ap-6-11.dvi Acta Polytechnica Vol. 51 No. 6/2011 Thermal Forming of Glass — Experiment vs. Simulation L. Sveda, M. Landová, M. Mı́ka, L. Ṕına, R. Havĺıková, V. Marš́ıková Abstract Thermal forming is a technique for forming glass foils precisely into a desired shape. It is widely used in the automotive industry. It can also be used for shaping X-ray mirror substrates for space missions, as in our case. This paper presents the initial results of methods used for automatic data processing of in-situ measurements of the thermal shaping process and a comparison of measured and simulatated values. It also briefly describes improvements of the overall experimental setup currently being made in order to obtain better and more precise results. Keywords: X-ray mirrors, Comsol Multiphysics, simulations, thermal forming, gravity forming, in-situ measurements, data processing. 1 Thermal forming process for lightweight optics Thermal forming is a technique for shaping light- weight precise space X-ray mirrors that has been under development for several years. Teams work- ing on the topic include the NASA team led by Dr. Zhang [7] working on the formerCON-Xproject, an Italian group [8], and the Czech group consist- ing of people from CTU, ICT, the Astronomical In- stitute and Rigaku company, working on the former XEUS/IXO/Athena mission [9]. The principle of the technique is rather straight- forward. A precise form (a mandrel) is prepared. A glass sheet is then placed atop the mandrel (or between two mandrels) and a temperature profile is applied. Since glass behaves like a viscous liquid above certain temperatures, forming can occur and the shape remains even after the sample cools down. Themethoddescribed in thispaper is very similar to the method outlined above, except that no man- drel is used. The glass sheets are held only on their edges, and shaping at high temperatures takes place onlydue togravity [6]. Theprinciple is demonstrated inFigure1. Asimilarmethod isused formanufactur- ing car frontwindows, howeverwith a lowerprecision requirement. 2 Metrology The shape of the formedglass sheetwasmeasuredby two principally different methods and provided com- plementary information about the forming process. In order to acquire information about the dy- namics of the forming process, a noncontact optical method was used. A scheme of the setup is shown in Figure 1. The sample remains in its position inside the furnacewhile it is observed via a camerawith an objective. Figure 2 shows a typical image obtained by the camera. The typical period between two con- secutive images was 5–10 minutes. A set of images with timestamps was obtained, and was further pro- cessed in order to obtain valid data. Fig. 1: Schematic view of the experiment. The glass was placed atop the support only by its edges. The tem- perature was increased and gravity performed the actual forming. The maximal bending as a function of time and temperature was then measured and compared with the simulations Fig. 2: Example of image processing of the in-situ optical measurement. The original image (upper, false colors) is processed in order to obtain reference points (second im- age), the reference line and the vertical line for glass posi- tionmeasurement is found (third, overlaid on the original image, false colors) and the position of the glass is finally found in this vertical line (fourth) 72 Acta Polytechnica Vol. 51 No. 6/2011 Several Matlab scripts were written in order to process the images automatically, if possible, or semi- automatically. The most important information for developing the forming process that can be obtained frommeasurementsof this kind is thedynamicsof the process. Thus, the total bend of the sample at each time was measured. The algorithm first located two fixed reference points within each image. This is be- cause the images can be shifted and inclined relative to each other, as the doors needed to be opened each time in order to take the image. Then the center of the sample was found, where the maximal bend was expected. The relative position of the center with respect to the reference points — the total bend — was identified. This was done either automatically, when the lighting conditions were optimal, or semi- automatically with some help from the user. Finally, the values were calibrated in order to obtain values in mm and not in pixels. A single step of the script is shown in Figure 2. The other semiautomatic script was also written, and was able to process the image in order to deter- mine the sample profile. It was used to detect the shape change of the sample between the end of the forming process and the end of the cooling process. As the method used standard commercial equip- ment only (camera, lenses, tripod, etc.) and the doors had to be opened each time, the precision of the method is low, typically 0.1—0.5 mm. It can therefore be used successfully only for a study of the process dynamics. The Taylor Hobson PGI PLUS contact profilometer was used to obtain more accu- rate data for shape fitting. This device is able to measure profiles up to 120 mm in length, sampling down to 0.25 micron and with vertical resolution at the nanometer scale. Three linesweremeasuredon each sample at each of the orthogonal directions, two close to the edges and one close to the center of the sample. The data from the device was initially processed by the Taylor Hobson software, and was further exported and pro- cessed in Matlab as well. The profiles were rotated in order to make them horizontal and concentric, as it is impossible to position the sample perfectly on the table. Varioius curves were then fitted onto the data, including polynomials of the order 2 and 4, a circle and a catenary curve. 3 Simulations Computer simulationof the formingprocess is a com- plex task [1–4]. Generally, a combination of a ther- mal model and a mechanical model has to be ap- plied. The thermal model should contain spatial and temporal temperature profiles, as well as the ther- mal interaction of the furnace and the form with the sample. This can be either measured or simulated. Mechanically, it can be modelled as a simple beam made of a viscoelasticmaterial, where several border conditions need to be met. The gravitational force is easily applied as a volume force. The dynamic viscosity of the material as a function of the tem- perature at a given location has to be known. The proper sample/support interaction must be entered. Either a rigid system or a system where some move- ments with defined friction are allowed, etc. Surface tension can be included. The simulation performed here used the Comsol Multiphysics software with the CFD module [5], but with several simplifications. The temperature of the sample was assumed to be homogeneous and identi- cal to the temperature of the furnace. This seems to be unrealistic, but it is a good starting point. More work needs to be done to justify and modify this condition. Rigid borders were applied, no slip, no squeezing of the glass, and the sample lay freely on the support. Uncertainty in border definition is ex- pected to produce differences in the bending speed of the order of 20 % [2]. Most of the forming time was spent above the transform temperature, thus a viscous Newton liquid model was used with the dy- namic viscosity given by the curve in Figure 3. The output of the Comsol Multiphysics simulations was only a velocity field as a function of time. The data was thus imported toMatlab and integrated in order to obtain the actual profile. Fig. 3: Dynamic viscosity as a function of the temper- ature used in the simulation in the form of the Vogel- Fulcher Tammann equation [6] 4 Processing experimental data The experiments were performed with glass sam- ples made from Desag D263 glass. Two different sample sizes were used: 75 × 25 × 0.75 mm3 and 100 × 100 × 0.4 mm3. The forming process, tem- peratures, measurements and fits by different curves as well as a description of the equipment used in the 73 Acta Polytechnica Vol. 51 No. 6/2011 experiment are described in greater detail in [6]. The average forming speed as a function of the forming temperature for one of the samples is shown in Fi- gure 4, together with the simulated data. Fig. 4: Average forming speed for a sample 100 × 100 × 0.4 mm3 in size as a function of temperature. The mea- sured values are compared with the simulation All the forming experiments were measured in- situ, and the results were processed by the noncon- tact method. The resulting data was then combined into the form of a unified plot where the change in shape (the bending) is a function of the forming time and the temperature (see Figure 5). Fig. 5: Measured bending as a function of time and form- ing temperature for 100 × 100 × 0.4mm3 samples Asimulationofall the experimentswasperformed and the data was processed in the same way as the experimental data in order to provide a comparison (see Figure 6). The overall shape and values are con- sistent, although not perfectly matching yet, see Fi- gure 4, which compares the average shaping speed as a function of temperature. Detailed contactprofilometermeasurements show that, due to short openings of the doors in order to make in-situ images for noncontact measurements, there are temperature gradients that result in inho- mogeneous forming. It can be seen that the shape is different at the edge closer to the doors. Further, we have no actual information about the temperatures and the temperature gradients in the glass, which is very important for the simulation. It is expected that, together with the poorly defined support, this leads to most of differences between the simulation and the experiment. Inaccurate noncontactmeasure- ment is not important for average forming speed de- tection, as least square fitting is used. Fig. 6: Simulated data based on the experiments from Figure 5. More data points are used for interpolating the colored surface 5 Expected improvements Several key points were identified during the ex- periments described above, which affect the mea- surements and the forming process itself, and are currently being upgraded. The shaping process is strongly affected by opening of the doors for mak- ing images, thus a sapphire window in the doors will be used. In addition, poorly defined support will be replaced by stable and perfectly defined mechanical support. Independent temperaturemeasurements in- side the furnace for greater precision will be per- formed. In order to be able to increase the preci- sion of the opticalmeasurements, a camera at a fixed point relative to the furnace will be used. It will be equippedwith a telecentric lens. The lightingwill be adjusted to enable automatic data processing for all images. 6 Conclusions Ageneralmethod for in-situmeasurements of a ther- mal free fall glass forming process has been demon- strated, including automatic data processing. The datawasused for improvements to a formingmethod suitable for space X-ray telescopes, and as an input 74 Acta Polytechnica Vol. 51 No. 6/2011 for computer simulations of the process. The simu- lation and the experiments are consistent, but more work needs to be done. Key points which need to be modified and corrected in future experiments have been identified, and experimental upgrades are cur- rently underway. Acknowledgement We acknowledge support from the Grant Agency of the Academy of Sciences of the CzechRepublic, Ma- terial andX-rayOpticalProperties of FormedSilicon Monocrystals project, grant number IAAX01220701. References [1] Starý, M.: Gravitačńı tvarováńı skla I. – Labo- ratorńı měreńı, Sklář a keramik, 59, 7–9, 2009, p. 150–154. [2] Starý, M.: Gravitačńı tvarováńı skla II. – Num- erická simulace,Sklář a keramik, 59, 10–12, 2009, p. 213–217. [3] Chen, Y.: Thermal forming process for precision freeform optical mirrors and micro glass optics, PhD thesis, The Ohio State University, 2010. [4] Stokes, Y. M.: Very Viscous Flows Driven By Gravity, PhDthesis,UniversityofAdelaide, 1998. [5] http://www.comsol.com/ [6] Landová, M.: Thermal forming of glass and Si foils for X-ray space telescopes, thesis, Institute of Chemical Technology Prague, 2011. [7] Zhang,W.W.: Lightweight and high angular res- olution X-ray optics for astronomy, Proceedings of the SPIE, 8076, 2011, p. 807602. [8] Prosperio, L., et al.: Thermal shaping of thin glass substrates for segmented grazing incidence active optics, Proceedings of the SPIE, 7803, 2010, p. 78030K. [9] Hudec, R., et al:, Advanced X-ray optics with Si wafers and slumped glass, Proceedings of the SPIE, 7437, 2009, p. 74370S. Libor Sveda Ladislav Pína Radka Havlíková CTU in Prague FNSPE Břehova 7, 115 19 Prague 1, Czech Republic Martina Landová Martin Míka ICT Prague Technická 5, 166 28 Prague 6-Dejvice Czech Republic Veronika Maršíková Rigaku Innovative Technologies Europe, s.r.o. Novodvorská 994, 142 21 Prague 4, Czech Republic 75