EXPERIMENTAL AND ANALYTICAL

INVESTIGATION OF

NONISOTHERMAL VISCOELASTIC

GLASS FIBER DRAWING

by

Xiaoyong Lu

A dissertation submitted in partial ful�llmentof the requirements for the degree of

Doctor of Philosophy(Mechanical Engineering)

in The University of Michigan1999

Doctoral Committee:

Assistant Professor Ellen M. Arruda, ChairpersonAssociate Professor William W. Schultz, Co-ChairpersonAssociate Professor John W. HolmesAssociate Professor David C. MartinAssociate Professor David Mead

c Xiaoyong Lu 1999All Rights Reserved

To my wife Ying and my daughter Yiyang

ii

ACKNOWLEDGEMENTS

I wish to express my gratitude to my advisor, Professor Ellen M. Arruda, for her

supervision, inspiration and encouragement throughout the course of this project and

for her guidance and support during the period of my graduate studies. I would also

like to thank Professor William W. Schultz, for his guidance and instructions and

constant encouragement in every stage of the project, especially in the theoretical

modeling aspect. I also wish to thank Professor David Mead and Professor David

C. Martin for serving on my thesis committee and their inspiration advice and also

for allowing me to use their equipment freely. I thank Professor John W. Holmes for

serving on my thesis committee. I would also like to thank Professor R. E. Robertson

for allowing me to use his equipment for my tests.

I would like also to thank my fellow colleagues and friends, Phil and Paris for

their help and discussions in my four year study and work. I am grateful for the

�nancial support provided by the National Science Foundation.

My special gratitude goes to my wife Ying and my daughter Yiyang for their sup-

port and sacri�ce, without them, I could not have succeeded. I can never adequately

express my thanks to them for all the help and encouragement they have given to

me.

iii

TABLE OF CONTENTS

DEDICATION : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : ii

ACKNOWLEDGEMENTS : : : : : : : : : : : : : : : : : : : : : : : : : : iii

LIST OF FIGURES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : vi

LIST OF TABLES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : xi

LIST OF APPENDICES : : : : : : : : : : : : : : : : : : : : : : : : : : : : xii

CHAPTER

I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . 41.1.1 Review of Experimental Work . . . . . . . . . . . . 41.1.2 Review of Fiber Drawing Modeling . . . . . . . . . 6

1.2 Objectives of the Research . . . . . . . . . . . . . . . . . . . 131.3 The Organization of the Dissertation . . . . . . . . . . . . . . 14

II. EXPERIMENTAL PROCEDURES AND RESULTS . . . . . 16

2.1 Drawing Apparatus . . . . . . . . . . . . . . . . . . . . . . . 162.2 Glass Creep Measurements . . . . . . . . . . . . . . . . . . . 192.3 Fiber Property Measurements . . . . . . . . . . . . . . . . . . 21

2.3.1 Fiber Diameter . . . . . . . . . . . . . . . . . . . . 212.3.2 Birefringence . . . . . . . . . . . . . . . . . . . . . . 252.3.3 Fiber Tensile strength . . . . . . . . . . . . . . . . . 26

2.4 The E�ect of Drawing Parameters on the Maximum Draw Ratio 272.5 Post-Processing . . . . . . . . . . . . . . . . . . . . . . . . . . 302.6 The E�ect of Heat Treatment on Fiber Tensile Strength . . . 302.7 The E�ect of Drawing Parameters on Fiber Birefringence . . 322.8 Fiber Birefringence Relaxation . . . . . . . . . . . . . . . . . 36

III. FIBER DRAWING MODELING . . . . . . . . . . . . . . . . . 40

iv

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . 413.3 Fiber Drawing Modeling . . . . . . . . . . . . . . . . . . . . . 473.4 The E�ect of Viscoelasticity . . . . . . . . . . . . . . . . . . . 523.5 The E�ect of Draw Ratio . . . . . . . . . . . . . . . . . . . . 533.6 The E�ect of Draw Temperature . . . . . . . . . . . . . . . . 563.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

IV. RELAXATION MODELING . . . . . . . . . . . . . . . . . . . . 69

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.2 Stretched Exponential Modeling . . . . . . . . . . . . . . . . 734.3 Je�rey Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 774.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

V. CONCLUSIONS AND RECOMMENDATIONS . . . . . . . . 83

APPENDICES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 86

BIBLIOGRAPHY : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 113

v

LIST OF FIGURES

Figure

1.1 A schematic diagram of ori�ce (a) and preform (b) �ber drawing. . 3

1.2 Gently convergent melt zone of glass containing stained glass owmarkers from [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Sharply convergent melt zone of glass containing stained glass owmarkers from [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 A typical relationship between the speci�c volume and temperatureof glass [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1 Schematic of �ber drawing apparatus. (All dimensions are in mm.) . 17

2.2 Two cases of temperature history of the heater . . . . . . . . . . . . 18

2.3 The heater temperature (in C) along the spinline. The temperaturemeasurements extend 10mm above and 10mm below the furnace. . . 20

2.4 Schematic of creep test �xture used to measure �ber viscosity. . . . 22

2.5 The viscosity versus temperature relation for Borosilicate glass. The�lled circles are creep test results and the solid line is from [2] . . . 23

2.6 Measured diameter variation along the length of a 120 cm �ber drawnat Tm=1215C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.7 Glass �ber birefringence measurement set-up. . . . . . . . . . . . . 26

2.8 Glass �ber tensile strength measurement �xture. . . . . . . . . . . . 27

2.9 The e�ect of drawing temperature on the maximum draw ratio. . . 29

2.10 The heat-up time history of the �bers in the furnace for birefringencemeasurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

vi

2.11 The e�ect of annealing temperature on the �ber Weibull mean tensilestrength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.12 The variation of the as-drawn glass �ber birefringence with draw-ing temperature Tm and cooling rate at constant draw ratio E =ww=wi = 4410. Cooling rate is increased by increasing wi and ww

by the same amount to maintain a constant draw ratio. (i): highcooling rate wi = 0:048mm=s, (ii): low cooling rate wi = 0:027mm=s. 34

2.13 The variation of as-drawn �ber birefringence with draw ratio at con-stant draw temperature (i) Tm = 1150C, (ii) Tm = 1215C. . . . . . . 35

2.14 The birefringence relaxation of glass �ber drawn at Tm = 1215C(E = 4410) for various annealing times and temperatures. Annealingtemperatures: (i) 309C, (ii) 360C, (iii) 387C, (iv) 408C, (v) 511C. . 37

2.15 Birefringence relaxation of glass �ber annealed at 309C from Figure2.14 including longer times. . . . . . . . . . . . . . . . . . . . . . . 38

2.16 The birefringence relaxation of glass �ber drawn at Tm = 1150C(draw ratio=4410) for various annealing times and temperatures.Annealing temperatures: (i) 309C, (ii) 360C, (iii) 387C, (iv) 408C,(v) 511C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.1 The cross-section view of the preform drawing system. . . . . . . . . 43

3.2 Viscosity-temperature correlation for borosilicate glass [3] The lineis the Walther correlation: exp(exp(18:0 � 2:18 ln �)); dashed line:simple exponential: 1:3293 � 1014 exp(�0:01769�); dotted line: Ar-rhenius correlation: exp(�22:8217 + 43113=�). . . . . . . . . . . . . 45

3.3 Comparison of the dimensionless axial temperature pro�les of theglass �ber (Tg = 565C) and the environment, solid line �env, dashedline �. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.4 The �ber radius change along the spinning line. . . . . . . . . . . . 50

3.5 The dimensionless axial and radial stress variation along the spinline. 51

3.6 The dimensionless Kelvin strain distribution. . . . . . . . . . . . . . 52

3.7 The e�ect of b on the dimensionless velocity w. . . . . . . . . . . . . 53

vii

3.8 The e�ect of b on the dimensionless axial stress Szz. . . . . . . . . . 54

3.9 The e�ect of the draw ratio E on dimensionless velocity w. . . . . . 55

3.10 The e�ect of the draw ratio E on dimensionless axial stress. . . . . 56

3.11 The e�ect of the draw ratio E on dimensionless radial stress. . . . . 57

3.12 The e�ect of the draw ratio E on dimensionless radius. . . . . . . . 58

3.13 The e�ect of the draw ratio E and draw temperature Tmax on theKelvin strain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.14 The e�ect of the dimensionless drawing force on Kelvin strain atTmax = 1215C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.15 The e�ect of the draw temperature Tmax on dimensionless velocity,E = 4410. Tmax=1100, 1150, 1200 and 1215C . . . . . . . . . . . . . 62

3.16 The e�ect of the draw temperature Tmax on axial stress, E = 4410.Tmax=1100, 1150, 1200 and 1215C . . . . . . . . . . . . . . . . . . . 63

3.17 The e�ect of the draw temperature Tmax on radius stress, E = 4410.Tmax=1100, 1150, 1200 and 1215C . . . . . . . . . . . . . . . . . . . 64

3.18 The e�ect of the drawing temperature Tmax on neck-down region,E = 4410. Tmax=1100, 1150, 1200 and 1215C . . . . . . . . . . . . . 65

3.19 The e�ect of the draw temperature Tmax and feed speed on Kelvinstrain, E = 4410. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.20 The e�ect of the draw temperature Tmax and feed speed on scaledKelvin strain, E = 4410. . . . . . . . . . . . . . . . . . . . . . . . . 67

4.1 Birefringence relaxation for �bers drawn at Tm = 1215C. Data fromFigure 2.14. Lines are plotted using (13) with b = 0:5, and (i)T = 309C, � = 1000min; (ii) T = 360C, � = 150min; (iii) T = 387C,� = 50min; (iv) T = 408C, � = 20min; (v) T = 511C, � = 5min. . . 74

4.2 Fictive temperature modeling of relaxation. (a) Temperature vari-ation during annealing: (i) Fictive temperature Tf and (ii) �bertemperature T ; (b) Relaxation time evolution; and (c) Birefringencerelaxation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

viii

4.3 Birefringence relaxation for �bers drawn at Tm = 1215C. Data fromFigure 2.14. Lines are plotted using (4.13) with b = 1, and (i)T = 309C, � = 50min, c(T ) = 0:65; (ii) T = 360C, � = 47min,c(T ) = 0:35; (iii) T = 387C, � = 45min, c(T ) = 0:05; (iv) T = 408C,� = 16:7min, c(T ) = 0:0; (v) T = 511C, � = 8:24min, c(T ) = 0:0. . . 78

4.4 Two Je�rey elements in parallel . . . . . . . . . . . . . . . . . . . . 79

4.5 �11 versus annealing temperatures. . . . . . . . . . . . . . . . . . . 80

4.6 Birefringence relaxation for �bers drawn at Tm = 1215C. Data fromFigure 2.14. Lines are plotted using the two Je�rey elements inparallel model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

A.1 Variation of steady-state stress vs. strain rate for a soda-lime-silicaglass at T = 596C showing deviation from Newtonian behavior athigh strain rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

A.2 Schematic representation of the uniaxial stress vs. strain responseof inorganic glass near Tg at various constant applied strain rates. . 89

A.3 Schematic two-dimensional representation of the structure of a hy-pothetical compound A2O3 . . . . . . . . . . . . . . . . . . . . . . . 93

A.4 Two-dimensional representation of the structure of a modi�ed oxideglass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

A.5 Schematic illustration of glass melts under shear . . . . . . . . . . . 96

A.6 Stress vs. time response of soda-lime silicate glass tension tests atvarious strain rates at T=836K. . . . . . . . . . . . . . . . . . . . . 98

A.7 Stress vs. time response of soda-lime silicate glass tension tests atvarious strain rates at T=866K. . . . . . . . . . . . . . . . . . . . . 99

A.8 Stress vs. time response of soda-lime silicate glass for tension testsat various strain rates at T=836K. Curves are model simulationsusing (A.9), and symbols are the data from [4]. . . . . . . . . . . . . 100

A.9 Stress vs. time response of soda-lime silicate glass for tension testsat various strain rates at T=866K. Curves are model simulationsusing (A.9), and symbols are the data from [4]. . . . . . . . . . . . . 101

ix

A.10 Stress vs. time response of soda-lime silicate glass for compressiontests at various displacement rates at T=869K. [5] and comparisonwith the predictions using (A.9). . . . . . . . . . . . . . . . . . . . . 103

A.11 Stress vs. time response of soda-lime silicate glass for compressiontests at various displacement rates at T=902K. [5] and comparisonwith the predictions using (A.9). . . . . . . . . . . . . . . . . . . . . 104

A.12 Stress vs. time response of soda-lime silicate glass for compressiontests at various displacement rates at T=930K. [5] and comparisonwith the predictions using (A.9). . . . . . . . . . . . . . . . . . . . . 105

A.13 Stress vs. time of soda-lime silicate glass for various strain rates atT=930K. [5] and comparison with the prediction using (A.10) and(A.11). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

A.14 Stress vs. time response of soda-lime silicate glass in tension for var-ious strain rates at T=836K. Curves are Marrucci model simulationsusing (A.12)-(A.15), and symbols are the data from reference [4]. . . 110

x

LIST OF TABLES

Table

3.1 Glass and air properties . . . . . . . . . . . . . . . . . . . . . . . . 49

A.1 Parameters used in tension simulations . . . . . . . . . . . . . . . . 99

A.2 Parameters used in compression simulations . . . . . . . . . . . . . 102

xi

LIST OF APPENDICES

Appendix

A. CONSTITUTIVE THEORIES OFGLASS NEAR THE GLASS TRAN-SITION RANGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

A.1 A Viscoplastic Theory . . . . . . . . . . . . . . . . . . . . . . 93A.1.1 Physical Description of Oxide Glasses . . . . . . . . 93A.1.2 The Constitutive Formation . . . . . . . . . . . . . 96A.1.3 Model Prediction for Elongational Tests . . . . . . . 97A.1.4 Model Prediction for Compression Tests . . . . . . . 102

A.2 Viscoelastic Theories . . . . . . . . . . . . . . . . . . . . . . . 107A.2.1 White-Metzner Model . . . . . . . . . . . . . . . . . 107A.2.2 Marrucci Network Model . . . . . . . . . . . . . . . 109

A.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

xii

CHAPTER I

INTRODUCTION

The continuous stretching of viscous molten polymer and glass to form �bers is

important in the textile and telecommunication industries. The current market for

glass �ber reinforcing materials of all types is about 1.8 million tons [6]. Glass �ber

composites are becoming environmentally friendly now that they can be recycled.

Recent advances in �ber-optic technology will continue to increase the worldwide

demand for �ber products. At the same time, the competition in the industry requires

glass �ber manufacturers to raise the e�ciency of production and quality of their

products. Better understanding and prediction of the product properties, such as

�ber tensile strength and anisotropic structure are important.

It is widely known that materials in the form of �bers have mechanical and

physical properties that are substantially di�erent from those in their bulk forms.

For instance, �laments typically have signi�cantly higher tensile strengths than in

the bulk form. The optical properties of the �ber and the bulk also di�er to some

extent. Many factors may contribute to these di�erences, but an important factor

may be the unique temperature-strain history of the �ber manufacturing process.

The �ber forming process usually involves extruding the melt through an ori�ce

and then drawing the melt in air using a high-speed winder. Preform �ber drawing

1

2

is another �ber drawing technique, often used in optical �ber manufacturing. A

schematic diagram of the two processes is shown in Figure 1.1. During melt spinning

molten polymer or glass is extruded through a bushing into cross- owing ambient

air or water spray. The solidi�ed material is taken up at a higher speed than the

velocity at the ori�ce resulting in the formation of a �lament. The preform �ber

drawing process involves feeding a solid glass rod into an axisymmetric furnace. The

softened glass rod is drawn into a �ber by a pair of winders rotating at a signi�cantly

higher speed than that of the feed rollers.

During the drawing process the material undergoes severe thermal gradients and

large mechanical deformations. Fibers made from preforms are both heated and

cooled during drawing, while ori�ce drawn �bers usually are only cooled. Although

the e�ect of processing on the physical properties of �bers di�ers in extent and

type for di�erent categories of materials, it is clear that processing parameters are

critically important. Process parameters a�ecting �ber properties include drawing

temperature, the feed and winder velocities, �ber cooling rate, and draw ratio. Post

processing, such as applying sizing and annealing, also play an important role in

determining �ber properties.

Glass �ber drawing is a well established technique. Tensile strength, diameter

uniformity and optical properties of glass �bers are the most critical. Most previ-

ous studies focus on the e�ect of processing parameters on the glass �ber mechanical

properties such as �ber tensile strength and elastic modulus. However the anisotropic

structure change during the glass �ber drawing plays an important role in the �ber

properties. Optical measurements of the anisotropy in glass �bers are seldom re-

ported in the open literature.

3

(a) (b)

Orifice

Preform

Heater

Fiber

Fiber

Figure 1.1: A schematic diagram of ori�ce (a) and preform (b) �ber drawing.

4

1.1 Literature Review

There have been signi�cant experimental and theoretical e�orts to study the

e�ects of processing parameters on �ber properties. Most published experimental

and theoretical work has been motivated by determining �nal �ber properties of

particular interest. This thesis is concerned with the preform glass drawing process.

The following is a brief review of the e�ects of �ber drawing on properties.

1.1.1 Review of Experimental Work

Early investigations of glass �ber were interested in the relationship between

diameter and strength. The experiments of Gri�th [7] show that when glass �ber

diameter decreases from 1mm to 3�m its tensile strength increases from 170 to 3,400

MPa. Gri�th suggested that the bulk glass surface contains numerous aws as small

as 5�m and that these aws become the starting points of cracks. Flaws of such size

are greatly reduced in thin �bers during the drawing process. Gri�th hypothesized

that the increase in tensile strength of �bers might be due to the orientation of the

molecules at the surface. Many investigators [8, 9, 10] supported this mechanism

for the dependence of strength on diameter. They suggested that the origin of the

aws might be in the molecular orientation in the glass during the drawing process.

Bartenev and Bovkunenko [11] proposed that structural orientation or orientation

of strong bonds increased the strength of �ne borosilicate glass �bers. Prebus et

al. [12] found evidence of ordered structure in super�ne E and C glass �bers (0.1�m

diameter) that they interpreted as indicating ordered molecular structures of up to

100 _A in length supported by electron microscope measurement.

Other investigators have rejected the assumption of structural orientation in

drawn glass �bers. Otto and Preston [13] heated drawn borosilicate glass �ber and

5

showed that thermal relative contraction in the longitudinal and radial directions

were equal. They suggested that �bers must be isotropic and the longitudinal and

transverse bonds must be of equal strength. Otto [14] also showed that the measured

strength of a glass �ber does not depend on �ber diameter under carefully controlled

experimental conditions.

Thomas [15, 16] systematically drew E glass �bers from an ori�ce and mea-

sured the tensile strength of the pristine �bers in an environment not exceeding 40%

relative humidity. He concluded that the strengths of pristine E glass �bers were

independent of the �ber diameter over the range of 5.3-15�m provided the molten

glass temperature was su�ciently high to permit a �ber of uniform diameter to be

produced. He also determined that the strength was independent of the �ber drawing

temperature (in the range of 1200 to 1340C). Thomas [16] later studied the strength

of borosilicate glass in the form of �bers and rods. Although his results showed that

the mean strength of glass rods with diameters of approximately 1.27mm was about

60% of that of �bers with a diameter about 50.8�m, he concluded that since the

maximum strength of glass rods was similar to that of glass �bers, there was little

di�erence between the intrinsic strength of glass rods and �bers. Thomas attributed

the di�erence in mean strength to the assumption that the rods contain sources of

weakness not present in the �bers. However, the draw ratio and �ber diameter in

the Thomas study are signi�cantly di�erent than those in the studies of others in

which large �ber strength changes were noted.

More recent works have shown that the physical properties of glass �bers have

a great deal to do with the processing parameters. Not only does the strength of

a glass �ber depends on its diameter, but its structure shows anisotropic properties

in the axial versus transverse directions of the �ber. Loewenstein and Dowd [17]

6

studied the tensile strengths of E glass, A glass, and a high-strength glass drawn from

an ori�ce. Their results indicate that all glasses show a tensile strength/diameter

relationship at a constant rate of loading, and drawing temperature is also likely

to be a signi�cant factor. Pahler and Br�uckner [18] determined the tensile strength

and elastic constants for pristine E glass �bers and both alkaline earth and alkali

metaphosphate glass �bers drawn from an ori�ce under well de�ned conditions. They

suggested that the Young's modulus, shear modulus and Poisson's ratio gave direct

evidence of orientation and anisotropy of the �ber structure when compared with

the isotropic structure of bulk glass.

Stockhorst and Br�uckner [19, 20] investigated the e�ect of processing parameters

on the physical properties of E glass and phosphate glass �bers drawn from an ori�ce.

By measuring the birefringence of the �bers, they concluded that glass �bers (E

glass) produced at di�erent temperatures but with the same drawing stress showed

an increasing optical anisotropy with increasing temperature. They also showed that

the alkali metaphosphate glass �bers had high birefringence values up to 104nm/mm,

indicating a high degree of orientation of linear phosphate chains.

1.1.2 Review of Fiber Drawing Modeling

Most of the previous theoretical studies of the �ber drawing process focus on

the ori�ce drawing method. In this method, glass melt is generally considered to

be Newtonian. The ow is usually taken to be one dimensional, such that axial

velocity and pressure are independent of the radial coordinate. A general evaluation

of the nonisothermal spinning process has been given by Andrews [21] and Kase

and Matsuo [22]. They have derived a set of simultaneous one-dimensional partial

di�erential equations for both steady-state and transient spinning conditions.

7

Glicksman [23] assumed the glass melt to be Newtonian with a viscosity varying

exponentially with temperature. Since most of the liquid jet attenuation occurs in

the region where the absolute value of the slope of the �ber free surface boundary

is less than one tenth, Glicksman directly formulated a one-dimensional momentum

balance equation. He concluded that the assumption of one-dimensional ow yields

excellent predictions for the behavior of a variable viscosity jet in the region where

the slope of the jet surface is less than one tenth. Matovich and Pearson [24] were

also among the �rst who derived the one-dimensional governing equations carefully.

They set up an expansion procedure, however they did not de�ne a small parameter

for the asymptotic expansion.

Schultz and Davis [25] carried out an asymptotic expansion analysis of the isother-

mal �ber drawing process for Newtonian uids. They applied lubrication type scaling

ideas to the governing equations and developed all expansions in powers of the ra-

tio of the radial to the axial-length scale. Two-dimensional numerical analysis and

visualization experiments have also shown that the one-dimension assumption is

acceptable in both ori�ce and preform �ber drawing. Huynh and Tanner [26] ana-

lyzed the nonisothermal �ber drawing process using a two-dimensional �nite-element

method. They assumed the melt to be Newtonian and found that the ow �eld is

predominantly one-dimensional except within one diameter of the ori�ce.

Geyling and Homsy [1] used visualization experiments to demonstrate that the

assumption of quasi one-dimensional extensional ow, commonly accepted for �bers

drawn from an ori�ce, also applies to �bers drawn from preforms. High temperature

ow markers consisting of stained glass were embedded in the longitudinal mid-

plane of a preform. The preform rod was assembled from two half rounds with

optically at mating surfaces, one of which had the ow markers a�xed by scoring

8

Figure 1.2: Gently convergent melt zone of glass containing stained glass ow mark-ers from [1].

and staining or by printing. The two halves were joined by fusing them at the

softening temperature. Figures 1.2 and 1.3 show the visualization results for gently

convergent and sharply convergent melt zones [1]. The ow markers indicate the

stretch history of a particular uid. Since the ow markers in the neck-down region

show relatively undistorted images, it is assumed that the ow is approximately

one-dimensional elongational ow.

Shah and Pearson [27, 28] studied the nonisothermal �ber spinning process for

Newtonian viscous uids with inertia, surface tension, and gravity e�ects included.

They also examined the e�ects of these factors on the stability of the drawing process

with a parameter representing convective heat transfer. Later Pearson and Shah [29]

extended the nonisothermal analysis to �ber spinning of power law uids. They

studied the relationship among the critical extension ratio, the power-law index, and

the shear thinning e�ect. Paek and Runk [30] modeled the nonisothermal preform

�ber drawing of a Newtonian uid. Convective and radiation heat transfer were con-

sidered in the model. The model predicted the preform neck shape and temperature

distribution within the neck-down region during the drawing of a high silica rod into

a �ber and compared with the measured neck shape.

9

Figure 1.3: Sharply convergent melt zone of glass containing stained glass ow mark-ers from [1].

10

Since the drawing process involves severe thermal gradients for either ori�ce or

preform �ber drawing, isothermal and Newtonian assumptions have little practical

signi�cance. The glass traverses from a Newtonian uid, through the transition range

in which the viscoelastic behavior governs to an elastic solid.

Figure 1.4 shows the typical relationship between speci�c volume (v) and tem-

perature (T) near glass transition [2]. In the �gure, point a is the glass in liquid form

at high temperature. On cooling, the volume gradually decreases along the line abc.

Point b corresponds to Tm, the melting point of the corresponding crystal, which

may be de�ned as the temperature at which the solid and the liquid have the same

vapor pressure or have the same Gibbs free energy. If crystallization does not occur

below Tm, the liquid mass moves into the supercooled liquid state along the line bcf,

which is an extrapolation of the line abc. No discontinuities in the v-T diagram

are observed. As the temperature and hence, energy is lowered the structure of the

liquid rearranges itself into a lower volume along the line bcf. As cooling continues,

the viscosity of the glass rapidly increases. At su�ciently low temperatures, the

state line starts a smooth departure from bcf and soon becomes a near-straight line,

ending at point g when cooled rapidly, or at h when cooled slowly. The smooth curve

between the onset of the departure from the supercooled liquid line and the glassy

state is the glass transition range. It is in this range that the viscoelastic behavior

governs the glass response. Fiber drawing involves heating above the viscoelastic

range and cooling to within the glassy state.

Both viscoelastic and nonisothermal modeling are necessary for glass �ber draw-

ing. Denn and others [31] solved the equations for steady isothermal spinning of a

viscoelastic liquid for a uid model with constant shear modulus and a single constant

relaxation time. They used a slight generalization of the classical Maxwell material

11

Figure 1.4: A typical relationship between the speci�c volume and temperature ofglass [2].

12

as the constitutive equation. Gupta et al [32] studied the polymer �ber-spinning

equations using the Oldroyd B uid constitutive equation. Sridhar and Gupta [33]

conducted isothermal �ber-spinning of an 1850 ppm solution of a polyisobutylene in

polybutene. The velocity pro�les and spinline stresses were measured. The results

are compared with the Oldroyd B constitutive model simulation. Good agreement

was obtained when the e�ects of gravity were properly taken into account. Phan-

Thien [34] also used the Oldroyd B uid to model isothermal polymer �ber spinning.

Schultz [35] presented the one-dimensional �ber-spinning analysis using a general-

ized convected Maxwell model. He found that the non-Newtonian behavior of the

uid must be severely limited for the one-dimensional equations. Nguyen et al [36]

experimentally examined the isothermal �ber-spinning of M1 uid and compared

the results with the Oldroyd B constitutive model. Recently Gupta and others [37]

examined ori�ce glass �ber drawing and its stability by using the generalized upper-

convected Maxwell model. Their results showed that cooling along the spin line

strongly stabilizes the drawing process.

Although some of the previous modeling includes non-isothermal and viscoelastic

e�ects in glass �ber drawing simulations, no attempt has been made to model the

anisotropic structural change introduced by the drawing process. In this study,

we use the Je�rey model as the constitutive equation to model the nonisothermal

viscoelastic preform glass �ber drawing and its property relaxation during post-

processing. Fiber structural anisotropy and other properties are predicted using this

model.

13

1.2 Objectives of the Research

In this study, both experimental and analytical preform �ber drawing investiga-

tions are conducted. Modeling of preform glass �ber drawing will be carried out by

using a nonisothermal viscoelastic model. Polymer or glass �bers must be produced

with speci�ed end-use properties, such as �ber dimensions, elastic modulus, tensile

strength, and refractive index. It is important to understand how processing pa-

rameters a�ect �ber mechanical and physical properties, including tensile strength,

and optical and thermal properties. Since most previous studies concentrate on the

e�ects on mechanical properties, a general understanding of these relationships has

been obtained. However little work has been done on investigating the e�ect of

drawing process on �ber birefringence (i.e. the structural anisotropy). Optical char-

acteristics of �bers such as birefringence are also important properties considered in

�ber products, especially for optical �bers. The structural orientation may a�ect the

transmission of the optical signal. The e�ect of post processing on �ber birefringence

will be conducted to study relaxation of anisotropic structure and its implications

and applications to related processes.

The proposed research has the following objectives: 1) To conduct a systematic

experimental investigation of the e�ect of processing parameters on �ber birefrin-

gence using borosilicate glass preforms. Furnace temperature, feed and drawing

speeds, and draw ratio will be varied to explore the responses of the glass �ber prop-

erties. 2) To characterize the state of as-drawn and thermally treated glass �bers

by means of standard optical and mechanical characterization techniques, i.e. the

�ber birefringence measurement and tensile strength measurement, and relate these

properties to processing parameters. The optical anisotropy of glass �bers will be

14

characterized by birefringence measurements. 3) To develop a nonisothermal, vis-

coelastic ow model of the preform �ber drawing process. This model will be based

on the 1-D ow approximation. Combined conduction, convection and radiation heat

transfer will be included. The model will be used to predict the temperature and

velocity gradients, as well as the axial and radial stresses along the spin line. The

model is also used to indirectly simulate birefringence relaxation during constant

temperature annealing.

1.3 The Organization of the Dissertation

Chapter 2 details the construction of the �ber drawing apparatus, the experimen-

tal procedures, and the results. Glass �ber birefringence measurements are described

and the e�ects of various processing parameters on �ber properties, including �ber

diameter, the maximum draw ratio and �ber as-drawn birefringence, as well as the

relaxation of birefringence under various annealing conditions are studied. The ef-

fect of annealing on �ber tensile strength is also discussed in Chapter 2. Most of the

results in Chapter 2 will soon appear in the Journal of Non-Newtonian Fluid

Mechanics [38].

The analytical study of the preform glass �ber drawing process is described in

Chapter 3. Nonisothermal viscoelastic one-dimensional theory models the drawing

process. A Je�rey model is employed as the constitutive equation and solved simul-

taneously with the conservation equations.

Glass �ber birefringence relaxation is modeled in Chapter 4. Existing theories of

structural relaxation for oxide glasses to model glass �ber birefringence relaxation

are discussed. The �ctive temperature, stretched exponential, and Je�rey models are

examined. Relaxation modeling using these theories are compared with birefringence

15

results. This chapter is also described in [38].

The dissertation concludes with a summary of new �ndings and their implications

and applications to �ber drawing related processes in textile and telecommunication

industries. Suggestions and recommendations for future work are also presented

in Chapter 5. Appendix A discusses alternative constitutive theory approaches,

a viscoplastic rate activation theory and a viscoelastic uid network theory. The

comparison of these theories with experimental results is conducted in Appendix A.

CHAPTER II

EXPERIMENTAL PROCEDURES AND

RESULTS

2.1 Drawing Apparatus

Our apparatus for continuously drawing �bers from cylindrical glass preforms is

shown in Figure 2.1. A cylindrical electric resistance heater softens and melts the

glass preforms. The heater coil made from Chromel-A wire is �tted between a Mullite

tube and a fused silica core. Insulation materials are cast around the Mullite tube.

Thermocouples embedded along the heater liner monitor the temperature during

�ber drawing. The thermocouple placed in the middle of the liner is used to control

the heater temperature. The heater power is supplied by a voltage transformer

(manually controlled) and a proportional controller (DP{26, Omega).

The glass preform is fed into the heater by a pair of feed rollers controlled by a

variable speed motor with a gear motor to reduce the rotation speed. The feed rollers

are designed to feed preforms with diameter between 1mm and 10mm. The preform

is fed into the heater and is heated (by conduction, convection, and predominantly

radiation) to soften and melt the glass. The winder is driven by a variable speed

motor with PID control (Computer Boards Inc.) to insure a constant speed under

variable load drawing conditions. The air-cooled �ber is collected after the winder or

16

17

Chromel-A coil

19.4

112

feed rollerglass preform

fused silica

insulation

winderfiber

152

112

mullite

TachometerPID ControlSystem

Thermometer

Thermometer

Thermometerwith AnalogOutput

ProportionalControlSystem

ComputerData Acquisition

System

Dimensions in mm

Figure 2.1: Schematic of �ber drawing apparatus. (All dimensions are in mm.)

by wrapping the �ber around the winder. The twin cylindrical three-inch diameter

winders are made of aluminum with a rubber coating. Insulating plates at the top

and the bottom of the heater with readily modi�ed ori�ce diameters adjust the

bulk convective ow through the furnace. The diameters of the ori�ce in the top

and bottom plates used in this study are both 10mm. The drawing distance (the

distance from the bottom of the furnace to the winder) can be varied between 10cm

and 50cm by moving the winder up or down on the frame.

One high temperature case and one low temperature case of heater temperature

history are displayed in Figure 2.2. After an initial transient, the temperature (using

a proportional controller Omega DP-26) is stable with a small variation of �2C

degrees as measured at the middle point of the heater liner.

Three millimeter diameter Borosilicate glass preforms (Corning 7740; 80.6% SiO2,

18

0 10 20 30 40 50 60 70 800

200

400

600

800

1000

1200

Hea

ter

Tem

pera

ture

s (C

)

Time (min.)

Figure 2.2: Two cases of temperature history of the heater

13.0% B2O3, 4.0% Na2O, 2.3% Al2O3, 0.1% miscellaneous by weight; ASTM anneal-

ing point (glass transition temperature) 565C) were drawn at various feed and winder

speeds and a range of drawing temperatures. Borosilicate glass was used instead of

E-glass because it is a silicate glass easily obtained in rod form. An optical micro-

scope is used to measure �ber diameters and their variation under stable drawing

conditions and to compare with theoretical values determined from the draw ratio.

The draw ratio is de�ned as the ratio of drawing speed to feed speed. The e�ect of

drawing temperature on the maximum draw ratio (draw ratio at which �ber breaks)

has also been examined. A typical temperature pro�le measured by a thermocouple

traversed down the centerline of the cylindrical heater when no �ber is being drawn

19

is illustrated in Figure 2.3. The drawing temperature Tm is de�ned as the maximum

temperature along the spinline. In subsequent tests the heater was controlled by Tm

as estimated by a thermocouple placed just inside the fused silica at the approximate

height of the maximum shown in Figure 2.3. Tm was varied in the range of 1100C to

1215C to study the e�ect of drawing temperature on birefringence. The location of

the temperature maximum can be controlled by convection suppression and radiation

shields at the top and bottom or variable winding spacing. For the uniform coil

electric heater, the maximum occurs slightly higher than the heater midpoint due to

natural convection.

The preforms were usually drawn at a feed speed of wi = 0:048 mm/s, and draw-

ing speed of ww = 212:0 mm/s to produce 45�m diameter �bers (Subscripts i and w

refer to quantities at the inlet and winder, respectively.). A similar series of tests at

wi = 0:027 mm/s and ww = 119 mm/s examined the e�ect of cooling rate on birefrin-

gence. A series of tests at Tm=1150C and 1215C for wi = 0:048 mm/s was conducted

at various draw speeds to study the e�ect of draw ratio on birefringence. Fibers were

carefully collected (without sizing) after the winder and stored in dessicators prior

to preparation for annealing and/or birefringence measurements.

2.2 Glass Creep Measurements

For modeling purposes, the rheological properties (primarily viscosity) of the

borosilicate glass �bers have been measured. Low temperature (511C and 618C)

viscosities were measured by using �ber creep tests.

Figure 2.4 is a schematic of the creep test experiment. Glass beaded ends were

formed at the two ends of a thin �ber (120�m in diameter) The �ber was suspended

from a steel frame by one bead and a weight was hung from the other bead. The

20 0

200

400

600

800

1000

1200

0 20 40 60 80 100 120 140 160 180

H

ea

ter

Te

mp

era

ture

(C

)

z (mm)

Figure2.3:

Theheatertemperature

(inC)alongthespinline.

Thetemperature

measurements

extend10mm

aboveand10mm

below

thefurnace.

21

apparatus was put into a constant temperature furnace at 511C and �ber length was

measured as a function of time for a 24 hour period. This procedure was repeated

at 618C. Calipers with an accuracy of 0.01mm measured the �ber length. Using this

simple elongational (Trouton) viscometer, the viscosity of the glass was calculated

(assuming small elongation) from

� =LF

3�r2v; (2.1)

where � is viscosity, L is the initial �ber length, F = mg is the force acting on the

�ber (the mass of the weight m is 17g), r is the original �ber diameter, and v is the

average creep velocity computed from �ber elongation and time. Typical changes

in L were about 10mm at 618C and 1mm at 511C. The measured viscosities of

Borosilicate glass were 5.3�1011P at 618C and 9.3�1013P at 511C. These are within

the 5% of the published values shown in Figure 2.5 obtained using a similar method

in the low temperature range.

2.3 Fiber Property Measurements

2.3.1 Fiber Diameter

Glass �bers, especially optical glass �bers, are best produced with uniform di-

ameter under stable conditions. To examine the uniformity of �bers drawn with

the preform �ber drawing apparatus, �bers were drawn from a 3mm preform. Fiber

diameter was measured using a microscope equipped with NIH Image software. The

resolution of this measuring method is 0.32�m, and its accuracy is �1.0�m. The

typical diameter variation along the �ber length for 59�m diameter �bers is shown

in Figure 2.6. Under stable drawing conditions the preform �ber drawing apparatus

draws uniform �bers to within �1�m.

22

Glass Fiber

17g Weight

Frame

Furnace Chamber

Steel

L=40mm

120µm

Figure 2.4: Schematic of creep test �xture used to measure �ber viscosity.

23

500 600 700 800 900 1000 1100 1200 1300 140010

2

104

106

108

1010

1012

1014

1016

Temperature (C)

Vis

cosi

ty (

P)

400

Figure 2.5: The viscosity versus temperature relation for Borosilicate glass. The�lled circles are creep test results and the solid line is from [2]

24

0 20 40 60 80 100 12050

55

60

65

Distance (cm)

Dia

met

er (

µm)

Figure 2.6: Measured diameter variation along the length of a 120 cm �ber drawnat Tm=1215C.

25

2.3.2 Birefringence

It is di�cult to measure the birefringence of a single glass �ber since the re-

tardation per unit thickness of glass is quite small (approximately 10�7 to 10�5).

Br�uckner [19] used an isotropic circular quartz tube �lled with many �bers to increase

the retardation signal. The �bers in the tube were immersed in a refractive index

matching liquid. This approach was followed here but with an optically-isotropic

square quartz tube to eliminate scattering and refraction from a round tube. Fiber

bundles were collected, packed and pulled through 2mm inner thickness square quartz

tubes to increase the retardation signal. Up to �ve �ber tubes were stacked to fur-

ther increase the retardation for measurement of low birefringence. The tube and

�bers were then immersed in an index of refraction matching uid to reduce light

scattering from the �ber surfaces.

A polarizing microscope equipped with a tint plate (530nm) �rst veri�ed the

existence of optical anisotropy in glass �bers from the color variation. Quantitative

�ber birefringence was measured with a polarizing microscope out�tted with a light

intensity meter as illustrated in Figure 2.7. The axial direction of the �ber sample

is aligned at 45 degrees to the polarizer axis. The light intensity transmitted is

measured and recorded by the intensity meter. The birefringence of the sample is

calculated from

4n = ��

h�sin�1

sI

Io(2.2)

where 4n is sample birefringence, � is a calibration coe�cient of the polarizing

microscope, � is the wavelength of the monochromatic light (589.6 nm), and h is the

e�ective thickness of the glass �bers in the tube. The e�ective thickness of the glass

�bers is computed from the weight, length and density of the �bers as h =�w�l

�1=2.

26

Photometer

Light SourceFilter

PolarizerSample

Analyzer

45

Figure 2.7: Glass �ber birefringence measurement set-up.

Io is the light intensity when the polarizer, analyzer and the �ber samples are all

aligned in the same direction. I is the light intensity when the sample is rotated 45

degrees and the polarizer and the analyzer are crossed as shown in Figure 2.7.

2.3.3 Fiber Tensile strength

Tensile strength is one of the most important characteristics of glass �ber products

used as reinforcement in composites and as optical waveguides in telecommunication.

Fiber tensile strength was measured using a setup shown in Figure 2.8. A �ber

sample was mounted on a pair of pulleys with a 35mm gage separation and �xed with

adhesive tape away from the point of initial roller contact. The test was conducted

at constant displacement rate on an Instron machine (Model TMS).

27

Gage Length35 mm

PP

Fiber

Figure 2.8: Glass �ber tensile strength measurement �xture.

2.4 The E�ect of Drawing Parameters on the Maximum

Draw Ratio

Draw ratio is de�ned as the ratio of the winder drawing speed to the preform feed

speed. One of the important issues in �ber drawing is to continuously draw �bers

without draw resonance or breakage. Draw resonance is a temporal instability in �ber

spinning caused by a constant force condition [25]. A regular and sustained periodic

variation in the drawn �lament diameter occurs when draw resonance is encountered.

This phenomenon can interrupt the production process and greatly reduce outputs of

production. Variation in diameter and breakage may result from many other factors,

such as unsteady feed and winding rates, temperature perturbations in the heater,

and heating and cooling variations.

Since the ow �eld of the drawing process can be modeled as a relatively simple

quasi-one-dimensional approximation, numerous investigations have concentrated on

the analytical and numerical analysis for draw resonance. A critical draw ratio

(the draw ratio at which draw resonance occurs) has been obtained under speci�ed

drawing conditions [39]. Experimental studies, however, are relatively rare in the

28

literature.

We brie y studied the drawing instability experimentally to explore if draw res-

onance occurs at �ber breaks. Maximum draw ratios are determined for a given

heater temperature by gradually increasing winder speed and hence, the draw ratio

until the �ber breaks. The draw ratio is increased incrementally, and the drawing of

�bers is restabilized; if there is no breakage, the draw ratio is again increased and

the process is repeated until the �ber breaks. The e�ect of drawing temperature on

the maximum draw ratio is shown in Figure 2.9. Each point represents the average

of three tests.

The maximum (or critical) draw ratio we obtained is generally higher than the

critical draw ratio (draw resonance) numerically acquired for nonisothermal viscoelas-

tic ow for ori�ce ow [37]. Since draw resonance causes large diameter variations,

the diameter of the broken �ber was examined to monitor for draw resonance. The

diameter variation remained small (as in Figure 2.6) even for �bers drawn to the

highest draw ratio in this study, suggesting that no draw resonance occurs for the

preform �ber drawing. The cause of �ber failure is fracture as the tensile strength is

exceeded.

The maximum draw ratio increases with drawing temperature as shown in Figure

2.9. As drawing temperature increases glass viscosity decreases, thus the drawing

force decreases, allowing greater extension ratios before breakage. As T increases

still further, the viscosity becomes very low and a capillary instability may occur.

Further work is required in this area to determine the ultimate tensile strength as a

function of temperature and to determine why draw resonance is suppressed.

29

1020 1040 1060 1080 1100 1120 1140 11602000

2500

3000

3500

4000

4500

5000

5500

Drawing Temperature (C)

Max

imum

Dra

w R

atio

Figure 2.9: The e�ect of drawing temperature on the maximum draw ratio.

30

2.5 Post-Processing

The e�ect of post-processing, i.e. annealing on glass �ber tensile strength and

birefringence is examined experimentally. The birefringence relaxation of �bers was

studied by annealing drawn �bers for various times and temperatures. The sam-

ples were prepared as described above for birefringence measurement and annealed

at constant temperature in a furnace maintained to within �2C. Fibers were air

quenched after annealing and held in a dessicator until tested. Two sets of �bers

drawn at temperatures Tm=1150C and 1215C at a draw ratio of 4410 were studied

for the birefringence relaxation. The annealing temperatures ranged up to 511C.

The annealing time was recorded as the residence time in the furnace without regard

to heat-up or cool-down times. The heat-up and cool-down times have been mea-

sured for some annealing processes by placing a thermocouple in the middle of the

�ber bundle. These results show that after placing the specimen in the furnace, the

temperature approaches the speci�ed temperature exponentially with time constants

in the range of 10 to 30 seconds shown in Figure 2.10. Cooling after removing the

�bers from the furnace shows a similar time constant.

2.6 The E�ect of Heat Treatment on Fiber Tensile Strength

The e�ect of post-processing by annealing on the strength of glass �bers was

brie y investigated in terms of annealing temperature and time. The as-drawn glass

�bers were heat treated in a furnace for one hour. The nominal annealing tempera-

tures were 100, 300 and 500C. The Weibull [40] mean tensile strength of glass �bers

was tested using the apparatus sketched in Figure 2.8. The Weibull mean tensile

strength is expressed as Ps = exp(�( ��0)m), where Ps is the survival probability, �

is the applied stress, �0 is the stress corresponding to a survival probability of 1eor

31

0 50 100 150 200 250 300 3500

100

200

300

400

500

600

Time (s)

Fib

er T

empe

ratu

re in

the

Fur

nace

(C)

Figure 2.10: The heat-up time history of the �bers in the furnace for birefringencemeasurements.

32

37%, and m is the Weibull modulus. A �ber sample was mounted on a pair of pul-

leys with a 35mm gage separation and �xed with adhesive tape away from the point

of initial roller contact. The test was conducted at constant displacement rate on

an Instron machine (Model TMS). Fiber samples which broke at the roller contact

point were discarded. Figure 2.11 shows the results of the Weibull mean strength

and modulus m versus the annealing temperature. Each point represents 12 �ber

samples. The tensile strength is shown to decrease with an increase in the annealing

temperature. A possible reason for this decrease may be that annealing causes aws

on the surface of glass �bers, resulting in the decrease of the tensile strength. The

as-drawn strength is similar to that previously reported [41].

2.7 The E�ect of Drawing Parameters on Fiber Birefrin-

gence

Strength measurements are di�cult to perform and the data show large scatter

as expected from tensile strengths of brittle materials. Therefore we sought and

found a related measurement, birefringence, that is more readily measured and re-

peatable. Figure 2.12 shows the variation of the as-drawn �ber birefringence with

drawing temperature for two cooling rates. The birefringence of the as-drawn glass

�bers decreases with increasing drawing temperature at constant draw ratio. Each

point represents the average of three measurements with a deviation �0:03 � 10�5.

The decrease in birefringence is moderate for drawing temperatures up to 1200C;

above 1200C (near the working point, viscosity of 103Pa�s) the birefringence of the

�bers drops signi�cantly with draw temperature. This suggests that as the drawing

temperature approaches the working point the drawing tension and stress decrease

quickly, causing the rapid reduction of the birefringence. The cooling rate during

33

Weibull mean Strength (MPa)

100 200 300 400 500

Annealing Temperature (C)

200.0

300.0

400.0

100.0

m=3.2m=4.2

m=2.4

m=5.8

m: Weibull Modulus

(as drawn)

Figure 2.11: The e�ect of annealing temperature on the �ber Weibull mean tensilestrength.

34

0

2e-06

4e-06

6e-06

8e-06

1e-05

1.2e-05

1.4e-05

1100 1120 1140 1160 1180 1200 1220

Bire

frin

genc

e

Drawing Temperature (C)

(i)

(ii)

Figure 2.12: The variation of the as-drawn glass �ber birefringence with drawingtemperature Tm and cooling rate at constant draw ratio E = ww=wi =4410. Cooling rate is increased by increasing wi and ww by the sameamount to maintain a constant draw ratio. (i): high cooling rate wi =0:048mm=s, (ii): low cooling rate wi = 0:027mm=s.

drawing was varied by proportionally changing the feed speed and winder speed to

maintain a constant draw ratio. Figure 2.12 results show that at constant draw tem-

perature Tm and draw ratio E, the as-drawn birefringence increases with increasing

cooling rate. This implies that a faster cooling rate is more e�ective at \freezing"

the anisotropic elastic strains developed during drawing deformation.

The e�ect of draw ratio on �ber birefringence was examined by varying the winder

speed at constant feed speed shown in Figure 2.13. The draw temperature was

constant at Tm = 1150C and Tm = 1215C and the feed speed at wi = 0:048mm/s.

35

0

2e-06

4e-06

6e-06

8e-06

1e-05

1.2e-05

0 500 1000 1500 2000 2500 3000 3500 4000 4500

Bire

frin

genc

e

Draw Ratio

(i)

(ii)

Figure 2.13: The variation of as-drawn �ber birefringence with draw ratio at constantdraw temperature (i) Tm = 1150C, (ii) Tm = 1215C.

Drawing speeds ww varied from 0.028m/s to 0.212m/s resulting in draw ratios E of

587 to 4410, and corresponding �ber diameters of 120�m to 45�m.

Fiber birefringence increases almost linearly with draw ratio in the range exam-

ined. At constant feed speed, the �ber diameter is inversely proportional to the

square of the drawing speed. Birefringence of glass �bers increases with increasing

draw ratio or with decreasing �ber diameter. These data con�rm the expected trend

of increasing birefringence with increasing tensile stresses or drawing forces. This

would indicate that high draw ratios and low draw temperatures are preferential in

developing birefringence.

36

2.8 Fiber Birefringence Relaxation

Figure 2.14 shows the scaled birefringence relaxation of �bers drawn at Tm =

1215C and draw ratio of 4410 (leading to an as-drawn birefringence of 4:72� 10�6)

annealed at various times and temperatures. Scaled birefringence is de�ned as the

relaxed birefringence divided by the as-drawn birefringence. Figure 2.14 includes

results for all temperatures and annealing times up to 180 minutes. Fibers were

annealed at 309C for longer times and the results are reported in Figure 2.15. At

annealing temperatures of 408C and above birefringence relaxation is rapid and com-

plete within 60 minutes annealing time. Fibers annealed at temperatures of 360C

and below appear to not fully relax as their birefringence asymptotes after 180 min-

utes. Fibers annealed at 309C for longer times show no further relaxation after 24

hours. Fibers annealed at 387C show an intermediate behavior whereby relaxation

is complete after an excess of 200 minutes.

Figure 2.16 contains the relaxation results for �bers drawn at Tm = 1150C and

a draw ratio of 4410 (leading to an as-drawn birefringence of 1:21 � 10�5). The

relaxation behavior is similar to that shown in Figure 2.14. Complete birefringence

relaxation occurs quickly at annealing temperatures of 408C and above; the birefrin-

gence does not fully relax at annealing temperatures below 360C, and intermediate

relaxation behavior is observed at 387C.

37

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200

Sca

led

Bire

frin

genc

e

Time (min)

(i)

(ii)

(iii)

(iv)(v)

Figure 2.14: The birefringence relaxation of glass �ber drawn at Tm = 1215C(E = 4410) for various annealing times and temperatures. Annealingtemperatures: (i) 309C, (ii) 360C, (iii) 387C, (iv) 408C, (v) 511C.

38

0

0.2

0.4

0.6

0.8

1

0 200 400 600 800 1000 1200 1400

Sca

led

Bire

frin

genc

e

Time (min)

Figure 2.15: Birefringence relaxation of glass �ber annealed at 309C from Figure2.14 including longer times.

39

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200

Sca

led

Bire

frin

genc

e

Time (min)

(i)

(ii)

(iii)

(iv)(v)

Figure 2.16: The birefringence relaxation of glass �ber drawn at Tm = 1150C (drawratio=4410) for various annealing times and temperatures. Annealingtemperatures: (i) 309C, (ii) 360C, (iii) 387C, (iv) 408C, (v) 511C.

CHAPTER III

FIBER DRAWING MODELING

3.1 Introduction

Kase and Matsuo [22] were among the �rst to analyze nonisothermal �ber spin-

ning. However, their studies did not take into account the viscoelastic behavior in

elongational ow. Glicksman [23] also assumed the glass melt to be Newtonian with

a viscosity varying exponentially with temperature. Matsui and Bogue [42] analyzed

melt spinning with a generalized nonisothermal viscoelastic theory. Recently Gupta

and others [37] conducted ori�ce glass �ber drawing and its stability by using the

generalized upper-convected Maxwell model.

It is important to choose the constitutive equation to appropriately represent the

material behavior during �ber drawing. We seek the simplest model possible that

allows for residual strain or stress to remain in drawn �ber. For completeness we ex-

amine in Appendix A several additional constitutive models that exhibit viscoplastic

and strain softening behavior found in other simple, elongational experiments [4].

In this chapter, the nonisothermal viscoelastic glass preform �ber drawing pro-

cess is analyzed. A temperature dependent viscosity is assumed, but density, shear

modulus and all other quantities are considered constant.

A generalized Je�rey viscoelastic model is used to simulate the preform glass

40

41

drawing process. The e�ect of drawing process parameters on the �ber properties are

studied. Anisotropic structural change, i.e. glass �ber birefringence, is also analyzed

using the elastic \frozen" strain due to rapid cooling. The momentum and energy

equations are solved simultaneously and the measured heater temperature pro�le is

the environmental temperature in the energy equation. The e�ect of viscoelasticity

on the �ber drawing is studied by comparing the results of Newtonian, Maxwell and

Je�rey models.

3.2 Governing Equations

Consider a cylindrical glass preform of initial radius R0 inside a heater, as shown

in Figure 3.1. The temperature pro�le of the heater is shown previously in Figure

2.3. The glass preform temperature at z = 0 is well below its transition temperature.

The point z = 0 is 10mm above the heater top ori�ce. (Since simulation shows that

most deformation occurs at around z = zm where T (zm) = Tm, it is not so important

where z = 0 is de�ned.). As the preform descends into the heater its temperature

increases until it reaches a maximum temperature Tmax close to where the maximum

heater temperature Tm occurs. The glass preform diameter begins to contract under

tension at temperatures above its softening point.

The radius of the elongating preform as a function of axial position is denoted

by R(z). As R(z) decreases, other variables such as the velocity, axial stress and

radial stress change along the spinline. The �ber starts to cool as it descends to

the lower edge of the heater. The glass �ber \freezes" into a solid at a temperature

below its transition range at z = L, a point 10mm below the heater bottom. As

with the origin, the de�nition of L is not so important as long as the temperature is

the temperature is close to the ambient temperature there. At this point, the �ber

42

is drawn past the winder at a given average velocity ww (the subscript w refers to

the quantities at the winder).

A di�erential Je�rey constitutive equation is chosen to simulate the glass �ber

drawing process in this chapter and the post processing birefringence relaxation of

glass �bers in the next chapter. The Cauchy stress tensor, T is

T = �pI+ S; (3.1)

where I is the identity tensor, p is the pressure and S is the extra Cauchy stress

tensor, given by

S + �1

DSDt � a(S �D+D � S)

!= 2�(�)

D+ �2

DDDt

!; (3.2)

Here � is the temperature dependent Newtonian viscosity, a is a rate parameter, �1

and �2 are the relaxation and retardation times, respectively, D is the deformation

rate tensor given by

D =1

2(rv +rvT ); (3.3)

where v is the velocity vector, and rv is the gradient of v; vT is the transpose of v.

and D

Dtis the Jaumann derivative such that

DSDt =

DS

Dt+

1

2(! � S� S � !); (3.4)

where ! is the vorticity tensor de�ned as

! =1

2(rv �rvT ): (3.5)

In the following model, the shear modulus of the glass is assumed constant unless

otherwise noted. Hence, the relaxation time �1 = �=G has the same temperature

dependence as the viscosity in accordance with the thermo-rheological simplicity

43

0

z=0

z=L

R(z

)R

r

z

0

200

400

600

800

1000

1200

0 20 40 60 80 100 120 140 160 180

H

eate

r T

empe

ratu

re (

C)

z (mm)

Figure

3.1:

Thecross-sectionviewof

thepreform

drawingsystem

.

44

(TRS) assumption [43]. For simplicity the retardation time �2 is also assumed to be

proportional to �1, �2 = b�1, where b is a constant.

The exponential temperature-viscosity correlation is widely used in glass �ber

drawing modeling [23]; the exponential correlation typically �ts viscosity data in the

moderate to high temperature range. An exponential correlation does not capture

the temperature dependence of borosilicate glass over a broad temperature range.

Although an Arrhenius function is valid for all temperatures (other than absolute

zero), it did not �t the data well in Figure 3.2. The often used modi�ed Arrhenius

model, where � is replaced with �� �ref �ts higher temperature data but is singular

at � = �ref and �ref is usually larger than ambient temperature. A Walther tempera-

ture viscosity correlation involves only two parameters [44] and �ts the experimental

viscosity data for borosilicate glass over a wide range of temperatures. The Walther

viscosity-temperature correlation is given by

� = �0 exp(exp(v0 � v1 ln �)); (3.6)

where � is the Newtonian viscosity and �0, v0 and v1 are correlation parameters, and

� is the temperature in degrees Kelvin. (Since dimensionless governing equations are

solved, �0 is canceled out when scaling the equations.) This correlation is better for

our purpose because it is not singular except at absolute zero temperature. Equation

(3.6) with v0 = 18:0 and v1 = 2:18 is compared with the data [3] in Figure 3.2. and

found to �t the data well over a large temperature range. The Walther correlation

is used to describe the temperature dependent viscosity in the following modeling.

The heat transfer coe�cient including convection and radiation is chosen in

lumped form [22] as

h =k12R

C

2�1wR

�1

!m

; (3.7)

45

600 800 1000 1200 1400 1600

105

1010

1015

1020

1025

1030

1035

1040

Temperature (K)

Vis

cosi

ty (

P)

Figure 3.2: Viscosity-temperature correlation for borosilicate glass [3] The line is theWalther correlation: exp(exp(18:0�2:18 ln �)); dashed line: simple expo-nential: 1:3293�1014 exp(�0:01769�); dotted line: Arrhenius correlation:exp(�22:8217 + 43113=�).

46

where k1, �1, and �1 are the thermal conductivity, density, and the viscosity of

the ambient air at room temperature, respectively. The coe�cients C and m are

determined from experiments [45]. To combine the e�ects of convection and radiation

in the heat transfer coe�cient, the constants C and m are modi�ed from those in

Kase and Matsuo [22].

Lubrication scaling [25] of the governing equations is employed for all expansions

in powers of � = Ri=L. The following dimensionless variables are de�ned:

R�=

RRi, z� = z

L , u�=

u�wi

, w�=

wwi, �� = �

�m

�� = ���1�m��1

, G�=

Gwi�m=L

, T� = T

wi�m=L

where the superscript * refers to the nondimensional quantities, and it is dropped

in the future equations for notational convenience. The subscript i represents the

initial value at z = 0, G is the shear modulus of the borosilicate glass, and �m is the

viscosity of the glass corresponding to the maximum temperature Tmax. The Peclet

and Biot numbers are de�ned as follows:

Pe =�wiRiCp

k�

Bi = 1

�2hRi

k

where �, Cp, and k are the density, speci�c heat, and thermal conductivity of borosil-

icate glass, respectively. The Peclet number Pe is an inverse dimensionless thermal

conductivity and the Biot number is a dimensionless heat transfer coe�cient. These

parameters are assumed to be constant for simplicity. Glass properties in the transi-

tion range are chosen for the model simulation. By requiring the Biot number to be

scaled in the above manner with Bi = O(1), the leading-order solution for � depends

47

only on the axial coordinate consistent with the one-dimensional assumption.

The unsteady one-dimensional dimensionless mass and momentum conservation

equations are obtained from the leading-order equations following the procedure of

Gupta et al [37].

@

@t(R2) +

@

@z(wR2) = 0; (3.8)

@

@z[R2(Tzz � Trr)] = 0 (3.9)

The unsteady energy equation includes axial conduction, advection and heat

transfer through the �ber surface to the environment due to convection. The dimen-

sionless energy equation gives:

Pe

@�

@t+ w

@�

@z

!=

1

R2

@

@z

R2@�

@z

!� 2Bi

R(� � �env(z)); (3.10)

where �env(z) is the dimensionless temperature of the environment. The above di-

mensionless conservation laws for mass, momentum, and energy along with the con-

stitutive equation relation (3.2) form the governing set of equations for the glass �ber

drawing analysis.

3.3 Fiber Drawing Modeling

This study only considers the steady-state drawing process; the governing equa-

tions are simpli�ed by removing the partial derivative with respect to time. The

mass conservation equation becomes:

R =1pw

(3.11)

The momentum conservation equation is simpli�ed to

Szz � Srrw

= F; (3.12)

48

where F is the dimensionless axial force independent of z, and the axial energy

conservation equation is

d2�

d2z� 1

w

dw

dz+ Pe w

!d�

dz� 2pwBi(� � �env) = 0; (3.13)

The constitutive equation (3.2) for elongational ow is simpli�ed to

�1w@Szz@z

+

1� 2a�1

@w

@z

!Szz � 2�

@w

@z� 2��2w

@2w

@z2= 0; (3.14)

�1w@Srr@z

+

1 + a�1

@w

@z

!Srr + �

@w

@z+ ��2w

@2w

@z2= 0; (3.15)

for the axial and radial directions, respectively.

There are �ve unknowns �(z), w(z), Szz(z), Srr(z) and R(z) in the above system

of equations. The boundary conditions are given as follows:

w = 1; R = 1; � = �env at z = 0;

w = E;d2�

dz2= 0 at z = 1; (3.16)

where E is the draw ratio. An upstream stress boundary condition completes the

problem statement. At z = 0, the momentum equation becomes Szz � Srr = F and

a combination of Szz =23F and Srr = �1

3F is employed as the boundary conditions

as in Schultz [25].

For the example presented below, the diameter of the glass preform is Ri =

3�10�3m. The feed speed is wi = 4:8�10�5m/s, the maximum drawing temperatures

is Tmax = 1215C, and T1 is 20C. The glass and air properties are listed in Table

3.1. The draw ratio was nominally taken as E = 4410. The values of C and m in

(3.7) are chosen to be 1.117 and 0.13, respectively [37].

Figure 3.3 shows the predicted dimensionless temperature pro�le of the glass �ber

and heater along the spinline. The environmental (heater) temperature pro�le is

49

Table 3.1: Glass and air properties

�(g=m3) cp(J=kg �K) k(w=m �K) G(GPa) �1(Pa � s)

Glass 2:23� 103 1:45� 103 1.31 26.67

Air(20C) 0.0392 1:11� 103 0.0616 36:85� 10�6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Z

Dim

ensi

onle

ss T

empe

ratu

re

Figure 3.3: Comparison of the dimensionless axial temperature pro�les of the glass�ber (Tg = 565C) and the environment, solid line �env, dashed line �.

50

00.1

0.20.3

0.40.5

0.60.7

0.80.9

10

0.2

0.4

0.6

0.8 1

1.2

Z

Radius R

Figu

re3.4:

The�ber

radiuschange

alongthespinninglin

e.

measu

redas

describ

edin

Chapter

2.The�ber

temperatu

repro�

leisapprox

imately

thesam

eas

�en

v .This

suggests

that

convection

andrad

iationheat

transfer

are

su�cien

tto

instan

taneou

slyheat

the�ber.

Thus,to

furth

ersim

plify

theanaly

sis,�

isset

equal

tothegiven

�en

vandtheenergy

equation

canbediscard

ed.

Figu

res3.4

disp

laysthe�ber

radiusvariation

intheneck

-dow

nregion

.The

glasspreform

radiuschanges

littleuntil

itreach

esthesoften

ingpoin

ttem

peratu

re.

Most

ofthedeform

ationoccu

rsnear

Tmax .

Asthetem

peratu

redecreases,

the�ber

radiusgrad

ually

decreases

until

thelarge

viscosity

makes

deform

ationnolon

ger

possib

le.Figu

res3.5

show

sthedim

ension

lessaxial

andrad

ialstress

distrib

ution

alongthespinlin

e.Therad

ialstress

isalm

ostnegative

one-h

alfof

theaxial

stress

asin

New

tonian

elongation

al ow

.Both

stressesapproach

constan

tas

theheater

51

00.1

0.20.3

0.40.5

0.60.7

0.80.9

110

1

102

103

104

105

106

Z

Axial Stress and Radial Stress

Szz

- Srr

Szz S rr

Figu

re3.5:

Thedim

ension

lessaxial

andrad

ialstress

variationalon

gthespinlin

e.

temperatu

redrop

sbelow

thesoften

ingpoin

t(at

z=0:7)

astheglass

�ber

e�ectively

becom

esan

elasticsolid

.

Tomodeltheanisotrop

icglass

structu

ralchange

durin

g�berdraw

ing,thedim

en-

sionless

strainin

theKelv

inelem

ent(th

e`Kelv

instrain

')issim

ulated

bysep

arating

itfrom

thetotal

strain.Thedi�eren

tialequation

fortheKelv

instrain

isgiven

as

@�1

@z=

Szz �

Srr

��

�1

�2 ;

(3.17)

where

�1isthedim

ension

lessstrain

intheKelv

inelem

ent.

This

equation

canbe

solvedas

apost-p

rocessin

gstep

toobtain

theKelv

instrain

,indicative

ofanisotrop

ic

structu

reandhence

birefrin

gence.

Figu

re3.6

show

stheKelv

instrain

alongthe

spinlin

e.It

isfrozen

into

the�ber

asthetem

peratu

redrop

sto

room

temperatu

re.

ThisKelv

instrain

may

relaxas

temperatu

reincreases

durin

gpost-p

rocessin

g,while

52

00.1

0.20.3

0.40.5

0.60.7

0.80.9

10

0.5 1

1.5 2

2.5 3x 10

−5

Z

Kelvin Strain ε1

Figu

re3.6:

Thedim

ension

lessKelv

instrain

distrib

ution

.

thelarge

deform

ationinduced

bytheviscou

s ow

does

not

recover.

3.4

TheE�ectofViscoelastic

ity

Thee�ect

ofvisco

elasticityon

thedraw

ingprocess

isstu

died

bycom

parin

gthe

New

tonian

,Maxwell,

andJe�

reymodels.

TheJe�

reyequation

becom

esMaxwell

equation

as�2 !

0andto

New

tonian

ow

when

both

therelax

ationandretard

ation

times

approach

zero.Theratio

ofretard

ationtim

eto

relaxation

timeb=�2 =�

1is

chosen

tobevery

small

tosim

ulate

Maxwell

ow

.Theran

geof

bbetw

een10�15

and10�1isexam

ined

toanaly

zetheKelv

in-elem

ente�ect.

For

thegiven

draw

ing

param

eters,theoverall

draw

ingvariab

lesof

velocity,

axial

andrad

ialstress,

and

�ber

radiusshow

littledi�eren

ceas

inFigu

res3.7

and3.8.

Thissuggests

that

since

53

00.1

0.20.3

0.40.5

0.60.7

0.80.9

10

500

1000

1500

2000

2500

3000

3500

4000

4500

Z

Velocity w

b=10

-15

b=10

-1

Figu

re3.7:

Thee�ect

ofbon

thedim

ension

lessvelo

cityw.

most

ofthedeform

ationoccu

rsinthehigh

temperatu

reregion

,thematerialb

ehavior

durin

gdraw

ingise�ectively

New

tonian

.Thevisco

elasticityhas

littlee�ect

onthe

overallresp

onse

before

the�ber

iscooled

toasolid

.How

ever,theJe�

reymodelis

required

tosim

ulate

theperm

anentanisotrop

icstru

ctural

change

ofthesolid

�ber.

This

realizationis

importan

tsin

cemost

visco

elasticmodels

of�ber

spinningare

variationsof

Maxwell

models.

3.5

TheE�ectofDrawRatio

Thee�ect

ofthedraw

ratioon

draw

ingvariab

lesisanaly

zedin

thissection

.The

draw

ingvariab

lesinclu

de�ber

velocity,

axial

andrad

ialstresses,

�ber

radiusandthe

frozen-in

strainintheKelv

inelem

ent.Thefrozen

-instrain

represen

tstheanisotrop

ic

54

00.1

0.20.3

0.40.5

0.60.7

0.80.9

10

0.5 1

1.5 2

2.5 3

3.5 4x 10

5

Z

Axial Stress Szz

b=10

-15

b=10

-1

Figu

re3.8:

Thee�ect

ofbon

thedim

ension

lessaxial

stressSzz .

55

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

500

1000

1500

2000

2500

3000

3500

4000

4500

Z

Vel

ocity

E=587

E=1150

E=3200

E=4410w

Figure 3.9: The e�ect of the draw ratio E on dimensionless velocity w.

structural change during the drawing process, characterized by birefringence. The

draw ratios are chosen as E=587, 1150, 3200 and 4410, corresponding to those of our

experiments. The dimensionless velocity pro�les are shown in Figure 3.9. For a given

draw ratio, the dimensionless velocity starts as w = 1 at z = 0 and increases rapidly

in the high temperature region near z = 0:45 where T = Tmax. As the temperature

decreases the velocity gradient tends to zero.

Figures 3.10 and 3.11 display the dependence of the axial and radial stresses

on the given draw ratios. Both the absolute axial and radial stresses increase with

an increase in the draw ratio, while they have the opposite signs, i.e., positive ax-

ial stress and negative radial stress. These stresses increase as E increases with a

corresponding decrease of �ber diameters shown in Figure 3.12. The dependence

56

10

10

10

10

10

10

1

2

3

4

5

6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Z

Axi

al S

tres

s

E=4410

E=3200

E=1150

E=587

Szz

Figure 3.10: The e�ect of the draw ratio E on dimensionless axial stress.

of the Kelvin strain shows a linear relationship on the draw ratio as displayed in

Figure 3.13. The birefringence measurement also shows an almost linear relation-

ship in Figure 2.11. This suggests that the Kelvin strain can e�ectively represent

the anisotropic structural change during the drawing process. Figure 3.14 shows the

e�ect of the dimensionless drawing force on the Kelvin strain.

3.6 The E�ect of Draw Temperature

The e�ect of drawing temperature (Tmax) is examined by choosing Tmax as 1100C,

1150C, 1200C and 1215C to match the experimental parameters. The feed and

winder speeds are held constant such that wi = 0:048 � 10�3m=s and the draw

ratio is E = 4100. Figure 3.15 shows the dimensionless velocity pro�les showing

57

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 16

5

4

3

2

−10

−10

−10

−10

−10

−101

Z

Rad

ial S

tres

s

E=587

E=1150

E=3200

E=4410

Srr

Figure 3.11: The e�ect of the draw ratio E on dimensionless radial stress.

58

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Z

Rad

ius

E=587

E=1150

E=3200

E=4410

R

Figure 3.12: The e�ect of the draw ratio E on dimensionless radius.

59

0

2

4

6

8x 10

−5

Tm=1150C →

Tm=1215C →

500 1000 1500 2000 2500 3000 3500 4000 4500

Draw Ratio

Kel

vin

Str

ain

ε 1

Figure 3.13: The e�ect of the draw ratio E and draw temperature Tmax on the Kelvinstrain.

60

90 95 100 105 110 115 120 1250

0.5

1

1.5

2

2.5

3x 10

−5

Dimensionless Drawing Force

Kel

vin

Str

ain

Figure 3.14: The e�ect of the dimensionless drawing force on Kelvin strain at Tmax =1215C.

61

changing slopes for various drawing temperatures. The absolute values of the axial

and radial stresses increase with the decreasing drawing temperature as shown in

Figures 3.16 and 3.17. As drawing temperature decreases the viscosity increases,

resulting in increasing drawing force. Figure 3.18 displays the �ber radius change for

various drawing temperatures. Low drawing temperature causes a sharp neck-down

convergence of the �ber radius. Fiber birefringence (Kelvin strain) decreases with

the increase of the drawing temperature shown in Figure 3.19. Figure 3.19 also shows

the e�ect of the cooling rate on �ber birefringence. The drawing stress increases with

the decrease of Tmax and so does the birefringence. Thus increasing drawing stress

results in larger �ber birefringence. If the Kelvin strain in Figure 3.19 is scaled by

the each of the feed speed wi, the two curves overlap and become a universal curve

shown in Figure 3.20.

3.7 Conclusion

A simple Je�rey model has been used as the constitutive equation, combined

with the mass, momentum and energy equations to model the glass �ber drawing

process. The Je�rey model simulates not only the overall drawing process, but also

the elastic strain corresponding to �ber birefringence. The model results show good

agreement with the experimentally determined birefringence data shown in Figure

2.13. The Kelvin strain has a linear relationship with the draw ratio. As the drawing

temperature increases, the Kelvin strain decreases. The viscoelastic behavior has lit-

tle e�ect on the kinematics and dynamics for high temperature during �ber drawing.

Under the same drawing conditions, the viscoelastic simulation (Maxwell and Je�rey

model) displayed little di�erence from that of the Newtonian ow for the drawing

variables such as velocity, axial stress, radial stress and the radius pro�le along the

62

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

500

1000

1500

2000

2500

3000

3500

4000

4500

Z

Vel

ocity

Increasing temperature

w

Figure 3.15: The e�ect of the draw temperature Tmax on dimensionless velocity, E =4410. Tmax=1100, 1150, 1200 and 1215C

63

00.1

0.20.3

0.40.5

0.60.7

0.80.9

10

0.5 1

1.5 2

2.5 3

3.5 4x 10

5

Z

Axial Stress Szz

Figu

re3.16:

Thee�ect

ofthedraw

temperatu

reTmaxon

axial

stress,E

=4410.

Tmax =

1100,1150,

1200and1215C

64

00.1

0.20.3

0.40.5

0.60.7

0.80.9

1−

2

−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2 0

x 105

Z

Radial Stress Srr

Figu

re3.17:

Thee�ect

ofthedraw

temperatu

reTmaxon

radiusstress,

E=

4410.Tmax =

1100,1150,

1200and1215C

65

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Z

Rad

ius

Increasing temperature

R

Figure 3.18: The e�ect of the drawing temperature Tmax on neck-down region, E =4410. Tmax=1100, 1150, 1200 and 1215C

66

1100 1120 1140 1160 1180 1200 12200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6x 10

−4

← wi=0.027x10 m/s

← wi=0.048x10 m/s

Drawing Temperature in C

Kel

vin

Str

ain

ε 1

-3

-3

Figure 3.19: The e�ect of the draw temperature Tmax and feed speed on Kelvinstrain, E = 4410.

67

1100 1120 1140 1160 1180 1200 12200.5

1

1.5

2

2.5

3

3.5

Drawing Temperature in C

Sca

led

Kel

vin

Str

ain

ε 1

Figure 3.20: The e�ect of the draw temperature Tmax and feed speed on scaled Kelvinstrain, E = 4410.

68

spinline. Hence, the viscoelastic e�ects can be determined after the kinematic are

obtained.

The large deformation during the drawing process occurs in the high temperature

range where the glass viscosity is very low and the ow is nearly Newtonian. The

simulation shows that all variables change mainly in the high temperature range.

Saturation of the Kelvin strain at high temperature occurs because the retardation

time is small compared to residence time during deformation at these higher tem-

peratures. As the �ber is rapidly cooled the variables stop changing and the state is

\frozen". However, viscous Newtonian ow and the viscoelastic Maxwell model are

unable to model the anisotropic structure change in the drawing process, and only

the Je�rey model is capable of simulating the frozen elastic strain and its relaxation

process successfully.

CHAPTER IV

RELAXATION MODELING

4.1 Introduction

The relaxation of �ber birefringence during annealing is investigated. Glass �ber

birefringence is the optically accessible manifestation of anisotropic structure that

relaxes in a time-and temperature-dependent manner. Experiments in this chapter

show that this relaxation near the glass transition range is similar to that of viscoelas-

tic strain relaxation, however at lower temperature annealing, �ber birefringence does

not fully relax.

Structural relaxation is the process by which material thermodynamic properties,

such as enthalpy, volume, and refractive index, gradually approach their equilibrium

values following changes in some external parameters, such as temperature and pres-

sure. In general, structural relaxation is described by

P (T; t)� P (T;1)

P (T; 0)� P (T;1)= M(T; t); (4.1)

where P (T; t) is any thermodynamic property of temperature T and time t, P (T;1)

is its equilibrium value, P (T; 0) is the initial value, M(T; t) is a relaxation function

that for an isothermal process may have the form

M(T; t) = e�t=�(T ); (4.2)

69

70

For oxide glasses in the annealing range, the �ctive temperature (the temperature

at which the present structure would be in equilibrium [46]) is commonly used to

describe structural relaxation such that

dTfdt

=T � Tf

�(4.3)

where the time constant � has the linearized Arrhenius dependence on temperature

and the �ctive temperature as

� = �0 exp(�A1T � A2Tf); (4.4)

where �0, A1, and A2 are constants. Tool's approach successfully captures the thermal

expansion of annealed glass by using

M = �g(T � T0)� �(Tf � Tf0); (4.5)

where �g is the glassy state thermal expansion coe�cient, T0 is a reference temper-

ature, � is a structural coe�cient, and Tf0 is the initial �ctive temperature. This

approach was unable to predict the thermal expansion data of quenched glass under

500C. Refractive index relaxation experiments by Ritland [47] have shown that a

single �ctive temperature evolution described by (4.3) and (4.4) is not su�cient to

describe the distributed nature of property relaxation. Since the as-drawn �ber is

anisotropic, it is not in equilibrium for any temperature and we have found that

only partial relaxation may occur, the use of a single �ctive temperature to describe

drawn �ber birefringence relaxation is questionable.

Narayanaswamy [48] proposed using a reduced time � to partition the temperature

and �ctive temperature e�ects. Linearity can be restored by using the reduced time

parameter de�ned by

� = �(T0)Z t

0

dt0

�; (4.6)

71

where T0 is an arbitrary reference temperature. Boltzmann's superposition principle

describes Tf by

Tf = T �Z �

0M(� � �0)

dT

d�0d�0: (4.7)

The thermodynamic property P (T; �) becomes

P (T; �) = P (T;1)� �s

Z �

0M(� � �0)

dT

d�0d�0; (4.8)

where M(� � �0) is a relaxation function most often described by a \stretched expo-

nential"

M(�) = e�(�=�)b

(4.9)

for a constant b, and �s is a structure parameter.

Based on the general Arrhenius relation � = �0 exp(4HRT

), to describe thermally

activated relaxation, Narayanaswamy suggested that the contribution of �ctive tem-

perature to structural relaxation could be included by partitioning the relaxation

time in the following empirical form

� = �0 exp

x4HRT

+(1� x)4H

RTf

!; (4.10)

where �0 is a constant, 4H is an activation energy, R is the ideal gas constant, and x

is a constant between 0 and 1. Later investigators [49, 50, 51] used the Adams-Gibbs

equation based on the suggestion that the ow of the structure involves the coop-

erative rearrangement of increasingly larger numbers of molecules as temperature

decreases. Thus the relaxation time depends on the con�gurational entropy as

� = �0 exp

A

T4S(Tk; Tf)

!; (4.11)

where 4S(Tk; Tf) is the con�gurational entropy with structure corresponding to Tf ,

the Kauzmann temperature Tk is de�ned by the vanishing of the equilibrium value of

72

4S, and �0 and A are constants. The phenomenological models based on the Tool-

Narayanaswamy (T-N) equations successfully describe relaxation of those isotropic

parameters such as density, refractive index and enthalpy in bulk glass in the glass

transition range. Recently, di�erential scanning calorimetry (DSC) experiments in

oxide glass �bers by Huang and Gupta [52] demonstrate that the T-N model is inad-

equate for describing enthalpy relaxation of drawn glass �bers at temperatures well

below the glass transition range. Relaxation well below the glass transition temper-

ature has also been observed in oxide glass [53], metal glass [54], and polymers [55].

Since birefringence in glass �bers is the result of frozen anisotropic elastic strain,

its relaxation is expected to behave similarly to that of viscoelastic strain recovery

of inorganic glasses. The experiments in this study demonstrate this behavior for

birefringence relaxation near the glass transition temperature Tg. The use of a re-

laxation spectrum to simulate the relaxation of glass �ber birefringence is explored.

Low temperature (sub { Tg) birefringence relaxes to a non{zero value, suggesting

that part of the frozen elastic strain remains. Over a very broad temperature range,

the relaxation process is not only nonlinear but also non{thermal rheologically sim-

ple, i.e. the form of the relaxation function changes with temperature. The stretched

exponential function is modi�ed to simulate the birefringence relaxation over a broad

range of annealing temperatures. A temperature dependent parameter is introduced

in the stretched exponential function to account for the unrelaxed part of the frozen

elastic strain. The Je�rey model used in the �ber drawing process is also explored

to model the birefringence relaxation. The Je�rey model has a relaxation time and

retardation time thus it contains the two characteristic times needed to capture the

relaxed and partially relaxed responses.

73

4.2 Stretched Exponential Modeling

Our experimental results show that glass �ber birefringence relaxation in the

glass transition range is qualitatively similar to that of the viscoelastic strain recov-

ery of oxide glasses [56] in the glass transition temperature range. Glass structure

and viscoelastic strain relaxation have been modeled using the stretched exponential

(sometimes called the Kohlrausch-Williams-Watts or KWW function).

M(t) = e�(t=�)b

(4.12)

where M(t) is the relaxation function, � is the relaxation time, and b is a constant.

Previous investigations have shown that b is around 0.5 for oxide glass stress relax-

ation or viscoelastic strain relaxation [43] in the glass transition range. We explore

the ability of the stretched exponential to simulate the glass �ber birefringence re-

laxation with the typical value of b = 0:5. The results for the scaled birefringence

relaxation are shown in Figure 4.1 using the relaxation times shown in the caption,

chosen to best �t the data. Here, � is assumed to be constant for the annealing time,

so that the heating and cooling times of the �ber are neglected.

We can compare these relaxation times to those determined from viscosity and

the elastic modulus as

� =�

G;

where � is viscosity and G is the shear modulus of glass (assumed constant). As

in Gupta [37], we �nd that the WLF and Arrhenius relationships are inadequate

to describe the viscosity over a very large temperature range. Instead we use the

Walther double exponential correlation

� = �0 exp[exp(v0 � v1 ln �)];

74

0 20 40 60 80 100 120 140 160 180 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (min)

Sca

led

Bire

frin

genc

e

(i)

(ii)

(iii)

(iv)

(v)

Figure 4.1: Birefringence relaxation for �bers drawn at Tm = 1215C. Data fromFigure 2.14. Lines are plotted using (13) with b = 0:5, and (i) T = 309C,� = 1000min; (ii) T = 360C, � = 150min; (iii) T = 387C, � = 50min;(iv) T = 408C, � = 20min; (v) T = 511C, � = 5min.

where �0, v0 and v1 are constants for varying absolute temperature T . The correlation

coe�cients for borosilicate glass are �0 = 3:0 Pa� s, v0 = 18 and v1 = 2:18 lnK�1 from

viscosity versus temperature data of Dormeus [3]. Using the Walther correlation

results in a ratio of time constants of 1016 between T = 309C and T = 511C rather

than the factor of 200 shown in Figure 4.1. Clearly, the relaxation phenomenon is

more complicated than what can be related by a single exponential relaxation time

and the glass cannot be viewed as thermo-rheologically simple.

75

As Figure 4.1 shows at high annealing temperatures (T = 408C and 511C) the

stretched exponential with b = 0:5 does not �t the data well as the birefringence

appears to relax to zero in �nite time, while the simulation over-predicts the time of

e�ectively complete relaxation by a factor of �ve. At low temperatures (T = 309C)

the birefringence does not fully relax for long times (24 hours in Figure 2.15). The

exponential constant b may be varied for di�erent relaxation temperatures without

solving the problem.

Similarly, one relaxation case is modeled using Tf and (4.3), (4.4), (4.6), and

(4.9) with b = 1. For this case, we use the experimentally-determined time constant

of 14 seconds for the temperature rise of the �bers at the beginning of the annealing

process. Tf0 = 1273K is assumed to correspond with temperatures slightly lower

than Tm, where most deformation will occur. The �ctive and �ber temperatures are

shown in Figure 4.2a for �0 = 1027s, a1 = 5�10�2K�1, and a2 = 2:3�10�2K�1 taken

from Tool [46]. The evolution of the relaxation time � and the scaled birefringence are

then shown on Figures 4.2b and 4.2c, respectively. As expected, this time-marching

simulation shows that the scaled birefringence decays to zero, and now, the relaxation

is slow at initial times. Clearly, this approach does not model the observed relaxation

either.

76

0 50 100 150 200200

400

600

800

1000

1200

1400

Time (s)0 50 100 150 200

100

102

104

106

108

Time (s)τ

(s)

0 50 100 150 20010

−2

10−1

100

Time (s)

Sca

led

Bire

frin

ge

nce

T &

T (K

)f

(i)

(ii)

(a) (b)

(c)

Figure 4.2: Fictive temperature modeling of relaxation. (a) Temperature variationduring annealing: (i) Fictive temperature Tf and (ii) �ber temperatureT ; (b) Relaxation time evolution; and (c) Birefringence relaxation.

77

To capture the relaxation at low temperatures the stretched exponential is mod-

i�ed by introducing a temperature dependent constant c(T ) such that

M(t) = [1� c(T )]e�(t=�)b

+ c(T ) (4.13)

Figure 4.3 shows the simulation using (4.13) in a simpli�ed form using b = 1. This

modi�ed relaxation function is capable of capturing all the relaxation data in Figure

2.14.

4.3 Je�rey Modeling

Equation (4.13) shows that relaxation can be modeled using two exponential

elements with at least one of them \frozen" at low annealing temperatures (i.e.

with very large relaxation times) to capture the incomplete relaxation. Equation

(4.13) o�ers limited physical interpretation and cannot simulate the development

of birefringence during the drawing process. A simple, two-component linearized

Je�rey element model is explored, shown in Figure 4.4, to simulate both the drawing

and relaxation processes. Here, either a tensile stress � or elongation rate _� can be

applied.

In the parallel Je�rey model, elements �12 and �22 represent the large viscous

deformation produced during the drawing process at high temperature. The two

Kelvin elements represent the anisotropic elastic strain that is frozen during �ber

cooling and relaxed during annealing. A minimum of two elements (two relaxation

times) is required for incomplete relaxation. Since birefringence is a measure of

anisotropic strain, we assume birefringence is proportional to the total strain in the

two Kelvin elements. This anisotropic strain may partially or fully relax depending

on the annealing temperature. The relaxation rate and extent of relaxation depend

78

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (min)0 20 40 60 80 100 120 140 160 180

Sca

led

Bire

frin

genc

e

(i)

(ii)

(iii)

(iv)(v)

Figure 4.3: Birefringence relaxation for �bers drawn at Tm = 1215C. Data fromFigure 2.14. Lines are plotted using (4.13) with b = 1, and (i) T = 309C,� = 50min, c(T ) = 0:65; (ii) T = 360C, � = 47min, c(T ) = 0:35; (iii)T = 387C, � = 45min, c(T ) = 0:05; (iv) T = 408C, � = 16:7min,c(T ) = 0:0; (v) T = 511C, � = 8:24min, c(T ) = 0:0.

79

µ11

µ12

µ21

µ22

G1

G 2

σ or ε

Figure 4.4: Two Je�rey elements in parallel

on the temperature dependent viscosities assigned to the viscous elements. The

moduli of the spring elements are assumed constant.

Birefringence relaxation simulations are conducted at various annealing temper-

atures using the quenched system after �ber drawing. Thermal expansion e�ects are

neglected. The simplest temperature versus relaxation study using the 2-element

Je�rey model varies one parameter with temperature. The two elastic elements are

assumed to have a constant modulus of 24.4GPa. Here for simplicity, three of the vis-

cous elements are assumed to have no temperature dependence in the annealing tem-

perature range and their values are held as �21 = 1:1�1013Pa�s, �12 = 2:2�1011Pa�s,

�22 = 2:2 � 109Pa�s. The viscosity of the �nal viscous element �11 varies with an-

nealing temperatures as shown in Figure 4.5. This simple simulation of the parallel

Je�rey model is shown in Figure 4.6 to model the birefringence relaxation of glass

�bers over the entire range of annealing temperatures.

4.4 Conclusion

The generally used stretched exponential for delayed elastic strain and struc-

tural relaxation is not able to accurately simulate observed birefringence relaxation.

80

300 320 340 360 380 400 420 440 460 480 50010

10

10

10

10

13

14

15

16

17

Temperature (C)

Vis

cosi

ty (

Pa

s)

Figure 4.5: �11 versus annealing temperatures.

81

0 20 40 60 80 100 120 140 160 1800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time(min)

Sca

led

Bire

frin

genc

e

Figure 4.6: Birefringence relaxation for �bers drawn at Tm = 1215C. Data fromFigure 2.14. Lines are plotted using the two Je�rey elements in parallelmodel.

82

A modi�ed exponential model with incomplete relaxation simulates relaxation satis-

factorily with an empirical temperature dependent constant. A parallel Je�rey model

with temperature dependent coe�cients simulates the glass �ber birefringence relax-

ation well not only for the high temperature annealing but also for low temperature

annealing.

The Je�rey model enables simulation of the development of birefringence dur-

ing �ber drawing as well as the relaxation process. The various elements in the

Je�rey model lend themselves to interpretation as viscous deformation elements or

viscoelastic elements storing anisotropic strain at low temperature. The complete

and incomplete relaxation behavior requires two temperature dependent relaxation

times. The empirical constant in the modi�ed exponential model represents the sec-

ond branch of the parallel Je�rey model and e�ectively provides a second retardation

time for modeling complete and incomplete relaxation.

CHAPTER V

CONCLUSIONS AND RECOMMENDATIONS

Our experiments and modeling show that optical anisotropy develops during �ber

drawing in glass �bers drawn from preforms. Optical anisotropy is the manifestation

of the structural anisotropy that develops during the drawing process. In oxide

glasses the structural and optical anisotropies are consequences of frozen anisotropic

elastic strain by rapid cooling under the drawing load. The anisotropy of glass

�bers has been determined from birefringence measurements for various drawing

parameters.

The amount of the anisotropy in as-drawn glass �bers is shown to be strongly

a�ected by the draw temperature, the draw ratio and the draw rate. Fiber bire-

fringence increases with decreasing drawing temperature, increasing draw ratio, and

increasing cooling rate. Increasing temperature decreases the glass melt viscosity

and hence the drawing force decreases, and �ber birefringence decreases accord-

ingly. Rapid cooling halts relaxation after the axial force is no longer applied. Thus

the structure is quenched in an anisotropically strained state and the anisotropic

structure exhibits birefringence. Increasing drawing speed increases draw ratio, this

increased deformation also results in increased structural anisotropy manifested as

birefringence.

83

84

A simple Je�rey model has been used as the constitutive equation for glass �ber

drawing, and combined with the mass, momentum and energy conservation equations

to model the glass �ber drawing process. The Je�rey model is able to simulate the

overall drawing process, and stored anisotropic elastic strain energy associated with

�ber birefringence.

The viscoelastic parameters have little e�ect on the kinematics and dynamics

of �ber drawing. Under the same drawing conditions viscoelastic simulations via

the Maxwell and Je�rey models displayed little di�erence from that of a Newtonian

model in drawing variables such as velocity, axial stress, radial stress and the radius

pro�le along the spinline. This is true because the majority of the large deformation

during drawing occurs at high temperatures wherein the glass viscosity is very low

and Newtonian viscous ow dominates. The �ber drawing simulations show that the

kinematic and dynamic variables change dramatically in the high temperature range.

Saturation of the Kelvin strain at high temperature occurs because retardation time

is small compared to residence-time during deformation at these high temperatures.

The �ber is cooled rapidly through the glass transition to the glassy (elastic solid)

state. However, the Newtonian model and the viscoelastic Maxwell model are unable

to store the anisotropic structure change during the drawing process, and the Je�rey

model is the simplest viscoelastic model capable of simulating frozen elastic strain.

The Je�rey model is used to predict the change in as-drawn �ber birefringence with

drawing temperature and draw ratio.

Glass �ber birefringence relaxation at temperatures well below the glass transi-

tion range in oxide glasses has been observed. At low temperatures, birefringence

relaxation is observed, but relaxation is incomplete. This temperature dependent

relaxation behavior suggests a broad spectrum of relaxation times. The generally

85

used stretched exponential for delayed elastic strain and structural relaxation is not

able to accurately simulate observed birefringence relaxation. An ad hoc modi�ed

exponential model without a stretching factor but with an empirically determined

temperature dependent constant models incomplete relaxation satisfactorily. A par-

allel Je�rey model with temperature dependent coe�cients simulates the glass �ber

birefringence relaxation well not only for high temperature annealing resulting in

complete relaxation but also for low temperature annealing resulting in incomplete

relaxation. The Je�rey model enables simulation of the development of birefringence

during �ber drawing as well as the relaxation process. The various elements in the

Je�rey model lend themselves to interpretation as viscous deformation elements or

viscoelastic elements storing anisotropic strain at low temperature.

The single Je�rey model is the simplest constitutive model capable of captur-

ing the kinematic and dynamic parameters during �ber drawing and the as-drawn

�ber birefringence. The complete and incomplete relaxation behavior requires two

temperature dependent relaxation times. The empirical constant in the modi�ed

exponential model represents the second branch of the parallel Je�rey model and

e�ectively provides a second retardation time for modeling complete and incomplete

relaxation.

Although it is shown that both glass �ber tensile strength and birefringence are

a�ected by the drawing process parameters, the correlation between these two prop-

erties is still not very clear. More work should be done to further study the correlation

between the anisotropic structural change and the �ber mechanical properties. It is

also important to understand how the anisotropic structural change induced during

the drawing process a�ects the optical �ber properties.

APPENDICES

86

87

APPENDIX A

CONSTITUTIVE THEORIES OF GLASS NEAR

THE GLASS TRANSITION RANGE

Inorganic glass melts are generally treated as Newtonian uids. However, at tem-

peratures around the glass transition, non-Newtonian behavior in inorganic glasses

has been reported for various glasses and by several authors [4, 5, 57, 58]. The steady-

state stress versus strain rate of a soda-lime-silica glass at T=596C (Tg = 570C) in

Figure A.1 [4] shows deviation from Newtonian behavior at high strain rates. The

transient stress versus strain response at constant strain rate contains a strain rate

and temperature dependent stress overshoot. The strain rate dependent uniaxial

stress-strain response of a silicate glass near the transition range is shown schemati-

cally in Figure A.2. At a low constant applied strain rate the stress rises monotoni-

cally to a steady state value. As the applied constant strain rate ( _�) is increased under

isothermal conditions, the stress reaches a maximum before it softens to a steady-

state value. The magnitudes of both the peak and steady-state stresses increase with

the applied strain rate.

Many previous studies of the rate dependent deformation response of inorganic

glasses have been concerned with the shear thinning behavior, i.e. the non-Newtonian

steady state behavior. Phenomenological models [4, 59, 60] have been developed to

88

00.5

11.5

22.5

33.5

44.5

50 2 4 6 8 10 12 14 16 18

Strain R

ate (10−

4/sec)

Steady−state Stress (10 9dyne/cm2)

Figu

reA.1:

Variation

ofstead

y-state

stressvs.strain

ratefor

asoda-lim

e-silicaglass

atT=596C

show

ingdeviation

fromNew

tonian

behavior

athigh

strainrates.

89

ε

Strain

Str

ess

Figure A.2: Schematic representation of the uniaxial stress vs. strain response ofinorganic glass near Tg at various constant applied strain rates.

90

describe the steady-state stress (or apparent viscosity) versus strain rate. Few at-

tempts have been made to describe the transient stress response, i.e. the stress

overshoot at high strain rates. In practice, high deformation rate processes are en-

countered in many situations, such as in �ber drawing. The �nal �ber properties such

as sti�ness, strength and optical anisotropy depend on the processing temperatures

and strain rates of the glass as it passes through its viscoelastic temperature range.

To fully characterize glass �ber drawing, it is important to consider the transient

deformation response of glass near Tg.

The strain rate or stress dependent non-Newtonian behavior of inorganic glasses

we seek to describe has been observed experimentally in various deformation states [4,

5, 57, 58]. Some authors [4] interpret these responses by comparing to molecular dy-

namic (MD) simulations on a Lennard-Jones potential glass. Both inorganic glass

experiments and the MD simulations show similar behaviors. The MD model indi-

cates that the material develops a layered structure under high shear strain rates

and that causes the drop in the true stress response.

A macroscopic constitutive description of both the transient and the steady state

responses of inorganic glasses near the transient region is explored, in which rate

and temperature dependent stress overshoot and shear thinning in the steady state

response are captured. In this appendix several constitutive modeling approaches are

examined for their ability to predict the rate and temperature dependent response

of glass near Tg in tension and compression.

Rekhson [61] proposed a model for non-linear viscoelastic relaxation based on the

hypothesis of stress-induced structural relaxation (an idea analogous to temperature-

induced relaxation) and introduced the concept of the �ctive stress based on the

phenomenological Tool's equation (replacing the �ctive temperature Tf in Equation

91

(4.3) with �ctive stress), to capture both the transient and the steady state stress.

The model is based on a spectrum of simple viscoelastic relaxations and has been

applied to the uniaxial tensile response of a silicate glass in comparison with experi-

mental data from Rekhson [4]. The relaxation times are functions of both stress and

�ctive stress. This model is phenomenological in that it assumes a �ctive stress and

incorporates a temperature e�ect by correcting the viscosity for di�erent tempera-

tures.

Argon [62] has proposed a model for the resistance of glass to inelastic deformation

that considers the rate and temperature dependent resistance to local transforma-

tions of the atoms in metallic and oxide glasses at low temperatures. This model pre-

dicts the rate and temperature dependent stress maximum in soda glass and E-glass.

Argon and Kuo [63] have studied the mechanism of plastic ow of atomic glasses by

using a disordered soap bubble raft. Their study revealed that under shear the rafts

are observed to change shape by a collection of very local shear transformations. The

2-D computer molecular dynamics approach by Deng and Argon and coworkers [64]

demonstrates that the principal mechanism of plastic strain is local shear transfor-

mations nucleated preferentially in the boundaries of liquid-like material separating

the small quasi-ordered domains. The sites of the maximum shear strain correlate

well with the sites of excess free volume, i.e. the fraction of matter having a lower

atomic coordination. The original rate and temperature dependent model of inelas-

tic deformation resistance for glassy materials proposed by Argon [62] serves as a

useful practical description of the actual deformation mechanism. A description of

the structural state in inorganic glass is examined based on the conceptualizations of

Argon and others [64] of a distribution of free volume within volume elements on the

order of several atomic distances. This distribution evolves with inelastic deforma-

92

tion in a manner that is rate and temperature dependent and results in an overshoot

in the transient stress versus strain response of inorganic glass.

The homogeneous response of glassy polymers also exhibits true stress softening

and strain rate and temperature dependent behavior. A number of investigators

have described this phenomenon near or below the glass transition temperature us-

ing phenomenological macroscopic, mesoscopic, or molecular models [65, 66, 67]. It

is clear that during deformation some structural rearrangement occurs in the form

of local rearrangements of the number and size of density variations in the polymer

microstructure. A micromechanically based constitutive model of this structural evo-

lution process has successfully predicted the rate dependent stress softening response

of glassy polymers [65, 66, 67].

Viscoelastic models such as the generalized Maxwell model are generally used to

describe the behavior of inorganic glasses in the transition range [68]. The White-

Metzner model is a viscoelastic model that involves rate dependent viscosity and has

been used to describe glass �ber drawing [37]. It is similar to the Je�rey model, and

at constant strain rate is equivalent to the Je�rey model. Structure related models

have rarely been explored to describe inorganic glass behavior in the transition range.

The Marrucci model [69] has demonstrated the ability to capture the stress overshoot

in polymers for one dimensional elongational ow. A elastic-viscoplastic model [62]

based on the local thermal transformation theory is examined. This approach is

modeled for tensile and compression tests and compared with the experimental data

to see if it can describe the strain rate and temperature dependent response in the

transient and steady state ranges.

93

Figure A.3: Schematic two-dimensional representation of the structure of a hypo-thetical compound A2O3

A.1 A Viscoplastic Theory

A.1.1 Physical Description of Oxide Glasses

The deformation of inorganic glass is closely related to its structure. Zachariasen's

random network hypothesis [2] is largely accepted as the classical theory to describe

the structure and properties of unmodi�ed oxide glasses in the solid state. Figure A.3

is a schematic 2D diagram showing the structure of a hypothetical oxide A2O3 in

the glassy form [2]. In this material the basic polyhedron is the AO3 triangle. The

triangles are joined only at their corners by bridging oxygens. In silica glass, the

basic polyhedron is the SiO4 tetrahedron.

For modi�ed oxide glasses, the introduction of a modi�er such as Na2O will

change the random network structure in which the alkali and alkaline earth ions are

94

Figure A.4: Two-dimensional representation of the structure of a modi�ed oxide glass

incorporated into the structure shown in Figure A.4 [2]. The cations C are situated

in the relatively large voids between the silica bonds and for each additional oxygen

anion introduced, one A-O-A bridge is broken so that two non-bridging oxygens are

formed.

Glass and glass forming melts are considered to have a large number of free volume

sites. Although glasses at low temperatures have little plastic ow before rupture

failure, it is expected that plastic ow prevails at the glass transition range. As in

the structure shown in Figure A.4, the modi�ers introduced into the glass create

larger free volume sites. 2-D MD simulations and soap bubble experiments [63, 64]

have shown that the free volume sites are the primary local transformation sites for

the mechanism of plastic deformation. These free volume sites can be characterized

as free energy barrier sites [62], and large free volume sites correspond to low free

95

energy barrier sites. The transformation of a series of bonded atoms from one con-

formation to another has to overcome the local free energy barrier. According to the

transition state theory, local rearrangements are thermally activated, and the local

transformation rate, !, can be expressed as an Arrhenius form

! = !0exp��F

kT

�; (A.1)

where, !0 is the attempt frequency; �F is the activation free energy for the trans-

formation; k is the Boltzmann constant and T is the absolute temperature. Under

an externally applied stress, the activation free energy is modi�ed as [70]

�F = �F0

�1�

��

�0

�p�q; (A.2)

where �F0 is the activation energy without external stress; � is the local e�ective

stress, �0 is the limiting stress, and p and q are material dependent parameters.

According to the Argon model which describes the transformation extent of inorganic

glasses as an exponential form of the free enthalpy change [71], the total inelastic

strain rate is given by

_ = _ 0exp���F0

kT

�1�

��

�0

�p�q�; (A.3)

where _ is the inelastic strain rate, and _ 0 is the pre-exponential factor determined

experimentally. Equation (A.3) can also be written in the following form [65]

_ = _ 0exp��AsT

�1�

��

s

�p�q�; (A.4)

In equation (A.4), s is the athermal shear strength of the material and it is a structure

related parameter. The quantity As is the zero stress level activation energy, and � is

96

unsheared sheared

Figure A.5: Schematic illustration of glass melts under shear

the e�ective ow stress. This inelastic deformation process is rate and temperature

dependent. We assume the athermal shear strength is evolving during plastic defor-

mation processes to a steady state sss as the structure becomes layered as shown in

Figure A.5. The proposed evolution law for s is given in equation (A.5) [72].

_s = h(1� s

sss) _ ; (A.5)

where h is the evolution slope. sss is given as sss = as0, here s0 =0:77�1��

[66], � is the

elastic shear modulus, and a is a rate and temperature dependent parameter. � is

the Poisson's ratio.

A.1.2 The Constitutive Formation

Various experiments show that the transient response of inorganic glasses is char-

acterized as initially elastic until yield occurs at a stress which depends upon tem-

perature and strain rate, and then plastic ow ensues. Therefore we seek an elastic-

viscoplastic constitutive framework for describing the response. The kinematics for

the �nite deformation are implemented as follows. The velocity gradient tensor L

97

can be expressed as

L = D+W; (A.6)

L = Le + FeLpFe�1; (A.7)

where D is the symmetric part of L, and W is the skew-symmetric part of L. Fe

is the elastic deformation gradient, Fe�1 is its inverse and Lp is the plastic velocity

gradient. For the one dimensional case of uniaxial ow, equation (A.7) has the form

L11 = Le11 + Lp

11; (A.8)

where L11 is the total velocity gradient, or the total strain rate _�. The elastic velocity

gradient Le11 can be written as Le

11 = _�e, where _�e represents the elastic strain rate.

If linear elastic behavior is assumed, �e = �E, where � is the applied axial stress for

one dimensional uniaxial ow and E is the Young's modulus. With equation (A.4)

for the inelastic strain rate where _ p =p3_�p and �e = �

E, equation (A.8) becomes

_� =_�

E+

_ 0p3exp

�AsT

1�

�p3s

!p!q!; (A.9)

A.1.3 Model Prediction for Elongational Tests

Previous experiments examining the inorganic glass deformation response have

been documented in the literature [4, 5, 57, 58]. The results show behavior similar

to that displayed in Figure A.2. Uniaxial tension and compression data from [4, 5]

will be used to compare with the model simulations.

Figures A.6 and A.7 show the uniaxial tension deformation responses of soda-lime-

silicate glass for two di�erent temperatures taken from [4]. The apparent viscosity

98

0

50

100

150

200

250

300

0 100 200 300 400 500 600 700 800

Str

ess

(MP

a)

Time (s)

Figure A.6: Stress vs. time response of soda-lime silicate glass tension tests at variousstrain rates at T=836K.

versus time graphs of [4] are transformed into uniaxial stress versus time using � =

3� _� in order to compare with the simulations of equation (A.9).

The material parameters in equation (A.9) are determined as follows. For inor-

ganic glasses, Eyring's approach is used, i.e. p and q are taken as q = 1 and p = 1

[67]. _ 0 and A are determined from the data in Figures A.6 using maximum stress

versus strain rate data pairs and equation (A.4). The elastic parameters, Young's

modulus for soda-lime silicate glass at elevated temperatures, have been measured

by McDraw [73]. Here we take the Young's modulus at a temperature of around

600C. The parameters needed for the computation are listed in Table A.1.

Figure A.8 shows the results of the simulation and comparison with the exper-

99

0

50

100

150

200

250

300

0 50 100 150 200

Str

ess

(MP

a)

Time (s)

Figure A.7: Stress vs. time response of soda-lime silicate glass tension tests at variousstrain rates at T=866K.

Table A.1: Parameters used in tension simulations

_ 0(1/s) A(K/MPa) E(MPa) � a h(MPa/s)

T=836K 1.1E+5 22.61 20,000 0.21 0.97-0.999 31,500

T=866K 1.1E+5 22.61 20,000 0.21 0.97-0.999 31,500

100

0

50

100

150

200

250

300

350

0 0.05 0.1 0.15 0.2 0.25 0.3

Str

ess

(MP

a)

Strain

ε= 0.0014 /sec

ε= 0.00072 /sec

ε= 0.00036 /sec

ε= 0.00018 /sec

ε= 0.00009 /sec

ε= 0.000036 /sec

Figure A.8: Stress vs. time response of soda-lime silicate glass for tension tests atvarious strain rates at T=836K. Curves are model simulations using(A.9), and symbols are the data from [4].

101

0

50

100

150

200

250

300

350

400

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Str

ess

(MP

a)

Strain

ε=0.0181 /sec

ε=0.0072 /sec

ε=0.0036 /sec

ε=0.0018 /sec

ε=0.00091 /sec

Figure A.9: Stress vs. time response of soda-lime silicate glass for tension tests atvarious strain rates at T=866K. Curves are model simulations using(A.9), and symbols are the data from [4].

102

Table A.2: Parameters used in compression simulations

_ 0(1/s) A(K/MPa) E(MPa) � m0 h(MPa/s)

T=869K 1232.96 13.16 20,580 0.21 6:58� 10�3 5000

T=902K 6.88 13.16 9800 0.21 6:58� 10�3 5000

T=930K 27.45 28.12 4900 0.21 6:58� 10�3 5000

imental data at T=836K. The curves at _�=0.0072/sec, 0.0036/sec and 0.0018/sec

were used to obtain the parameters in Table A.1, and the remaining curves are pre-

dictions. It is shown that the quality of the �t and the predictive capability are

quite good for the strain rate range 0.000036-0.0072/sec. The model overpredicts

the stress overshoot at 0.0014/sec. The simulations at T=866K are shown in Figure

A.9 in which all curves are predictions. The theory captures the data fairly well for

the three highest rates; the transient and steady state responses are overpredicted

at the lower rates.

A.1.4 Model Prediction for Compression Tests

Equation (A.9) is also used to simulate the uniaxial compression responses at

constant displacement rates for soda-lime-silicate glass at three di�erent tempera-

tures from reference [5]. The relation between strain rate and displacement rate is

_� = _h=h, where h is the current height of the sample.

We have found that the model is not predictive if we use the same approach as in

the tensile test simulation; using the data at one temperature to determine the model

parameters is not predictive of the response at other temperatures. The reason for

this is not clear but the data show increasing rate dependence with decreasing tem-

103

0 0.2 0.4 0.6 0.8 1 1.2 1.40

50

100

150

200

250

300

Str

ess

(MP

a) h=0.022mm/s

h=0.045mm/s

h=0.096mm/s

h=0.067mm/s

h=0.031mm/s

Figure A.10: Stress vs. time response of soda-lime silicate glass for compression testsat various displacement rates at T=869K. [5] and comparison with thepredictions using (A.9).

104

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

50

100

150

200

250

300

350

400

Str

ess

(MP

a) h=3.1mm/s

h=2.2mm/s

h=1.5mm/sh=1.0mm/s

h=0.75mm/s

Strain

Figure A.11: Stress vs. time response of soda-lime silicate glass for compression testsat various displacement rates at T=902K. [5] and comparison with thepredictions using (A.9).

105

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

20

40

60

80

100

120

140

Str

ess

(MP

a)

h=4.0mm/s

h=3.0mm/s

h=2.0mm/s

h=1.0mm/sh=0.7mm/s

Strain

Figure A.12: Stress vs. time response of soda-lime silicate glass for compression testsat various displacement rates at T=930K. [5] and comparison with thepredictions using (A.9).

106

perature, which contradicts modeling assumptions. Thus we have adopted another

approach to attempt to predict the compression data. The compression tests are

carried out in three temperatures as 869K, 902K and 930K. It is found that in this

temperature range the elastic shear modulus is not constant [73], but temperature

dependent. We have used the temperature dependent shear modulus from refer-

ence [73], and have determined other parameters in the model ( _ 0 and A) individually

for various temperatures. In addition, the softening parameter a is also temperature

and displacement rate dependent. The relationship between a and temperature T

and displacement rate _h is given as a = (1�m(T )) exp���

_h0:3063(T�869)+1:2

��+m(T )

by curve �tting, where m(T ) = m0[(T � 869)3 + 27:42(T � 869)2] and m0 is shown

in Table A.2. The model predictions and comparison with the data for three tem-

peratures and various displacement rates are shown in Figures A.10, A.11 and A.12.

The parameters used in the simulation are listed in Table A.2.

Since the compression data are carried out with constant displacement rate in-

stead of constant strain rate, the stresses are expected �rst to decrease after yield

and then to increase with the increasing strain rate during compression. The sim-

ulation successfully captures this feature. However, the elastic-viscoplastic model

does not include viscoelastic behavior before yield which is shown in the experi-

ments, especially at the lower displacement rates. Overall the model successfully

captures the strain rate and temperature dependence of the deformation response of

the glass and the transient behavior (stress overshoot) is also captured by using a

phenomenological evolution structure related parameter model.

107

A.2 Viscoelastic Theories

Various viscoelastic models based on continuum mechanics or molecular struc-

tures are developed to describe the mechanical response of glass and polymer melts.

Two of these models, the White-Metzner model [74] and the Marrucci model [69]

are chosen here to simulate the glass response near Tg and compare with the tension

data at 836K.

A.2.1 White-Metzner Model

The White-Metzner constitutive equation [74] is often used to describe the vis-

coelastic response of polymer and glass melts. The equation for uniaxial ow with a

single relaxation time has the following form.

_�zz +1 + � _��

p3

��zz = 2G _�; (A.10)

_�rr +1 + � _��

p3

��rr = �G _�; (A.11)

where �zz and �rr are the stress components along the axial and radial directions

respectively, _� is the elongational strain rate, � and G are the relaxation time and

the shear modulus of the glass respectively, and � is an adjustable parameter. The

parameters used here are � = 18; � = 18:91; G = 8130MPa to �t the data.

Figure A.13 shows the prediction of the White-Metzner model and comparison

with experimental data in tension at T=836K. The White-Metzner model can not

predict the transient deformation response, i.e. the stress overshoot at high strain

rate, and the rate dependence is also not accurately predicted for the range of strain

rates prescribed.

108

0

50

100

150

200

250

300

350

0 0.05 0.1 0.15 0.2 0.25 0.3

Str

ess

(MP

a)

Strain

= 0.00072 /secε

= 0.00036 /secε

= 0.00009 /secε

= 0.00018 /secε

= 0.000036 /secε

Figure A.13: Stress vs. time of soda-lime silicate glass for various strain rates atT=930K. [5] and comparison with the prediction using (A.10) and(A.11).

109

A.2.2 Marrucci Network Model

Marrucci et al. have proposed a network polymer model [69] in which a structure

related variable is introduced to describe the formation and destruction of the net-

work entanglements. This model has demonstrated the ability to capture the stress

overshoot in polymers for one dimensional elongational ow. We expect that the

deformation of inorganic glasses accompanies structure rearrangement. The defor-

mation response approaches a steady state when these processes reach an equilibrium

state (layered structure). We employ this model to predict the inorganic glass re-

sponse and compare with the data at T=836K. The Marrucci model for elongational

ow with a single relaxation time has the form:

_�zz = x3�zz _x + 2_��zz � 1

�x1:4�zz + 2xG _�; (A.12)

_�rr = x3�rr _x� _��rr � 1

�x1:4�rr � xG _�; (A.13)

_x =x�1:4

�(1� x)� �x�1:4

�(�zz + 2�rr

2Gx)1

2 ; (A.14)

�zz(0) = �rr(0) = 0; x(0) = 1; (A.15)

where x is a structure related variable which is evolving with deformation. All the

other variables are as de�ned before. The parameters are taken as � = 7:0; � =

18:95; G = 8130MPa to �t the data.

The results of simulations using the Marrucci model and comparison with the

tension data for glass at T=836K are shown in Figure A.14.

110

0

50

100

150

200

250

300

350

0 0.05 0.1 0.15 0.2 0.25 0.3

Str

ess

(MP

a)

Strain

= 0.00072 /secε

= 0.00036 /secε

= 0.00009 /secε

= 0.00018 /secε

= 0.000036 /secε

Figure A.14: Stress vs. time response of soda-lime silicate glass in tension for variousstrain rates at T=836K. Curves are Marrucci model simulations using(A.12)-(A.15), and symbols are the data from reference [4].

111

The Marrucci model successfully predicts the stress overshoot at high strain rates

and the reduction extent of the overshoot decreases with decreasing strain rate. How-

ever the strain rate dependence of the steady-state response has not been satisfacto-

rily predicted. Temperature dependence is not included in this model.

A.3 Conclusion

An approach for modeling the mechanical response of inorganic glasses in the

transition range based on a viscoplastic deformation theory has been examined. A

phenomenological model for the structural evolution during deformation has been

proposed to simulate the strain rate and temperature dependent response of glass

near Tg. The simulation of the model and comparison with the uniaxial experimental

data have shown that the model can predict the uniaxial deformation response of

soda-lime-silicate glass around the transition range. The transient stress overshoot

and the rate dependence of the glass are successfully predicted for various temper-

atures for uniaxial tension deformation. In compression the response features are

captured by the model but the temperature dependent response is not predicted;

model constants were chosen for each temperature individually. The model assumes

a linear elastic behavior before the glass yields, while the experiment shows a vis-

coelastic characteristic in the transient range.

The viscoelastic White-Metzner and Marrucci models explored did not capture

the rate dependence response well, although the Marrucci model is able to predict

the stress overshoot. These were examined here because of their similarity to the

Je�rey model used in �ber drawing. Most glass �ber drawing processes occurs in

the high temperature range (above glass softening point), the glass shows Newtonian

behavior in this temperature range. Large deformation during �ber drawing occurs

112

in the high temperature range, thus viscoelasticity has little e�ect on the overall

kinematic and dynamic drawing variables. Although the Je�rey model captures the

�ber drawing process well, it is insu�cient to describe the viscoelastic behavior at

longer times.

BIBLIOGRAPHY

113

114

BIBLIOGRAPHY

[1] F. T. Geyling and G. M. Homsy. Glass Tech., 21(2):95, 1980.

[2] A. K. Varshneya. Fundamentals of Inorganic Glasses. Academic Press, Inc.,1994.

[3] R. H. Doremus. Glass Science. John Wiley and Sons, 1973.

[4] J.H. Simmons; R.K. Mohr and C.J. Montrose. J. Appl. Phys., 53:4057, 1982.

[5] P. Manns and R. Br�uckner. Glastechn. Ber, 61:46, 1988.

[6] K. L. Loewenstein. The manufacturing Technology of Continuous Glass Fibers.Elsevier, 1993.

[7] A. A. Gri�th. Phil. Trans. Roy. Soc. A, 221:163, 1920.

[8] A. Smekel. J. Soc. Glass Tech., 20:4322{453T, 1936.

[9] O. Reinkober. Phys. Z., 32:243, 1931.

[10] J. B. Murgtroyd. J. Soc. Glass Tech., 28:368{405T, 1944.

[11] T. A. Bartenev. Zh. Fizi. Khim., 29 (3):508, 1955.

[12] A. F. Prebus and J. V. Michener. Industr. Eng. Chem., 46:146, 1954.

[13] W. H. Otto and F. W. Preston. J. Soc. Glass Tech., 34 (15):63, 1950.

[14] W. H. Otto. J. Am. Ceram. Soc., 38 (3):38, 1955.

[15] W. F. Thomas. Glass Tech., 1(1):4, 1960.

[16] W. F. Thomas. Glass Tech., 12(2):42, 1971.

[17] K. L. Loewenstein and J. Dowd. Glass Tech.Glass Tech., 9(6):164, 1968.

[18] V. G. Pahler and R. Br�uckner. Glastech. Ber., 58(2):33, 1985.

[19] H. Stockhorst and R. Br�uckner. J. Non-Cryst. Solids, 49:471, 1982.

[20] H. Stockhorst and R. Br�uckner. J. Non-Cryst. Solids, 86:105, 1986.

115

[21] E. H. Andrews. Brit. J. Appl. Phys., 10:39, 1959.

[22] S. Kase and T. Matsuo. J. Poly. Sci. Part A, 3:2541, 1964.

[23] D. A. McGraw. J. Basic Eng., 90:343, 1968.

[24] M. Matovich and J. R. A. Pearson. Ind. Eng. Chem. Fund., 8:512, 1969.

[25] W. W. Schultz and S. H. Davis. J. Rheol., 40:285, 1982.

[26] B. P. Huynh and R. I. Tanner. Rheology Acta, 22:482, 1983.

[27] Y. T. Shah and J. R. A. Pearson. Ind Eng. Chem. Fund., 11:145, 1972.

[28] Y. T. Shah and J. R. A. Pearson. Ind Eng. Chem. Fund., 11:150, 1972.

[29] J. R. A. Pearson and Y. T. Shah. Ind Eng. Chem. Fund., 13:134, 1974.

[30] U. C. Paek and R.B. Runk. J. Appl. Phys, 49(8):4417, 1978.

[31] M. M. Denn; C. J. Petrie and P. Avenas. J. AIChE., 21:791, 1975.

[32] R. K. Gupta; J. Puszynski and T. Sridhar. J. Non-Newtonian Fluid Mech.,21:99, 1986.

[33] T. Sridhar; R.K. Gupta; D. V. Boger and R. Binnington. J. Non-NewtonianFluid Mech., 21:115, 1986.

[34] N. Phan-Thien. J. Non-Newtonian Fluid Mech., 25:129, 1987.

[35] W. W. Schultz. J. Rheol., 31:733, 1987.

[36] D. A. Nguyen; R.K. Gupta and T. Sridhar. J. Non-Newtonian Fluid Mech.,35:207, 1990.

[37] G. K. Gupta; W. W. Schultz; E. M. Arruda and X. Lu. Rheol. Acta, 35(6):584,1996.

[38] X. Lu; E. M. Arruda and W. W. Schultz. J Non-Newtonian Fluid Mech., toappear, 1998.

[39] C. J. Petrie and M. M. Denn. J. AIChE., 22 (2):209, 1976.

[40] W. Weibull. J. App. Mech., 18:293, 1951.

[41] Horst Scholze. Glass. Springer-Verlag, 1991.

[42] M. Matsui and D. C. Bogue. Polymer. Eng. Sci., 16:735, 1976.

[43] G. W. Scherer. Relaxation in Glass and Composites. John Wiley and Sons,1986.

116

[44] R. J. O'Donnell and J. A. Zakarian. Ind Eng Chem Process Des Dev, 23:491,1984.

[45] G. K. Gupta and W. W. Schultz. Int J Nonlinear Mech 33: (1) 151-163 JAN1998, 33:151, 1998.

[46] A. Q. Tool. J. Am. Ceram. Soc., 29:240, 1946.

[47] H. N. Ritland. J. Am. Ceram. Soc., 39:403, 1956.

[48] O.S. Narayanaswamy. J. Am. Ceram. Soc., 54(10):491, 1971.

[49] G. W. Scherer. J. Am. Ceram. Soc., 67:504, 1984.

[50] I. M. Hodge. J. Non-Cryst. Solids, 131-133:435, 1991.

[51] I. M. Hodge. J. Res. Natl. Inst. Stand. Technol., 102:195, 1997.

[52] J. Huang and P. K. Gupta. J. Non-Cryst. Solids, 151:175, 1992.

[53] H. S. Chen and C. R. Kurkjian. J. Am. Ceram. Soc., 66(9):613, 1983.

[54] H. S. Chen. J. Non-Cryst. Solids, 46:289, 1981.

[55] A. R. Berens and I. M. Hodge. Macromolecules, 15:756, 1982.

[56] D. H. Uhlmann and N. J. Kreidl, editors. Glass: Science and Technology. Aca-demic Press, 1986.

[57] J.H. Li and D.R. Uhlman. J. Non-Cryst. Solids, 22:127, 1970.

[58] J.H. Simmons; R. Ochoa; K.D. Simmons and J.J. Mills. J. Non-Cryst. Solids,105:313, 1988.

[59] R. Br�uckner and Y. Yue. J. Non-Cryst. Solids, 175:118, 1994.

[60] Y. Yue and R. Br�uckner. J. Non-Cryst. Solids, 180:66, 1994.

[61] S. Rekhson. J. Non-Cryst. Solids, 131-133:467, 1991.

[62] A. S. Argon. Glass: Science and Technology Vol. 5. Academic Press, Inc., 1980.

[63] A. S. Argon and H. Y. Kuo. Materials Science and Eng, 39:101, 1979.

[64] D. Deng; A. S. Argon and S. Yip. Phil. Trans. R. Soc. Lond., A 329:613, 1989.

[65] M. C. Boyce and E. M. Arruda. Polym. Eng. Sci., 30:1288, 1990.

[66] O. A. Hasan and M. C. Boyce. Polym. Eng. Sci, Feb.:�nd, 1995.

[67] P.H. Mott; A. S. Argon and U.W. Suter. Philos. Mag., 67:931, 1993a.

117

[68] H. Burlet L. Du�rene; R. Gy and R. Piques. J. Rheol., 41(5):1021, 1997.

[69] D. Acierno; F. P. La Mantia; G. Marrucci and G. Titomanlio. J. Non-NewtonianFluid Mech., 1:125, 1976.

[70] U.F. Kocks; A.S. Argon and M.F. Ashby. Thermodynamics and kinetics of slip,progress in materials science. 1975.

[71] A.S. Argon. Phil. Mag., 28:839, 1973.

[72] M.C. Boyce; D.M. Parks and A.S. Argon. Mech. of Materl., 7:15, 1988.

[73] D. A. McGraw. J. of the Amer. Ceram. Soc., 35(1):22, 1952.

[74] J. L. White and A. B. Metzner. J. Appl. Polym. Sci., 8:1367, 1963.

ABSTRACT

EXPERIMENTAL AND ANALYTICAL INVESTIGATION OF

NON-ISOTHERMAL VISCOELASTIC GLASS FIBER DRAWING

by

Xiaoyong Lu

Chairperson: Ellen M. Arruda, Co-Chairperson: William W. Schultz

A preform glass �ber drawing apparatus is designed and built to investigate non-

isothermal viscoelastic glass �ber drawing. The e�ect of the drawing parameters on

glass �ber properties is studied experimentally and analytically.

Birefringence is a measure of anisotropy in glass structure that can in uence

�ber properties and performance. Birefringence is produced during �ber drawing

as the �ber is rapidly stretched in the viscoelastic glass transition range, and is

\frozen" into the glass during rapid cooling. With our drawing apparatus using

Borosilicate glass (Corning code 7740) preforms, we produce glass �bers for a range

of process conditions and measure their as-drawn birefringence. The development of

birefringence in glass �bers is found to depend on the amount of deformation, the

deformation rate, and temperature. Results for various process parameters show that

increasing draw ratio, increasing elongation rate, and decreasing draw temperature

increase birefringence.

Post-process annealing is used to examine the time and temperature dependence

of glass �ber birefringence relaxation and its e�ect on glass �ber tensile strength.

Birefringence is found to completely relax in the temperature range close to the glass

transition range as expected, but it is also noted that birefringence shows substantial

(although incomplete) relaxation in a temperature range well below the glass tran-

sition. This low temperature relaxation indicates that the relaxation process is due

to a very wide distribution of relaxation times.

A nonisothermal one-dimensional model is examined to simulate the glass �ber

drawing and �ber birefringence relaxation using a generalized Je�rey model. The

relaxation time and retardation time are both functions of the temperature. The

conservation equations of mass, momentum and energy are solved simultaneously

with the viscoelastic constitutive equation to determine the spatial distribution of

�ber dimensions, velocities, stresses and frozen in elastic strains. The e�ects of

process parameters on the �ber properties are simulated. The model predictions are

compared with the experiments and are found to be in qualitative agreement with

data.