Density, Volume, and Packing: Part 2 Thursday, September 4, 2008 Steve Feller Coe College Physics Department

Transcription

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

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

3 Short range borate units, Short anges B and Si Structures F i unit Structure value F 1 trigonal boron with three bridging oxygen 0 0 F 2 tetrahedral boron with four bridging oxygen 1 0 F 3 trigonal boron with two bridging oxygen (one NBO) 1 0 F 4 trigonal boron with one bridging oxygen (two NBOs) 2 0 F 5 trigonal boron with no bridging oxygen (three NBOs) 3 0 Short range silicate units, Q i unit Structure J value Q 4 tetrahedral silica with four bridging oxygen 0 0 Q 3 tetrahedral silica with three bridging oxygen (one NBO) 0 5 Q 2 tetrahedral silica with two bridging oxygen (two NBOs) 1 0 Q 1 tetrahedral silica with one bridging oxygen (three NBOs) 1 5 Q 0 tetrahedral silica with no bridging oxygen (four NBOs) 2 0

4 Lever ule Model for Silicates Fraction of Q-Unit Q 4 Q 3 Q 2 Q 1 Q J-Value

5 Density (g/cc) Model Peters et al. Literature Compilation J-Value

6 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 )

7 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 g/cc

8 Density (g/cc) Li Na K b Cs Mg Ca Sr Ba Figure 1 Borate Glasses Value

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

10 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 ( = 0) is defined to be 1. The = 0 glass is boron oxide with density of g/cc

11 Barium Calcium V f V f V f V f V Q V Q V Q

12 Li and Ca Silicates Li Ca V Q4 = 1.00 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 g/cc

13 Silicates

14 Densities of Barium Borate Glasses = x/(1-x) Density (g/cc) Use these data and the borate model to find the four borate volumes. Note this model might not yield exactly the volumes given before.

15 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]. System Fraction Unit Least Squares Volumes Packing Li 0.34 f f f f Na 0.35 f f f f K 0.35 f f f

16 System Unit Least Squares Volumes Packing Fraction Mg f f f f Ca f f f f Sr f f f f Ba f f f f

17 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. System Unit Least Squares Volumes Packing Fraction Li Q Q Q Q Q Na Q Q Q Q Q K Q Q Q Q b Q Q Q Q Cs Q Q Q Q

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

19 Alkali Thioborate data

20 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. System Unit Least Squares Volumes Corresponding Oxide Volume (compared with volume of BO1.5 unit in B2O3 glass.) Na f f f f * K f f f f * b f f f f * Cs f f f f * *extrapolated

21 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.

22 Density (g/cc) Li Na K b Cs Mg Ca Sr Ba Figure 1 Borate Glasses Value

23 Molar Volume 55 Borate Glasses Lithium Sodium Potassium ubidium Cesium Magnesium Calcium Strontium Barium Value

24 45 40 Li Na K b Cs Molar Volume (cc/mol) Borate Glasses Kodama Data

25 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 and there are +1 moles of glassy materials. You could also use x to do the calculation.

26 Li-Vanadate (yellow) Molar Volumes campared with Li- Phosphates (black) Molar Volume (cc/mole)

27 Molar Volume (cc/mol) Molar volumes of Alkali Borate Glasses Top to bottom: Li, Na, K, b, Cs

28 Stiffness vs in Alkali Borate Glasses Stiffness (GPa) Top to bottom: Li, Na, K, b, Cs

29 Stiffness of Borates vs Molar Volume Li Na K b Cs Stiffness (GPa) Molar V olum e (cc/m ol)

30 Stiffness vs Molar Volume -Cs Stiffness (GPa) Molar Volume (cc/mol)

31 b and Cs Stiffness using Cs Volumes Stiffness, G (GPa) Molar Volume (cc/mol) b -purple squares Cs -yellow triangles

32 Stiffness (cc/m ol) Stiffness vs normalized Cs molar volume molar volume (cc/mol) K- open circles b -purple squares Cs -yellow triangles

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

34 Implies:, ) ( ) ( ) '( ) ( ) ( ) '( v f v f v v f v f v (0). ) ( ) ( ' ) ( ) ( (0) ) ( ) ( ' ) ( ) ( m v f v f m v f v f

35 Solve for v 1 and v 2 simultaneously and assign it to = ( )/2 This is accurate if 1 and 2 are close to each other; in Kodama s data this condition is met. (0). ) ( ) ( ' ) ( ) ( (0) ) ( ) ( ' ) ( ) ( m v f v f m v f v f

36 elative Volumes of the f 2 Unit 2.50 e lativ e V o lu m e Cs b K Na Li

37 Volume of f 1 and f 2 Units of b and Cs (Extended to = 0.6) f 2 Units 68 Volume (in Å 3 ) f 1 Units (atio of Alkali Oxide to Boron Oxide)

38 elative Volumes of the f 1 Unit 1.10 elative Volum e Cs b K Na Li

39 elative Volumes of the f 1 & f 2 Units of Ba-B and Na-B 1.45 f 2 -Unit 1.35 e lativ e V o lu m e f 1 -Unit f1-ba f2-ba f1-na f2-na

40 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 moles of modifier.

41 Volume per Mol Silica of Lithium Silicates using Bansaal and Doremus Volume per Mol Silica (cc/mol) J

42 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.

43 Differential Volume by J for JLi2O-SiO2 Glasses dv/dj J Average

44 Lead Silicate System: Bansaal and Doremus Data Differential Volume by JPb Average Volume(cc/mol/J) J Average

45 Sodium Borogermanates Volumes per Mol B 2 O 3 Compared to Sodium Borosilicates V B 2 O 3 (cc/mol) Si, K=0.5 Si, K=1 Si, K=2 Si, K=4 Ge, K=0.5 Ge, K=1 Ge, K=2 Ge, K=4 Na 2 O*B 2 O 3 3.5

46 LiB LiSi LiGe J or Volume per mole glass former (cc/mol)

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

48 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. pf 4 3 V f r 3 i N i