CORRELATION BETWEEN VOGEL FULCHER

TAMMAN & AVRAMOV EQUATION FOR GLASS VISCOSITY

A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF BACHELOR OF TECHNOLOGY

By

PRATIK PATTANAYAK

( Roll No: 10508025)

DEPARTMENT OF CERAMIC ENGINEERING

NATIONAL INSTITUTE OF TECHNOLOGY

ROURKELA

1

CORRELATION BETWEEN VOGEL FULCHER TAMMAN & AVRAMOV EQUATION FOR

GLASS VISCOSITY

A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF BACHELOR OF TECHNOLOGY

By

PRATIK PATTANAYAK

Under the Guidance of

Prof. SUMIT KUMAR PAL

DEPARTMENT OF CERAMIC ENGINEERING

NATIONAL INSTITUTE OF TECHNOLOGY

ROURKELA

2009

2

NATIONAL INSTITUTE OF TECHNOLOGYROURKELA

2009

CERTIFICATE

This is to certify that the thesis entitled , “Correlation of Vogel Fulcher Tamman

and Avramov equations for glass viscosity” submitted by Mr Pratik Pattanayak

in partial fulfillments of requirements of the award of Bachelor of Technology

degree in ceramic engineering at National Institute Of Technology, Rourkela is an

authentic work carried out by him under my supervision and guidance.

To the best of my knowledge the matter embodied in the thesis has not been

submitted to any other university/institute for the award of any degree or diploma.

Prof S. K. Pal

Date: Department of Ceramic Engineering

National Institute Of Technology,

Rourkela-769008

3

ACKNOWLEDGEMENT

I wish to express my deep sense of gratitude and indebtedness to Prof. Sumit Kumar Pal, Department of Ceramic Engineering, NIT Rourkela for introducing the

present topic and for his inspiring guidance, constructive criticism and valuable suggestion throughout this project work.

I also want to thank my teachers Prof S.Bhattacharya, Prof J.Bera, Prof S.K.Pratihar, Prof B.B.Nayak, Prof R.Mazumdar and Prof A.Choudary for their encouragement, teaching and in helping me to successfully complete my B.Tech

degree and also all the members of the technical staff of my department.

I would also like to thank to all my friends, seniors and other staffs who have patiently extended all sorts of help for accomplishing this work.

Date: 12.05.2009 Pratik Pattanayak

Department Of Ceramic Engineering

NIT Rourkela

Contents4

Page noA.ABSTARCT 2 B.CHAPTERS

1.Introduction 3 2. Literature Review 5 2.1 Temperature dependence of viscosity 5

2.2 Compositional dependence of viscosity 10

2.3 VFT Equation 11

2.4 Avramov Equation 11

3. Statement of the Problem 12 4. Experimental Work 13

4.1 Softwares used 13

4.2 Graph plots 13

4.2.1 VFT & Avramov plots for SiO2(52.75 mol%)& PbO(47.25 mol%) 14

4.2.2VFT & Avramov plots for varying composition 19

4.3 Tabulation 27

4.3.1 VFT fitting parameters 30

4.3.2 Avramov fitting parameters 33

5.. Results & Discussions 35

6. Conclusions 36

C. References. 37

D Annexure 38

Abstract

5

Among many equations the VFT and Avramov equations can fit the glass viscosity data ranging

from 101 to 1015 poise. The VFT equation needs 3 empirical constants (fitting parameter) to fit

the viscosity data over a wide range. Whereas, Avramov equation is based on entropy –

temperature correlation. When fitting a particular viscosity data over a wide range of viscosity it

has been observed that both equation fall over each other with minimum deviation. So it would

be logical to find out some correlation from that characteristic. In the present work an attempt

has been made to find the origin of VFT constants by help of Avramov equation. Viscosity data

of SiO2 based glass system has been taken in to account to find some correlations.

Chapter1: Introduction

6

The VFT equation is as follows :

0exp(BT0/T-T0)

The best way to evaluate the VFT constants is to perform a nonlinear regression analysis of

several experimentally obtained logη – T data. As there are 3 constants, so a minimum of three

viscosity – temperature data points will suffice. Obviously, the values of log η0, B, and T0 will

depend on the choice of the data sets. For instance, if one holds more than three sets of logη – T

data, the values of log η0, B, and T0 will depend on which three sets are in use. In such cases one

opts for the best fit through all data points. For carrying regression analysis, it is required to

guess three values for log η0, B, and T0 to start with. The convergence of the fitting greatly

depends on how close these guess values are to the actual values. In fact, in iterative processes

somewhat off guess values often lead to non convergence and the procedure fails to determine

the constants. In the present case, however, one can predict the guess values reasonably well

form the plot of experimental data points..It is evident from VFT equation that the logη versus T

plot is a rectangular hyperbola. Extending both sides of a coarsely fitted curve through the data

points one may obtain reasonably good guesses for log η 0 and T0. This is because as T → ∞,

logη → log η0 and as T → T0, logη → ∞ .Obtaining initial guess values for log η 0 and T0, one

may obtain initial guess values for B using Avramov. As for example, a plot of logη – T data

points for Soda Lime Silica Glass – SRM 710a.The dashed curve is a coarse fit through these

data points, which is extended both sides until they become parallel to the axes. Backward

extrapolation of these parallel regions cut the axes at values equal to log η0 and T0.

Fig. 1. Plot of log η versus temperature of soda lime silica glass 710a

7

From the Fig. 1, one obtains initial guess values for log η0 and T0 to be 1 and 300. From the

above discussion it is evident that log 0 and T0 in VFT equation are similar quality to viscosity

and temperature.

The Avramov model, an entropy-based description of the effects of temperature and pressure on

structural relaxation times, assumes separability of these two dependences. This implies that the

fragility of glass-formers is independent of pressure. Herein we show that experimental results

for polymethyltolylsiloxane are at odds with this assumption. By introducing a linear increase of

the coordination number of the liquid state with pressure, the model can be modified, enabling

good agreement with experiment to be achieved.The equation is generally expressed as:

Where 0, T0 & α are the fitting parameters of the equation.

0=pre exponential constant

Tg= glass transition temperature

α = fragility index

CHAPTER2: LITERATURE REVIEW

Understanding the nature of viscous flow in glasses and the estimation of viscosity over the

range of temperatures and compositions encountered in practice remain two of the most

challenging goals in glassmaking technology [2]. In glasses, viscosity regulates melting

conditions, rate of removal of bubbles, annealing temperature, crystallization rate, and many

other phenomena. Numerous studies have been reported in the literature on the dependence of

viscosity on temperature and composition of glass. A discussion on essential features in this area

is necessary before we present our investigation.The glass transition phenomenon is one of the

most important phenomena in the world of soft condensed matter. Despite decades of study,

8

many aspects of the glass forming liquids remains elusive. That the viscosity of a glass forming

liquids diverges to infinity at some finite temperature above absolute zero, at least is one things

on which most experts agree. The dynamic divergence or “SUPER ARHENIUS” behaviour of

glass forming systems is commonly represented by VFT equation.

2.1 Temperature dependence of viscosity:

The necessity for an appropriate model for describing the viscosity of a glass melt is

predominantly twofold. First, the nature of the relaxation processes which govern the flow of a

glass melt as it approaches the glass transition temperature is yet essentially unknown. The effort

of fitting viscosity data with Arrhenius-type relation[2], viz:

(T) 0exp(E/kBT)

does not generally succeed over a wide range of temperature. The viscous flow has low

activation energies at high temperatures and high activation energies at low temperatures.

Accordingly, a two-exponent formula was proposed to describe the variation of viscosity of

glasses,

(T)= Aexp(B/RT)[1+Cexp(D/RT)],

which uses four temperature-independent constants, viz., A,B, C and D. In this model, the low

viscosity range high temperature region_ has the activation energy B while the high viscosity

range _low-temperature region_ has the activation energy (B+D). Glasses such as silica, sodium

disilicate(Na2O–2SiO2), albite (NaAlSi3O8), etc., show such viscosity pattern with a clear

demarcation point between the two regions. In other glass-forming liquids such as B2O3,there are

intermediate regions with gradually changing activation energies.. The viscous flow has low

activation energies at high temperatures and high activation energies at low temperatures. To

describe the viscosity of non-Arhenian liquids several models have been proposed with limited

validity. Among them three models are commonly used they are:

(i) VFT(Vogel-Fulcher-Tamman)

9

(ii) Adam-Gibbs

(iii) Avramov

Of these, the VFT equation has the maximum usage possibly due to its relatively simple form.

Several attempts have been made to provide a theoretical basis for each of these models. For the

VFT model, one such attempt is based on the free-volume concept in the melt,which relates the

relaxation time and the probability distribution of free volume with percolation theory. The AG

model has a quasi theoretical basis that links macroscopic transport properties directly to

thermodynamic properties via the configurational entropy, and the model proposed by Avramov.

relates the fit parameter to the fragility of the melt.The other impetus for the search for an

appropriate model lies in the estimation of viscosity of a glass melt at any temperature, which is

much needed in process technology. This requires a reasonably good fitting equation with least

number of parameters _constants_. In this perspective, a three-parameter equation any of

equations is more preferred than a four-parameter one. Moreover, to determine the constants of

Eq. 2, two reference points are to be chosen from the high-temperature region and two from the

low temperature region.

10

FIG. 2. A plot showing goodness of fit of different model equations on the experimental data

points of SiO2–PbO (57.93:42.07 molar ratio glass).Values of the constants and other parameters

are given in Table 1.

11

TABLE I. Best-fit values for the constants of different model equations as obtained by fitting

experimental data points of SiO2–PbO (57.93:42.07 molar ratio) glass.

The three parameter equation however, require three reference points from any wherein the

temperature region under consideration. In this context, it is worth mentioning that as far as

fitting of an equation and estimation of viscosity are concerned, no three-parameter equation is

distinctly better than the others. To elucidate this, on a set of representative log _vs T data points

for a SiO2–PbO glass (57.93 mol % SiO2+42.07 mol % PbO), we have fitted Eqs. . Figure 1

shows the fit. Along with Eqs. , we have also fitted another equation, viz,

log=log0+B/(logT-logT0)

the significance of which will be explained later. Equation is similar to the VFT equation except

for the fact that while VFT relates log_ with T directly, Eq. relates log with temperature via log

12

T. We can address Eq. as a modified-VFT equation .Figure 1 shows that all the four equations fit

the data points reasonably well. This may be also understood from a comparison of the

goodness-of-fit parameters, which are listed in Table I. Hence, the problem that prevails is not

about choosing any particular fitting equation as better than the other, rather to acquire the values

of the constants either from accurate data from direct measurements or from other sources, if

possible. Efforts have been made to search for possible roots of these constants in the

composition of the melt or in any other parameters derived from the composition. A few attempts

have been made to correlate parameters such as polarization or basicity of the melt with viscosity

,but the results had not been much useful.

2.2 Compositional dependence of viscosity

Studies on the compositional dependence of viscosity have been as common as those on

temperature dependence[2]. The compositional dependence also has been approximated by

various empirical equations. These approaches are primarily based on the idea that the

concentration dependence of a given parameter may be described by an equation, which is linear

or polynomial (or, in some cases, more complicated functional type) with respect to the

concentration (expressed in mole fraction) of the constituent components. A comparative

characterization of these methods is given by Priven. However, contrary to what is observed for

temperature dependence, the models describing the compositional dependence of viscosity are

not equivalent; the adequacy of description largely depends on the choice of a specific equation.

Given this backdrop, we are set to establish a simple relationship of viscosity with a

compositional parameter of the melt, viz., optical basicity (OB). In the following sections we

first describe OB briefly and then show how log_ varies with the OB of a melt. We have

presented here results on four glass systems, viz., SiO2–PbO, SiO2–PbO–Na2O,SiO2–PbO–

K2O, and SiO2–Na2O. At the end, we establish a formula for the viscosity versus temperature

behaviour based on our findings.

13

2.3 VFT Equation:

[3]Two mathematical expressions,the Arrhenian equation and the Vogel-Fulcher-Tamman

equations are commonly used to express the temperature dependence of the viscosity of glass

forming melts. At one extreme we find that the viscosity can often be fitted, at least over limited

temperature ranges by an Arrhenian expression of the form:

0e(Hn/RT)

Where 0 is constant, Hn is the activation energy for viscous flow, R is the gas constant and T is

temperature in Kelvin. In general Arrhenian behaviour is observed within the glass

transformation range and at high temperatures where melts are very fluid. The activation energy

for viscous flow is much lower for the fluid melt than for the high viscosity for the glass

transformation region. The temperature dependence between these limiting regions is decidedly

non-Arrhenian with a continually varying value of Hn over this intermediate region.

A relatively good fit to viscosity data over the entire viscosity range is provided by a

modification of the equation, which effectively includes a varying activation energy for viscous

flow. This expressioin was derived by several workers and is usually called the Vogel-Fulcher-

Tamman(or VFT) equation in recognition of each of their contribution. The VFT equation adds a

third fitting variable T0.to the Arrhenian expression to account for the variability of the activation

energy for viscous flow, and replaces the Hn with a less defined variable B as indicated by the

expression:

0exp(BT0/T-T0)

Where the terms have their usual meaning.

The value of T0 for a given composition is always considerably less than the value of Tg for that

composition. While the VFT equation provides a good fit to viscosity data over a wide

temperature range, it should be used with caution for temperatures at the lower end of the

transformation region, where Hn becomes constant. The VFT equation always overestimates the

viscosity in the temperature regime. The degree of curvature of viscosity/temperature plots can

vary over a wide range due to variations in the value of T0 with respect to Tg. if T0 is equal to

zero the viscosity/temperature curve will exhibit Arrhenian behaviour over the entire viscosity

3

14

region, from very fluid to the transformation range, with a single value for Hn. on the other

hand as T0 approaches Tg, the curvature will increase and the difference between Hn for the

fluid melt and in the transformation region will become very large.

2.4 Avramov Equation:

[4]The Avramov model, an entropy-based description of the effects of temperature and pressure

on structural relaxation times, assumes separability of these two dependences. This implies that

the fragility of glass-formers is independent of pressure. Herein we show that experimental

results for polymethyltolylsiloxane are at odds with this assumption. By introducing a linear

increase of the coordination number of the liquid state with pressure, the model can be modified,

enabling good agreement with experiment to be achieved.

The equation is generally expressed as:

=0exp(Tg/T)^α

Where 0, T0 & α are the fitting parameters of the equation.

0=pre exponential constant

Tg= glass transition temperature

α = fragility index

15

CHAPTER 3: STATEMENT OF THE PROBLEM

1. Analysis of glass viscosity data using VFT and Avramov equation.

2. Finding a correlation between fitting parameters of VFT and Avramov Equation

Chapter 4: EXPERIMENTAL WORK:

16

4.1 Softwares used:

1. MATLAB (6.0 & 6.5):

[5]MATLAB is a numerical computing environment and programming language.

Maintained by The Math Works, MATLAB allows easy matrix manipulation, plotting of

functions and data, implementation of algorithms, creation of user interfaces, and

interfacing with programs in other languages. Although it is numeric only, an optional

toolbox uses the MuPAD symbolic engine, allowing access to computer algebra

capabilities. An additional package, Simulink, adds graphical multi domain simulation

and Model-Based Design for dynamic and embedded systems.

2. ORIGIN 6.0:

[6]Origin is a scientific graphing and data analysis software package, produced by OriginLab

Corporation, that runs on Microsoft Windows. Origin supports various 2D/3D graph types. Data

analyses in Origin include statistics, signal processing, curve fitting and peak analysis. Origin's

curve fitting is performed by the nonlinear least squares fitter which is based on the Levenberg–

Marquardt algorithm (LMA).

4.2 Graphical plots:

Then the graphs were plotted using the above two softwares. At first the graphs were

plotted using the MATLAB 6.0 for the above datas. Following are some of the graphs

plotted by using MATLAB 6.0:

4.2.1 VFT & AVRAMOV PLOT FOR SiO2(52.75 mol%) & PbO(47.25 mol%)

17

VFT Plot for SiO 2(52.75 mol %) & PbO (42.25 mol%)

Figure3. Plot of log(viscosity) in Pascal on Y axis vs Temerature in Kelvin on X axis

18

AVRAMOV Plot for SiO 2(52.75 mol %) & PbO (47.25 mol%):

Figure4. Plot of log(viscosity) in Pascal on Y axis vs Temerature in Kelvin on X axis

19

Simultaneous plot of VFT & AVRAMOV for SiO 2(52.75 mole%) & PbO (47.25 mole%)

Figure5. Plot of log(viscosity) in Pascal on Y axis vs Temerature in Kelvin on X axis.

After the plotting of the graphs the values of the respective fitting parameters were recorded

from the MATLAB command window. Following are the datas of the fitting parameters that are

obtained for the above composition:

20

VFT fitting parameters:

General model:

f(x) = a*exp(b*c/(x-c))

Coefficients (with 95% confidence bounds):

a = 0.004587 (-0.001465, 0.01064)

b = -15.37 (-15.74, -15.01)

c = -743.6 (-1024, -463)

Goodness of fit:

SSE: 0.04651

R-square: 0.9997

Adjusted R-square: 0.9997

RMSE: 0.0682

21

AVRAMOV fitting parameters:

General model:

f(x) = a*exp((b/x).^c)

Coefficients (with 95% confidence bounds):

a = 0.001515 (-0.007652, 0.01068)

b = 9.096e+004 (-3.615e+005, 5.434e+005)

c = 0.4522 (0.1275, 0.7768)

Goodness of fit:

SSE: 0.146

R-square: 0.9992

Adjusted R-square: 0.999

RMSE: 0.1208

Generally the goodness of fit or the ease of fitting of datas into the equation is determined by

the R-square value. The R-square value is computed taking base as 1. The more the R-square

value remains closer to 1, the better the fit is. In the above case R-square value VFT equation is

0.9997, whereas the R-square value for AVRAMOV equation is 0.9992, so VFT equation gives a

good fit as compared to the AVARMOV equation.

22

4.2.2 VFT & AVRAMOV PLOTS FOR VARYING COMPOSITION:

VFT & AVRAMOV for SiO 2(55.36 mole%) & PbO (44.34 mole%):

Figure6. Plot of log(viscosity) in Pascal on Y axis vs Temerature in Kelvin on X axis.

23

VFT & AVRAMOV for SiO 2(51.27 mole%) & PbO (48.73 mole%)

Figure7. Plot of log(viscosity) in Pascal on Y axis vs Temerature in Kelvin on X axis

24

VFT & AVRAMOV for SiO 2(70 mole%) , PbO (10 mole%) & K 2O (20 mol%)

Figure8. Plot of log(viscosity) in Pascal on Y axis vs Temerature in Kelvin on X axis

25

VFT & AVRAMOV for SiO 2(70 mole%) , PbO (20 mole%) & K 2O (10 mol%)

Figure9. Plot of log(viscosity) in Pascal on Y axis vs Temerature in Kelvin on X axis

26

VFT & AVRAMOV for SiO 2(60 mole%) , Na 2O (30 mole%) & CuO (10 mol%)

Figure10. Plot of log(viscosity) in Pascal on Y axis vs Temerature in Kelvin on X axis

27

VFT & AVRAMOV for SiO 2(60 mole%) , Na 2O (20 mole%) & SrO (20 mol%)

Figure11. Plot of log(viscosity) in Pascal on Y axis vs Temerature in Kelvin on X axis

28

VFT & AVRAMOV for SiO 2(65 mole%) , Na 2O (30 mole%) & CaO (5 mol%)

Figure12. Plot of log(viscosity) in Pascal on Y axis vs Temerature in Kelvin on X axis

29

VFT & AVRAMOV for SiO 2(75 mole%) & Na 2O (25 mole%)

Figure13. Plot of log(viscosity) in Pascal on Y axis vs Temerature in Kelvin on X axis

From the graphs it is evident that there should be a strong correlationship between different

fitting parameters of VFT and Avramov equation. Now the question is which parameter should

be taken into account. From the VFT & Avramov equation it is clear that two preexponential

constants are equivalent to viscosity value, but the b of VFT equation which somehow indicates

the activation energy cannot be correlated to (Tg)^ because the latter term is time dependent

property. Now the fragility index which is a function of Cp does not depend on time. The fragility

index depends on to what extent glass structure can be disrupted, so it is logical to find a

correlationship between activation energy and fragility index.

30

4.3 Tabulation:

The VFT & Avramov parameters for different glass compostions are listed below.

4.3.1 VFT Fitting parameters:

After the completion of the graphs, the fitting parameters of VFT are recorded and tabulated.

There are four sets of datas for which the three fitting parameters designated as a, b, c are

calculated. All the sets of datas have different compositions, represented in mole percentage

Set 1:.

SiO2 PbO a b c

59.73 40.27 0.08586 -19.91 -243.6

57.93 42.07 0.007277 -13.61 -872.4

55.36 44.64 0.003408 -14.69 -903.9

52.75 47.25 0.004587 -15.37 -743.6

51.27 48.73 0.00222 -16.34 -775.5

50.46 49.54 0.001233 -16.09 -924.2

49.76 50.24 7.30E-05 -17.78 -1440

48.81 51.19 0.000151 -17.76 -1200

47.75 52.25 0.000362 -17.42 -1013

44.98 55.02 8.56E-05 -18.82 -1139

42.47 57.53 0.000283 -18.87 -850.1

39.84 60.16 1.97E-06 -22.88 -1395

37.71 62.29 2.53E-08 -26.44 -1975

Table 2

31

Set 2:

SiO2 PbO K2O a b c

80 - 20 0.2335 -10.3 -489.4

70 10 20 0.05614 -9.211 -985.6

60 20 20 0.03486 -11.01 -750.4

50 30 20 0.03239 -17.53 -332.4

80 10 10 0.01011 -9.382 -2317

70 20 10 0.08156 -9.131 -859.8

60 30 10 0.001485 -12.7 -1731

50 40 10 0.003908 -15.45 -1344

70 - 30 0.03542 -8.763 -1458

60 10 30 0.02051 -10.72 -994.6

70 30 - 0.5932 1.88E+04 0.1172

60 40 - 0.3603 1.78E+04 0.1385

Table 3

Set 3:

SiO2 Na2O CaO a b c

75 20 5 0.1902 -12.1 -413.5

75 15 10 0.1749 -13.55 -380.7

70 25 5 0.2211 -15.7 -263.1

70 20 10 0.1489 -14.06 -365.4

70 15 15 0.2197 -28.81 -136.9

65 30 5 0.0207 -16.01 -257

65 25 10 0.1685 -15.46 -298.6

75 25 - 0.2838 -13.24 -297.7

70 30 - 0.3219 -18.89 -177

Table 4

32

Set 4:

SiO2 N

a2O

Cu

O

Mg

O

Ca

O

Zn

O

Sr

O

Cd

O

Ba

O

Mn

O

Co

O

Pb

O

a b c

70 20 100.52 -2337

-0.96

7

60 20 100.36

-3.51E+

3 -0.1360 30 10 0.408

1 -3512 -0.7070 20 10 0.278

6 -31.72 -11

60 3010

0.3633

-1.66E+

4 -0.15

70 20 10 0.3825

-1.74E+

4 -0.15

60 20 20 0.2729

-1.56E+

3 -1.92

60 30 10 0.5153

-1.89E+

3 -116

70 20 0.4446

-1.74E+

3 -1.44

70 20 10 0.1596

-1.46E+

1 -319

60 30 10 0.4123

-1.67E+

4 -0.14

70 20 10 0.3601

-2.22E+

1 -144

70 20 10 0.4418

-1.76E+

4 -0.13

60 30 10 0.3456

-1.79E+

4 -0.13 Table 5

33

4.3.2 Avramov Fitting parameters:

After the completion of the graphs, the fitting parameters of VFT are recorded and tabulated.

There are four sets of datas for which the three fitting parameters designated as a, b, c are

calculated. All the sets of datas have different compositions, represented in mole percentage.

Set 1:

SiO2 PbO a b c

59.73 40.27 0.02114 1.70E+04 0.5888

57.93 42.07 0.001988 1.36E+05 0.4143

55.36 44.64 0.001653 1.08E+05 0.4366

52.75 47.25 0.000242 4.33E+05 0.3712

51.27 48.73 0.001146 8.30E+04 0.466

50.46 49.54 0.000989 1.05E+05 0.4468

49.76 50.24 0.001021 9.34E+04 0.4562

48.81 51.19 0.000879 8.38E+04 0.4694

47.75 52.25 0.000829 7.90E+04 0.4743

44.98 55.02 0.000769 6.60E+04 0.4942

42.47 57.53 5.86E-04 6.98E+04 0.4918

39.84 60.16 0.00039 6.08E+04 0.5143

37.71 62.29 0.000343 5.29E+04 0.5321

Table 6

34

Set 2:

SiO2 PbO K2O a b c

80 - 20 0.02972 5.79E+04 0.4176

70 10 20 0.005204 4.12E+05 0.3213

60 20 20 0.003316 2.02E+05 0.3682

50 30 20 0.0038 4.32E+04 0.4978

80 10 10 0.004868 1.86E+06 0.2628

70 20 10 0.001851 1.83E+06 0.2766

60 30 10 0.001969 3.29E+05 0.3526

50 40 10 8.79E-05 1.14E+06 0.3317

70 - 30 0.003146 1.87E+06 0.2688

60 10 30 0.000645 1.68E+06 0.2925

70 30 - 0.001219 2.08E+06 0.2796

60 40 - 0.000504 1.77E+06 0.2953

Table 7

Set 3:

SiO2 Na2O CaO a b c

75 20 5 0.01816 6.63E+04 0.4227

75 15 10 0.02936 4.07E+04 0.4707

70 25 5 0.06566 1.36E+04 0.5752

70 20 10 0.02004 3.98E+04 0.4758

70 15 15 0.1286 6.20E+03 0.7604

65 30 5 0.06235 1.32E+04 0.5798

65 25 10 0.03606 2.10E+04 0.5348

35

75 25 - 0.07202 1.66E+04 0.528

70 30 - 0.1568 6.54E+03 0.6761

Table 8

Set 4:

SiO2 Na2O CuO MgO CaO ZnO SrO CdO BaO MnO CoO PbO a b c

70 20 10 0.532 2231 1.00560 20 10 0.6046 1680 1.20660 30 10 0.652 1782 1.20370 20 10 0.1887 4847 0.799860 30 10 0.6545 1730 1.25970 20 10 0.3846 2725 0.99860 20 20 0.3654 2466 1.10160 30 10 0.6562 1642 1.28470 20 0.5283 2207 1.067

70 20 100.0275

1 2740 0.503560 30 10 0.8085 1500 1.3370 20 10 0.2147 5099 0.735470 20 10 0.9031 1426 1.3660 30 10 0.7737 1391 1.374

Table 9

36

CHAPTER 5: RESULTS AND DISCUSSIONS:

0 0.1 0.2 0.3 0.4 0.5 0.60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1f(x) = 0.00136159709969215 exp( 13.9426029662373 x )R² = 0.747186439388629

Series1Exponential (Series1)

Figure 14. plot of pre-exponential term(a) of VFT on Y axis vs the pre-exponential term(a) of

Avramov on X axis

In the picture it is evident that the pre exponential constants between two equations are

exponentially correlated

37

y = -37.39x + 0.8865R² = 0.8638

-35

-30

-25

-20

-15

-10

-5

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Series1

Linear (Series1)

Plot of activation energy parameter(b) on Y- axis of VFT & fragility parameter(c) on X axis of

Avramov

It is clear from the graph that fragility index bears a linear relationship with the b value of the

VFT equation, till now the constant b was considered as empirical constant now we can postulate

that the b value is a function of fragility index

38

CHAPTER 6: CONCLUSIONS:

It has been shown from that VFT & Avramov equation simultaneously fit glass viscosity data

over wide range of viscosity for several components of silicate based glasses. Two carrelations

have been obtained from these two equations:

1. The relationships between pre exponential constants between VFT & Avramov equation

bears a exponential nature which is :

η0,VFT=0.0014exp(13.943η0.Avramov)

2. The fragility index is linearly correlated with b value of VFT equation & the relationship

between fragility index and b value is as follows:

bVFT= -37.39α+0.8865

39

D. References:

1. Journal of Physics: Condensed Matter, a rheological model for glass forming silica melts

in the system CAS,MAS, MCAS. Daniele Giordano et al 2007 J. Phys.: Condens. Matter

19.

2. Journal of applied physics 100, on the prediction of viscosity of glasses from optical

basicity. P Choudhary, S.K Pal & H.S Ray. Central glass & ceramic research institute,

196 Raja Subodh Chandra Mullick road Kolkata 700 032, India.

3. Viscosity of glass forming melts, page no: 120,121. Introduction to glass science and

technology by J.E Shelby.

4. The Avramov model of structural relaxation.Journal of Non-Crystalline Solids, volume

316, issue 2-3 february 2003, page 413-417. By M.Paluch & C.M Roland. Naval

Research Laboratory, Chemistry Division, Code 6120, 4555 Overlook Avenue, SW,

Washington, DC 20375-5342, USA.Institute of Physics, University of Silesia,

Uniwersytecka 4, 40-007, Katowice, Poland.

5. http://en.wikipedia.org/wiki/MATLAB .

6. http://en.wikipedia.org/wiki/Origin_(software) .

40

E. ANNEXURE:

Following are the composition of the glasses:

Set 1:

SL NO. SiO2 CONTENT(mole %age) PbO CONTENT(mole %age)

9036 59.73 40.27

9037 57.93 42.07

9038 55.36 44.64

9039 52.75 47.25

9040 51.27 48.73

9041 50.46 49.54

9042 49.76 50.24

9043 48.81 51.19

9044 47.75 52.25

9045 44.98 55.02

9046 42.47 57.53

9047 39.84 60.16

9048 37.71 62.29

Table 10

41

Set 2:

Sl No SiO2(mole %age) PbO(mole %age) K2O(mole %age)

22927 80 - 20

22928 70 10 20

22929 60 20 20

22930 50 30 20

22931 80 10 10

22932 70 20 10

22933 60 30 10

22934 50 40 10

22935 70 - 30

22936 60 10 30

22937 70 30 -

22938 60 40 -

22927 80 - 20

Table 11

42

Set 3:

Sl No SiO2(mole %age)) Na2O(mole %age) CaO(mole %age)

2168 80 15 5

2169 80 10 10

2170 75 20 5

2171 75 15 10

2172 75 10 15

2173 75 5 20

2174 77 4.6 18.4

2175 70 25 5

2176 70 20 10

2177 70 15 15

2178 70 10 20

2179 65 30 5

2180 65 25 10

2181 65 20 15

2182 65 15 20

2183 60 35 5

2184 60 30 10

2185 60 25 15

132521 85 15 -

132522 80 20 -

132523 75 25 -

132524 70 30 -

132525 65 35 -

Table 12

43

Set 4:

Sl

No

SiO2 Na2O CuO MgO CaO ZnO SrO CdO BaO MnO CoO PbO

2186 80 20 - - - - - - - - - -

2187 70 30 - - - - - - - - - -

2188 70 20 10 - - - - - - - - -

2189 60 20 20 - - - - - - - - -

2190 60 30 10 - - - - - - - - -

2191 70 20 - 10 - - - - - - - -

2192 60 20 - 20 - - - - - - - -

2193 60 30 - 10 - - - - - - - -

2194 70 20 - - 10 - - - - - - -

2195 60 20 - - 20 - - - - - - -

2196 60 30 - - 10 - - - - - - -

2197 70 20 - - - 10 - - - - - -

2198 60 20 - - - 20 - - - - - -

2199 60 30 - - - 10 - - - - - -

2200 70 20 - - - - 10 - - - - -

2201 60 20 - - - - 20 - - - - -

2202 60 30 - - - - 10 - - - - -

2203 70 20 - - - - - 10 - - - -

2204 60 20 - - - - - 20 - - - -

2205 60 30 - - - - - 10 - - - -

2206 70 20 - - - - - - 10 - - -

2207 60 20 - - - - - - 20 - - -

2208 60 30 - - - - - - 10 - - -

2209 70 20 - - - - - - - 10 - -

2210 60 20 - - - - - - - 20 - -

44

2211 60 30 - - - - - - - 10 - -

2212 70 20 - - - - - - - - 10 -

2213 60 20 - - - - - - - - 20 -

2214 60 30 - - - - - - - - 10 -

2234 70 20 - - - - - - - - - 10

2235 60 20 - - - - - - - - - 20

Table 13

45