EXPERIMENTAL AND NUMERICAL ANALYSIS OF THERMAL FORMING PROCESSES FOR PRECISION OPTICS
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University
By
Lijuan Su
Graduate Program in Industrial and Systems Engineering
The Ohio State University
2010
Dissertation Committee:
Dr. Allen Y. Yi, Adviser
Dr. Jose M. Castro
Dr. Betty L. Anderson
Copyright by
Lijuan Su
2010
ABSTRACT
Glass has been fabricated into different optical elements including aspherical lenses and
freeform mirrors. However, aspherical lenses are very difficult to manufacture using
traditional methods since they were specially developed for spherical lenses. On the other
hand, large size mirrors are also difficult to make especially for high precision
applications or if designed with complicated shapes. Recently developed two closely
related thermal forming processes, i.e. compression molding and thermal slumping, have
emerged as two promising methods for manufacturing aspherical lenses and freeform
mirrors efficiently. Compression molding has already been used in industry to fabricate
consumer products such as the lenses for digital cameras, while thermal slumping has
been aggressively tested to create x-ray mirrors for space-based telescopes as well as
solar panels. Although both process showed great potentials, there are a quite few
technical challenges that prevent them from being readily implemented in industry for
high volume production.
This dissertation research seeks a fundamental understanding of the thermal forming
processes for both precision glass lenses and freeform mirrors by using a combined experimental, analytical and numerical modeling approach. First, a finite element method ii
(FEM) based methodology was presented to predict the refractive index change of glass
material occurred during cooling. The FEM prediction was then validated using
experimental results. Second, experiments were also conducted on glass samples with
different cooling rates to study the refractive index variation caused by non-uniform
cooling. A Shack-Hartmann Sensor (SHS) test setup was built to measure the index
variations of thermally treated glass samples. Again, an FEM simulation model was
developed to predict the refractive index variation. The prediction was compared with the
experimental result, and the effects of different parameters were evaluated.
In the last phase of this dissertation research, an FEM simulation model was developed to
study the thin glass slumping processes on both concave and convex mandrels.
Simulation of thin glass sheet slumping on convex mandrel was performed to study the
effects of different process parameters, i.e. thickness of the glass sheet, cooling and
heating rate, soaking time and soaking temperature. Finally, experiments of thermal
slumping glass plates on a parabolic concave mandrel were performed to study the
thickness effect on slumping process and the final surface contour of the upper surface of
the glass plate. Simulation was again conducted to predict the surface contour. The
comparison between simulation and experiments showed that the FEM simulation is adequate for predicting the surface contour if the glass was fully slumped. It was also
discovered that for process conditions used, thinner glass sheets were not fully slumped.
ii
DEDICATION
This document is dedicated to my family.
iii
ACKNOWLEDGEMENTS
I would like to express my gratitude to my adviser, Prof. Allen. Yi for providing me with an opportunity to work in a very exciting field of precision optical engineering. I would like to thank Prof. Yi for his trust, guidance, enthusiasm and insight during my research. I also would like to thank Dr. William W. Zhang from Goddard Space Flight Center at
NASA for providing financial support on the x-ray mirror development project. I also appreciate the suggestions, assistance and comments of other faculty members at OSU whom I had the opportunities to work with during the course of this research: Prof. Jose
Castro, Prof. Betty Lise Anderson, and Prof. Rebecca Dupaix. I am especially in debt to
Prof. Castro for the time he spent helping with my questions and patiently guiding me with explanation. I also would like to thank Prof. Anderson for her assistance to listen and suggestions for optical methods to test the lens performance.
I would like to thank Dr. Thomas Raasch for assistance in setting up the Shack-Hartmann
Sensor (SHS) measuring system.
iv
I want to thank Prof. Fritz Klocke, Axel Demmer, Dr. Olaf Dambon, Guido Pongs, and
Fei Wang at the Institute for Production Technology (IPT), Aachen, Germany for providing some of the experimental support reported in this research.
I acknowledge the help from Mary Hartzler at the Department of Integrated Systems
Engineering for assisting in the Coordinate Measuring Machine operation.
Sincere thanks are extended to all my colleagues and fellow PhD students, for their suggestions and assistance in different parts of this research. I would like to thank Dr.
Anurag Jain and Dr. Yang Chen for suggestions in glass forming simulation and their assistance in using different finite element method software programs. Special thanks also go to Dr. Lei Li, Dr. Chunning Huang, Dr. Wei Zhao, Peng He, and Likai Li.
In the end, I would also like to take this opportunity to acknowledge the encouragement and moral support provided by my family.
v
VITA
April 1981 ...... Born, WuHu, China
July 2003 ...... B. S. Measuring and Controlling Technique
and Instrumentation, University of Science
and Technique of China, Hefei, China
June 2006 ...... M.S. Optical Engineering, Xi'an Institute of
Optics and Precision Mechanics of China
Academic of Science, Xi'an, China
June 2006 to 2010 ...... Graduate Research Associate, Integrated
System Engineering Department, The Ohio
State University, Columbus, Ohio
vi
PUBLICATIONS
JOURNAL PUBLICATIONS:
1. L. Su, Y. Chen, A. Y. Yi, F. Klocke, G. Pongs, “Refractive Index Variation in Compression Molding of Precision Glass Optical Components”, Applied Optics, 47 (10), 1662-1667, 2008.
2. Y. Chen, L. Su, A. Y. Yi, F. Klocke, G. Pongs, “Numerical Simulation and Experimental Study of Residual Stresses in Compression Molding of Precision Glass Optical Components,” Journal of Manufacturing Science and Engineering, 130 (5), 051012-1-8, 2008.
FIELDS OF INTERESTS
Major Field: Industrial and Systems Engineering
Studies in: Precision engineering, Precision optics, Glass thermal forming
vii
LIST OF CONTENTS
ABSTRACT ...... ii
DEDICATION ...... iii
ACKNOWLEDGEMENTS ...... iv
VITA ...... vi
LIST OF CONTENTS ...... ii
LIST OF TABLES ...... vi
LIST OF FIGURE...... viii
Chapter 1: INTRODUCTION ...... 1
1.1 Compression Molding of Glass Lenses ...... 2
1.2 Thermal Slumping of Glass Mirror ...... 4
Chapter 2: RESEARCH OBJECTIVE ...... 7
Chapter 3: STATE OF THE ART ...... 10
3.1 Glass Rheology ...... 10
ii
3.1.1 Viscosity ...... 10
3.1.2 Viscoelasticity in Glass Transition Region ...... 13
3.1.3 Viscoelastic models ...... 16
3.2 Numerical Modeling of Glass Thermal Forming Processes ...... 18
Chapter 4: GLASS INDEX CHANGE AS A RESULT OF THERMAL TREATMENT
22
4.1 Theory ...... 23
4.1.1 Structural Relaxation ...... 23
4.1.2 Density Change ...... 28
4.1.3 Relation between the Density and the Refractive Index ...... 31
4.2 FEM Modeling of Glass Cooling ...... 33
4.3 Simulation Results and Discussions ...... 36
4.3.1 The Effect of Thermal Expansion Coefficient Ratio ...... 38
4.3.2 The Effect of Cooling Rate ...... 41
4.4 Conclusion ...... 43
Chapter 5: GLASS REFRACTIVE INDEX VARIATION CAUSED BY THERMAL
TREATMENT ...... 44
5.1 Theory ...... 44
iii
5.1.1 Refractive Index Variation ...... 44
5.1.2 Shack-Hartmann Test...... 48
5.2 Glass Thermal Treatment Process ...... 50
5.3 FEM Modeling of Glass Cooling Process ...... 52
5.4 Results ...... 58
5.4.1 Measured Index Variation ...... 58
5.4.2 Simulated Index Variation ...... 62
5.5 Discussion ...... 64
5.5.1 Effect of the Cooling Rate ...... 65
5.5.2 Effect of the Heat Capacity ...... 69
5.5.3 Effect of the Thermal Expansion Coefficient ...... 71
5.6 Conclusion ...... 76
Chapter 6: FEM MODELING OF THERMAL SLUMPING OF GLASS MIRRORS 80
Chapter 7: FEM INVESTIGATION OF THE THERMAL SLUMPING PROCESS
WITH CONVEX MANDREL ...... 86
7.1 FEM Modeling ...... 88
7.2 FEM Results and Discussion ...... 93
7.2.1 Influence of the Glass Thickness ...... 94
iv
7.2.2 Influence of the Glass Sheet Length ...... 98
7.2.3 Influence of Heating Rates...... 99
7.2.4 Influence of the Cooling Rate ...... 107
7.2.5 Influence of the Soaking Temperature and Soaking Time ...... 111
7.3 Conclusions ...... 114
Chapter 8: INVESTIGATION OF GLASS THICKNESS EFFECTS ON SLUMPING
WITH CONVACE MANDREL USING BOTH EXPERIMENTAL AND NUMERICAL
APROACHES ...... 116
8.1 Slumping Process ...... 117
8.2 FEM Modeling ...... 122
8.3 Simulated and Experimental Results...... 124
8.4 Discussion ...... 130
REFERENCE ...... 143
v
LIST OF TABLES
Table 4.1 Mechanical and thermal properties of glass P-SK57 ...... 34
Table 4.2 Structural relaxation parameters of glass P-SK57 required for simulation ..... 35
Table 4.3 Predicted index change caused by cooling process with different liquid
thermal expansion coefficient, *based on the refractive index nd @ wavelength 587.6 nm
...... 38
Table 4.4 Predicted index change at different rates and calculated index change *based
on reference [Schott, 2007] ...... 41
Table 5.1 Mechanical and thermal properties of BK7 glass ...... 54
Table 5.2 Structural relaxation parameters used in numerical simulation ...... 54
Table 5.3 Heat capacity and density of BK7 glass and Iron mold ...... 66
Table 7.1 Mechanical and thermal properties of D263 glass ...... 90
Table 7.2 Viscosity of D263 glass at different Temperature ...... 90
Table 7.3 Final length and thickness of the slumped glass sheet at the end of the cooling
...... 108
Table 8.1 Thickness information of the glass workpieces [Source: S.I Howard Galss] 121
Table 8.2 Mechanical and thermal properties of soda lime glass and MACOR® ...... 123 vi
Table 8.3 Viscosity of the soda lime glass at different temperatures ...... 123
Table 8.4 Curvature parameter P fitted from simulated and experimental results ...... 127
vii
LIST OF FIGURE
Figure 1.1 Schematic illustration of the glass compression molding machine and the complete molding process [Jain, 2006b]...... 3
Figure 1.2 Illustration of the thermal slumping process ...... 5
Figure 3.1 Fitted viscosity vs. temperature curve for glass D263 (VFT equation) ...... 12
Figure 3.2 The stress σ(t) change of glass when a constant a strain ε0 is imposed at t0, redrawn from [Scherer, 1986] ...... 14
Figure 3.3 The strain ε(t) change of glass when a constant uniaxial stress σ0 at t0, redrawn from [Scherer, 1986] ...... 14
Figure 3.4 The property change after a sudden change in temperature in glass transition
region, redrawn from [Scherer, 1986] ...... 15
Figure 3.5 Viscoelastic models: (a) Maxwell model (b) Voigt element (c) Burger model
...... 16
Figure 3.6 Generalized Maxwell model ...... 17
Figure 4.1 Property change of glass during cooling ...... 24
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Figure 4.2 Volume versus temperature during cooling of glass liquid, redrawn from
[Scherer, 1986] ...... 28
Figure 4.3 The thermal expansion coefficient of the glass volume during cooling, the curve is the derivative of figure 4.2, redrawn from [Scherer, 1986] ...... 29
Figure 4.4 Volume versus temperature at two different cooling rates during cooling ... 30
Figure 4.5 Predicted volume versus temperature curves obtained by structure relaxation model at two cooling rates qref = 2 °C/hr and q1 = 3500 °C/hr ...... 36
Figure 4.6 Predicted volume versus temperature curves obtained by structure relaxation model at two cooling rates qref = 2 °C/hr and q1 = 3500 °C/hr with different rα ...... 39
Figure 4.7 Thermal expansion coefficient αV(T) versus temperature during cooling ..... 40
Figure 4.8 Predicted volume versus temperature curves obtained by structure relaxation model at three cooling rates q1 = 3500 °C/hr, q2 = 350 °C/hr and q3 = 35 °C/hr ...... 42
Figure 4.9 Thermal expansion coefficient α(T) versus temperature during cooling at two cooling rate q1 = 3500 °C/hr, and q2 = 350 °C/hr ...... 42
Figure 5.1 Principle of Shack-Hartmann sensor ...... 48
Figure 5.2 Temperature histories of three different cooling rates ...... 51
Figure 5.3 Meshed numerical simulation model ...... 53
Figure 5.4 Heat capacity versus temperature for 0.4Ca(NO3)2·0.4KNO3 [Copy from
Moynihan, 1976] ...... 55
Figure 5.5 Heat capacity Cp of BK7 glass used in the FEM simulation ...... 55
Figure 5.6 The experiment arrangement of sample and molds ...... 56
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Figure 5.7 Simulated temperature changes caused by convection at mold bottom surface
and convection at glass side surface ...... 57
Figure 5.8 Schematic of the Shack-Hartmann testing system: 1. He-Ne Laser; 2.
Polarizer; 3. Beam expander; 4. Sample in matching liquid; 5. Lens 1; 6. Lens 2; 7.
Shack-Hartmann Sensor...... 59
Figure 5.9 Reconstructed wavefront variation using Shack-Hartmann sensor
(0.225°C/sec) ...... 60
Figure 5.10 Measured refractive index variations along the radial direction for three
different cooling rates and an untreated glass lens (blank) ...... 61
Figure 5.11 Predicted index variations for three different cooling rates ...... 63
Figure 5.12 Comparison of measured and simulated results of the refractive index
variation curves of three cooling rates ...... 64
Figure 5.13 Illustration of the heat flux through the surface ...... 65
Figure 5.14 Simulated temperature history near the glass and mold surfaces caused by
force convection ...... 67
Figure 5.15 Measured and simulated results of the refractive index variation curves at
cooling rate q1 and simulated result with adjusted cooling rate qa ...... 68
Figure 5.16 Temperature gradients along radical direction of the glass with/without
considering heat capacity Cp changes as temperature changes ...... 70
Figure 5.17 Simulated results of the refractive index variation curves with/without considering heat capacity Cp changes as temperature changes ...... 71
x
Figure 5.18 Induced stress σzz at node 122 with different solid thermal expansion αg
(same thermal expansion coefficient ratio rα = 4) ...... 73
Figure 5.19 Induced stress σzz at node 122 with different thermal expansion coefficient
-6 -1 ratio rα (same solid thermal expansion coefficient αg =8.3x10 K ) ...... 73
Figure 5.20 Simulated results of the refractive index variation curves with two different
value of solid thermal expansion αg (same thermal expansion coefficient ratio rα = 4) ... 75
Figure 5.21 Simulated results of the refractive index variation curves with different
thermal expansion coefficient ratio rα (same solid thermal expansion coefficient αg
=8.3x10-6 K-1) ...... 75
Figure 5.22 Flowchart of optimizing the lens designing and manufacturing with FEM
simulation ...... 79
Figure 6.1 Schematic of the setup and thermal slumping process ...... 81
Figure 6.2 Simulation stages in the thermal slumping process ...... 84
Figure 7.1 The simulation model ...... 88
Figure 7.2 Glass sheet deformed on its own weight ...... 89
Figure 7.3 Fitted viscosity vs. temperature curve of D263 glass ...... 91
Figure 7.4 The y displacement of the end of the glass sheet during the slumping stage . 92
Figure 7.5 Final deformed glass sheet at the end of cooling ...... 93
Figure 7.6 Inner surface profile of sag variation of glass sheets with different
thicknesses (a) t = 0.2 mm, (b) t = 0.4 mm, (c) t = 0.8 mm, (d) Comparison between the
predicted curve shape of three glass thicknesses ...... 95
xi
Figure 7.7 Comparison of radius variation Δr of the inner surface profile for different thicknesses ...... 97
Figure 7.8 Radius variation comparison of glass sheets for two different lengths ...... 99
Figure 7.9 Three tested heating rates ...... 100
Figure 7.10 Definition of Angle θ ...... 100
Figure 7.11 Comparison of Y displacements of the end point between q1 and q2 ...... 101
Figure 7.12 Comparison of Temperature vs. Angular Velocity θ& curves between q1 and
q2 ...... 102
Figure 7.13 Comparison of sag variation Δy vs. x between q1 and q2 ...... 103
Figure 7.14 Shape difference δy vs. x between q1 and q2 ...... 103
Figure 7.15 Comparison of radius variation Δr between q1 and q2 ...... 104
Figure 7.16 Comparison of angular velocity θ& vs. Temperature curves between q2 and q3 ...... 105
Figure 7.17 Comparison of radius variation Δr between q2 and q3 ...... 106
Figure 7.18 Three tested cooling rates ...... 107
Figure 7.19 Comparison of radius variation Δr among c1, c2 and c3 ...... 108
Figure 7.20 Cooling rates c3 and c4 ...... 109
Figure 7.21 Comparison of radius variation Δr between c3 and c4 ...... 110
Figure 7.22 Comparison of radius variation ∆r after being held one hour at different soaking temperature ...... 112
xii
Figure 7.23 Comparison of radius variation ∆r after being held two hours at different soaking temperature ...... 112
Figure 7.24 Comparison of radius variation ∆r when glass sheet was held at 555 °C for
two hours and at 560 °C for one hour respectively ...... 113
Figure 7.25 Comparison of radius variations ∆r of 0.8 mm thick glass sheet held at 565
°C for 2, 3 and 4 hours ...... 114
Figure 8.1 Procedures of the thermal slumping process ...... 117
Figure 8.2 A typical thermal history for glass thermal slumping process ...... 118
Figure 8.3 Parabolic shape of the MACOR® mold ...... 119
Figure 8.4 Mold curvature error caused by machining error ...... 120
Figure 8.5 FEM model and the result of the slumping predicted by the FEM model .. 122
Figure 8.6 Fitted viscosity vs. temperature curve of soda lime glass ...... 124
Figure 8.7 Scheme of zero order compensation assumption ...... 125
Figure 8.8 Sag variation between the simulated upper surface of glass with different
thickness and the mold surface ...... 126
Figure 8.9 Fitted curvature parameter Pm error in the measurements with different glass
thicknesses ...... 127
Figure 8.10 Sag variations between experimental and simulated results of different glass
thicknesses ...... 128
Figure 8.11 Surface contour RMS errors between experimental and simulated results of
different glass thicknesses...... 129
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Figure 8.12 Surface roughness of the upper surface of different glass thicknesses ...... 130
Figure 8.13 Illustration of incomplete slumped glass workpiece ...... 130
Figure 8.14 Strain response to applied stress of Voigt-Kelvin model ...... 131
Figure 8.15 Viscosity vs. temperature curves with different reference points ...... 135
Figure A.1 Simulation model with deformable mandrel ...... 138
Figure A.2 Radius change of the D263 glass mandrel after two cycles ...... 139
Figure A.3 Relative radius variation of glass sheet at the end of heating process ...... 141
Figure A.4 Radius variation of the glass sheet at the end of cooling ...... 142
xiv
CHAPTER 1: INTRODUCTION
Glass has been used to fabricate different optical elements. Although polymers have also
been adopted as materials for precision optics, glass still has many advantages, such as
higher stability, transparency and scratch resistance. The current optical fabrication
methods are based on traditional abrasive techniques and with assistance of numerical
computer control systems. These methods can manufacture spherical lenses with high quality very productively. However aspherical lenses are more difficult to manufacture using conventional methods. Although the computer control technology has improved
process, it is still time consuming and costly to manufacture aspherical lenses than spherical lenses. It is also difficult to manufacture optics components with much larger
size, which are almost always mirrors.
New techniques developed for manufacturing aspherical lenses include single-point
diamond turning for low to medium volume production and glass thermal forming for
high volume production [Malacara, 2001]. The compression molding and thermal
slumping methods are considered typical thermal forming processes since both of these
processes involve thermal treatment of glass during the manufacturing processes. The
glass workpiece is soften when heated to a certain temperature, which allows it to be 1
formed into a mold cavity and replicate the mold shape. The thermal forming processes have been customized for mass production of spherical, aspherical, and freeform optical components. Both compression molding and thermal slumping (replicating) processes are the topics investigated in this dissertation due to similarity between two processes.
1.1 Compression Molding of Glass Lenses
Compression molding process is a hot forming method which is used to directly press heated raw glass gob or blank between optically polished molds to obtain the lens with the desired surface shape or pattern, especially aspherical or freeform surface
[Maschmeyer, 1983]. The molds are manufactured with exact opposite shape of the designed lenses. The process has been adopted to manufacture precision spherical and aspherical lenses, which are highly demanded by the growing electronics industry (e.g. pocket cameras, projectors, and CD/DVD player) [Pollicove, 1988].
The procedures of compression molding method have been well developed and established through past three decades [Milton, 1974; Menden-Piesslinger, 1983;
Maschmeyer, 1983; Yi, 2005]. Figure 1.1 shows a machine used for the process, and a schematic illustration of the process. The process starts by heating the molds and raw glass material to a working temperature above the transition temperature of glass. The glass viscosity will be in a range of 107-109 Pa·sec at the working temperature. Once the glass and mold temperature are steady, the molds are pressed together with the soft glass inside the cavity. The temperature is maintained during the “molding” procedure. After
2
maintaining the pressing position for a short time period, controlled cooling of the formed glass optic is carried out by blowing nitrogen around the still closed molds. The finished lens is released from the molds at a temperature close to room temperature. Compression molding process is a one-step net-shape method, while traditional glass fabrication methods need several steps to finalize a lens to a designed shape. The lens shape is formed during molding procedure and finished when released at the end of the cooling process. Molds with different shapes can be made for different lens shape designs, and molds with multiple cavities can be designed to accommodate for high volume production. Furthermore, the compression molding process is more environment friendly since no polishing and grinding are needed to finalize the lens shape.
Figure 1.1 Schematic illustration of the glass compression molding machine and the
complete molding process [Jain, 2006b].
3
In the past few years, some experiments and simulations have been done to investigate
compression molding precision glass lenses to meet higher optical performance
requirements. Aspherical and freeform glass lenses were fabricated by using compression
molding process for precision optical applications [Yi, 2005; Vogt, 2007]. In addition,
compression molding process is also used to create diffractive optical elements [Yi, 2006]
and microlens arrays [Chen, 2008], which can be used in optical data storage, optical
communication, and digital displays. However, there are still quite a few technique
challenges preventing this process from being fully implemented in industry. These
challenges includes thermal expansion of the molds, mold life, lens curvature shrinkage,
residual stresses and refractive index change induced into the molded lens after cooling.
Some of these challenges have been investigated, such as, curvature compensation [Wang,
2008], residual stress [Chen, 2008b] and refractive index change [Su, 2008; Zhao, 2009;
Zhao, 2009b]. However, more research is needed to solve these problems to prepare the process for wider applications in industry.
1.2 Thermal Slumping of Glass Mirror
In optics fabrication, the size of the optics will also affect the manufacturing cost and the level of difficulty. For mirrors with large dimensions, it is much more difficult to manufacturing by using traditional methods. Glass thermal replicating (slumping) method provides a more affordable and less time consuming alternative for manufacturing spherical, aspherical, and freeform mirrors. Glass thermal slumping process is another thermal forming process, which has similar process steps such as heating, forming and 4
cooling. As shown in figure 1.2 , in glass thermal molding process, the glass sheet is
placed on a mold and both of the glass and mold are heated to working temperature (or
soaking temperature), then the temperature is held while the glass sheet is deformed
under its own weight or vacuum pressure. The slumped glass is slowly cooled (annealed)
to ensure the shape of the mirror and minimize the residual stresses when reaches the
room temperature.
Figure 1.2 Illustration of the thermal slumping process
The low cost, easy fabrication of non spherical shape and high volume productivity make
the thermal slumping process a good candidate for fabricating large size mirrors for
different applications. Labov started using this technique to form a thin glass sheets to x- ray mirrors with a concave stainless steel mold [Labov, 1988]. The same setup was used to produce x-ray optics for the HEFT project [Craig, 1998]. More recently, a similar slumping technique with convex mandrels was developed for fabricating x-ray mirrors
for the Constellation-X project [Zhang, 2003; Zhang, 2004]. Chen started to research on
fabrication of optical components for solar energy systems using thermal slumping
process [Chen, 2010]. 5
Finite element methods (FEMs) have been widely used in the industry to study, analyze,
develop and optimize different manufacturing processes to improve products quality.
With the improved computing capabilities of personal computers and the progress of commercial FEM codes, it became possible to realistically simulate the glass thermal forming process with established numerical model and experimental data. The FEM assisted modeling provides a promising approach for observing some parameters and variables during the process which are difficult or impossible to measure in experiments.
These variables include: a) Temperature distribution during and after forming, refractive index distribution, residual stress distribution, and lens shrinkage in compression molding of glass lenses. b) Glass sheet behavior as stress relaxation and structural relaxation at any given time and temperature during thin glass sheet slumping process. This research is mainly focused on developing a reliable numerical simulation model to study the effects of cooling on the compression molded lenses and the process parameters that affects the thermal slumping process, and predicting both the refractive index of the molded glass lenses and the surface contour of a slumped glass sheet.
6
CHAPTER 2: RESEARCH OBJECTIVE
The development of glass thermal forming technique provides a high volume, low cost
approach for producing precision aspherical lenses, micro lens arrays, and even freeform
optical mirrors, which are difficult to fabricate by using traditional abrasive lens
manufacturing techniques. However, there are still several technical issues that prevent
the technique from being readily implemented in the industry.
The overall objective of this dissertation research is to determine the refractive index
performance of compression molded glass lenses and the feasibility of using thermal slumping for production of precision freeform mirrors. Extensive experiments and Finite
Element Method (FEM) simulation are combined as a practical approach to studying
some of the challenges and developing glass slumping into a potential manufacturing
process for aspherical/freeform mirrors. Customized experiments are designed and
conducted to study the refractive index variation induced into glass optical components in
the compression molding process. The research provides an FEM based methodology for
lens designer and manufacturer to identify the cooling rate effects on the refractive index
of compression molded glass optics and optimize their design and manufacturing process
7
to achieve the required optical performance. An FEM model is presented to analyze thermal slumping thin glass mirrors process and different parameters are applied into the model to evaluate their effects on the molding process. This methodology can help manufacturer to identify an optimal procedure for fabricating aspherical/freeform mirrors with desired optical performance. Experiments and corresponding FEM simulation are performed to evaluate the thickness effects of glass workpiece for the thermal slumping process with a concave mandrel.
The specific objectives of the proposed research are:
i. Develop a methodology to predict the refractive index change caused by cooling
process. Perform 2D FEM simulations incorporated with viscoelastic stress and
structural relaxation phenomenon of glass to study the volume change which
leads to the refractive index change prediction. Verify the model by comparing
simulated results with value calculated from empirical equations provided by the
manufacturer. Study the effects of thermal expansion and cooling rates on the
refractive index change by adjusting the values of parameters. (Chapter 4)
ii. Conduct cooling test of a simple cylindrical glass at different cooling rates and
measure the refractive index variation caused by temperature gradient during the
cooling process with a setup based on Shack-Hartmann Sensor (SHS). Perform
2D FEM simulation based on the previous developed methodology with glass
material properties and measured cooling rates during experiments to predict the
8
refractive index variation induced in the glass. Investigate the difference by
comparing experimental and simulated results and discuss the possible causes for
the discrepancy. (Chapter 5) iii. Demonstrate the basic FEM simulation models that will be used to simulate the
thermal slumping glass mirrors processes. (Chapter 6) iv. Perform 2D FEM simulation on thermal slumping of a thin glass sheet on a
convex mandrel. Adjust values of parameters as thickness, heating rate, cooling
rate, soaking temperature and soaking times to study the effects of these values on
the slumping process. Investigate the possibility of optimizing the procedure with
implementing FEM simulation prior performing real experiments. (Chapter 7)
v. Design and conduct experiments on a commercial machine to thermally slumped
glass workpieces with different thicknesses on a concave mandrel. Measure the
upper surface of the slumped glass workpiece and predict the surface contour with
established 2D FEM model. Compare the measured and simulated results and
discuss the possible causes for discrepancies between the results. (Chapter 8)
9
CHAPTER 3: STATE OF THE ART
3.1 Glass Rheology
3.1.1 Viscosity
When a liquid undergoes a shearing force, there is a resistance to the deformation.
Viscosity is a measure of the resistance and can be defined as:
Fd η = (3.1) Av
where, F is the tangential force difference applied to two parallel planes of area A separated by a distance d, v is the relative velocity of the two planes. The SI unit of
viscosity is Pascal-second.
The viscosity and viscoelastic behavior of glass are very important subjects to investigate
for glass thermal forming processes. The viscosity of glass determines the various
forming conditions such as: melting conditions, working temperature, annealing
temperature to reduce internal stress, turning temperature point for changing heating and
cooling rates. The composition of glass determines the viscosity of glass at the first place,
and the viscosity of glass strongly depends on temperature. 10
Different models have been used to describe the temperature dependence of the viscosity of glass material. Among these models, two most commonly used equations are the
Arrhenius equation and the Vogel-Fulcher-Tamman(VFT) equation. The Arrhenius equation is given by [Scherer, 1986]:
⎛ ΔH ⎞ =ηη 0 exp⎜ ⎟ (3.2) ⎝ RT ⎠
Where η0 is a constant, ΔH is the activation energy for viscous flow, R is a gas constant and T is the glass temperature in K. Arrhenius equation provides a good fit in the glass transition temperature range (1013 to 109 Pa-s) and is widely used to calculate fitting parameters required for simulating the glass structural relaxation in the cooling stage. The temperature dependence becomes non-Arrhenius at temperature higher than transition range. The VFT equation provides a relative better fit over a wider temperature range, and the most used expression is given as [Fulcher, 1925]:
B logη A +−= (3.3) −TT 0
where A, B and T0 are the fitting constants. T and T0 are given in °C, and To is always less than the Tg of the given glass composition.
Figure 3.1 shows a viscosity curve of a commercial glass D263 fitted from the VFT equation with parameters from reference provided by the glass manufacturer (Schott). As shown in this picture, a series of specific viscosity have been characterized as reference
11
points of the viscosity temperature curve. The temperature at a viscosity of 103 Pa-s is known as the working point of glass, since it is the typical temperature when glass melt is delivered to a forming device. The softening point of glass is the temperature when the viscosity is 106.6 Pa-s. The temperature range between the softening and working point is referred to as working range. The glass transition temperature (Tg) is defined as the temperature when the viscosity approximately equals to 1011.3 Pa-s. The annealing point where the internal stresses relax within a few minutes happens at temperature when the viscosity is 1012.4 Pa-s. The temperature when the viscosity of glass is 1013.6 Pa-s and the internal stresses relax in several hours is known as the strain point. The typical viscosity for compression molding process is in the order of 107 Pa-s, while it is in the order of
109~1010 Pa-s for glass thermal slumping process.
Figure 3.1 Fitted viscosity vs. temperature curve for glass D263 (VFT equation)
12
3.1.2 Viscoelasticity in Glass Transition Region
“The glass transition is a region of temperature in which molecular rearrangements occur on a scale of minutes or hours, so that the properties of the liquid change at a rate that can be easily observed” [Scherer, 1986]. The glass transition temperature (Tg) is a characterized temperature in the transition range used to indicate that glass is in the glass transition region during the thermal treatment. Generally, the glass transition temperature is a temperature determined by changes in either enthalpy or volume versus temperature curve in glass transition region. The chemistry composite of a glass material determines the transition temperature of the glass. Different glass materials have different transition temperatures. For a given glass material, the transition temperature is slightly influenced by the cooling rate. The enthalpy or volume of a glass departs from equilibrium state early at a higher cooling rate, which causes a higher transition temperature. The glass transition temperature is generally determined by a cooling rate of 10K/min.
In the glass transition region, the glass exhibits both viscous and elastic behavior when a deformation force is applied. The response of glass material to a mechanical load is time- dependent and known as viscoelasticity. Figure 3.2 shows the stress response of a viscoelastic material, when an instantaneous strain ε0 is applied at time t0 and held constant. There is an instant stress response Eε0, but the stress decreases to zero over time. This phenomenon is called stress relaxation.
13
Figure 3.2 The stress σ(t) change of glass when a constant a strain ε0 is imposed at t0,
redrawn from [Scherer, 1986]
Figure 3.3 The strain ε(t) change of glass when a constant uniaxial stress σ0 at t0,
redrawn from [Scherer, 1986]
14
When a constant stress σ0 is applied to a glass material, a time-dependent strain response occurs as shown in figure 3.3. The response has three components: an instantaneous elastic strain εΕ, a delay elastic strain εD and a viscous flow at the rate σ0/3η. This behavior can be modeled in FEM software by using different viscoelastic models.
In the glass transition region, the property of the glass shows a time-dependent response when it is subjected to a sudden change in temperature, as shown in figure 3.4. This phenomenon is called structural relaxation. Several mathematical models can be used to describe the structural relaxation of glass. The Tool-Narayanaswamy model [Tool, 1948;
Narayanaswamy, 1971] is the mostly used model in numerical analysis which was adopted in this research as well.
Figure 3.4 The property change after a sudden change in temperature in glass transition
region, redrawn from [Scherer, 1986]
15
3.1.3 Viscoelastic models
The viscoelastic behavior of glass can be presented by mechanical models as shown in figure 3.5. These mechanical models consist of springs and dashpots and help to understand the relation between stress and strain in the material.
Figure 3.5 Viscoelastic models: (a) Maxwell model (b) Voigt element (c) Burger model
Applying a shear stress σ12 to the Maxwell model as shown in figure 3.5-(a), the total strain ε12 is given by:
NE += εεε 121212 (3.4)
E D where, ε 12 and ε 12 are the strains in the spring and dashpot respectively. Solving the equation with a constant strain, the time-dependent relation of stress is given as:
σ 12 ()t = σ 12 ( )exp0 (− t τ s ) (3.5)
16
where, τs is the stress relaxation time given by η/G. G is the shear modulus of the spring,
η is the viscosity of dashpot. A generalized Maxwell model as shown in figure 3.6 is used to simulate the stress relaxation of viscoelastic glass in MSC/MARC software code.
Figure 3.6 Generalized Maxwell model
The Voigt element has a dashpot in parallel with a spring, as shown in figure 3.5-(b).
When a constant stress is applied, the time-dependent strain in the element is given by:
σ ⎡ ⎛ t ⎞⎤ 12 ⎜ ⎟ ε12 ()t ⎢ exp1 ⎜−−= ⎟⎥ (3.6) 2G ⎣ ⎝ τ s ⎠⎦
This represents the delayed elasticity, with neither an instantaneous elastic response nor a viscous flow. This is a creep equation, and τs is called the retardation time.
The Burger model shown in figure 3.5-(c) is the simplest mechanical model that has all the characteristics of the shear response of a viscoelastic material. The total strain when the glass is subjected to a constant stress is given by:
17
σσ ⎡ ⎛ t ⎞⎤ σ t 12 12 ⎜ ⎟ 12 ε12 ()t ⎢ exp1 ⎜−−+= ⎟⎥ + (3.7) 1 22 GG 2 ⎣ ⎝ τ s ⎠⎦ 2η
This model provides a complete description of a viscoelastic material behavior shown in figure 3.3.
3.2 Numerical Modeling of Glass Thermal Forming Processes
FEM has been widely applied in the industry for designing, optimizing and developing the manufacturing procedures. Major issues related to numerical modeling of glass thermal forming include: high deformation rate, large free surface deformation, non- linear contact problem, non-linearity of glass material properties, and temperature boundary conditions. Analytical models, which describe temperature-dependent of glass viscosity (VFT model), time-dependent of viscoelastic behavior (Maxwell model), and structural relaxation behavior (Tool-Narayanaswamy model), have been implemented in numerical methods to simulate above issues.
Cesar de Sa used the Newtonian fluid model with an in-house FEM code to simulate the beverage container forming process [Cesar de Sa, 1986]. The simulation results provided the position of glass melt and temperature distribution during the entire forming process.
Weidmann et al. also simulated the pressing of a ‘Drinking Glass’ using a commercial program FIDAP based on modeling glass as Newtonian fluid [Weidmann, 2002]. A good comparison between the predicted and analytical results of flow was presented. Hoque and Druma demonstrated using commercial FEM software DEFORM to simulate
18
pressing and cooling stages in television panel forming process [Hoque, 2003; Druma,
2004]. The temperature-dependent viscosity was modeled using VFT equation and glass was also modeled as Newtonian fluid.
Hyre presented the numerical simulation of different stages of glass container manufacturing process using the FEM program POLYFLOW [Hyre, 2002]. Williams-
Landel-Ferry (WLF) equation was used to describe the temperature-dependent of the glass viscosity. Non-Newtonian behavior, viscoelastic behavior, and surface tension were also included in the model. The simulation results showed that numerical modeling method could be useful for designing process, optimizing parameters and analyzing the impact of different stages of forming on the quality of the finished container. Tsai et al. proposed an elasto-viscoplastic model for glass material at the thermal forming temperature and confirmed the model with experiment results [Tsai, 2008].
Soules et al. used a commercial FEM program MSC/MARC to calculate the stresses inside glass for components like a simple sandwich seal and a bead seal under conditions of uniform cooling, reheating and an isothermal hold and a tempered glass plate [Soules,
1987]. Stresses induced inside sandwich seal and bead seal glass were due to the thermal expansion mismatch between different materials. The residual stresses inside tempered glass caused by non-uniform cooling were simulated by using Narayanaswamy theory to describe the stress and structural relaxation behavior. A good agreement of residual stresses between the predicted and experimental results was obtained. Carre et al. also used MAS/MARC to simulate the residual stresses in tempered soda-lime glass plate by 19
using an FEM software [Carre, 1996]. Narayanaswamy model was also implemented in the simulation to describe the viscoelastic behavior and structural relaxation of glass during the cooling process. The numerical simulation results of stress in the thickness direction of plate were in good agreement with their experimental results. Dang et al. used FEM program ANSYS to calculate the residual stresses during annealing process of glass bottles based on the Narayanaswamy model [Dang, 2005]. Na et al. also investigated the birefringence distribution from stress-optic relation by using a commercial FEM program ABAQUS [Na, 2007]. The simulation model was verified by comparing predicted results with results from Bruckner’s experiments.
Sellier et al. developed an iterative algorithm to optimize mold design by using FEM code ABAQUS to simulating glass molding process [Sellier, 2007]. Wang et al. also presented an iterative algorithm to compensate the mold curvature for glass compression molding process by using FEM software ANSYS [Wang, 2008]. A multi-Maxwell element was used as the viscoelastic model for the structural relaxation behavior. The curvature deviation of a finished lens from original design was less than 2μm after compensation, while it was 12μm before compensation. Gaylord et al. presented an
ABAQUS model to predict the final shape of the molded glass lens. Viscosity, friction coefficient, and structural relaxation parameters of glass material were measured from experiments and used in the numerical simulation [Gaylord, 2008].
Tuck and Stokes et al. developed the numerical algorithm to simulate the sagging of molten glass [Tuck, 1997; Stokes 1998; Stokes, 2000; Agnon, 2005]. They considered the 20
molten glass as Newtonian fluid and used zero-order solution to predict the upper surface of glass. Hunt also studied the slumping of a thin glass sheet under gravity by using numerical solution [Hunt, 2002]. A simplified Navier-Stokes equation was implemented in-house FEM code to describe the glass viscoelastic behavior at the slumping temperature.
Recently, our group also investigated the glass lens deformation, glass viscosity, and residual stresses induced in the glass molding process by using the FEM program
MSC/MARC to [Yi, 2005; Jain, 2005; Jain, 2005b; Jain, 2006; Jain, 200b; Jain, 2006c;
Chen, 2008]. Jain simulated molding stage based on Newtonian fluid model and cooling stage with Narayanaswamy model to describe the structural relaxation behavior. The simulation was performed by using both MSC/MARC and DEFORMTM-3D programs.
Chen simulated the residual stresses inside the molded glass lens after different cooling rates and compared the simulation results with experiment data. Chen also performed glass slumping experiments to produce mirrors for solar energy systems. Chen used FEM program MSC/MARC to simulate the slumping and cooling stages of glass thermal slumping process and compensated the mold for thermal shrinkage to produce mirror with required upper surface contour.
21
CHAPTER 4: GLASS INDEX CHANGE AS A RESULT OF THERMAL
TREATMENT
Glass compression molding process is a promising method for fabricating glass lenses which are in high demand by the ever growing electronic industry. However the properties of glass will change as the glass undergoes thermal treatment, especially during cooling. The refractive index of glass is one of the properties that would be altered in the molding process. “The faster a glass is cooled, the lower its refractive index and density will be [Scherer, 1986]”. On the other hand, the refractive index is a very important property that governs the performance of an optical lens along with irregularity, surface shape and roughness. If the refractive index change caused by the manufacturing process was not considered during the lens designing stage, the optical performance of the compression molded lenses would be different. Therefore it is important to investigate the refractive index change caused by cooling. Research has been performed to study the refractive index change due to thermal treatments e.g. cooling, annealing. Haken et al. performed a series of experiments to study the dependence of refractive index on the fictive temperature [Haken, 2000]. Kakiuchida et al. investigated the refractive change with various fictive temperatures [Kakiuchida, 2004]. Fotheringham et al. studied the group index drop at different cooling rates [Fotheringham, 2008]. 22
In this chapter, the structural relaxation behavior of glass which causes the properties changes is simulated by a finite element method software MSC/MARC using the
Narayanaswamy model. The refractive index change was predicted by applying the simulation results into a density-index relation function. The predicted results were confirmed by calculated the results from empirical equation according reference [Schott,
2007]. The effects of thermal expansion coefficients at liquid state and cooling rates were also discussed.
4.1 Theory
4.1.1 Structural Relaxation
Structural relaxation is a non-linear time dependent response of glass material properties
(e.g., volume, and enthalpy) to temperature change. The structural response depends on the thermal history, both the current temperature and direction of the change. Figure 4.1 shows a plot of temperature dependent property of a glass liquid being cooled at a certain rate. When cooled below the melt temperature, a non-glass material will freeze into a crystalline state and form a long range, periodic atomic structure. As a result, the property will change abruptly. However, the structure of glass liquid will continue to rearrange as the temperature decreases. In the liquid state, the viscosity of glass is so low that the time required for rearranging its structure is small enough to keep up with the temperature change rate. It means that the structure can attain the new structure equilibrium state simultaneously as the temperature changes. As the liquid is cooled further, the viscosity
23
gradually increases along with the time required to reach a new equilibrium structure.
The property starts to deviate from the equilibrium line due to the lack of time to completely rearrange the structure, following a curve with gradually decreasing slope, until the structure is frozen into a fixed configuration. The new state is called glassy/solid state because it processes the rigidity of a solid but has a liquid like internal structure.
The temperature region lying between liquid state and glassy state limits (segment AC in figure 4.1) is the glass transition region. The extensions of the glassy and liquid lines intersect at a temperature that is defined as the transition temperature Tg. The value of Tg is also a function of the temperature change rate.
Figure 4.1 Property change of glass during cooling
24
In the transition region, the change of property due to a temperature change from T1 to T2 is time dependent, and it can be described with a response function Mp(t) [Scherer, 1986]:
PtP 2 ∞− )()( fp )( −TtT 2 p tM )( = = (4.1) 2 PP 2 ∞− )()0( −TT 21 where, the subscripts 0 and ∞ represent the instantaneous and steady state values of the property p following a sudden temperature change. Tfp is the fictive temperature which is defined to measure the structural relaxation at time t due to the temperature change. As shown in figure 4.1, the fictive temperature Tfp(Ti) corresponding to the temperature Ti, is found by extrapolating a straight line from p(Ti) with slope of glassy state to intersect with the extension of liquid equilibrium state line.
The property response function Mp(t) can also be described as an exponential function
[Scherer, 1986]:
b ⎡ ⎛ t ⎞ ⎤ (tM exp) ⎢−= ⎜ ⎟ ⎥ (4.2) p ⎢ ⎜τ ⎟ ⎥ ⎣ ⎝ p ⎠ ⎦
Where, τp is the structural relaxation time of property and b is the empirical parameter obtained by fitting the response curve with the experimental data. The value of b lies between 0 and 1. For the volume, the response function and relaxation time become Mv(t) and τv.
25
In the finite element method, the experimental data can be accurately fit with a series of exponential functions [Soules, 1987]:
n ⎛ t ⎞ = wtM exp)()( ⎜− ⎟ (4.3) p ∑ ig ⎜ ⎟ i=1 ⎝ τ pi ⎠
where, (wg)i are weighing factors and τpi are the associated structural relaxation times.
The structural relaxation time is strongly temperature dependent so that relaxation happens very faster at higher temperature and much slower at lower temperature. Since the nonlinearity of structural relaxation arises as τp increases, Narayanaswamy introduced the reduced time ξ to restore the linearity [Narayanaswamy, 1971]:
t dt' =τξ pr (4.4) ∫0 τ p
where, τpr is the relaxation time at an arbitrary reference temperature Tr. This is based on the assumption that the Themorheological Simplicity Behavior applies in the glass transition region. Then equation 4.3 can be rewritten as:
n ⎛ ξ ⎞ = wtM exp)()( ⎜− ⎟ (4.5) p ∑ ig ⎜ ⎟ i=1 ⎝ τ pi ⎠
The relaxation time τp at any given time and temperature can be calculated by the
Narayanaswamy model [Narayanaswamy, 1973]:
26
⎡ ΔH ⎛ x 11 − x ⎞⎤ = ττ exp⎢− ⎜ −− ⎟⎥ (4.6) ,refpp ⎜ TR T T ⎟ ⎣⎢ ⎝ ref fp ⎠⎦⎥
where, τp,ref is the structural relaxation time of property at reference temperature Tref. Tfp is the fictive temperature of property, H is the activation energy and R is the idea gas constant. This is based on the model, that relaxation time is Arrhenius temperature dependent with the activation energy H at high temperature above the transition range where Tfp ≈ T, and the activation energy xH at temperature below the transition range.
The value of the fraction parameter x lies between 0 and 1, and typically x ≈ ½. The fictive temperature can be obtained by solving equation 4.1 by using the Boltzmann superposition principle and integrating over the thermal history:
t tdT )'( fp −= p []−ξξ ttMtTtT )'()()()( dt' (4.7) ∫0 dt'
Markovsky and Soules also presented an efficient finite element algorithm to calculate the fictive temperature [Soules, 1987]. Once the fictive temperature is obtained, the property of glass at a given time during cooling can be calculated with following equation:
1 Tdp )( ⎛ dT ⎞ ⎜ fp ⎟ pg [fppl −+= ααα TTT fppg )()()( ]⎜ ⎟ (4.8) p )0( dT ⎝ dT ⎠
27
where, αpg and αpl are the characteristic coefficients of glass property at glassy and solid states respectively. For the volume of glass, they are the thermal expansion coefficients
αVg and αVl.
4.1.2 Density Change
The density of a glass sample can be simply expressed as:
m ρ = (4.9) V where, m is the mass and V is the volume of the sample. For a given sample, the mass is considered constant during the cooling, but the volume will change, subsequently resulting in density change after thermal treatment.
Figure 4.2 Volume versus temperature during cooling of glass liquid, redrawn from
[Scherer, 1986]
28
Figure 4.2 shows the volume change during cooling through glass transition range. As mentioned before, the volume starts to deviate from the liquid equilibrium state line as the temperature drops into the transition range. The slope of curve continues to decrease until the viscosity becomes so great that the liquid is frozen to glassy state. The slope dV/dT is defined as the thermal expansion coefficient αV(T) of the glass material, which is the derivative of the curve in figure 4.2. It can be represented as:
1 TdV )( ⎛ dT ⎞ ⎜ f ⎟ αV = Vg []−+= ααα TTT fVgfVl )()()( ⎜ ⎟ (4.10) V )0( dT ⎝ dT ⎠
Figure 4.3 The thermal expansion coefficient of the glass volume during cooling, the
curve is the derivative of figure 4.2, redrawn from [Scherer, 1986]
Figure 4.3 shows the thermal expansion coefficients αV(T) at the liquid state and the solid state as constants αVl and αVg respectively during the cooling. Normally, the ratio rα = αVl /
29
αVg ≈ 3-5 [Scherer, 1986]. The value of thermal expansion coefficient changes gradually in the transition range. The linear thermal strain can be calculated by integrating equation
4.10 over the temperature range:
T th 1 ΔV 1 2 ε = = αV )( dTT (4.11) V )0(3 3 ∫T1
where, T1 and T2 are the temperature at the beginning and the end of cooling respectively,
V(0) is the volume at the beginning of the cooling stage.
Figure 4.4 Volume versus temperature at two different cooling rates during cooling
For the same amount of glass melt, the density at the room temperature depends on the cooling rates. Figure 4.4 schematically shows the volume change at two different cooling
30
rates. The volume deviates from the equilibrium line earlier at a faster cooling rate q1, and results in higher transition temperature Tg1. This means lower density at higher cooling rate. The raw glass sample used for compression molding is usually produced by cooling at a very slow rate. If the glass was cooled very slowly after the heating-pressing process, we can assume that the glass volume first returns to its original value, complete relax state, and then there are no changes in density and other properties. However, in glass compression molding process, the cooling rate is much higher. As a result, the density of the molded lens will be lower than its original value.
4.1.3 Relation between the Density and the Refractive Index
The relation between refractive index and density of a transparent material especially glass has been studied experimentally and theoretically for a long time. A number of formulae have been proposed to represent the change of refractive index as its density is changed. The most often used equation is the Lorentz-Lorenz equation:
n2 −1 4 N ρπ = A α (4.12) n2 + 2 3 M
where, n is the refractive index, M is the molecular weight, NA is the Avogadro number, ρ is the density and α is the electronic polarizability.
Ritland modified the Lorentz-Lorenz formula with an adjustable parameter b, as given in the following equation [Ritland, 1955]:
31
n2 −1 N ρ = A α (4.13) π nb 2 −+ )1(4 M
Various empirical density and refractive index formula can be identified with corresponding values of b. When b = 4π/3, the Lorentz-Lorenz equation is obtained. One can also obtain the Newton-Drude relation (n2 1) ρ =− constant by setting b = 0. Based on this equation, assuming that the polarizability α is independent of the density ρ, the relation between refractive index change dn and density change dρ can be obtained by differentiating equation 4.11:
dn n 2 4)(1( π 2 −+− bbn ) = (4.14) dp 8 nρπ
The density change can be expressed as:
m m ρ −=Δ (4.15) Vq V0
where, Vq is the volume at the end of cooling process at rate q, and V0 is the original volume of the glass sample before heating-cooling treatment. Here, m/V0 is the initial density ρ. Then the index change can be calculated by:
n2 4)(1( π 2 −+− bbn ) ⎛V ⎞ n =Δ ⎜ 0 −1⎟ (4.16) ⎜ ⎟ 8 nρπ ⎝Vq ⎠
32
This equation will be used for predicting the index change caused by cooling process using the simulation results.
4.2 FEM Modeling of Glass Cooling
In this research, the properties of glass P-SK57 are used. Reference [Schott, 2007] provided empirical data and equation that can be used to cross check the simulation results. The cooling process of the glass sample was performed using the commercial
FEM software MSC/MARC. The Narayanaswamy model was adopted to simulate the structural relaxation behavior of glass material during the cooling process.
To simplify the modeling effort, only one small four-node isoparameteric quadrilateral element is used to model the sample as we envision only a small volume of a glass liquid is cooled. In this simulation, the element is cooled uniformly with a time dependent temperature curve at a constant rate. The glass sample is cooled from 600 °C to room temperature 20 °C with four different cooling rates, 2 °C/hr, 3500 °C/hr, 350 °C/hr, and
35 °C/hr. The cooling rate 2 °C/hr is the reference annealing rate that used in the reference [Schott, 2007]. The result at this rate will be used as reference volume or original volume V0. The results from other cooling rates are used to predict the index change.
Table 4.1 shows the mechanical and thermal properties of glass P-SK57 used in the simulation. The properties required for structural relaxation model in the simulation are given in Table 4.2. The structural relaxation time τv,ref at reference temperature Tref was 33
calculated from stress relaxation time τs,ref with a ratio of τv/ τs =10.6 for a similar glass
[Scherer, 1986]. The stress relaxation time τs,ref at Tref can be calculated by equation:
η τ = ref (4.17) ,refs G where, G is the shear modulus which is given by ( + 21 vE ).
Material Properties Value
Elastic modulus, E [Mpa] 93,000
Poisson’s ratio, v 0.249
Density, ρ [kg/m3] 3,010
Thermal conductivity, kc [W/m ºC] 1.01
Specific heat, Cp [J/kg ºC] 760
Transition temperature, Tg [ºC] 493
Solid linear coefficient of thermal expansion, 8.9x10-6 αg [/ºC] (20-300 ºC)
Viscosity, η [MPa-sec] (at 494 ºC) 106
Table 4.1 Mechanical and thermal properties of glass P-SK57
34
The ratio of activation energy over gas constant was calculated from fitting viscosity available from reference [Schott, 2010] with Arrhenius equation. Due to lack of information, the fraction parameter x is set to its typical value 0.5.
Typically, the thermal expansion coefficient ratio rα = αl / αg lies between 3 and 5
[Scherer, 1986]. However, a value of 4 of the ratio is selected for this research based a measurement of BK7 glass which has similar composition to glass P-SK57 [Jain, 2006d].
This ratio will be confirmed and discussed in the next section by comparing the simulation results with the reference.
Material Properties Value
Reference Temperature, Tref [ºC] 494
Activation energy/gas constant, ΔH/R [ºC]* 71,988
Fraction parameter, x * 0.5
Weighing factor, wg 1
Structural relaxation time, τv [sec] (at Tref) 268.8
Stress relaxation time, τs [sec] at Tref 26.88
Table 4.2 Structural relaxation parameters of glass P-SK57 required for simulation
35
4.3 Simulation Results and Discussions
Figure 4.5 shows the predicted volume versus temperature curves by the FEM simulation method, with thermal ratio rα = 4. According reference [Schott, 2007], the listed refractive index of glass P-SK57 is based on a reference annealing rate qref = 2 °C/hr. So the predicted volume at the end of cooling with this rate is considered as the original volume V0. The final volume of the sample at the cooling rate q1 = 3500 °C/hr isV . q1
Figure 4.5 Predicted volume versus temperature curves obtained by structure relaxation
model at two cooling rates qref = 2 °C/hr and q1 = 3500 °C/hr
36
For glass P-SK57, no specific experiments results to fit the empirical value b for equation
4.13. Therefore, b is set as 4π/3, and then the equation 4.16 can be rewritten as an equation based on the Lorentz-Lorenz relation:
2 nn 2 +− )2)(1( ⎛ V ⎞ n =Δ ⎜ 0 −1⎟ (4.18) 6n ⎜V ⎟ ⎝ qi ⎠
As mentioned in section 4.2.3, the value of the thermal expansion coefficient ratio r was selected as 4. Two other values of r 3 and 5 were used for simulations separately for comparison. The corresponding index changes were calculated, as shown in table 4.3.
According to reference [Schott 2007], a reliable formula is used to calculate the refractive index after a given cooling rate qi:
q += mqnqn log)()( i (4.19) did 0 nd q0
where, q0 is the reference annealing rate of 2 °C/hr as mentioned before, and mnd is the
annealing coefficient for the refractive index depending on the glass type. The d (qn 0 ) of glass P-SK57 is 1.587 and m is -9.5x10-4 according to [Schott 2007]. Applying the nd cooling rate qi = 3500 °C/hr to equation 4.17, the new index would be 1.5839 with change ∆nd = -0.0031. From the experimental results, the real index nd after the molding- cooling process is 1.5843 with change ∆nd = -0.0027. The differences might be caused by geometry, molding process and other factors. Compared to the values in table 4.3, it is
37
confirmed that the thermal expansion coefficient ratio rα = 4 provides a better agreement with the reference value.
Ratio Linear Liquid Thermal Expansion Index change
rα Coefficient αVl [/ºC] ∆nd *
3 2.67x10-5 -0.0022
4 3.56x10-5 -0.0033
5 4.45x10-5 -0.0044
Table 4.3 Predicted index change caused by cooling process with different liquid thermal expansion coefficient, *based on the refractive index nd @ wavelength 587.6 nm
4.3.1 The Effect of Thermal Expansion Coefficient Ratio
Figure 4.6 shows the predicted volume versus temperature curves with different thermal expansion coefficient ratios rα1 and rα2. It is obvious that higher thermal expansion coefficient ratio has lower original volume V0 (final volume of cooling rate qref = 2
°C/hr). For a given cooling rate, higher thermal expansion coefficient ratio has smaller volume which means the sample has higher density. As shown in figure 4.7, the thermal expansion coefficient αV(T) during cooling can be obtained by differentiating curves in figure 4.6 and the volume at the end of cooling can then be expressed as:
38
V T2 qi −= αVqi )(1 dTT (4.20) V )0( ∫T1
where, V(0) is the sample volume at the beginning of cooling process, T1 and T2 are the temperature at the begin and end of cooling respectively. In this case, T1 = 600 °C, while
T1 = 20 °C.
Figure 4.6 Predicted volume versus temperature curves obtained by structure relaxation
model at two cooling rates qref = 2 °C/hr and q1 = 3500 °C/hr with different rα
As mentioned before, the volume of reference annealing rate qref is considered as the original volume of glass sample before the heating-pressing-cooling process. Therefore, the real volume change ∆Vr of a given sample after cooling is:
39
⎡T1 T1 ⎤ =−=Δ VVVV ⎢ α dTT − α )()()0( dTT ⎥ (4.21) r qi 0 ∫∫V0 Vqi ⎣⎢T2 T2 ⎦⎥
Then equation 4.18 can be rewritten as:
2 2 nn +− )2)(1( ⎛ Δ− Vr ⎞ n =Δ ⎜ ⎟ (4.22) 6n ⎝ Δ+ VV r0 ⎠
Define real relative change as volume change ratio V = Δ VVr 0r , then:
2 2 nn +− )2)(1( ⎛ − rV ⎞ n =Δ ⎜ ⎟ (4.23) n ⎝16 + rV ⎠
Figure 4.7 Thermal expansion coefficient αV(T) versus temperature during cooling
40
As shown in figure 4.7, the real volume change ∆Vr is the shadowed area and the higher thermal expansion coefficient ratio has larger ∆Vr. Furthermore, from figure 4.6 the higher thermal expansion coefficient ratio corresponds to a lower original volume V0.
Thus, the volume change ratio rV will be greater at higher thermal expansion coefficient ratio, which will lead to greater index change. Here, rV << 1. Hence, the thermal expansion coefficient ratio has a great impact on the refractive index and density changes.
4.3.2 The Effect of Cooling Rate
Table 4.4 shows the index changes calculated based on the simulation results and from equation 4.16 with data from reference [Schott, 2007]. Figure 4.8 shows the simulation results at three different cooling rates: q1 = 3500 °C/hr, q2 = 350 °C/hr and q3= 35 °C/hr.
The predicted results match the reference index changes. This validates the simulation methodology and the selection of the thermal expansion coefficient ratio rα = 4, which was determined previously.
Cooling Rate Reference index change* Simulated index change
qi (ºC/hr) ∆nd ∆nd 3500 -3.08x10-3 -3.33x10-3
350 -2.13x10-3 -2.25x10-3
35 -1.18x10-3 -1.23x10-3
Table 4.4 Predicted index change at different rates and calculated index change *based
on reference [Schott, 2007]
41
Figure 4.8 Predicted volume versus temperature curves obtained by structure relaxation
model at three cooling rates q1 = 3500 °C/hr, q2 = 350 °C/hr and q3 = 35 °C/hr
Figure 4.9 Thermal expansion coefficient α(T) versus temperature during cooling at two
cooling rate q1 = 3500 °C/hr, and q2 = 350 °C/hr 42
As shown in figure 4.9, the real volume change ∆Vr is the shadowed area and the higher cooling rate has larger ∆Vr. Thus, the volume change ratio rV will be greater at higher cooling rate, which will lead to greater index change.
4.4 Conclusion
This chapter describes the structural relaxation phenomenon, and the theory that is used in the FEM simulation modeling. The simulated volume change after cooling is used to predict the index change caused by the structural relaxation with the density-index relation formula. The simulation results are confirmed with the empirical data from reference [Schott, 2007]. The effects of the cooling rates and thermal expansion coefficients are also discussed.
The results suggested that higher cooling rate would lead to lower refractive index after cooling in compression molding process, and the thermal expansion coefficient at the liquid equilibrium state plays an important role in the volume change during the cooling process. It is important to measure the value of thermal expansion coefficient α more accurately with experiments to improve the prediction. Further, FEM simulation model can be used to predict the index change of the compression molding process if enough material properties were provided. This approach allows optical designer to precisely identify the correct lens design and the associated manufacturing process to minimize the aberration caused by the refractive index change.
43
CHAPTER 5: GLASS REFRACTIVE INDEX VARIATION CAUSED BY
THERMAL TREATMENT
In the cooling stage of the compression molding process, the molded glass lens undergoes structural relaxation due to temperature change. In reality, the entire lens will not be cooled uniformly. There is a temperature gradient inside the molded glass during cooling.
The temperature history inside the glass is different from that of the exterior of glass.
Therefore residual stresses are induced in the glass. As a result, the refractive index change at different points of the glass will be different, which introduces inhomogeneous refractive index distribution inside the lens. If a wavefront coming through the thermally treated plate it will be distorted due to the refractive index variation. The wavefronts after the glass samples in this research were tested by Shack-Hartmann Sensor (SHS). In addition FEM software was used to simulate the refractive index variation based on the structural relaxation model.
5.1 Theory
5.1.1 Refractive Index Variation
Thermal strain εth is imposed on the glass when there is a temperature change ∆T. As a result, there is a stress response following with a stress relaxation. In compression 44
molding process, the glass lens cannot be cooled uniformly due to the finite dimension.
There is a temperature gradient inside the glass during the entire cooling stage. When the exterior of glass shrinks as the temperature decreases, the stresses induced at this part are compression stresses. Because the exterior of glass is cooled much faster than the interior, the interior always has higher temperature and continues shrinking even the structure of exterior is already frozen. Since the outside region is frozen, tensile stresses are induced inside glass due to strains caused by thermal shrinkage. The induced stresses bring in the inhomogeneous refractive index change that leads to the refractive index variation inside the glass.
The relative volume change due cooling can be expressed as:
ΔV ++= εεε (5.1) V )0( 332211
where, ε11, ε22 and ε33 are the actual strains along three axes in a Cartesian coordinate system, ∆V is the volume change caused by the cooling process, V(0) is the sample volume at the beginning of cooling, and is given by:
)0( = RVV 0 (5.2)
where, V0 is the initial sample volume before the heating-cooling process, and R is the volume expansion ratio after the glass sample is heated to the given temperature. Then the real volume change after cooling is:
45
r [])0( −=Δ Δ −VVVV 0 (5.3)
Combining equation 5.1-5.3, the real relative volume change after heating-cooling process is given by:
ΔVr 1 RR (++−−= εεε 332211 ) (5.4) V0
The actual strains are the sum of the thermal strains and elastic strains of the nodes and can be expressed as:
σ − v(σ +σ ) εε th += 11 3322 (5.5a) 11 E
σ − v(σ +σ ) εε th += 22 3311 (5.5b) 22 E
σ − v(σ +σ ) εε th += 33 1122 (5.5c) 33 E
where, σ11, σ22 and σ33 are the normal stresses along the axes, E is the Young’s modulus, v is the Poisson’s ratio, and the thermal strain εth is given by:
T th = 2 αε )( dTT (5.6) ∫T 1 where, α T )( is the thermal expansion coefficient of the glass material in the temperature range [T2,T1].
46
If the glass is cooled uniformly, there is only thermal strain εth, and then equation 5.1 becomes:
ΔV T2 th == αε )(33 dTT (5.7) V )0( ∫T1
This equation is an alternative expression of equation 4.11.
However, the residual stresses are induced since there is a temperature gradient in the glass during the cooling stage. Then combining equation 5.1, 5.2 and 5.6, the relative volume change can be expressed as:
ΔV (σ + σ + σ )( − 21 v) 3ε th += 332211 (5.8) V )0( E
Soules [Soules, 1987] proposed an elastic-thermal analysis equation to calculate the total stress change of viscoelastic material during the cooling:
⎡ ⎛ Δt ⎞⎤ th ⎜ ⎟ D[]εεσ ∑⎢ exp1 ⎜−−−Δ−Δ=Δ ⎟⎥σ k (Δ− tt ) (5.9) ⎣ ⎝ λk ⎠⎦ where, D is the elastic modulus matrix, ∆ε and ∆εth are the tensors of the actual strains change and the thermal strains change between time − Δtt and t respectively, σk is a partial stress component. The stress induced by a sudden change in the strain and the relaxed portion of previous stress during ∆t are converted to nodal forces and subtracted from any external force. Then the changes in displacements, strains and stresses in the
47
viscoelastic analysis can be solved in the same way those are solved in the elastic analysis.
The relative volume changes as given in equation 5.8 at different points of the glass are different since the stresses induced are different at different points. The corresponding real relative volume changes given by equation 5.4 will also be different at different points, and as a result the index changes calculated using equation 4.23 will be different.
The difference of the refractive index between one point and another is considered as the refractive index variation in this dissertation research.
5.1.2 Shack-Hartmann Test
Figure 5.1 Principle of Shack-Hartmann sensor
The refractive index variation was measured by detecting the distorted wavefront coming through the thermally treated glass plate with a Shack-Hartmann sensor. Platt and Shack proposed a modified Hartman setup using a lenticular screen instead of a screen with an
48
array of holes [Platt, 1971]. This method is called the Shack-Hartmann test (Hartmann-
Shack test). The principle of the Shack-Hartmann test to measure the wavefront is shown in figure 5.1. The lenticular screen is made as a lenslet-array now. This method can measure any negative or positive power. The incident wavefront is collected by an array of lenslets. Each lenslet focus a small part of the wavefront onto the charge-coupled device (CCD), which is placed on the focal plane of the lenslets. If the wavefront is flat, the wavefront passing through lenslets will produce a regular array of focal spots. When the wavefront is distorted, the wavefront tilt across each lenslet results in a shift of the respective focal spot. The displacements of the spots are proportional to the wavefront slopes across the aperture of the lenslets. Then the wavefront can be reconstructed by integrating the calculated wavefront slopes.
The diffraction spot radius ρ is given by:
fλ ρ = (5.10) d where, d and f are the diameter and the focal length of the lenslet respectively, and λ is the wavelength of the light. The spot displacement δ is given by:
δ = f tanθ (5.11) where, θ is the wavefront slope. The maximum allowed spot displacement is about d/2-ρ, so the maximum possible angular aberration (dynamic angular range) of the wavefront slope is given by: 49
d 2 − ρ d λ θ = −= (5.12) max f 2 df
The angular sensitivity of this test is determined by:
σ θ = (5.13) min f where, σ is the size of each pixel of the detector. According to equation 5.11, a shorter focal length of lenslet allows larger wavefront tilt, but the angular sensitivity will be reduced as given in equation 5.13.
The spot displacements are used to calculate the wavefront slopes, which are integrated to retrieve the measuring wavefront. A common method to retrieve the wavefront is centroid method, which yields the Zernike coefficients of the wavefront. Fourier transform can also be used for wavefront reconstruction.
5.2 Glass Thermal Treatment Process
The experiments were performed on a Toshiba GMP 211V machine [Yi, 2005; Yi, 2006] at Fraunhofer Institute for Production Technology in Germany. Since the focus of this research is to study the refractive index variations inside glass lenses caused by the structural relaxation during cooling, the glass plates were heated to a certain temperature and cooled down at different cooling rates. The compression operation for glass molding process was eliminated to simplify the problem. The BK7 glass blank plates used in the experiment was placed manually on the lower mold. The thermal histories of the 50
experiments are shown in figure 5.2. The temperature was measured by the sensors buried inside of the molds, which was not the actual temperature at the surface of the glass plate. This will cause the discrepancy between the simulation results and the measured experiment results, and the effect will be discussed late in this chapter.
Figure 5.2 Temperature histories of three different cooling rates
1) The experiment began with placing a glass blank plate at the lower mold, then the
entire mold assembly system with the glass lens were heated to the molding
temperature of 680ºC at a heating rate around 3.0 ºC/sec. The heating rate was the
same for all experiments with different cooling rates.
2) The temperature was maintained at 680 °C for 400 seconds before cooling.
3) Cooling of the glass plate was performed at three different cooling rates, e.g., q1 =
1.60 °C/sec, q2 = 0.60 °C/sec, and q3 = 0.225 °C/sec (these are all nominal values of
measured temperature histories by the embedded sensors in the molds).
51
4) Once the temperature of the molds and the glass plate was lowered to approximately
200 °C, the glass plate was cooled to the room temperature by natural cooling. At the
end of cooling, the glass plate was removed from the molding machine manually.
During the experiments, one side polished glassy carbon wafers were placed between the glass plate and the molds. The air remained in the gaps among glassy carbon wafer, molds and the glass plate was removed by applying vacuum at the beginning of each experimental cycle. Oxygen residual was removed by nitrogen purge to protect glass plate and molds from oxidation at high temperature. Nitrogen was also used to maintain the constant cooling rates. Both the lower and upper mold maintained contact with the glass lenses during the entire cooling stage.
5.3 FEM Modeling of Glass Cooling Process
Taking advantage of the simplicity of the glass plate geometry, a two-dimensional (2D) axisymmetric model was used for FEM simulation. The lower mold was a 2 mm thick glassy carbon wafer that was simplified as rigid bodies in this simulation. The original glass blank plate was a 25 mm diameter and 10 mm thick double-side polished cylinder, which was defined as the deformable part. Four-node isoparametric quadrilateral element was used to mesh the glass sample into 8,000 elements, as shown in figure 5.3. The simulation includes two major steps: 1) The glass blank plate and molds were heated to a certain temperature above the transition temperature. 2) The heated glass plate was cooled to room temperature under one of the three pre-determined cooling rates.
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Figure 5.3 Meshed numerical simulation model
The thermal boundary condition applied in the simulation is given by:
∂T − k ∞ (−= TTh ∞ ) (5.14) ∂nu
where, nu is the unit vector normal to the surface, k is the thermal conductivity of glass, T is the temperature of sample, h∞ is the convection coefficient, T∞ is the sink temperature which is measured from experiments as shown in figure 5.2. The important material properties of BK7 glass were summarized in Table 5.1 and Table 5.2 [Schott, 2010; Jain,
2006]. Based on measurement results [Jain, 2006d], the thermal expansion coefficient ratio for BK7 was selected as 4.
53
Elastic modulus, E [Mpa] 82,500
Poisson’s ratio, v 0.206
Density, ρ [Kg/m3] 2,510
Thermal conductivity, kc [W/m ºC] 1.1
Specific heat capacity, Cp [J/Kg ºC] 858
Transition temperature, Tg [ºC] 557
Viscosity, η [MPa-sec] (at 685ºC) 60
Table 5.1 Mechanical and thermal properties of BK7 glass
Reference Temperature, T [ºC] 685
Activation energy/gas constant, ΔH/R [ºC] 47,750
Fraction parameter, x 0.45
Weighing factor, wg 1
-6 Solid coefficient of thermal expansion, αg [/ºC] 8.3x10
-5 Liquid coefficient of thermal expansion, αl [/ºC]* 3.32x10
Structural relaxation time, τv [sec] (at 685ºC) 0.019
Stress relaxation time, τs [sec] at 685ºC 0.0018
Table 5.2 Structural relaxation parameters used in numerical simulation
54
Figure 5.4 Heat capacity versus temperature for 0.4Ca(NO3)2·0.4KNO3 [Copy from
Moynihan, 1976]
Figure 5.5 Heat capacity Cp of BK7 glass used in the FEM simulation
As mentioned in chapter 4, enthalpy is another glass property of which the temperature dependence changes in the transition region. The specific heat capacity Cp is the characteristic value of enthalpy. Figure 5.4 shows the measured heat capacity Cp versus temperature for 0.4Ca(NO3)2·0.4KNO3 [Moynihan, 1976]. Normally, CC pgpl ≈ 3.1 for
55
multicomponent oxides [Scherer, 1986]. As a result, the heat capacity Cp used in the simulation is modified as shown in figure 5.5. The effect of the heat capacity value used in the model will be discussed later.
Figure 5.6 The experiment arrangement of sample and molds
In the experiments, the cooling rate was maintained by controlling the flow rate of the 40 °C Netrogen gas around the molds and inside the chamber, as shown in figure 5.6. The glass was cooled by thermal convection at the side surface and conduction between the top / bottom surfaces and the inserted glassy carbon wafers. However, simulations showed that the convection at the side surface domninated the cooling of glass plate. 56
First, eliminate the inserted glassy carbon wafers and only consider the conduction between the molds (e.g. Tungsten Carbide) and the glass plate. Apply convection at the bottom surface of the bottom mold with a convection coefficient h = 2900 W/(m2K), and sink temperature T∞ = 40 °C without considering the upper mold. The points N1 and point
N2 are the points of glass and mold at the contact area as shown in figure 5.4. The temperature change at two points can be simulated and are shown in figure 5.7..
Second, only apply convection at side surface of the glass with same convection coefficient and sink temperature. The point N3 is near the side surface of the glass plate.
The temperature change at this point can also be simulated and is given in figure 5.7.
Figure 5.7 Simulated temperature changes caused by convection at mold bottom surface
and convection at glass side surface
57
As shown in figure 5.7, for the contact thermal conduction, the temperature change at the glass side is much slower than that at mold side. At temperature above 200 °C, the mold side point N2 was cooled about 9 times faster than the glass side point N1. For the thermal convection cooling, the temperature at the glass side point N3 drops drastically, the temeprature change is even faster than that the point N2. Furthermore, there was 2 mm thick glassy carbon placed between the glass sample and the molds during the experiment as insulation. Then the temperature change caused by conduction at the top / bottom surfaces of the glass plate should be even slower than the change caused by convection at the glass side surface. From these simulation results, we can assume the convection at the glass side surface dominates the cooling of the glass plate. Thus, in this research as shown in figure 5.3, the simulation model was simplified by only applying thermal convection boundary condition at the side surface of glass without considering the conduction between galss and the molds during cooling.
5.4 Results
5.4.1 Measured Index Variation
Figure 5.8 shows the schematic of the measuring system using the Shack-Hartmann sensor. The incident wavefront to the glass plate was a plane wave (variation less than
λ/20, where λ = 632.8 nm). The output wavefront was distorted due to the refractive index variation of the sample. The smaple was immersed into a box filled with optical index matching liquid (Cargill Laboratories, Cedar Grove, New Jersey) to eliminate the
58
effects of the surface variation and the thickness variation. The lenslets array of the
Shack-Hartmann sensor was placed in a plane that is conjugate to the output wavefront from the sample. The system magnification M is equal to -f2/f1.
Figure 5.8 Schematic of the Shack-Hartmann testing system: 1. He-Ne Laser; 2.
Polarizer; 3. Beam expander; 4. Sample in matching liquid; 5. Lens 1; 6. Lens 2; 7.
Shack-Hartmann Sensor
The optical path distribution through the thermally treated glass lens is defined by:
= ⋅ ,(),(),( yxtyxnyxL ) (5.15) where n(x,y) is the refractive index distribution of the sample. Since the glass lenses are flat plates in this experiment, the thickness of the sample t(x,y) = t.
Assuming the refractive index of the center of glass lenses is nc, then the reference optical path Lr is defined by :
r = ctnL (5.16)
Thus, the wavefront variation as the optical path difference can be defined by: 59
V =Δ − r = ),(),(),( ⋅ − ctntyxnLyxLyxL (5.17)
The wavefront variation ΔLv(x, y) could be reconstructed by using the Shack-Hartmann sensor for measuring the position of spots. So when the wavefront variation ΔLv(x, y), and thickness t of the sample is known, the refractive index variation Δnv(x, y) can be calculated from the following equation:
V =Δ ),(),( − = Δ Vc ),( tyxLnyxnyxn (5.18)
Figure 5.9 Reconstructed wavefront variation using Shack-Hartmann sensor
(0.225°C/sec)
The wavefront variations of glass lenses at three different cooling conditions were measured and reconstructed by the setup shown in figure 5.8. Figure 5.9 shows the reconstructed wavefront variation at cooling rate 0.225°C/sec. The wavefront variation of
60
a blank glass lens, which represented initial refractive index distribution, was also measured and reconstructed. The reconstructed wavefront variation values along circles with different radiuses were averaged to obtain the average wavefront variation along the radial direction of the glass lens.
Figure 5.10 Measured refractive index variations along the radial direction for three
different cooling rates and an untreated glass lens (blank)
The refractive index variations were calculated by equation 5.18. Figure 5.10 shows the refractive index variations of glass lenses treated with three different cooling rates and an untreated blank glass. The influence of the cooling rates on index variation is clearly shown in figure 5.10. The blank glass also has refractive index variation as shown in the same figure, but it is much smaller since it was annealed by a very slow cooling rate.
61
As shown in figure 5.10, the refractive index variations of the thermally treated glass lenses are much higher than that of the blank glass plate. This indicates that the refractive indices at different positions of the thermally treated glass plates are no longer the same.
Therefore, the refractive index variations will affect the performance of the compression molded glass lenses. So when designing a precision optical system with lenses made by compression molding process, the group refractive index or a shift of the group refractive index alone is not adequate unless the cooling rate is slower than a critical value [Chen,
2008] or the lens is relatively thin.
5.4.2 Simulated Index Variation
Applying the simulation results in equation 4.16, the index change ∆ni,j of element (i,j) in figure 5.3 can be calculated. Then, the refractive index changes along the radial direction of the glass were calculated by averaging the refractive index changes of the finite elements along the same axial lines, as given by:
N n Δ=Δ Nn (5.19) i (∑ j=1 , ji )
The refractive index variation is defined as the refractive index differences among different points in a glass lens. In this research, the refractive index change ∆n0 at the center of the glass was used as a reference refractive index change to calculate the refractive index variation δnv along the radial direction as in:
δ iV Δ−Δ= nnn 0 (5.20)
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Figure 5.11 shows the predicted refractive index variations of three different cooling rates along the radial direction using the structural relaxation model for the FEM simulations.
The effect of the cooling rate on the refractive index variation is also clearly illustrated.
From both the experimental results and the FEM simulated results shown in figure 5.10 and figure 5.11, the cooling operation indeed introduces refractive index variations into the thermally treated glass plates. Moreover, different cooling rates will result in different refractive index variations. The refractive index variation increases as the cooling rate increases. Both the experimental and simulated results confirmed that the refractive index variation inside the compression molded glass lenses depends on the cooling rate.
Figure 5.11 Predicted index variations for three different cooling rates
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5.5 Discussion
Figure 5.12 Comparison of measured and simulated results of the refractive index
variation curves of three cooling rates
However, as shown in figure 5.12, the refractive index variation curves, which were predicted by FEM simulation, do not appear to completely agree with the values measured by Shack-Hartmann sensor system. The predicted values of the refractive index variations are generally lower than measured results. The discrepancy (between simulation and measurements) is believed to have been caused mainly by idealizing the cooling process in the FEM simulation. For example, the measured cooling rates used as the thermal history were not the exactly temperature at the side surface of the glass sample. It could also due to the material properties used in the simulation are simplified and based on some previous studies. Some assumptions are close to the real values, but
64
do not exactly match the true values, e.g. the heat capacity Cp, the thermal expansion coefficient α. In addition, eliminating the conduction between the molds and the glass plate also caused the discrepancy. In this section, the effects of the inaccuracy of some idealizations and assumptions will be presented by simulated results.
5.5.1 Effect of the Cooling Rate
As mentioned before, the temperature histories used in the simulation were measured by the sensors embodied inside of the molds. However, the measured temperature was different from the surface of the glass undergoing the force convection. Therefore, the actual cooling rates would be different from those used in the FEM simulation.
Figure 5.13 Illustration of the heat flux through the surface
Assuming the flow rate of the Nitrogen gas was maintained at a constant rate, then the heat flux φq caused by convection through the glass surface and mold surface should be the same. This means the heat Qglass coming out of the surface area A of glass at a given time t is the same as the heat Qmold coming out of the mold at the same area for the same
65
amount of time. Meanwhile, as shown in figure 5.13, the heat Q through a given volume of the material can be expressed as:
p ρ Δ= TAdCQ (5.21)
where, Cp and ρ is the heat capacity and density of the material respectively, d is the thickness of the volume, ∆T is the temperature change in given time which equals q·∆t, q is the cooling rate. For a given same volume of glass and mold, the relation between the temperature change rate q of the glass and the mold can be expressed as:
q C ρ glass = ,moldp mold (5.22) qmold C ,glassp ρ glass
Heat Capacity, Cp Properties Desnsity, ρ [Kg/m3] [J/Kg·ºC]
Glass 858 2,510
Tungsten Carbide mold 314 14,650
Table 5.3 Heat capacity and density of BK7 glass and Iron mold
Assuming the molds are made of steel, the properties of BK7 glass, and the molds are summarized in table 5.3. Applying these data to equation 5.22, the relation between temperature changes rate in glass and mold is given by:
66
qglass ≈ 14.2 qmold (5.23)
The equation 5.23 is an estimated equation, because only convection at surface is considered and the heat capacity of glass changes as the temperature changes.
Furthermore, the thermal conductions between the glass and molds, and larger dimension of the molds than that of glass will also complicate the relation. However, the result still shows that the temperature measured by sensor in experiments did not represent the temperature change at the glass surface during cooling.
Figure 5.14 Simulated temperature history near the glass and mold surfaces caused by
force convection
Figure 5.14 shows the simulated temperature changes near the surfaces of the glass and the molds. A force convection with a convection coefficient h = 300 W/(m2K), and the
67
sink temperature T∞ = 40 °C were applied to the surfaces of glass and molds respectively.
The cooling rate at glass surface is around 3 times of that at mold surface in the temperature range 400 °C to 680 °C. The cooling rate at this range is very critical for the index change, because the glass transition region of BK7 glass is in this temperature range. Applying a cooling temperature history with a cooling rate qa which is two times as the cooling rate q1 showed in figure 5.2 in the simulation, the predicted refractive index variation is shown in figure 5.15.
Figure 5.15 Measured and simulated results of the refractive index variation curves at
cooling rate q1 and simulated result with adjusted cooling rate qa
The value of simulated result with adjusted cooling rate at radius at 10 mm matches the measurement result better. This suggests that the real temperature at the glass has an
68
important effect on the simulation accuracy. The simulated result still does not match the measurement result perfectly along the entire radius, that might be due to the elimination of the thermal conduct between the glass and the molds during the cooling process in the simulation model.
5.5.2 Effect of the Heat Capacity
Generally, the stresses induced in the glass resulted from the temperature gradient inside the glass during cooling. As given in equation 5.21, the heat flow Q is a function of heat capacity Cp. Thus the heat flow rate dQ/dt given in equation 5.23 for thermal conduction is affected by the heat capacity Cp.
dQ dT = kA (5.23) dt dx where, k is the thermal conductivity of material, A is the cross-section area of conducting surface, dT/dx is the temperate gradient along the x direction. As a result, the temperature gradient at right hand of the equation 5.23 will be different with different Cp value. As mentioned in section 5.3, the heat capacity Cp also changes as the temperature changes, and the value of the heat capacity used in this research is given in figure 5.5.
Figure 5.16 shows the temperature gradient from node 122 to node 8060 in figure 5.3, with a fixed heat capacity Cp value at 858 J/Kg·ºC, and the changing Cp given in figure
5.5. The temperature gradients at cooling time t = 200 sec and t = 300 sec of cooling rate q1 are given in figure 5.16. As shown in figure 5.16, the temperature gradient is different 69
with constant and changing values of Cp. Thus, the induced stresses at a given point must be different, which will lead to different refractive index change and different refractive index gradient inside the glass.
Figure 5.16 Temperature gradients along radical direction of the glass with/without
considering heat capacity Cp changes as temperature changes
Figure 5.17 shows the predicted index variations with fixed heat capacity Cp value at 858
J/Kg·ºC, and the changing Cp. The simulation results prove that heat capacity does affect the index variation and cannot be neglected for simulation. In order to improve the simulation accuracy, the heat capacity Cp value needs to be measured accurately in the temperature range at a given cooling rate.
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Figure 5.17 Simulated results of the refractive index variation curves with/without
considering heat capacity Cp changes as temperature changes
5.5.3 Effect of the Thermal Expansion Coefficient
The solid thermal expansion coefficient αg used in the simulation is based on the catalog
-6 -1 value from Schott [Schott, 2000]. The value of αg = 8.3x10 K used in the simulation is an average value from 20 ~ 300 °C. The value is 7.1x10-6 K-1 in -30 ~ 70 °C temperature range. The liquid thermal expansion coefficient αl used in this chapter is 4 times of αg.
The values used in the simulation are idealized and might not be the exactly the case in experiments. Therefore, errors are introduced to simulation results because of the assumption and idealization of the thermal expansion coefficients.
The thermal strain change ∆εth imposed to glass as the temperature changes is given by:
71
th αε )( Δ=Δ TT (5.24) where, α(T) is the thermal expansion coefficient, ∆T is the temperature change from time
Δ− tt to time t. It is the obvious that the value of the thermal expansion coefficient determine magnitude of the thermal strain that imposed to glass. As a result, the stresses that response to the strain change will be different, thus the stresses changes calculated from equation 5.9 will be different. Figure 5.18 shows the stress σzz at node 122 in figure
5.3 with different thermal expansion coefficient values.
As illustrated in figure 5.18, the glass is cooled from 680 °C to 20 °C at a constant rate
1.6 °C/sec. The temperature at node 122 is also shown in the figure. At the beginning, the stress relaxation time is so short in the temperature range of AC that relaxation of the glass material can keep up with the cooling rate and no significant stress change occurred in range BD. Then the stress starts to increase because it can no longer relax to zero as the relaxation time increases when temperature continues to decrease. The structure of the exterior of the glass is frozen when the temperature reach room temperature at point
E. However the temperature at node 122 is still around 150 °C, thus the interior of glass keeps shrinking while the atoms at exterior are already frozen and more tensile stress is induced. The stress keeps increasing from point F/G until the temperature at node 122 approaching the room temperature. It clearly shows that the thermal expansion has a significant effect on the induced stress. The higher solid thermal expansion coefficient αg is, the higher stress is induced inside the glass.
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Figure 5.18 Induced stress σzz at node 122 with different solid thermal expansion αg
(same thermal expansion coefficient ratio rα = 4)
Figure 5.19 Induced stress σzz at node 122 with different thermal expansion coefficient
-6 -1 ratio rα (same solid thermal expansion coefficient αg =8.3x10 K ) 73
As discussed in chapter 4, the value of thermal expansion coefficient of glass changes as the temperature changes. It is a function of temperature. The stress is induced during the entire temperature range during cooling, which means the value of the thermal expansion coefficient in the entire temperature range has an impact on the final stresses. Normally, the liquid thermal expansion coefficient αl is 3 – 5 times of the solid thermal expansion coefficient αg, and the selection of the thermal expansion coefficient ratio rα will determine the value of the liquid thermal expansion coefficient αl.
As shown in figure 5.19, the induced stress component σzz at node 122 with different thermal expansion coefficient ratio rα for a given solid thermal expansion coefficient αg =
-6 -1 8.3x10 K . The liquid thermal expansion coefficient αl is higher with a higher value of rα, which means a higher thermal strain is imposed to node 122 in the high temperature range OD. The responding stresses would be higher and the residual stresses will be higher even as part of responding stresses relaxed. The stress increase during the time range t1 to t2 is larger at higher liquid thermal expansion αl. As the temperature decreases and is no longer in the glass transition region, the thermal expansion becomes the same as the solid thermal expansion αg. Therefore same thermal strain is imposed to glass and the induced stress increases at the same rate, which is shown as the parallel lines BE and CF.
Then stress continues to increase as the temperature approaching room temperature while exterior of the glass is already frozen.
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Figure 5.20 Simulated results of the refractive index variation curves with two different
value of solid thermal expansion αg (same thermal expansion coefficient ratio rα = 4)
Figure 5.21 Simulated results of the refractive index variation curves with different
thermal expansion coefficient ratio rα (same solid thermal expansion coefficient αg
=8.3x10-6 K-1)
75
Figure 5.18 and figure 5.19 clearly show that the thermal expansion coefficient of glass
α(T) during cooling range has an impact on the induced residual stresses in the glass. The volume changes are different with different residual stresses, since the volume change is a function of stresses. The effect of thermal expansion coefficient on the refractive index variation distribution is simulated and shown in figure 5.20 and 5.21. The cooling rate q1 in figure 5.2 is used for both simulation comparisons. Figure 5.20 shows the simulation results based on two different values of solid thermal expansion coefficient αg. The simulation results for different liquid thermal expansion coefficient αl are shown in figure
5.21. Based on the simulation results, the value of the thermal expansion coefficient affects the simulation results. Therefore, more accurate information about the thermal expansion coefficient is required to improve the accuracy of the simulation results.
5.6 Conclusion
In this chapter, the refractive index variation caused by the temperature gradient during cooling in the glass compression molding process was discussed and investigated by both the experiments and FEM simulation. The experiments with the different cooling rates used in simulation were performed on a Toshiba GMP 211V machine. The thermally treated glass plates were measured to reconstruct the refractive index variation by using a
Shack-Hartmann sensor system. The experimental results were also compared with the simulated results, which showed a reasonable agreement.
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Some of the key contributions of this chapter are summarized below:
1) The refractive index variation of a thermally treated glass plate caused by the
temperature gradient during cooling can be predicted by FEM simulation. A
comparison between numerical simulation and experiment results was
demonstrated in this research. The results reasonably agreed each other. Therefore
the methodology established in this chapter can be adapted to the optimization of
the glass lenses designing in industry.
2) Both numerical simulation and experiment results confirmed that a faster cooling
rate would result in a higher refractive index variation, if the other conditions
were the same. This can be used to optimize the heating-cooling cycle of
compression molding precision glass lenses.
3) The experiment results showed that the refractive index variations induced into
the glass lenses by the cooling operation were much higher than that of an
untreated blank glass. The refractive index variations are too large to be ignored.
A group refractive index may no longer be sufficient to be used to design a
precision optical system with optical components that are made by compression
molding process at high cooling rate.
4) The FEM simulation methodology is based on some assumptions and
idealizations of the material properties of glass. The effects of the cooling rates
assumption, value of the heat capacity Cp and the thermal expansion coefficient α
were discussed and presented. The simulation results suggested that more
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5) Future work would include: a) Investigate the real cooling rate at glass surface. b)
Develop a more sophisticated simulation model that includes heat conduction
between the glass workpiece and molds to predict the refractive index variation
more precisely. c) Design an experiment setup with a closed cavity to contain the
glass sample that is close to the real setup of the compression molding process.
The results from chapter 4 and this chapter showed that the refractive index of compression molded glass lenses could be predicted by the structural relaxation model. Thus, the FEM model can be used to predict the refractive index distribution inside a thermally treated glass lens. The value can be applied to the lens designing stage to help the designer to adjust the lens design for improved performance. The simulation results also showed the effect of the cooling rate that a slower cooling rate resulted in less refractive index variation. This analysis can help optimizing the manufacturing process. Figure 5.22 is the flowchart of a proposed process integrating the FEM simulation to optimize the lens design and manufacturing capability.
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Figure 5.22 Flowchart of optimizing the lens designing and manufacturing with FEM
simulation
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CHAPTER 6: FEM MODELING OF THERMAL SLUMPING OF GLASS
MIRRORS
Precision optical mirrors are mostly made of glass. There is a growing demand for mirrors with spherical, aspherical, and freeform surfaces over a very large dimension, e.g. precision reflecting mirrors for x-ray telescope and freeform mirrors as concentrators for solar systems. These mirrors cannot be fabricated easily and efficiently by using traditional manufacturing methods. The cost is too high that applications like solar energy systems cannot compete with traditional electricity generation methods. Thermal slumping thin glass sheet technique is a newly developed method for producing precision glass mirrors with aspherical or freeform surfaces efficiently at low cost. Even though thermal slumping process has been used to fabricate segments of x-ray telescopes mirrors successfully [Jimenez-Garate, 2003; Zhang 2003], it still needs too many experiments to achieve the required quality for precision products. Chen proposed this method as a potential approach for manufacturing freeform mirrors for a solar energy system [Chen,
2010]. Chen has performed some preliminary experiments and analysis on the process. In this chapter, the steps and principles of modeling thermal slumping glass mirrors using
FEM software MSC/MARC is briefly described.
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Figure 6.1 Schematic of the setup and thermal slumping process
Heating and slumping glass under gravity into a mold to replicate the pre-fabricated mold surface profile is called thermal replication or thermal slumping. The thermal slumping procedures are schematically shown in figure 6.1. First a thin glass sheet is placed on a mandrel which can be either convex or concave. Then both the mandrel and the glass sheet are heated slowly to working temperature which is slightly above the annealing point of the glass material. The working temperature is corresponding to the glass viscosity in the order of 109-1010 Pa·s. At this temperature, the viscosity should allow the workpiece to slow deform down to the designed mandrel by gravity. The viscosity of the glass sheet has to be high enough that the thickness of the glass workpiece does not change appreciably during forming and the surface tension can stretch the glass surface to achieve the desired surface roughness. The temperature is held while the glass sheet
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slowly deforms under gravity. Then the entire setup is slowly cooled (annealed) to ensure that the final shape of the mirror and the residual stresses can be kept at a minimum level when reaching room temperature. The edges are trimmed sometimes due to large deformations.
As shown in figure 6.1-(a), the reflecting surface touches the mandrel if the glass is formed on a convex mandrel. As a result, over time the surface might deteriorate and affect the x-ray reflecting ability of the mirror. However, the replicated surface matches the designed mandrel shape closely as compared to using concave mandrel that this setup is chosen by Zhang [Zhang, 2003] to produce the x-ray mirrors for the space based x-ray telescope.
On the contrary figure 6.1-(b) shows that the reflecting surface never touches the mandrel if a concave mandrel is used [Jimenez-Garate, 2003]. In this case, the surface roughness of the upper surface will be preserved as the raw glass sheet. However, the contour of the upper surface is not exactly the shape of the inner surface of the concave mandrel. The relationship between the profiles of the mandrel surface and the slumped glass mirror surface is quite nonlinear. Since the contour of the reflecting surface is a critical requirement for mirror performance, it is necessary to alter the mandrel shape in order to achieve the desired reflecting contour.
In previous chapters , the finite element methods have been utilized to study glass components manufactured by compression molding process which utilizes similar
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process conditions as thermal slumping process, e.g. heating, molding and cooling. FEM has been proven to be an effective tool for understanding and optimizing the hot glass forming process. Therefore, FEM can be a useful alternative for studying parameters that affect thermal slumping process, and evaluating some crucial values like the final contour profile of the slumped glass.
The slumping process is a highly non-linear process because of the nonlinear material behavior, nonlinear contact boundary conditions and nonlinear geometry during the process. For the structural modeling, eight-node isoparametric quadrilateral elements were used to mesh the thin glass sheet that would be deformed under its own weight at high temperature. The higher order elements are preferred because they use biquadratic interpolation functions to represent the coordinate and displacements which allows an accurate representation of the nonlinear contact surface and the deformed shape of the slumped glass workpiece.
As mentioned before, glass material has very strong non-linear temperature dependent material properties. The viscosity of glass is highly sensitive to the temperature changes and cannot be neglected in the processes of heating and cooling. The glass material retains linearity between the applied load and deformation at a constant temperature in real time. However, the temperature load of the slumping process varies with time; the current state of deformation is a nonlinear function of the temperature and depends on the entire loading history. This temperature dependence known as Thermo-Rheologically
Simple behavior (TRS) is modeled by shift functions in MSC/MARC software. The main 83
shift functions used in this dissertation to model the thermal slumping process are the
Williams-Landel-Ferry (WLF) equation [Williams, 1955] and the Narayanaswamy model.
Figure 6.2 Simulation stages in the thermal slumping process
As shown in figure 6.2 the slumping process was modeled into two major steps: heating and slumping and cooling. The heating and slumping stage includes modeling initial structural and thermal parameter of the material. The shift function WLF equation applied in this stage is given by:
η Tref )( − 1 −TTc ref )( αT = loglog = (6.1) η T)( 2 −+ TTc ref
where, c1 and c2 are constants for selected glass material and can be calculated with glass viscosity at given temperatures. The typical reference temperature Tref in the equation is the transition point of the glass material. This equation provides a good fit of glass viscosity in both the glass transition temperature range and temperature above the glass
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transition range. With this equation, the software can calculate the viscosity of the glass at any temperature.
The slumped glass result from heating and slumping stage was transferred to cooling stage to obtain the final geometry profile of the reflecting surface. The Narayanaswamy model was used to simulate the structural relaxation behavior during the cooling stage.
The structural relaxation model has been described in detail in chapter 4. Taking the advantage of the geometry of the glass sheet and the slow cooling rates, a uniform temperature distribution was applied to both the glass sheet and mandrel during cooling.
The above FEM modeling algorithm was used to investigate the thermal slumping glass mirrors with both convex and concave mandrels in the following two chapters.
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CHAPTER 7: FEM INVESTIGATION OF THE THERMAL SLUMPING
PROCESS WITH CONVEX MANDREL
The performance of x-ray telescope mirrors is critical for high-energy astronomy, e.g. x- ray telescopes that require optimal sensitivity and imaging resolution. High angular resolution mirrors have been fabricated by grinding-and-polishing technique [Young,
1979; Aschenbach, 1988; Speybroeck, 1997], which was time consuming and costly. In addition to higher fabrication cost, heavier x-ray mirrors also increase the cost of launching. To this end, different techniques have been tested to produce light mirrors with lower angular resolution. One is called the epoxy replication technique, which was developed and used for the EXOSAT mirrors [Korte, 1988] and x-ray mirrors for Astro-
E2 mission [Tsusaka, 1995]. Another is the Ni electroforming technique that was adopted for the XMM/Newton mirrors [Citterio, 1993]. They are categorized as the replication technique as they have the same goal of replicating the figure of precision mandrels with very thin substrates, e.g. aluminum foils.
Recently, thermal slumping thin glass sheets technique has been developed to form x-ray mirrors. Labov started this technique by slumping a thin glass sheet to form x-ray mirrors with a concave stainless steel mold [Labov, 1988]. The same setup was used to produce
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x-ray optics for the High Energy Focusing Telescope (HEFT) [Craig, 1998]. Later on, a similar slumping technique with a convex mandrel, as shown in figure 6.1(a), was developed to fabricate x-ray mirrors for the Constellation-X project as well[Zhang, 2003;
Zhang, 2004].
Zhang has led his group in Goddard Space Flight Center at NASA in developing the precision glass slumping technique. Although excellent results have been obtained, a systematic understanding and quantitative analysis method is still needed to further develop and industrialize the process. It will be impractical to develop an appropriate scheme for manufacturing x-ray mirrors with proper performance solely based on experiments. FEM has been proven to be an effective tool for fundamental understanding of the slumping process. The object of this chapter is to develop a numerical model based methodology to simulate the thermal slumping process. The simulation model can provide a critical insight about the glass slumping phenomenon around the glass transition temperature to help us further understanding the process. There are various parameters such as glass thickness and length, soaking temperature and soaking time, which have crucial effects on the final surface contour of the slumped glass sheet. The
FEM analysis provides an alternative to identifying and evaluating the effects of these parameters thus helps further optimizing the process. With predictions from FEM, the number of experiments can be reduced.
In this study, three thicknesses and two lengths were used to study the influence on the slumped glass sheet shape. Three heating rates and four cooling rates were tested to study
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the influence on the final shape of the slumped glass sheet. Furthermore, the soaking temperature and soaking time were also studied by applying different values of these parameters in the simulation model.
7.1 FEM Modeling
The thin rectangular glass sheet (glass D263 by Schott) with 432 mm length was placed on a cylindrical mandrel with 250 mm radius. Taking the advantage of geometry simplicity of glass and mandrel, a 2D plan-strain model was used to simulate the slumping and annealing process of the x-ray mirrors. Eight-node isoparametric quadrilateral elements were used to mesh the glass sample. The mandrel was modeled as a rigid curve (Appendix A). Since the heating rate and the cooling rate are very slow that it take more than 10 hours to finish each process, a uniform temperature change was applied to the model to represent the temperature change.
Figure 7.1 The simulation model
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As shown in figure 7.1, only half of the glass sheet was model to reduce elements and save time since the geometry is symmetrical. The simulation consists of two major steps:
1) the glass sheet and the mold (cylindrical mandrel) were heated to the soaking temperature. The temperature was held at the soaking temperature to allow the glass sheet gradually deform under gravity since the viscosity of the glass was lowered at higher temperature. The viscosity of glass in this process was obtained by using shift function Williams, Landel and Ferry (WLF) equation [William, 1955]. 2) After slumping, the glass sheet was cooled to room temperature under a pre-determined cooling rate, which can be simulated by using the Tool-Narayanaswamy model.
In the thermal slumping process, the glass sheet will deform under its own weight when placed on the mandrel as shown in figure 7.2. In the simulation model, the glass sheet will vibrate before it settles at the equilibrium position. Thus at the beginning of the simulation thermal history, the temperature is held at 20 ºC for 1 hour before the heating process started.
Figure 7.2 Glass sheet deformed on its own weight
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Elastic modulus, E [Mpa] 72,900
Poisson’s ratio, v 0.208
Density, ρ [Kg/m3] 2,510
Thermal conductivity, kc [W/m ºC] 0.96
Specific heat capacity, Cp [J/Kg ºC] 820
Transition temperature, Tg [ºC] 557
-6 Solid coefficient of thermal expansion, αg [/ºC] 7.2x10
-5 Liquid coefficient of thermal expansion, αl [/ºC] 2.16x10
Activation energy/gas constant, ΔH/R [ºC] 58,130
Fraction parameter, x 0.45
Weighing factor, wg 1
Table 7.1 Mechanical and thermal properties of D263 glass
Viscosity η dPa·s Temperature (ºC)
Strain Point, 1014.5 529
Annealing Point, 1012 557
Softening Point, 107.65 736
Table 7.2 Viscosity of D263 glass at different Temperature 90
Figure 7.3 Fitted viscosity vs. temperature curve of D263 glass
The important mechanical and thermal properties of glass material D263 are summarized in table 7.1. Table 7.2 shows the catalog viscosity of D263 glass at different temperature which are applied in equation 6.1 to fit the shift function. By selecting 529 °C as the reference temperature, two coefficient c1 and c2 were obtained for WLF equation. Figure
7.3 shows the viscosity curve of D263 glass over the temperature obtained by fitting
WLF model with calculated c1 and c2.
The y displacement of the corner node of the sheet during the heating and slumping stage at a given heating rate was shown in figure 7.4. The glass sheet deformed under its own weight right after it was placed on the mandrel. Then the sheet would not deform further until the temperature reaches around 500 ºC. Therefore technically the setup could be heated quickly to 480 ºC to save time. As shown in figure 7.4-(a), the sheet started to deform before the temperature reached the soaking temperature which is usually above the transition temperature of glass.
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(a) Y displacement vs. Temperature and deformed glass at different temperature stages
(b) Y displacement and temperature vs. Time
Figure 7.4 The y displacement of the end of the glass sheet during the slumping stage
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The cooling stage was simulated by using the Narayanaswamy model. The glass sheet expanded during the slumping stage and shrank back during the cooling stage. The result of the cooling process is shown in figure 7.5. The end of the glass sheet continues deforming on its own weight at the early stage of the cooling process as shown in figure
7.5-(b).
Figure 7.5 Final deformed glass sheet at the end of cooling
7.2 FEM Results and Discussion
There are varies parameters needed to be determined before finalizing a stable manufacturing setup for fabricating the x-ray mirrors in mass production. The parameters can be categorized into two main groups: process parameters and material properties. The process parameters include glass type, thickness and dimensions, heating rates, cooling rates, soaking (holding) temperature and time. The material properties include thermal 93
expansion coefficients of glass and mandrel, friction between glass and mandrel. The simulated results will provide systematical understanding of the effects of process parameters for a proper manufacturing scheme. The simulated effects of process parameters are demonstrated and discussed in this section.
7.2.1 Influence of the Glass Thickness
The thickness of the glass sheet will affect the slumping process and the final slumped shape. Three different thicknesses 0.2 mm, 0.4 mm and 0.8 mm were used for simulation to predict the effect of the glass thickness.
Assuming the coordinate of a point on the reflecting surface of the deformed glass sheet is (x1, y1), and the coordinate of the point along at exact cylinder circle is (x0, y0). For x1= x0 (the apex of the mandrel), the sag variation Δy along the y direction is given by:
2 2 01 −−=Δ xryy 1 (7.1)
where r0 is the designed or the nominal radius of the mandrel.
The variations (Δy) of the deformed glass sheet for three different thicknesses are shown in figure 7.6. The dotted lines in figure 7.6-(a), (b), (c) are the shapes of glass sheet after same heating and cooling history given in 7.4(b).
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(a) Thickness t = 0.2 mm
(b) Thickness t = 0.4 mm
Figure 7.6 Inner surface profile of sag variation of glass sheets with different thicknesses (a) t = 0.2 mm, (b) t = 0.4 mm, (c) t = 0.8 mm, (d) Comparison between the
predicted curve shape of three glass thicknesses 95
(c) Thickness t = 0.8 mm
(d) Sag variation comparison
Figure 7.6 Inner surface profile of sag variation of glass sheets with different thicknesses (a) t = 0.2 mm, (b) t = 0.4 mm, (c) t = 0.8 mm, (d) Comparison between the
predicted curve shape of three glass thicknesses 96
Figure 7.7 Comparison of radius variation Δr of the inner surface profile for different
thicknesses
Another parameter that will be influenced by the glass thickness is the radius variation.
As the coordinate of a point at the bottom surface of the deformed glass sheet is (x1, y1), the corresponding radius is:
2 2 11 += yxr 1 (7.2)
If the designed radius at the point is r0, then the radius variation is given by:
−=Δ rrr 01 (7.3)
The radius variation comparison of different thicknesses is shown in figure 7.7. The x- axis is the corresponding angle of the point (x1, y1), and the y-axis is the radius variation.
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The variations increase from center to outside edge of the curve. From figure 7.6-(d) and figure 7.7, it is obvious that the thickness of the glass sheet has a strong effect on the inner surface profile of the slumped glass sheet. Thinner glass can provide surface profile with limited variation in a much larger range. The profile variation increases along the length direction since the torsion decreases as sheet keeps deforming and contacts the mandrel. From the above comparison, the thinner the glass sheet is the better profile will match the design mandrel shape. It is shown in figure 7.7 that thickness 0.4 mm provides almost the same profile as the 0.2 mm in angle range [-20°, 0°]. If this range was the required working range of the x-ray mirror, then thickness 0.4 mm would be enough for production. As for future discussion of other parameters, thickness 0.4 mm is chosen as the primary thickness for simulation unless it is noted as other thickness.
7.2.2 Influence of the Glass Sheet Length
The nominal length of the glass sheet is 432 mm. To study the length effect on the profile, assuming a glass sheet with length 300 mm is also slumped under the same condition, and simulated using the same material parameters as the nominal design. Figure 7.8 shows the radius variation comparison for glass sheets with different lengths. The longer glass sheet provides larger torsion during the slumping process, thus it has good curvature profile over a larger range. Since most of the finished glass sheets will be trimmed after slumping, the length can be increased to obtain a good profile.
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Figure 7.8 Radius variation comparison of glass sheets for two different lengths
7.2.3 Influence of Heating Rates
The heating process was mainly divided into two stages. First, the glass was heated quickly to 480 ºC at the pre-slumping stage. Then the glass was slowly heated to 565 ºC and held at this temperature for 1 hour, which is considered as the slumping stage. As shown in figure 7.9, three heating rates were used for simulating glass sheet with 0.4 mm thickness to study the influences of the heating rate. As mentioned before, the temperature was held at 20 ºC for 1 hour before the heating process started in the simulation. Heating rate q2 is half of the heating rate q1 at both the pre-slumping and the slumping stages. Heating rate q3 has the same value in the pre-slumping stage as the heating rate q1, while in the slumping stage it is the same as the heating rate q2.
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Figure 7.9 Three tested heating rates
Two main parameters of the glass sheet end corner point are used to describe and compare the slumping process at different heating rates. One is the y direction displacement; the other is the angular velocity. The angular velocity θ& is the change of the angle θ versus time. Angle θ is defined as an angle of the end point to the center point as shown in figure 7.10.
Figure 7.10 Definition of Angle θ
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(a) Time vs. Y displacement
(b) Temperature vs. Y displacement
Figure 7.11 Comparison of Y displacements of the end point between q1 and q2
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Figure 7.12 Comparison of Temperature vs. Angular Velocity θ& curves between q1 and
q2
Figure 7.12 shows the angular velocity θ& at these two different heating rates. Due to the faster heating rate from 480 ºC to 565 ºC at q1, the glass sheet deforms much faster and affects the final shape of the glass sheet at the end of the slumping process.
Figure 7.13 shows the sag variation Δy at two different heating rates. The faster the heating rate is, the larger the difference is. The sag difference δy between two heating rates are given by:
δ ()Δ= 1 ()− Δ 2 (xyxyxy ) (7.4)
where, Δy1 and Δy2 are the sag variation of heating rates q1 and q2 respectively. Figure
7.14 shows the shape different of the glass sheets between q1 and q2. 102
Figure 7.13 Comparison of sag variation Δy vs. x between q1 and q2
Figure 7.14 Shape difference δy vs. x between q1 and q2
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Figure 7.15 Comparison of radius variation Δr between q1 and q2
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Figure 7.15 shows the radius variation comparison between heating rates q1 and q2.
Figure 7.15-(a) shows the overall radius variations, while figure 7.15-(b) shows the variation at the inside of the glass sheet. Figure 7.15-(c) gives more clear view of the radius variations.
Figure 7.16 Comparison of angular velocity θ& vs. Temperature curves between q2 and
q3
The comparisons of the slumping process and the final shape variation between heating rate q2 and q3 are shown in figure 7.16 and figure 7.17. As mentioned before, the glass sheet is heated faster to 480 ºC at rate q3 and then heated with the same rate in the slower heating stage (480 ºC to 565 ºC) in both processes. As shown in figure 7.16, the glass sheets slumped at the same angular velocity θ& . According to the radius variation
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comparison that is shown in figure 7.17, the final surface contour of the slumped glass sheet under the slumping processes at heating rate q2 and q3 are the same.
Figure 7.17 Comparison of radius variation Δr between q2 and q3
Based on these comparisons, the fast heating rate at the pre-slumping stage does not seem to affect the slumping process and the final surface contour of the slumped glass sheet.
The glass sheet does not deform until the temperature reaches a point when the viscosity of the glass decreases to the value that allow the glass sheet to deform under gravity. The heating stage before this point is defined as the pre-slumping stage. When the temperature reaches this turning point, the heating rate afterwards affects the slumping process and the final surface of the slumped glass sheet. This heating stage is defined as the slumping stage. Although heating rate q3 is twice as the heating rate q2 at the pre- slumping stage, the slumping processes and the final surface contour of the slumped glass 106
sheet are the same, because they have the same heating rate at the slumping stage.
Therefore, in order to save time, higher heating rate can be applied at the pre-slumping stage. The heating rate at the slumping stage should be determined by the requirement of the final shape.
7.2.4 Influence of the Cooling Rate
Figure 7.18 Three tested cooling rates
The shape of the glass with 0.4 mm thickness slumped at heating rate q2 was used as the initial shape for the cooling process. As shown in figure 7.18, three cooling rates c1, c2 and c3 were applied in simulation to study the influences of the cooling rate. As mentioned before, Tool-Narayanaswamy model was used for simulating the cooling process. The setup was slowly cooled from 565 ºC to 500 ºC, and then cooled to 400 ºC at a faster cooling rate, and finally cooled to 20 ºC with the fastest cooling rate. 107
Figure 7.19 Comparison of radius variation Δr among c1, c2 and c3
Cooling Rate Length (mm) Average Thickness (mm)
c1 433.172759 0.40176
c2 433.115177 0.40102
c3 433.059990 0.40097
Table 7.3 Final length and thickness of the slumped glass sheet at the end of the cooling
Figure 7.19 shows the radius variation comparison among the three different cooling rates. The faster the heating rate is, the larger the radius variation will be. From the above comparison, it is obvious that different cooling rates also lead to different final surface profiles of the slumped glass sheet. Due to the structural relaxation behavior, the final
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length and thickness of the glass sheet is no longer the same as those at the beginning of the slumping and cooling process. The final length and thickness at different cooling rates are given in table 7.3. The results are consistent with the theory that faster cooling rate results in larger final volume after cooling.
Another cooling rate c4 as shown in figure 7.20 was applied in the simulation. The cooling rate c4 has the same rate as cooling rate c3 before the temperature reaches 400 ºC.
Then c4 becomes twice as c3 from 400 ºC to 20 ºC.
Figure 7.20 Cooling rates c3 and c4
The radius variation comparison of cooling rates c3 and c4 is shown in figure 7.21. The two curves overlap each other, which means these two rates lead to the same final surface profile of the slumped glass sheet at the end of the cooling process. The final length and
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thickness are also the same at these two cooling rates. This indicates that the cooling process also has a critical temperature point. Before this critical temperature point is reached, different cooling rates will lead to different final shapes. After the critical temperature, the cooling rates will not affect the final surface profile of the slumped glass sheet, thus faster rate can be used after that point to save time. Further research may also include finding the turning temperature point of the glass material for the cooling process in order to optimize the cooling process. The FEM model can predict a reasonable temperature range of the turning point, and experiments can be performed in this range to verify the prediction so that selecting an appropriate turning point temperature can be identified.
Figure 7.21 Comparison of radius variation Δr between c3 and c4
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7.2.5 Influence of the Soaking Temperature and Soaking Time
Even though the glass sheet started to deform before the temperature reaches the soaking time, the deformation of the entire glass sheet mainly happened at the soaking temperature. Generally, the higher soaking temperature and the longer soaking time, the better the inner surface profile of the slumped glass sheet matches the mandrel shape.
However, the inner surface will deteriorate if the soaking temperature is too high or soaking time is too long since the surface makes contact with the mandrel directly.
Therefore, lower soaking temperature and shorter soaking time are preferred. The FEM simulation results can help to narrow down the soaking temperature and the soaking time ranges in order to minimize the need for experimental verification.
Figure 7.22 shows the radius variation comparison of the 0.4 mm thickness glass sheets which were slumped at four arbitrary soaking temperatures, 545°C, 550°C, 555 °C, and
560 °C for one hour respectively. The higher the temperature is, the better the slumped inner surface profile will be. At lower soaking temperature, the profile improves drastically as the soaking time increases. The profile at temperature 555 °C was improved significantly. Figure 7.23 shows the radius variation of the glass sheets which were held for two hours at given soaking temperatures.
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Figure 7.22 Comparison of radius variation ∆r after being held one hour at different
soaking temperature
Figure 7.23 Comparison of radius variation ∆r after being held two hours at different
soaking temperature
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As shown in figure 7.22 and 7.23, the surface profile at 555 °C is improved in the azimuth direction from -20 degree to the center when the soaking time was increased.
Figure 7.24 shows that acceptable inner surface profile can be obtained either at higher soaking temperature (560 °C) with shorter soaking time (one hour) or at lower soaking temperature (555 °C) with longer soaking time (two hours). As mentioned before, higher soaking temperature and longer soaking time both lead to the surface defects, it is necessary to weigh the effects of these two parameters to obtain required profile with limited surface defects.
Figure 7.24 Comparison of radius variation ∆r when glass sheet was held at 555 °C for
two hours and at 560 °C for one hour respectively
In some cases, thicker glass sheet maybe preferred since the workpiece will be less fragile. In this case, a higher temperature is not recommended because the viscosity of 113
glass decreases drastically as temperature increases. Since a long soaking time allows better inner surface profile, simulation was performed on thicker glass sheet (0.8 mm) with different soaking time to test the required soaking time. Figure 7.25 shows 4-hour soaking time provides an acceptable surface profile of a 0.8 mm thick glass sheet.
Figure 7.25 Comparison of radius variations ∆r of 0.8 mm thick glass sheet held at 565
°C for 2, 3 and 4 hours
7.3 Conclusions
In this chapter, FEM simulation was performed to study and analyze the effects of the process parameters of slumping thin glass sheet on a convex mandrel. The simulation results showed that the slumping process started at a temperature below the transition temperature of the glass material. The heating rate after this point affects the final contour 114
of the glass sheet’s inner surface, while the heating rate before this point does not. Hence the fast heating rate can be applied to heat the glass and mandrel quickly to this turning point.
For cooling rate, there is also a temperature turning point. After this point, the cooling rate does not change the final surface profile of the slumped glass sheet. However before the glass is cooled to this temperature, different cooling rates will lead to different thickness, length and curvature. A fast cooling rate after this point is suggested to shorten the manufacturing cycle time.
The thickness and length of the glass sheet, soaking time and soaking temperature have great impacts on the final contour of the slumped glass inner surface. A thinner glass sheet matches the mandrel over a larger range than a thicker glass sheet. On the other hand, higher soaking temperature and longer soaking time provide better curvature profile for a given glass sheet. If thicker glass was chosen, the curvature profile of slumped optics can be improved by increasing the soaking temperature or/and soaking time. Simulation results can help to narrow down the proper ranges of thickness, soaking temperature and soaking time in order to find an optimal process to manufacture x-ray glass mirrors with the required surface performance.
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CHAPTER 8: INVESTIGATION OF GLASS THICKNESS EFFECTS ON
SLUMPING WITH CONVACE MANDREL USING BOTH
EXPERIMENTAL AND NUMERICAL APROACHES
The glass slumping process can also be performed on a concave mold (mandrel), which has been applied to fabricate x-ray mirror for telescope [Jimenez-Garate, 2003] and freeform mirrors as concentrators for solar systems [Chen, 2010]. In these reports, there was no deterioration at the reflecting surface, since it is the upper surface of the slumped glass sheet, which does not make contact with the mold. The profile of the upper surface is a proximate replication of the mold shape. This transformation is nonlinear because the mold curvature profile is nonlinear. The nonlinearity increases as the thickness of the glass increases. On the other hand, the contour of the reflecting surface is an important design criterion required for proper performance of the optics. It is important to study the curvature deviation (or curvature change) and compensate the error in order to obtain the designed curve with required optical performance. In this chapter, both experiments and
FEM simulations were performed to study the final inner surface contour of the glass with different thickness. The surface roughness of the slumped glass was also measured by a profilerometer.
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8.1 Slumping Process
The major procedures of the thermal slumping process are shown in figure 8.1. All the thermal slumping experiments in this research were performed in a thermal furnace
(Grieve, BF-12128-HT thermal furnace). MACOR® (Corning Inc., New York) ceramic was used to fabricate the mold due to its excellent chemical stability at high temperature up to 1,000°C which is above the soaking temperature in this research. MACOR® ceramic was selected for its ability to be cut to designed shape by using commercial cutting tools.
Figure 8.1 Procedures of the thermal slumping process
The following procedures described the slumping process of glass mirrors.
1) A circular soda lime glass workpiece with 100 mm diameter and 0.55 mm
thickness was placed on a parabolic concave MACOR® ceramic mold. Then both
the glass workpiece and the mold were placed into the furnace. Glass workpieces
with different thicknesses were used for research purpose.
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2) The furnace was heated to the strain temperature Tstrain, and the temperature was
held for a fixed amount of time to allow the glass workpiece and the mold reaches
the thermal equilibrium state.
3) Then the furnace temperature was heated to the soaking temperature Tsoaking at a
lower heating rate. The temperature was held for another fixed amount of time to
allow the glass workpiece deforms to the mold shape. This stage is called soaking
or slumping stage.
4) The furnace was shut down after the soaking stage. The glass was slowly cooled
inside the furnace the temperature was far below the strain point Tstrain before
being taken out of the furnace.
Figure 8.2 A typical thermal history for glass thermal slumping process
In this research, the soaking temperature was 640 °C and the soaking time was 120 minutes. The thermal history for the entire process is shown in figure 8.2. Glass 118
workpieces with different thicknesses were treated under the same process condition for research purpose.
A coordinate measurement machine (CMM, Sheffield Cordax RS-30 DCC) was used to measure the upper surface contour to evaluate the experiment results and compare with the predictions by FEM simulation. Since the geometry of the slumped mirror is asymmetric parabolic surface, only a small area along the radial direction was measured to gather the geometry profile of the glass upper surface.
Figure 8.3 Parabolic shape of the MACOR® mold
The mold shape was designed as parabolic curvature as shown in figure 8.3. The sag of the surface is a function of radius, as given by:
1 Z 2 ⋅== rPr 2 (8.1) 2 p where, p is the parabolic parameter and was designed as 262.9 mm in this research. The alterative parabolic parameter P is 1.90186x10-3 mm-1. For parabolic curve, the larger p is, the flatter the curvature will be. In another word, larger P value corresponds to the curve with deeper sag. In this study, P value is used to describe degree of the curvature.
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However, due to the machining inaccuracy, the finished mold shape was different from designed shape. The curvature was also measured by CMM, and the sag variation
-3 measured −=Δ ZZZ designed is shown in figure 8.4. The corresponding fitted P is 1.93224x10 mm-1, and p is 258.8 mm. The fitting of measured mold surface curvature was implanted into the FEM model as the mold shape for simulation prediction.
Figure 8.4 Mold curvature error caused by machining error
Glass workpieces with four different thicknesses, as given in table 8.1, were slumped with same conditions to the same mold and studied by FEM simulation. The difficulty of replication steep curvature for different thicknesses can be calculated by [Smith 1988]:
Δ⋅= PtK (8.2)
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where, t is the thickness, K is the "degree of difficulty" which should be less than 25 for high accuracy replication production [Smith 1988]. ∆P is the characteristic parameter of the curvature defined as ΔP = Δ(130.5 R) . Δ(1 R) is the change of surface curvature which can be obtained by differentiating the curvature function. Since the curvature of parabolic surface is given by:
1 p 2 = 23 (8.3) R ()+ rp 22
The change of curvature at any point is:
31 p 2 =Δ 25 r (8.4) R ()+ rp 22
Thickness t (mm) Thickness Variation δ (mm) Degree of Difficulty Kmax
-5 t1 = 0.55 ±0.05 3.56x10 -5 t2 = 0.95 ±0.05 6.15x10 -4 t3 = 1.90 ±0.10 1.23x10 -4 t4 = 3.00 ±0.10 1.94x10
Table 8.1 Thickness information of the glass workpieces [Source: S.I Howard Galss]
For glass workpieces with 100 mm diameter in this study, the maximum degrees of difficulty Kmax at different thicknesses were calculated and given in table 8.1. Since the
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values are much less than 25, the glass workpiece can easily accommodate the curvature change of the mold designed for this research.
8.2 FEM Modeling
Figure 8.5 FEM model and the result of the slumping predicted by the FEM model
Taking advantage of the simplicity of the glass workpiece geometry, two-dimensional
(2D) axisymmetric simulation of the glass thermal slumping process was conducted using a commercial FEM software MSC/MARC. Eight-node isoparametric quadrilateral elements were used to mesh the glass sample. The simulation includes two major steps:
1) The glass workpiece and mold were heated to a soaking temperature above the
transition temperature of the glass material, and then the temperature was held for
a given amount of time to allow the glass workpiece to deform to the mold under
gravity. This is considered as the slumping stage, which is shown in figure 8.5.
2) The slumped glass workpiece was slowly cooled to a temperature less than 200
°C in our experiments which was well below the strain point of the material, and
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then was taken out of the furnace and cooled to room temperature at a rather
faster cooling rate.
Soda lime MACOR® glass
Elastic modulus, E [MPa] 68,500 66,900
Poisson’s ratio, v 0.22 0.29
Density, ρ [Kg/m3] 2,483 2,520
Thermal conductivity, kc [W/m ºC] 0.98 1.46
Specific heat, Cp [J/Kg ºC] 1,140 790
-6 -6 Solid coefficient of thermal expansion, αg [/ºC] 8.36x10 11.4x10
Table 8.2 Mechanical and thermal properties of soda lime glass and MACOR®
Viscosity η MPa·s Temperature (ºC)
107.5 490
106.4 535
104.0 575
100.65 720
Table 8.3 Viscosity of the soda lime glass at different temperatures
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The important material properties of soda lime glass and MACOR® used in the simulation were summarized in Table 8.2. Table 8.3 shows the catalog viscosity of soda lime glass at different temperatures. In this research, the viscosity at 490 °C, 575 °C and
720 °C are applied in equation 6.1 to fit the shift function. By selecting 575 °C as the reference temperature, two coefficient c1 and c2 were obtained for WLF equation. Figure
8.6 shows the viscosity curve of soda lime glass over the temperature range in thermal slumping obtained by fitting the WLF model with calculated c1 and c2.
Figure 8.6 Fitted viscosity vs. temperature curve of soda lime glass
8.3 Simulated and Experimental Results
Agnon [Agnon, 2005] presented a compensation method called zero order compensation by assuming that the slumped glass forms a layer of uniform thickness t over the mold surface as shown in figure 8.7. Based on the zero order assumption, for any point of the 124
mold surface where the radius of curvature is Rf, the corresponding radius of the glass top surface curvature can be simplified as Rk = Rf – t. Since for parabolic surface the curvature is given by equation 8.3, the curvature at the bottom point (r = 0) becomes:
11 == 2Pf (8.5) pR ff
Figure 8.7 Scheme of zero order compensation assumption
Then the curvature at the bottom point of the glass upper surface is given:
11 1 = 2Pk =>= 2Pf (8.6) fk − tRR R f
From equation 8.6, the parameter P at the lowest point of the glass upper surface is larger than that of mold surface. Even though the curvature of the entire surface contour is more complicated than this specific simplified case; it represents the fact that the curvature parameter Pk at the top surface is larger than that of the mold surface. In other words, the upper glass surface has deeper sag than the mold surface. Furthermore, based on equation
8.6, a thicker glass plate has a deeper sag. This conclusion is based on the fact that the glass workpiece is fully slumped and the bottom surface makes complete contact with the mold surface. 125
Figure 8.8 Sag variation between the simulated upper surface of glass with different
thickness and the mold surface
The measured and simulated data were used to fit a parabolic curve with the parabolic parameter P to evaluate the experimental and simulated results. The curvatures of different thicknesses were compared with curvature of the mold. The sag variation is given by:
()=Δ f ()− m (rZrZrZ ) (8.7)
where Zf(r) and Zm(r) are the sags of the slumped glass upper surface and the mold surface. Figure 8.8 shows the simulated sag variation for different thicknesses.
Obviously, the thicker the glass is, the larger the variation is. As shown in figure 8.8, the
126
thinner the glass workpiece is, the better the replication will be if the glass was fully slumped and made full contact with the mold.
Thickness t Simulated Ps Average Measured Pm ∆P =| Ps – Pm | -1 -1 -1 (mm) (mm ) (mm ) (mm ) -3 -3 -5 t1 = 0.55 1.93354x10 1.90421x10 2.933 x10 -3 -3 -5 t2 = 0.95 1.93655x10 1.92001x10 1.654 x10 -3 -3 -5 t3 = 1.90 1.94356x10 1.93212x10 1.144 x10 -3 -3 -5 t4 = 3.00 1.95376x10 1.95717x10 0.341 x10
Table 8.4 Curvature parameter P fitted from simulated and experimental results
Figure 8.9 Fitted curvature parameter Pm error in the measurements with different glass
thicknesses
The curvature parameters Ps and Pm obtained by fitting the simulated data and measured data are summarized in table 8.4. Figure 8.9 shows the error range of the measured
127
curvature parameter Pm. Both the simulation and measured results show that thicker glass will result in a deeper sag.
Figure 8.10 Sag variations between experimental and simulated results of different glass
thicknesses
The curvature variation between the simulated and the measured results is evaluated by the sag variation:
−=Δ ZZZ ,, simii (8.8)
where, Zi,m and Zi,s are the measured and simulated sag at point i respectively. The root mean square (RMS) of the sag variation is given by:
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n 1 2 RMS ∑Δ= Zi (8.9) n i=1 where, n is the total number of points. The comparison of sag variations between measured curves and simulation results of different thickness is shown in figure 8.10. The corresponding RMS comparison is shown in figure 8.11. The RMS value can be considered as a characteristic parameter to evaluate the surface contour error between the experimental and simulated results
Figure 8.11 Surface contour RMS errors between experimental and simulated results of
different glass thicknesses
Figure 8.12 shows the upper surface roughness with different thicknesses measured by
Mitutoyo S-300 profiler at similar position of glass plates. The surface roughness Ra at the MACOR® mold surface is around 0.84μm. As shown in this result, glass workpieces with different thicknesses have no significant impact on the surface roughness
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Figure 8.12 Surface roughness of the upper surface of different glass thicknesses
8.4 Discussion
Figure 8.13 Illustration of incomplete slumped glass workpiece
As summarized in table 8.4, one can see that the fitted measurement curvature parameter
P of the thin glass is much less than the designed P value of the mold surface, which is not consistent with the conclusion based on zero order compensation assumption that the glass upper surface has larger P. On the other hand, as shown in figure 8.10, the
130
measured sag of the thinner glass surface is also much less than the predicted value. The reason might be that the thin glass workpiece was not fully slumped, therefore was not making full contact with the mold surface, as illustrated in figure 8.13. Since the glass bottom does not make ful contact with mold surface, the curvature parameter Pbottom for the bottom surface is much less than that of the mold surface Pmold. Even though the curvature parameter Pup of the upper surface is larger than Pbottom, it may still be less than
Pmold.
Figure 8.14 Strain response to applied stress of Voigt-Kelvin model
This behavior can be represented by the Kelvin Solid model (Voigt or Kelvin-Voigt
Model), which was described in chapter 3. Figure 8.14 shows the strain response of the
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glass material based on this model when a constant stress (force) is applied. The strain reaches a steady state asymptotically over time as predicted by the Voigt model, which is different from the constant increasing strain in the Maxwell model as long as the stress is applied. Therefore, if the applied stress (force) is not high enough, the deformation of glass has a limitation which prevents the glass from being fully slumped to the mold.
In this study, no extra force, e.g. vacuum, was applied to glass to enhance deformation.
The glass was deformed under gravity G. Then the applied stress at the bottom surface of glass workpiece should be:
G ρgAt σ === ρgt (8.10) A A where, ρ is the material density, g is the acceleration of gravity and t is the thickness of glass.
From equation 8.10, the bottom surface of thicker glass undergoes higher stress which means larger maximum strain as given by the Voigt model. As a result, a thicker glass workpiece will experience a larger deformation, which is proven by experimental results as well. As summarized in table 8.4, the difference between simulated Ps and measured
Pm decreases as the thicknesses increases. It is also shown in figure 8.11, the surface contour error between measured curves from experiments and predicted curves by simulation decrease as the thickness increases.
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There are several other reasons that also contributed to the incomplete deformation, e.g., inadequate temperature, sticking between the glass workpiece and mold, and surface tension.
The real temperature distribution of glass in the experiments was unknown. The temperature given in figure 8.2 was the temperature of ambient air inside the furnace. The furnace used for this study is an industrial electric tempering furnace which is not specially designed for glass slumping process. The glass plate and mold were mainly heated by convection. The temperature of glass workpiece might be lower than the temperature of the atmosphere in the chamber. In addition, the oven uniformity is ±20 °F
(Grieve Corp.) which also gives ±11.11 °C temperature uncertainty of the atmosphere in the chamber. The viscosity is higher at lower temperature. Thus, higher forces are required for deformation. A glass workpiece with thickness 0.55 mm was slumped at a much higher temperature. The glass workpiece was fully slumped, which means the glass plate can be fully slumped if the viscosity is low enough. However, the test at this temperature failed to generate valid information as the glass plate adhered to the mold surface.
There was sticking between the glass workpiece and the mold, which was demonstrated by the imprints at the edge of bottom surface of glass workpiece where the glass initially made the contact with the mold edge. Thus during the slumping process, the edge of the glass workpiece was sticking to the mold edge, then the required force for slumping process became larger than the theoretic value. 133
When the glass is heated above its transition temperature, the viscosity of glass decreases and glass state is considered to be between that of liquid and solid. Thus, glass possesses some liquid behaviors, e.g. surface tension. Surface tension is caused by the cohesive forces among the liquid molecules. The molecules at the liquid surface adhere strongly to each other on the surface. This also increases the required force for stretching the surface of glass workpiece during the slumping process.
In the simulation, the glass workpiece with any thickness value is fully slumped, which is not the real case for experiment result. Four possible reasons are summarized here: 1)
Temperature inaccuracy was not considered in design. 2) Sticking between glass and mold was not included in the simulation model. 3) Surface tension of liquid-solid state glass was not included in the simulation model. 4) The data used to fit the WLF equation may not accurate. Thus the viscosity calculated from the WLF equation with the fitted coefficients might be lower than the real values.
In section 8.2, three of the four catalog values of viscosity provided by the manufacturer were used to fit the coefficients for the WLF equation. However, if the viscosity values at temperature 490 °C, 535 °C and 720 °C were chosen instead, the fitted coefficients will be slightly different. Figure 8.15 shows the fitted viscosity versus temperature curves with different selections of data. The solid line was selected for simulation because it was closer to the viscosity temperature dependence trend than the dashed line. The soaking temperature for this research was also chosen based this curve. However, the difference between the two fitted lines suggests that the catalog data is not accurate. The true 134
viscosity of the selected soaking temperature might be higher than the calculated value from the WLF equation.
Figure 8.15 Viscosity vs. temperature curves with different reference points
As summarized in Table 8.4, the curvature parameter P for measured curve of glass workpiece at thickness 3.0 mm is larger than the value predicted by FEM simulation.
This is because the glass and mold have different thermal expansion coefficients. In simulation the bottom surface of the glass shrinks back slightly, therefore separating from the mold surface when cooled to room temperature. However, in the experiments, sticking occurred between the bottom surface of the glass workpiece and the mold which prevented the glass from shrinking back.
Also shown in figure 8.10 and 8.11, the simulation predictions provided reference values of curvature parameter P to evaluate whether the glass workpieces were fully slumped.
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Based on the comparison , the manufacturing process can be adjusted, e.g. applying extra force such as vacuuming during slumping, increasing the soaking temperature to make sure the glass deforms fully and selecting other material for molds to avoid sticking at high temperature.
Furthermore, the prediction for fully slumped glass workpiece (e.g. glass workpiece with
3.00 mm thickness) matches the experiment results well. This suggests that if the experiment conditions are carefully designed to ensure that glass workpiece is fully slumped, the upper surface of the glass could be predicted by FEM software. Then the predicted countour can be used for compensating the contour design of the mold that makes the final upper surface contour on the glass satisfies the required shape for proper optical performance.
Finally, based on the experimental and simulated results, here are some suggestions for future research:
1) Measure the real temperature of glass workpiece during the soaking stage and
apply the data in simulation model.
2) Obtain more accurate viscosity information of the material to make sure the
viscosity calculated from the WLF equation is close to the true value, and perform
experiments to test the viscosity of selected soaking temperature to confirm the
accuracy.
136
3) Increase soaking time to test if the thin glass workpiece really reach the
deformation limit. Select new material of mold to eliminate the sticking if a
higher soaking temperature is chosen.
4) Add surface tension and sticking condition in to the FEM model to represent the
experiment conditions more accurately.
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APPENDIX A. MODELING OF CONVEX MANDREL
Considering the thermal effect of the convex mandrel, the mandrel was modeled as deformable contact body meshed by Four-node isoparametric quadrilateral elements, as shown in figure A.1.
Figure A.16 Simulation model with deformable mandrel
It was proposed to use same material of glass sheet for mandrel, so that no slipping between glass sheet and mandrel during the cooling caused by the thermal expansion coefficient difference. However, the mandrel shape will still change during cooling.
Because the soaking temperature is around the annealing point of the material, the volume of the mandrel changes as well. To properly simulate the glass behavior around the transition temperature, the Narayanaswamy-Tool model is used to simulate the 138
mandrel shape change. The gravity influence is also included during heating and cooling.
If the material of the mandrel is assumed to be D263 glass, and the mandrel was heated to
560 °C then cooled to the room temperature at 20 °C. The initial outside radius is R0 =
250 mm for the mandrel. The radius change of the mandrel dr is determined by equation
A.1:
22 ii 0 ii −+=−= RyxRrdr 0 (A.1)
where, xi and yi are the x and y coordinates of the point after one manufacturing cycle respectively.
Figure A.17 Radius change of the D263 glass mandrel after two cycles
139
The radius change of the mandrel after one heating-cooling cycle is shown in figure A.2.
Furthermore, the mandrel will be heated and cooled repeatedly for production. For the material of this particular problem and thermal history, the shape of the mandrel will keep changing in every manufacturing cycle. The mandrel shape change after the second cycle is also shown in figure A.2.
In real production, the mandrel is made of fused silica or fused quartz glass which has much higher transition temperature than D263 glass. Thus the thermal expansion coefficient of fused silica glass is considered as constant during the glass slumping process. The solid thermal expansion coefficient of fused silica glass α = 5.5e-7 ºK-1. The mandrel would expand during the slumping process. Then the reference radius in equation 7.3 is no longer the initial radius r0 of the mandrel. The reference radius at temperature T is given by:
00 (1' αΔ+= Trr ) (A.2) where, ∆T is the temperature change. The radius of the mandrel would have changed to
250.0865mm when the mandrel was heated from room temperature of 20 ºC to the soaking temperature of 565 ºC.
Figure A.3 shows the radius variation at the end of the heating-slumping process, the heating rate is q2 given in chapter 7. The dashed line is the simulated result based on modeling the mandrel as rigid curve (chapter 7). The dashed-dotted line and the solid line are the results based on modeling the mandrel as a deformable part. The solid line is the
140
contour profile of the glass sheet inner surface comparing to the initial shape of the mandrel, while the dashed-dotted line is the contour profile comparing to the real shape of the mandrel at temperature T. As shown in figure A.3, the surface profile of the inner surface of the glass sheet is much different from the initial design shape of the mandrel at the end of heating-slumping cycle. However, the contour is still close to the changed shape of the mandrel.
Figure A.18 Relative radius variation of glass sheet at the end of heating process
The slumped glass sheet and the mandrel were cooled to room temperature at rate c2 given in chapter 7. Figure A.4 shows the final radius variation of the inner surface of glass sheet at the end of cooling process. As the mandrel is cooled to room temperature, the reference radius r0 returns to 249.9999955 mm. At this point, the reference radius r0 can be considered as the initial radius 250 mm. As shown in figure A.4, the thermal 141
expansion coefficient of the mandrel does not show significant impact on the final profile in the angular range [-20 °, 0°] of the glass inner surface, as long as the mandrel shrinks back to its initial radius at the end of the cooling process. As a result, the mandrel was modeled as a rigid curve in chapter 7 to save time in simulation.
Figure A.19 Radius variation of the glass sheet at the end of cooling
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