separation. The coefficient values of struc-tural units were evaluated from the experimental data exceptthe higher Tg. It seemed to be the reason that the temperaturecoefficient of Q4 unit was 818 and 412 K lower than that ofQ3 unit. It will be easy to understand that the value obtainedfrom the multiple regression analysis and the value of R2 andRMSD are changed, when changing the search condition ofthe data base. Another important factor on the precision ofthe data is the distribution of the composition of the mea-sured glass, which has an influence on the coefficient of deter-mination. As there were few points of the physical propertieson the Na2O-rich side in the Na2O–B2O3–SiO2 system, thereliability of the coefficient of the structural units such asBO2/2ONa, BO1/2O2/2Na2, Q

2 and Q1 units is relatively low.From this point of view, it is necessary to measure highly reli-able physical properties of Na2O rich glass and record themin the database.26,28 Nevertheless, we found that if the struc-tural units of the system can be modeled well, in which thecoordination number of cations changes and the physicalproperties shows nonlinearity to the chemical composition, itwill be possible to estimate the physical properties to withinabout 5% of relative RMSD. As a further support, structuralstudies, such as thermodynamics or molecular dynamics sim-ulation, are effective ways to verify the calculation method.In the near future, the model, which can be applied to variousmulti-component glass, will be constructed.

IV. Conclusion

A method to directly calculate physical properties usingstructural units of glass was applied to a Na2O–B2O3–SiO2

ternary glass system and its validity was investigated. Thestructural units used in the calculation are classified intoBO3/2, BO4/2Na, BO2/2ONa, BO1/2O2Na2, SiO4/2, SiO3/2ONa,SiO2/2O2Na2, SiO1/2O3Na3 units, on the basis on the struc-tural study using Budhwani and Feller’s NMR analysis.Many properties, including density, refractive index, meandispersion, thermal expansion coefficient, Young’s modulus,Shear modulus, Bulk modulus, Poisson’s ratio, and glasstransition temperature of glass were calculated directly fromthe chemical composition of the glass. Less than 10% of rela-tive root mean square deviation of the calculated values wasobtained for experimental data extracted from INTER-GLAD. It was found that even the simplest model was a use-ful and powerful way to predict the physical properties ofthe sodium borosilicate glass.

Acknowledgments

This work was performed as a project for Research and Development to Pro-mote the Creation and Utilization of Intellectual Infrastructures of New

Energy and Industrial Technology Development Organization in Japan(NEDO) from 2005 to 2007.

References

1M. B. Volf, Mathematical Approach to Glass. Elsevier Publishing Company,1988 and references therein.

2K. Budhwani and S. Feller, “A Density Model for the Lithium, Sodiumand Potassium Borosilicate Glasses Systems,” Phys. Chem. Glasses, 36 [4]183–90 (1995).

3B. A. Shakhmatkin, N. M. Vedishcheva, M. M. Shultz, and A. C. Wright,“The Thermodynamics Properties of Oxide Glasses and Glass-FormingLiquids and Their Chemical Structure,” J. Non-Cryst. Solids, 177, 249–56(1994).

4N. M. Vedishcheva, B. A. Shakhmatkin, and A. C. Wright, “Thermody-namic Modelling of the Structure and Properties of Glasses in the SystemNa2O-B2O3-SiO2 and Na2O-CaO-SiO2,” Phys. Chem. Glasses, 46 [2] 99–107(2005).

5N. M. Vedishcheva, B. A. Shakhmatkin, and A. C. Wright, “The Structureof Sodium Borosilicate Glasses: Thermodynamic Modelling vs. Experiment,”J. Non-Cryst. Solids, 345 & 346, 39–44 (2004).

6N. M. Vedishcheva, B. A. Shakhmatkin, and A. C. Wright, “Thermo-dynamic Modeling of the Structure of Sodium Borosilicate Glasses,” Phys.Chem. Glasses, 44 [3] 191–6 (2003).

7A. I. Priven, “Evaluation of the Fraction of Fourfold-Coordinated Boronin Oxide Glasses From Their Composition,” Glass Phys. Chem., 26 [5] 441–54(2000).

8A. I. Priven, “Calculation of the Structure Linear Expansion Coefficientfrom the Chemical Composition of Glasses in the R2O(RO)-Al2O3-B2O3

(R=Li, Na, Ca, and Ba) System,” Glass Phys. Chem., 29 [1] 60–8 (2003).9A. I. Priven and O. V. Mazurin, “Comparison of Methods Used for the

Calculation of Density, Refractive Index and Thermal Expansion of OxideGlasses,” Phys. Chem. Glasses, 44 [4] 156–66 (2003).

10A. I. Priven, “General Method for Calculating the Properties of OxideGlasses and Glass Forming Melts from Their Composition and Temperature,”Glass Technol., 45 [6] 244–54 (2004).

11A. I. Priven, “A New Equation Describing the Temperature Dependenceof the Electrical Resistivity of Alkali Containing Silicate Melts,” Glass Tech-nol., 46 [3] 263–70 (2005).

12M. E. Milberg, J. G. O’Keefe, R. A. Verhelst, and H. O. Hooper, “BoronCoordination in Sodium Borosilicate Glasses,” Phys. Chem. Glasses, 13 [3]79–84 (1972).

13Y. H. Yun and P. J. Bray, “Nuclear Magnetic Resonance Studies of theGlasses in the System Na2O-B2O3-SiO2,” J. Non-Cryst. Solids, 27, 363–80(1978).

14P. J. Bray, S. A. Feller, G. E. Jellison Jr., and Y. H. Yun, “B10 NMRStudies of the Structure of Borate Glasses,” J. Non-Cryst. Solids, 38 & 39,93–8 (1980).

15Y. H. Yun and P. J. Bray, “B11 Nuclear Magnetic Resonance Studies ofLi2O-B2O3 Glasses,” J. Non-Cryst. Solids, 44, 227–37 (1981).

16W. J. Dell, P. J. Bray, and S. Z. Xiao, “11B NMR Studies and StructureModeling of Na2O-B2O3-SiO2 Glasses of High Soda Content,” J. Non-Cryst.Solids, 58, 1–16 (1983).

17B. C. Bunker, D. R. Tallant, R. J. Kirkpatrick, and G. L. Turnrt, “Multi-nuclear Nuclear Magnetic Resonance and Raman Investigation of SodiumBorosilicate Glass Structure,” Phys. Chem Glasses, 31, 30–41 (1990).

18D. Feil and S. Feller, “The Density of Sodium Borosilicate GlassesRelated to Atomic Arrangements,” J. Non-Cryst. Solids, 119, 103–11 (1990).

19R. Boekenhauer, H. Zhang, S. Feller, D. Bain, S. Kambeyanda, K.Budhawani, P. Pandikuthira, F. Alamgir, A. M. Peters, S. Messer, and K.L. Loh, “The Glass Transition Temperature of Lithium Borosilicate GlassesRelated to Atomic Arrangesments,” J. Non-Cryst. Solids, 175, 137–44(1994).

20L.-S. Du and J. F. Stebbins, “Nature of Silicon-Boron Mixing in SodiumBorosikicate Glasses: A High-Resolution 11B and 17O NMR Study,” J. Phys.B, 107, 10063–76 (2003).

Fig. 5. Ternary contour graph of (a) density, (b) Young’s modulus, and (c) glass transition temperature, calculated from the coefficientsobtained from multiple regression analysis.

January 2012 Calculation of the Physical Properties of Sodium Borosilicate Glass 215

21L.-S. Du and J. F. Stebbins, “Site Preference and Si/B Mixing in Mixed-Alkali Borosilicate Glasses: A High-Resolution 11B and 17O NMR Study,”Chem. Mater., 15, 3913–21 (2003).

22M. M. Smedskjaer, J. C. Mauro, S. Sen, and Y. Yue, “QuantitativeDesign of Glassy Materials Using Temperature-Dependent ConstraintTheory,” Chem. Mater., 22, 5358–65 (2010).

23M. M. Smedskjaer, J. C. Mauro, and Y. Yue, “Prediction of Glass Hard-ness Using Temperature-Dependent Constraint Theory,” Phys. Rev. Lett., 105,115503 (2010).

24A. Pedone, G. Malavasi, A. N. Cormack, U. Segre, and M. C. Menziani,“Elastic and Dynamical Properties of Alkali-Silicate Glasses from ComputerSimulations Techniques,” Theor. Chem. Account, 120, 557–64 (2008).

25New Glass Forum, International Glass Database System INTERGLAD Ver.7.1, New Glass Forum, Tokyo, Japan. Available at http://www.newglass.jp/(accessed on 1 October 2011).

26H. Maekawa, T. Makawa, K. Kawamura, and T. Yokokawa, “The Struc-ture Groups of Alkali Silicate Glasses Determined from 29Si MAS-NMR,”J. Non-Cryst. Solids, 127, 53–64 (1991).

27G. Bhasin, A. Bhatnagar, S. Bhowmik, C. Stehle, M. Affatigato, S. Feller,J. Mackenzie, and S. Martin, “Short Range Order in Sodium BorosilicateGlasses Obtained via Deconvolution of 29 Si MAS NMR Spectra,” Phys.Chem. Glasses, 39 [5] 269–74 (1998).

28O. V. Mazurin, SciGlasses 7.5, Institute of Theoretical Chemistry Inc,Available at http://www.sciglass.infor/ (accessed on 1 October 2011). h

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