EXPERIMENTAL AND NUMERICAL ANALYSIS OF THERMAL FORMING PROCESSES FOR PRECISION OPTICS DISSERTATION. Lijuan Su

Transcription

1 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

2 Copyright by Lijuan Su 2010

3 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

4 (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

5 DEDICATION This document is dedicated to my family. iii

6 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

7 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

8 VITA April Born, WuHu, China July B. S. Measuring and Controlling Technique and Instrumentation, University of Science and Technique of China, Hefei, China June M.S. Optical Engineering, Xi'an Institute of Optics and Precision Mechanics of China Academic of Science, Xi'an, China June 2006 to Graduate Research Associate, Integrated System Engineering Department, The Ohio State University, Columbus, Ohio vi

9 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), , 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), , FIELDS OF INTERESTS Major Field: Industrial and Systems Engineering Studies in: Precision engineering, Precision optics, Glass thermal forming vii

10 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 Compression Molding of Glass Lenses Thermal Slumping of Glass Mirror... 4 Chapter 2: RESEARCH OBJECTIVE... 7 Chapter 3: STATE OF THE ART Glass Rheology ii

11 3.1.1 Viscosity Viscoelasticity in Glass Transition Region Viscoelastic models Numerical Modeling of Glass Thermal Forming Processes Chapter 4: GLASS INDEX CHANGE AS A RESULT OF THERMAL TREATMENT Theory Structural Relaxation Density Change Relation between the Density and the Refractive Index FEM Modeling of Glass Cooling Simulation Results and Discussions The Effect of Thermal Expansion Coefficient Ratio The Effect of Cooling Rate Conclusion Chapter 5: GLASS REFRACTIVE INDEX VARIATION CAUSED BY THERMAL TREATMENT Theory iii

12 5.1.1 Refractive Index Variation Shack-Hartmann Test Glass Thermal Treatment Process FEM Modeling of Glass Cooling Process Results Measured Index Variation Simulated Index Variation Discussion Effect of the Cooling Rate Effect of the Heat Capacity Effect of the Thermal Expansion Coefficient Conclusion Chapter 6: FEM MODELING OF THERMAL SLUMPING OF GLASS MIRRORS 80 Chapter 7: FEM INVESTIGATION OF THE THERMAL SLUMPING PROCESS WITH CONVEX MANDREL FEM Modeling FEM Results and Discussion Influence of the Glass Thickness iv

13 7.2.2 Influence of the Glass Sheet Length Influence of Heating Rates Influence of the Cooling Rate Influence of the Soaking Temperature and Soaking Time Conclusions Chapter 8: INVESTIGATION OF GLASS THICKNESS EFFECTS ON SLUMPING WITH CONVACE MANDREL USING BOTH EXPERIMENTAL AND NUMERICAL APROACHES Slumping Process FEM Modeling Simulated and Experimental Results Discussion REFERENCE v

14 LIST OF TABLES Table 4.1 Mechanical and thermal properties of glass P-SK Table 4.2 Structural relaxation parameters of glass P-SK57 required for simulation Table 4.3 Predicted index change caused by cooling process with different liquid thermal expansion coefficient, *based on the refractive index n wavelength nm Table 4.4 Predicted index change at different rates and calculated index change *based on reference [Schott, 2007] Table 5.1 Mechanical and thermal properties of BK7 glass Table 5.2 Structural relaxation parameters used in numerical simulation Table 5.3 Heat capacity and density of BK7 glass and Iron mold Table 7.1 Mechanical and thermal properties of D263 glass Table 7.2 Viscosity of D263 glass at different Temperature Table 7.3 Final length and thickness of the slumped glass sheet at the end of the cooling 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 vi

15 Table 8.3 Viscosity of the soda lime glass at different temperatures Table 8.4 Curvature parameter P fitted from simulated and experimental results vii

16 LIST OF FIGURE Figure 1.1 Schematic illustration of the glass compression molding machine and the complete molding process [Jain, 2006b] Figure 1.2 Illustration of the thermal slumping process... 5 Figure 3.1 Fitted viscosity vs. temperature curve for glass D263 (VFT equation) Figure 3.2 The stress σ(t) change of glass when a constant a strain ε 0 is imposed at t 0, redrawn from [Scherer, 1986] Figure 3.3 The strain ε(t) change of glass when a constant uniaxial stress σ 0 at t 0, redrawn from [Scherer, 1986] Figure 3.4 The property change after a sudden change in temperature in glass transition region, redrawn from [Scherer, 1986] Figure 3.5 Viscoelastic models: (a) Maxwell model (b) Voigt element (c) Burger model Figure 3.6 Generalized Maxwell model Figure 4.1 Property change of glass during cooling viii

17 Figure 4.2 Volume versus temperature during cooling of glass liquid, redrawn from [Scherer, 1986] 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 4.4 Volume versus temperature at two different cooling rates during cooling Figure 4.5 Predicted volume versus temperature curves obtained by structure relaxation model at two cooling rates q ref = 2 C/hr and q 1 = 3500 C/hr Figure 4.6 Predicted volume versus temperature curves obtained by structure relaxation model at two cooling rates q ref = 2 C/hr and q 1 = 3500 C/hr with different r α Figure 4.7 Thermal expansion coefficient α V (T) versus temperature during cooling Figure 4.8 Predicted volume versus temperature curves obtained by structure relaxation model at three cooling rates q 1 = 3500 C/hr, q 2 = 350 C/hr and q 3 = 35 C/hr Figure 4.9 Thermal expansion coefficient α(t) versus temperature during cooling at two cooling rate q 1 = 3500 C/hr, and q 2 = 350 C/hr Figure 5.1 Principle of Shack-Hartmann sensor Figure 5.2 Temperature histories of three different cooling rates Figure 5.3 Meshed numerical simulation model Figure 5.4 Heat capacity versus temperature for 0.4Ca(NO 3 ) 2 0.4KNO 3 [Copy from Moynihan, 1976] Figure 5.5 Heat capacity C p of BK7 glass used in the FEM simulation Figure 5.6 The experiment arrangement of sample and molds ix

18 Figure 5.7 Simulated temperature changes caused by convection at mold bottom surface and convection at glass side surface 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 Figure 5.9 Reconstructed wavefront variation using Shack-Hartmann sensor (0.225 C/sec) Figure 5.10 Measured refractive index variations along the radial direction for three different cooling rates and an untreated glass lens (blank) Figure 5.11 Predicted index variations for three different cooling rates Figure 5.12 Comparison of measured and simulated results of the refractive index variation curves of three cooling rates Figure 5.13 Illustration of the heat flux through the surface Figure 5.14 Simulated temperature history near the glass and mold surfaces caused by force convection Figure 5.15 Measured and simulated results of the refractive index variation curves at cooling rate q 1 and simulated result with adjusted cooling rate q a Figure 5.16 Temperature gradients along radical direction of the glass with/without considering heat capacity C p changes as temperature changes Figure 5.17 Simulated results of the refractive index variation curves with/without considering heat capacity C p changes as temperature changes x

19 Figure 5.18 Induced stress σ zz at node 122 with different solid thermal expansion α g (same thermal expansion coefficient ratio r α = 4) Figure 5.19 Induced stress σ zz at node 122 with different thermal expansion coefficient ratio r α (same solid thermal expansion coefficient α g =8.3x10-6 K -1 ) Figure 5.20 Simulated results of the refractive index variation curves with two different value of solid thermal expansion α g (same thermal expansion coefficient ratio r α = 4) Figure 5.21 Simulated results of the refractive index variation curves with different thermal expansion coefficient ratio r α (same solid thermal expansion coefficient α g =8.3x10-6 K -1 ) Figure 5.22 Flowchart of optimizing the lens designing and manufacturing with FEM simulation Figure 6.1 Schematic of the setup and thermal slumping process Figure 6.2 Simulation stages in the thermal slumping process Figure 7.1 The simulation model Figure 7.2 Glass sheet deformed on its own weight Figure 7.3 Fitted viscosity vs. temperature curve of D263 glass 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 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 xi

20 Figure 7.7 Comparison of radius variation Δr of the inner surface profile for different thicknesses Figure 7.8 Radius variation comparison of glass sheets for two different lengths Figure 7.9 Three tested heating rates Figure 7.10 Definition of Angle θ Figure 7.11 Comparison of Y displacements of the end point between q 1 and q Figure 7.12 Comparison of Temperature vs. Angular Velocity & θ curves between q 1 and q Figure 7.13 Comparison of sag variation Δy vs. x between q 1 and q Figure 7.14 Shape difference δy vs. x between q 1 and q Figure 7.15 Comparison of radius variation Δr between q 1 and q Figure 7.16 Comparison of angular velocity & θ vs. Temperature curves between q 2 and q Figure 7.17 Comparison of radius variation Δr between q 2 and q Figure 7.18 Three tested cooling rates Figure 7.19 Comparison of radius variation Δr among c 1, c 2 and c Figure 7.20 Cooling rates c 3 and c Figure 7.21 Comparison of radius variation Δr between c 3 and c Figure 7.22 Comparison of radius variation r after being held one hour at different soaking temperature xii

21 Figure 7.23 Comparison of radius variation r after being held two hours at different soaking temperature 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 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 Figure 8.1 Procedures of the thermal slumping process Figure 8.2 A typical thermal history for glass thermal slumping process Figure 8.3 Parabolic shape of the MACOR mold Figure 8.4 Mold curvature error caused by machining error Figure 8.5 FEM model and the result of the slumping predicted by the FEM model Figure 8.6 Fitted viscosity vs. temperature curve of soda lime glass Figure 8.7 Scheme of zero order compensation assumption Figure 8.8 Sag variation between the simulated upper surface of glass with different thickness and the mold surface Figure 8.9 Fitted curvature parameter P m error in the measurements with different glass thicknesses Figure 8.10 Sag variations between experimental and simulated results of different glass thicknesses Figure 8.11 Surface contour RMS errors between experimental and simulated results of different glass thicknesses xiii

22 Figure 8.12 Surface roughness of the upper surface of different glass thicknesses Figure 8.13 Illustration of incomplete slumped glass workpiece Figure 8.14 Strain response to applied stress of Voigt-Kelvin model Figure 8.15 Viscosity vs. temperature curves with different reference points Figure A.1 Simulation model with deformable mandrel Figure A.2 Radius change of the D263 glass mandrel after two cycles Figure A.3 Relative radius variation of glass sheet at the end of heating process Figure A.4 Radius variation of the glass sheet at the end of cooling xiv

23 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

24 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 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

25 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

26 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

27 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

28 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

29 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

30 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

31 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

32 CHAPTER 3: STATE OF THE ART 3.1 Glass Rheology 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

33 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 η = η 0 exp (3.2) RT Where η 0 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 (10 13 to 10 9 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) T T 0 where A, B and T 0 are the fitting constants. T and T 0 are given in C, and T o is always less than the T g 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

34 points of the viscosity temperature curve. The temperature at a viscosity of 10 3 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 Pa-s. The temperature range between the softening and working point is referred to as working range. The glass transition temperature (T g ) is defined as the temperature when the viscosity approximately equals to Pa-s. The annealing point where the internal stresses relax within a few minutes happens at temperature when the viscosity is Pa-s. The temperature when the viscosity of glass is 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 10 7 Pa-s, while it is in the order of 10 9 ~10 10 Pa-s for glass thermal slumping process. Figure 3.1 Fitted viscosity vs. temperature curve for glass D263 (VFT equation) 12

35 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 (T g ) 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 timedependent and known as viscoelasticity. Figure 3.2 shows the stress response of a viscoelastic material, when an instantaneous strain ε 0 is applied at time t 0 and held constant. There is an instant stress response Eε 0, but the stress decreases to zero over time. This phenomenon is called stress relaxation. 13

36 Figure 3.2 The stress σ(t) change of glass when a constant a strain ε 0 is imposed at t 0, redrawn from [Scherer, 1986] Figure 3.3 The strain ε(t) change of glass when a constant uniaxial stress σ 0 at t 0, redrawn from [Scherer, 1986] 14

37 When a constant stress σ 0 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 ε Ε, a delay elastic strain ε D and a viscous flow at the rate σ 0 /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

38 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 σ 12 to the Maxwell model as shown in figure 3.5-(a), the total strain ε 12 is given by: ε + E N 12 = ε12 ε12 (3.4) where, ε E 12 and ε D 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 () t = σ ( 0) exp( t τ ) s (3.5) 16

39 where, τ s 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 () = 12 t ε 12 t 1 exp (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 τ s 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

40 ε 12 () t σ σ t σ 12t = + 1 exp + 2G1 2G 2 τ (3.7) s 2η This model provides a complete description of a viscoelastic material behavior shown in figure 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, nonlinear 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

41 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

42 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

43 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 DEFORM TM -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

44 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

45 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 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

46 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 T g. The value of T g is also a function of the temperature change rate. Figure 4.1 Property change of glass during cooling 24

47 In the transition region, the change of property due to a temperature change from T 1 to T 2 is time dependent, and it can be described with a response function M p (t) [Scherer, 1986]: M p P( t) P ( ) 2 ( ) T fp t T2 ( t) = = (4.1) P (0) P ( ) T T where, the subscripts 0 and represent the instantaneous and steady state values of the property p following a sudden temperature change. T fp 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 T fp (T i ) corresponding to the temperature T i, is found by extrapolating a straight line from p(t i ) with slope of glassy state to intersect with the extension of liquid equilibrium state line. The property response function M p (t) can also be described as an exponential function [Scherer, 1986]: M p b t ( t) = exp (4.2) τ p Where, τ p 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 M v (t) and τ v. 25

48 In the finite element method, the experimental data can be accurately fit with a series of exponential functions [Soules, 1987]: M p ( t) = n i= 1 ( w g t ) i exp τ pi (4.3) where, (w g ) i are weighing factors and τ pi 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 τ p increases, Narayanaswamy introduced the reduced time ξ to restore the linearity [Narayanaswamy, 1971]: t dt' ξ = τ pr (4.4) 0 τ p where, τ pr is the relaxation time at an arbitrary reference temperature T r. 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: M n p ( t) = ( w g i= 1 ξ ) i exp τ pi (4.5) The relaxation time τ p at any given time and temperature can be calculated by the Narayanaswamy model [Narayanaswamy, 1973]: 26

49 ΔH 1 x 1 x τ p = τ p, ref exp (4.6) R Tref T T fp where, τ p,ref is the structural relaxation time of property at reference temperature T ref. T fp 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 T fp 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 dt ( t') T fp ( t) = T ( t) M p[ ξ ( t) ξ ( t') ] 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: p 1 (0) dp( T ) dt fp [ ] = α T + T T pg ( ) α pl ( fp ) α pg ( fp ) (4.8) dt dt 27

50 where, α pg and α pl are the characteristic coefficients of glass property at glassy and solid states respectively. For the volume of glass, they are the thermal expansion coefficients α Vg and α Vl 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

51 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 α V (T) of the glass material, which is the derivative of the curve in figure 4.2. It can be represented as: 1 dv ( T ) dt f [ ] α = = T + T T V αvg ( ) αvl ( f ) αvg ( f ) (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 α V (T) at the liquid state and the solid state as constants α Vl and α Vg respectively during the cooling. Normally, the ratio r α = α Vl / 29

52 α Vg 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: 1 Δ 1 T th V 2 ε = = αv ( T ) dt (4.11) 3 V (0) 3 T1 where, T 1 and T 2 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

53 rates. The volume deviates from the equilibrium line earlier at a faster cooling rate q 1, and results in higher transition temperature T g1. 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 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: n n π N Aρ = α M (4.12) where, n is the refractive index, M is the molecular weight, N A 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

54 2 n 1 = 2 4π + b( n 1) N Aρ α M (4.13) 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 2 can also obtain the Newton-Drude relation ( n ) ρ = 1 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 dp 2 2 ( n 1)(4π + bn b) = (4.14) 8πnρ The density change can be expressed as: m m Δρ = (4.15) V q V 0 where, V q is the volume at the end of cooling process at rate q, and V 0 is the original volume of the glass sample before heating-cooling treatment. Here, m/v 0 is the initial density ρ. Then the index change can be calculated by: 2 2 ( n 1)(4π + bn b) Δ = V0 n 1 8πnρ V q (4.16) 32

55 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 V 0. 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 τ v,ref at reference temperature T ref was 33

56 calculated from stress relaxation time τ s,ref with a ratio of τ v / τ s =10.6 for a similar glass [Scherer, 1986]. The stress relaxation time τ s,ref at T ref can be calculated by equation: ηref τ s, ref = (4.17) G where, G is the shear modulus which is given by E ( 1+ 2v). Material Properties Value Elastic modulus, E [Mpa] 93,000 Poisson s ratio, v Density, ρ [kg/m 3 ] 3,010 Thermal conductivity, k c [W/m ºC] 1.01 Specific heat, C p [J/kg ºC] 760 Transition temperature, T g [ºC] 493 Solid linear coefficient of thermal expansion, α g [/ºC] ( ºC) 8.9x10-6 Viscosity, η [MPa-sec] (at 494 ºC) 10 6 Table 4.1 Mechanical and thermal properties of glass P-SK57 34

57 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 r α = α l / α g 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, T ref [ºC] 494 Activation energy/gas constant, ΔH/R [ºC]* 71,988 Fraction parameter, x * 0.5 Weighing factor, w g 1 Structural relaxation time, τ v [sec] (at T ref ) Stress relaxation time, τ s [sec] at T ref Table 4.2 Structural relaxation parameters of glass P-SK57 required for simulation 35

58 4.3 Simulation Results and Discussions Figure 4.5 shows the predicted volume versus temperature curves by the FEM simulation method, with thermal ratio r α = 4. According reference [Schott, 2007], the listed refractive index of glass P-SK57 is based on a reference annealing rate q ref = 2 C/hr. So the predicted volume at the end of cooling with this rate is considered as the original volume V 0. The final volume of the sample at the cooling rate q 1 = 3500 C/hr is V q1. Figure 4.5 Predicted volume versus temperature curves obtained by structure relaxation model at two cooling rates q ref = 2 C/hr and q 1 = 3500 C/hr 36

59 For glass P-SK57, no specific experiments results to fit the empirical value b for equation 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 2 ( n 1)( n + 2) Δ = V0 n 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 q i : qi nd ( qi ) = nd ( q0) + mn log (4.19) d q 0 where, q 0 is the reference annealing rate of 2 C/hr as mentioned before, and m nd is the annealing coefficient for the refractive index depending on the glass type. The n d ( q 0 ) of glass P-SK57 is and m nd is -9.5x10-4 according to [Schott 2007]. Applying the cooling rate q i = 3500 C/hr to equation 4.17, the new index would be with change n d = From the experimental results, the real index n d after the moldingcooling process is with change n d = The differences might be caused by geometry, molding process and other factors. Compared to the values in table 4.3, it is 37

60 confirmed that the thermal expansion coefficient ratio r α = 4 provides a better agreement with the reference value. Ratio r α Linear Liquid Thermal Expansion Coefficient α Vl [/ºC] Index change n d * x x x Table 4.3 Predicted index change caused by cooling process with different liquid thermal expansion coefficient, *based on the refractive index n wavelength nm The Effect of Thermal Expansion Coefficient Ratio Figure 4.6 shows the predicted volume versus temperature curves with different thermal expansion coefficient ratios r α1 and r α2. It is obvious that higher thermal expansion coefficient ratio has lower original volume V 0 (final volume of cooling rate q ref = 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 α V (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

61 V q T2 i = 1 Vqi ( T dt V (0) α ) (4.20) T 1 where, V(0) is the sample volume at the beginning of cooling process, T 1 and T 2 are the temperature at the begin and end of cooling respectively. In this case, T 1 = 600 C, while T 1 = 20 C. Figure 4.6 Predicted volume versus temperature curves obtained by structure relaxation model at two cooling rates q ref = 2 C/hr and q 1 = 3500 C/hr with different r α As mentioned before, the volume of reference annealing rate q ref is considered as the original volume of glass sample before the heating-pressing-cooling process. Therefore, the real volume change V r of a given sample after cooling is: 39

62 T 1 T1 Vr = Vq V0 = V (0) α V ( T ) dt α ( ) 0 Vqi T dt (4.21) i T2 T2 Δ Then equation 4.18 can be rewritten as: 2 2 ( n 1)( n + 2) ΔV r Δn = (4.22) 6n V0 + ΔVr Define real relative change as volume change ratio r V = ΔV r V0, then: ( n Δn = 2 1)( n 6n 2 + 2) rv 1+ r V (4.23) Figure 4.7 Thermal expansion coefficient α V (T) versus temperature during cooling 40

63 As shown in figure 4.7, the real volume change V r is the shadowed area and the higher thermal expansion coefficient ratio has larger V r. Furthermore, from figure 4.6 the higher thermal expansion coefficient ratio corresponds to a lower original volume V 0. Thus, the volume change ratio r V will be greater at higher thermal expansion coefficient ratio, which will lead to greater index change. Here, r V << 1. Hence, the thermal expansion coefficient ratio has a great impact on the refractive index and density changes 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: q 1 = 3500 C/hr, q 2 = 350 C/hr and q 3 = 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 r α = 4, which was determined previously. Cooling Rate q i (ºC/hr) Reference index change* n d Simulated index change n d x x x x x x10-3 Table 4.4 Predicted index change at different rates and calculated index change *based on reference [Schott, 2007] 41

64 Figure 4.8 Predicted volume versus temperature curves obtained by structure relaxation model at three cooling rates q 1 = 3500 C/hr, q 2 = 350 C/hr and q 3 = 35 C/hr Figure 4.9 Thermal expansion coefficient α(t) versus temperature during cooling at two cooling rate q 1 = 3500 C/hr, and q 2 = 350 C/hr 42

65 As shown in figure 4.9, the real volume change V r is the shadowed area and the higher cooling rate has larger V r. Thus, the volume change ratio r V 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

66 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 Refractive Index Variation Thermal strain ε th 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

67 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 = ε 11 + ε 22 + ε V (0) 33 (5.1) where, ε 11, ε 22 and ε 33 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: V ( 0) = RV 0 (5.2) where, V 0 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

68 [ V ( 0 ΔV ] V0 Δ V (5.3) r = ) Combining equation , the real relative volume change after heating-cooling process is given by: ΔVr = R 1 R( ε 11 + ε 22 + ε 33) (5.4) V 0 The actual strains are the sum of the thermal strains and elastic strains of the nodes and can be expressed as: ( σ + σ ) th σ 11 v ε11 = ε + (5.5a) E ε 22 ( σ + σ ) th σ 22 v = ε + (5.5b) E ε 33 ( σ + σ ) th σ 33 v = ε + (5.5c) E where, σ 11, σ 22 and σ 33 are the normal stresses along the axes, E is the Young s modulus, v is the Poisson s ratio, and the thermal strain ε th is given by: ε th T = 2 T 1 α( T ) dt (5.6) where, α (T ) is the thermal expansion coefficient of the glass material in the temperature range [T 2,T 1 ]. 46

69 If the glass is cooled uniformly, there is only thermal strain ε th, and then equation 5.1 becomes: Δ V T th = 3ε = 3 2 α( T ) dt (5.7) V (0) T1 This equation is an alternative expression of equation 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: ( σ + σ + σ )( 1 v) ΔV th = 3ε + (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: th Δt [ Δε Δε ] 1 exp ( t Δt Δσ = D σ k ) (5.9) λk where, D is the elastic modulus matrix, ε and ε th are the tensors of the actual strains change and the thermal strains change between time t Δt and t respectively, σ k 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

70 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 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

71 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

72 d 2 ρ d λ θ max = = (5.12) f 2 f d The angular sensitivity of this test is determined by: σ θ min = (5.13) 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 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

73 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., q 1 = 1.60 C/sec, q 2 = 0.60 C/sec, and q 3 = C/sec (these are all nominal values of measured temperature histories by the embedded sensors in the molds). 51

74 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. 52

75 Figure 5.3 Meshed numerical simulation model The thermal boundary condition applied in the simulation is given by: T k n u = h ( T T ) (5.14) where, n u 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

76 Elastic modulus, E [Mpa] 82,500 Poisson s ratio, v Density, ρ [Kg/m 3 ] 2,510 Thermal conductivity, k c [W/m ºC] 1.1 Specific heat capacity, C p [J/Kg ºC] 858 Transition temperature, T g [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, w g 1 Solid coefficient of thermal expansion, α g [/ºC] 8.3x10-6 Liquid coefficient of thermal expansion, α l [/ºC]* 3.32x10-5 Structural relaxation time, τ v [sec] (at 685ºC) Stress relaxation time, τ s [sec] at 685ºC Table 5.2 Structural relaxation parameters used in numerical simulation 54

77 Figure 5.4 Heat capacity versus temperature for 0.4Ca(NO 3 ) 2 0.4KNO 3 [Copy from Moynihan, 1976] Figure 5.5 Heat capacity C p 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 C p is the characteristic value of enthalpy. Figure 5.4 shows the measured heat capacity C p versus temperature for 0.4Ca(NO 3 ) 2 0.4KNO 3 [Moynihan, 1976]. Normally, C 1. 3 for 55 pl C pg

78 multicomponent oxides [Scherer, 1986]. As a result, the heat capacity C p 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

79 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/(m 2 K), and sink temperature T = 40 C without considering the upper mold. The points N 1 and point N 2 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 Second, only apply convection at side surface of the glass with same convection coefficient and sink temperature. The point N 3 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

80 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 N 2 was cooled about 9 times faster than the glass side point N 1. For the thermal convection cooling, the temperature at the glass side point N 3 drops drastically, the temeprature change is even faster than that the point N 2. 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 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 = 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

81 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 -f 2 /f 1. 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: L( x, y) = n( x, y) t( x, y) (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 n c, then the reference optical path L r is defined by : L r = n t c (5.16) Thus, the wavefront variation as the optical path difference can be defined by: 59

82 Δ L ( x, y) = L( x, y) Lr = n( x, y) t n t (5.17) V c The wavefront variation ΔL v (x, y) could be reconstructed by using the Shack-Hartmann sensor for measuring the position of spots. So when the wavefront variation ΔL v (x, y), and thickness t of the sample is known, the refractive index variation Δn v (x, y) can be calculated from the following equation: Δ n ( x, y) = n( x, y) n = ΔL ( x, y t (5.18) V c V ) 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 C/sec. The wavefront variation of 60