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Density, Volume, and Packing: Part 2 Thursday, September 4, 2008 Steve Feller Coe College Physics Department Lecture 3 of Glass Properties Course

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Some Definitions x = molar fraction of alkali or alkaline-earth oxide (or any modifying oxide) 1-x = molar fraction of glass former R = x/(1-x) This is the compositional parameter of choice in developing structural models for borates. J = x/(1-x) for silicates and germanates

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Short Ranges B and Si Structures Short-range borate units, F i unitStructureR value F1F1 trigonal boron with three bridging oxygen 0·00·0 F2F2 tetrahedral boron with four bridging oxygen 1·01·0 F3F3 trigonal boron with two bridging oxygen (one NBO) 1·01·0 F4F4 trigonal boron with one bridging oxygen (two NBOs) 2·02·0 F5F5 trigonal boron with no bridging oxygen (three NBOs) 3·03·0 Short-range silicate units, Q i unitStructureJ value Q4Q4 tetrahedral silica with four bridging oxygen 0·00·0 Q3Q3 tetrahedral silica with three bridging oxygen (one NBO) 0·50·5 Q2Q2 tetrahedral silica with two bridging oxygen (two NBOs) 1·01·0 Q1Q1 tetrahedral silica with one bridging oxygen (three NBOs) 1·51·5 Q0Q0 tetrahedral silica with no bridging oxygen (four NBOs) 2·02·0

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Lever Rule Model for Silicates

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Method of Least Squares Take (ρ mod – ρ exp ) 2 for each data point Add up all terms Vary volumes until a least sum is found. Volumes include empty space. ρ mod = M/(f i V i )

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Example: Li-Silicates V Q4 = 1.00 V Q3 = 1.17 V Q2 = 1.41 V Q1 = 1.69 V Q0 = 1.95 V Q4 (J = 0) defined to be 1. The J = 0 glass is silicon dioxide with density of 2.205 g/cc

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Borate Glasses

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Borate Structural Model R < 0.5 F 1 = 1-R, F 2 = R 0.5 <R <1.0 F 1 = 1-R, F 2 = -(1/3)R +2/3, F 3 = +(4/3)R -2/3 1.0 <R < 2.0 F 2 = -(1/3)R +2/3, F 3 = -(2/3)R +4/3, F 4 = R-1

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Another Example: Li-Borates V 1 = 0.98 V 2 = 0.91 V 3 = 1.37 V 4 = 1.66 V 5 = 1.95 V 1 (R = 0) is defined to be 1. The R = 0 glass is boron oxide with density of 1.823 g/cc

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BariumCalcium V f1 0·960·99 V f2 1·160·96 V f3 1·541·29 V f4 2·161·68 V Q4 1·441·43 V Q3 1.921.72 V Q2 2.542.09

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Li and Ca Silicates LiCa V Q4 = 1.00 V Q3 = 1.17 V Q3 = 1.20 V Q2 = 1.41 V Q2 = 1.46 V Q4 (J = 0) defined to be 1. The J = 0 glass is silicon dioxide with density of 2.205 g/cc

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Silicates

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Densities of Barium Borate Glasses R = x/(1-x)Density (g/cc) 0·01·82 0·22·68 0·22·66 0·43·35 0·43·29 0·63·71 0·63·68 0·83·95 0·83·90 0·94·09 1·24·22 1·34·31 1·54·40 1·74·50 2·04·53 Use these data and the borate model to find the four borate volumes. Note this model might not yield exactly the volumes given before.

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Volumes and packing fractions of borate short-range order groups. The volumes are reported relative to the volume of the BO1.5 unit in B2O3 glass. Packing fractions were determined from the density derived volumes and Shannon radii[i],[ii].[i][ii] SystemUnitLeast Squares VolumesPacking Fraction Lif10.98 0.34 f20.94 0.65 f31.28 0.39 f41.61 0.41 Naf10.95 0.35 f21.24 0.62 f31.58 0.41 f42.12 0.46 Kf10.95 0.35 f21.66 0.69 f31.99 0.52 f42.95 0.59 Rbf10.98 0.34 f21.92 0.67 f32.27 0.52 f43.41 0.59 Csf10.97 0.34 f22.28 0.67 f32.62 0.54 f44.13 0.61 [i] Barsoum, Michael, Fundamentals of Ceramics (McGraw-Hill: Columbus OH,1997), 85- 89. [i] [ii] R.D. Shannon, Acta. Crystallogr., A32 (1976), 751.[ii]

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SystemUnitLeast Squares VolumesPacking Fraction Mgf10.980.34 f20.950.63 f31.260.39 f41.460.44 Caf10.980.34 f20.950.71 f31.280.44 f41.660.48 Srf10.940.36 f21.080.68 f31.410.45 f41.920.48 Baf10.970.35 f21.130.73 f31.550.47 f42.160.51

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Volumes and packing fractions of silicates short-range order groups. The volumes are reported relative to the volume of the Q4 unit in SiO2 glass. Packing fractions were determined from the density derived volumes and Shannon radii4,23. SystemUnitLeast Squares VolumesPacking Fraction LiQ41.000.33 Q31.170.38 Q21.410.41 Q11.670.42 Q01.920.43 NaQ41.000.33 Q31.340.42 Q21.740.46 Q12.170.48 Q02.630.49 KQ40.990.33 Q31.580.52 Q22.270.58 Q12.970.61 RbQ41.000.33 Q31.720.53 Q22.630.57 Q13.530.59 CsQ40.980.33 Q31.960.56 Q22.900.64 Q14.250.62

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Alkali Thioborates Data from Prof. Steve Martin x M 2 S.(1-x)B 2 S 3 glasses Unusual F2 behavior

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Alkali Thioborate data

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Volumes and packing fractions of thioborate short-range order groups. The volumes are reported relative to the volume of the BS1.5 unit in B2S3 glass. SystemUnitLeast Squares Volumes Corresponding Oxide Volume (compared with volume of BO1.5 unit in B2O3 glass.) __________________________ Naf11.000.95 f21.371.24 f31.511.58 f52.952.66* Kf11.000.95 f21.651.66 f31.781.99 f53.533.91* Rbf11.000.98 f21.791.92 f31.972.27 f53.904.55* Csf11.000.97 f21.832.28 f32.142.62 f54.465.64* *extrapolated

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Molar Volume Molar Volume = Molar Mass/density It is the volume per mole glass. It eliminates mass from the density and uses equal number of particles for comparison purposes.

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Borate Glasses

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Kodama Data

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Problem: Calculate the molar volumes of the barium borates given before. You will need atomic masses. Also, remember that the data are given in terms of R and there are R+1 moles of glassy materials. You could also use x to do the calculation.

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Li-Vanadate (yellow) Molar Volumes campared with Li- Phosphates (black)

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Top to bottom: Li, Na, K, Rb, Cs

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Stiffness of Borates vs Molar Volume

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Rb -purple squares Cs -yellow triangles

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K- open circles Rb -purple squares Cs -yellow triangles

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Differential Changes in Unit Volumes To begin the calculation, we define the glass density in terms of its dimensionless mass relative to pure borate glass (R=0), m′(R), and its dimensionless volume relative to pure borate glass, v′(R):

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Implies:

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Solve for v 1 and v 2 simultaneously and assign it to R = (R 1 +R 2 )/2 This is accurate if R 1 and R 2 are close to each other; in Kodama’s data this condition is met.

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The Volume per Mole Glass Former Volume per mole glass former = Mass (JM 2 O.SiO 2 )/density It is the volume per mole glass former and R moles of modifier.

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The Volume per Mole Glass Former Why is there a trend linear? JM 2 O.SiO 2 (1-2J)Q 4 +2JQ 3 J<0.5 So slope of volume curve is the additional volume needed to form Q 3 at the expense of Q 4. This is proportional to J.

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Lead Silicate System: Bansaal and Doremus Data

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Sodium Borogermanates Volumes per Mol B 2 O 3 Compared to Sodium Borosilicates

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Problem For the barium borate data provided earlier plot the volume per mol boron oxide as a function of R. How do you interpret the result? Lecture 2 ends here

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Packing in Glass We will now examine the packing fractions obtained in glasses. This will provide a dimensionless parameter that displays some universal trends. We will need a good knowledge of the ionic radii. This will be provided next.

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