WO2005055284A2

WO2005055284A2 - Method of fabricating low-warp flat glass

Info

Publication number: WO2005055284A2
Application number: PCT/US2004/039820
Authority: WO
Inventors: Gautam Meda; May Yanmei Xun
Original assignee: Corning Incorporated
Priority date: 2003-11-28
Filing date: 2004-11-29
Publication date: 2005-06-16
Also published as: KR101073783B1; WO2005055284A3; CN1902045A; CN1902045B; KR20060103478A

Abstract

A method of fabricating a glass sheet comprises modifying the thermal stress in the glass such that it is a tensile stress or substantially zero stress in a particular temperature zone of the glass, with that zone selected such that the glass sheet is formed with reduced levels of curtain warping. In an example embodiment, the modifying of the thermal stress is effected by non-uniform cooling of the glass as it passes through its glass transition temperature range. This non-uniform cooling may, for example, be applied in cooling segments that are linear with at least two of the segments have differing slopes.

Description

METHOD OF FABRICATING LOW-WARP FLAT GLASS

I. FIELD OF THE INVENTION This invention relates to the manufacture of glass sheets such as the glass sheets used as substrates in display devices such as liquid crystal displays (LCDs). More particularly, the invention relates to methods for reducing a problem known as "curtain warp" which occurs in the manufacture of such glass sheets by, for example, the fusion downdraw process.

II. BACKGROUND OF THE INVENTION A. DISPLAY DEVICES Display devices are used in a variety of applications. For example, thin film transistor liquid crystal displays (TFT-LCD) are used in notebook computers, flat panel desktop monitors, LCD televisions, and Internet and communication devices, to name only a few. Some display devices such as TFT- LCD panels and organic light-emitting diode (OLED) panels are made directly on flat glass sheets. With many display devices, the glass used in the panels must be flat to within approximately 150 and approximately 250 micrometers over the surface of the glass. Any warping or ripple in the glass will have deleterious effects on the display quality. For purposes of illustration, in many display devices, such as those referenced above, it is useful to incorporate electronic components onto a glass sheet (glass substrate) used in the display device. Often, the electronic components are complementary metal oxide semiconductor (CMOS) devices including TFT's. In these applications, it is beneficial to form the semiconductor structure directly on the glass material of the display. Thus, many LCD displays often comprise a layer of liquid crystal (LC) material associated with a glass substrate upon which transistors have been formed. The transistors are arranged in a patterned array and are driven by peripheral circuitry to provide (switch on) desired voltages to orient the molecules of the LC material in the desired manner. The transistors are essential components of the picture elements (pixels) of the display. As can be readily appreciated, any variation in the flatness of the glass panel may result in a variation of the spacing of the transistors and the pixels. This can result in distortion in the display panel. As such, in LCD and other glass display applications, it is exceedingly beneficial to provide glass substrates that are within acceptable tolerances for flatness to avoid at least the problems of warped glass discussed above. B. WARP Warp is a glass sheet defect characterized by deviation from a plane. It has been one of the most troublesome and persistent problems in the manufacture of LCD glass substrates. Various types of warp are known, the present invention being concerned with curtain warp. As illustrated in Figure 1, curtain warp is characterized by a sine wave like deviation across the sheet width. In particular, for glass sheets made from a drawn ribbon of glass, e.g., glass sheets made from a ribbon of glass produced by, for example, a downdraw fusion process (see Figure 2), curtain warp is a sine wave like deviation in a direction transverse to the direction in which the ribbon is drawn. As is evident from Figure 1, the term "curtain warp" is apropos since the ripple in the glass resembles, and for a vertical draw is in, the direction of a hanging curtain. To date, there has been no fundamental understanding of the origin of curtain warp, and thus no systematic approach for reducing/controlling it. What is needed therefore is a method of forming substantially flat glass that overcomes at least these drawbacks in the art. in. SUMMARY OF THE INVENTION In accordance with a first aspect, the invention provides methods of fabricating a glass sheet which comprise modifying the thermal stress in the glass such that it is a tension (tensile) stress or substantially zero over a particular temperature zone in the glass formation process, with that temperature zone (hereinafter the "TZ") being pre-selected by the user so the curtain warp of the final, fully-solidified, glass sheet is below a designated (i.e., specified) level. Preferably, the final, fully-solidified, glass sheet is designated to be substantially free of curtain warp. As will be understood by persons skilled in the art, the level of curtain warp which can be accepted in the final glass sheet will depend on the intended, application for the sheet. As general guidelines, the level of peak-to-peak curtain warp across the width of the sheet is preferably less than 1000 microns, more preferably less than 600 microns, and most preferably around 200 microns or less. In the preferred embodiments of this aspect of the invention, the TZ is pre-selected based on the glass' non-linear coefficient of thermal expansion (CTE) in its glass transition temperature range (GTTR). As discussed fully below, in accordance with the invention, it has been found that: (1) glasses of the type used as display substrates have non-linear CTE's in their GTTR's; and (2) these non-linearity's interact with the glass' cooling pattern during manufacture to produce thermally-induced compression and tension bands in the glass. By taking these non-linearity's into account, cooling patterns, specifically, non-linear cooling patterns, can be pre-selected which result in reduced curtain warp because the cooling pattern/non-linear CTE interaction produces a tension band or a band with substantially zero compression at the portion of the cooling process where the final shape of the glass is determined. In accordance with another aspect, the invention provides methods of fabricating glass sheets that comprise providing (selecting) one or more substantially non-linear cooling sequences (cooling patterns) over a glass transition temperature range (GTTR) and using the one or more sequences to obtain modeled stress data for a glass sample that represents at least a portion of the sheet. The method also can include selecting one of the one or more substantially non-linear cooling sequences which results in substantially no compression stress in the glass sample or results in tension stress in the glass sample in a desired zone, e.g., in the TZ. This substantially non-linear cooling sequence can then be employed in manufacturing glass sheets whose curtain warp is below a designated (specified) level. In accordance with a further aspect, the invention provides a method of fabricating glass sheets using a fusion forming apparatus (e.g., an isopipe) that has a root, said glass having a glass transition temperature range (GTTR) and said method comprising applying a cooling pattern to the glass as a function of distance from the root and insuring that the cooling pattern includes at least one non-linearity in the glass' GTTR which is sufficient to result in glass sheets whose curtain warp is below a designated (specified) level, e.g., the glass sheets can have substantially no curtain warp. In certain preferred embodiments, the at least one non-linearity includes a non-linearity which produces a tension band or a band of substantially zero compression in the region of the lower temperature end of the GTTR. In accordance with each of the above aspects of the invention, the non- linearity in the glass' cooling pattern can include either a reduction or an increase in the glass' rate of cooling as a function of distance (or, equivalently, time) from, for example, the root of an isopipe. Reductions in the rate of cooling can be visualized as upward "kinks" in a temperature versus distance (or time) plot for the glass, while increases in the rate of cooling can be visualized as downward "kinks" in such a plot. By purposely introducing such kinks and taking into account the non-linear CTE's which display glasses exhibit in their GTTR's, curtain warp can be controlled, e.g., substantially eliminated. Additional features and advantages of the invention are set forth in the detailed description which follows, and in part will be readily apparent to those skilled in the art from that description or recognized by practicing the invention as described herein. It is to be understood that both the foregoing general description and the following detailed description are merely exemplary of the invention, and are intended to provide an overview or framework for ^• understanding the nature and character of the invention as claimed below. Also, the above listed aspects of the invention, as well as the preferred and other embodiments of the invention discussed and claimed below, can be used separately or in any and all combinations. The accompanying drawings are included to provide a further understanding of the invention, and are incorporated in and constitute a part of this specification. The drawings illustrate various embodiments of the invention, and together with the description serve to explain the principles and operation of the invention. It should be noted that the various features illustrated in the figures are not necessarily drawn to scale. In fact, the dimensions may be arbitrarily increased or decreased for clarity of discussion. IV. BRIEF DESCRIPTION OF THE DRAWINGS Figure 1 is a schematic diagram illustrating curtain warp. Figure 2 is a schematic view of a fusion glass fabrication apparatus in accordance with an example embodiment. Figure 3 is a plot illustrating the generation of induced stress bands as a result of linear cooling of a glass sheet having a non-linear coefficient of thermal expansion (CTE). Tref for the secant plot was 25°C. Figure 4 is a plot of CTE versus temperature for Corning Incorporated's Code 1737 glass. Figure 5 is a graphical representation of the stress of glass subjected to various cooling rates, with the graphical representation being used to optimize the glass thermal stress in accordance with an example embodiment. Figure 6 is a graphical representation of the stress of glass subject to various cooling rates in the range from 300-650°C. Figure 7 is a flow diagram of a process of fabricating substantially flat fusion glass in accordance with an example embodiment. Figure 8 is a plot of various cooling patterns tested in Example 1. Figure 9 is a plot of three cooling patterns discussed in Example 2. Figure 10 is a plot comparing estimated expansion curves of Corning Incorporated's Code 1737 and Eagle 2000 glasses. V. DETAILED DESCRIPTION OF THE INVENTION AND ITS PREFERRED EMBODIMENTS In the following detailed description, for purposes of explanation and not limitation, example embodiments disclosing specific details are set forth in order to provide a thorough understanding of the present invention. However, it will be apparent to one having ordinary skill in the art having had the benefit of the present disclosure, that the present invention may be practiced in other embodiments that depart from the specific details disclosed herein. Moreover, descriptions of well-known devices, methods and materials may be omitted so as not to obscure the description of the present invention. As discussed above, the present invention is concerned with warp and, in particular, curtain warp, in display glasses. Such glasses can be produced by various drawing processes, including, the downdraw fusion process. Figure 2 is a schematic diagram of glass fabrication apparatus 100 of the type typically used in the fusion process. The apparatus includes a forming apparatus (isopipe) 101, which receives molten glass (not shown) in a cavity 102. The root of the isopipe is shown at 103, and the sheet of glass 104, after leaving the root, traverses edge rollers 105. The root 103 of the isopipe 101 refers to the location where molten glass from both outer sides of isopipe 101 join together. Finally, a portion of a set of tubes (doctari) 106 is shown. The tubes are used to control the local temperature of the glass sheet (glass ribbon) and thus its thickness. As fusion apparatus is known in the art, details are omitted so as to not obscure the description of the example embodiments. It is noted, however, that other types of glass fabrication apparatus may be used in conjunction with the invention. Such apparatus is within the purview of the artisan of ordinary skill in glass manufacture. In a fusion or other type of glass manufacturing apparatus, as a glass sheet (glass ribbon) travels down the drawing portion of the apparatus, the sheet experiences intricate structural changes, not only in physical dimensions but also on a molecular level. The change from a supple 50 millimeter thick liquid form at, for example, the root of an isopipe to a stiff glass sheet of approximately a half millimeter of thickness is achieved by a carefully chosen temperature field that balances delicately the mechanical and chemical requirements to complete the transformation from a liquid to a solid state. Less than perfect temperature gradients cause sheet deviations from a plane, specifically, curtain warp. Although trial and error changes in glass cooling rates can be employed as one approach to reducing the level of curtain warp or non-flatness in glass sheets, such trial and error changes are entirely empirical and thus commonly result in reduced yields and higher costs for the final product. Also, the trial and error approach must be repeated each time a glass composition and/or the design of glass manufacturing equipment is changed. To understand the genesis of curtain warp, in accordance with the invention, a model was developed which captured the effect of temperature distribution on the temporary thermal stresses in the sheet. In particular, in accordance with an exemplary embodiment of the invention, a model was built using ANSYS finite element software. The sheet as included in the model started some distance from the root (the "upper boundary") and extended to the point where the glass ribbon would be cut during production. SHELL63 elements were used to represent the glass sheet and the location of the upper boundary was specified as a model input. The temperature distribution on the sheet was also specified to the model, along with temperature dependent elastic moduli and a temperature dependent coefficient of thermal expansion (CTE). The ANSYS software then calculated the thermal stresses in the sheet, taking the sheet to be elastic. The specific equations used are set forth in Appendix A. Out-of-plane buckling of the sheet and membrane stress effects were taken into account, but not viscoelasticity. The calculated stresses thus serve as a measure of the stresses in the sheet, but are not the actual stresses. The upper boundary was chosen to be far enough below the root so that the glass would have attained its final thickness before reaching the upper boundary. Also, the location of the upper boundary was chosen to exclude regions where the glass is so soft that stresses are immediately relieved, but to include the region of the glass in which out-of-plane deformations become fixed in the glass. Although the foregoing ANSYS elastic model has been found to work successfully in the practice of the invention, it should be understood that a wide variety of other models, modeling techniques, and modeling software can be used. For example, a model that includes the effects of viscoelasticity can be used if desired. To investigate the cause of curtain warp, the above thermal stress model was initially run for a linear temperature profile down the length of the sheet. The results are shown in Figure 3. As can be seen in this figure, at temperatures of about 650-830°C (specifically, 650-800°C in Figure 3), the linear temperature profile produces a stress pattern in the glass which is characterized by a tension peak followed by a compression peak followed by a tension peak, i.e., a positive- negative-positive stress pattern as shown in Figure 3. The compression stress band in this pattern is particularly important as it occurs in the glass transition zone where the sheet is soft enough to deform, yet stiff enough for the deformation to stay as the sheet travels down the draw, resulting in a permanently distorted glass sheet. Above the glass transition zone, glass is basically a Newtonian fluid, where stress is relaxed quickly; below this zone, glass is stiff enough to resist buckling. It is the presence in the glass transition zone of compression stress of the type illustrated in Figure 3 which, in accordance with the invention, has been found to be the source of curtain warp. Put in other words, it has been found that a glass sheet will buckle and develop curtain warp if allowed to cool linearly through its glass transition temperature region (GTTR). The stress pattern of Figure 3 is the result of the glass having a non-linear coefficient of thermal expansion (CTE) in its GTTR. Figure 4 shows a typical CTE versus temperature plot for Corning Incorporated's Code 1737 glass. The non-linear dependence of CTE on temperature in the GTTR shown in this figure results in uneven contraction of the glass sheet as it cools through this range. This creates the tension-compression-tension stress pattern of Figure 3. The compression portion of this pattern, in turn, produces buckling and thus curtain warp in the finished glass. It is important to note that the non-linear CTE of Figure 4 is not unique to 1737 glass, but is a property exhibited generally by display glasses. In accordance with the invention, it has been discovered that the problem of curtain warp can be controlled by controlling the glass sheet cooling rates in the glass forming process. In particular, as discussed above, the invention provides methods of fabricating a glass material which include modifying the thermal stress such that the thermal stress is a tensile stress, or substantially zero in a specific region, so that the resulting glass sheet has a controlled level of curtain warp, e.g., the resulting glass sheet is substantially free of curtain warp. In accordance with example embodiments described herein, over the glass transition temperature range (GTTR) the cooling versus distance from the root is substantially non-linear in order to substantially avoid internal compression stress at undesired locations that creates curtain-warped glass. The non-linear cooling over the GTTR may be comprised of a plurality of cooling segments. Each segment may be substantially linear, but does not have to be completely linear. To wit, one or more of these segments may itself be non-linear. Ultimately, the slope and duration of these cooling segments are chosen to reduce or substantially eliminate compression stress at undesired locations. Illustratively, the glass of the example embodiments is flat glass having a thickness on the order of approximately 0.1 to 2.0 mm. The glass beneficially has a flatness across the substrate on the order of approximately 150 microns to approximately 250 microns, depending on the size of the substrate. The glass may be used in glass displays such as those referenced above, or in other applications where a flat, substantially ripple-free glass surface is beneficial. As representative examples, the glass may be Corning Incorporated's Code 1737 or Code Eagle 2000 glass, or glasses for display applications produced by other manufacturers. Figure 5 shows the impact of representative non-linear cooling patterns on the stress pattern calculated using the above ANSYS computer model. As shown in Figure 5, non-linear cooling makes the temperature profile deviate from its linear dependence on distance down the draw and therefore creates a stress pattern in the glass that is altered from the one shown in Figure 3. Specifically, as can be seen in Figure 5, a tension band surrounded by compression bands is added to the stress pattern at every upward inflection point in the temperature curve, and a compression band surrounded by tension bands is added at each downward inflection point. An upward inflection point corresponds to a cooling rate change from faster to slower as the sheet travels down the draw, while a downward inflection point corresponds to a cooling rate change from slower to faster. By selecting the locations of the changes in cooling rate, whether those changes be from faster to slower or slower to faster, tensile stress bands (or bands having substantially zero compression) can be placed at one or more desired locations along the glass's cooling curve. In particular, a tensile stress band or a band having substantially zero compression can be placed in, for example, the TZ so that the final glass has less than a specified level of curtain warp, e.g., the final glass can be substantially free of curtain warp. Graph 300 of Figure 5 in particular shows stress (left hand vertical axis) and temperature (right hand vertical axis) versus distance from the root of an isopipe. Alternatively, the horizontal axis of the graph may be time from the root instead of distance. Graph 300 is mathematically modeled using a computer model, such as the model discussed above. The glass sample used in the modeling is illustratively a glass material that is formed into a glass sheet for use as a glass display of the type reference above. It should be noted that graph 300 may be used to perform the selection of the optimal cooling method described in conjunction with the exemplary embodiments of Figure 7 discussed below. To wit, of the three modeled cooling sequences (cooling curves 301, 302, 303), one may be chosen to optimize flatness in the glass sheet made of the chosen glass material. Graph 300 includes three separate modeled cooling sequences for a particular glass material that is being formed by, for example, a known fusion technique. The cooling sequences of Figure 5 are in the glass transition region of -l ithe particular glass being processed, illustratively from approximately 850°C to approximately 650°C. The first sequence shown as first curve 301 is substantially linear. The second sequence, a non-linear curve, is shown as second curve 302 and has a reduced initial cooling rate compared to the first curve. A third sequence is shown as curve 303 and has an increased initial cooling rate compared to the first curve 301. The first curve 301 produces stress curve 304 of Figure 5. Glass manufactured using this cooling sequence would have some positive (tension) stress, but has a significant amount of compression stress. The glass sheet in this example is subject to an unacceptable amount of compression. Accordingly, the modeling method of the example embodiments allows the elimination of this cooling sequence without having to actually perform the cooling of a sample. In keeping with the benefits of these embodiments of the invention, this reduces waste and improves yield in manufacture. The second curve 302 has three segments of cooling. A first segment 305 and a second segment 306, and a third segment 307. In the present embodiment, the first segment 305 and the second segment 306 have substantially the same slope, while the cooling curve of the third segment has a slope that differs from that of segments 305 and 306. The cooling segments result in stress in the glass represented by curve 309. The final (fourth) segment 308 is also linear and is equal in slope to the final (fourth) segment of curves 301 and 303. The initial cooling rate of the second curve 302, cooling segment 305, is reduced compared to that of the slope of curve 301. This results in some tension initially, as shown in the stress curve 309. However, the third segment 307, which has a relatively steep slope compared to the second segment 306, results in significant compression in the glass. This is manifest as a stress peak 310. As such, the second sequence represented by the second curve is also not useful as the glass will have curtain warp that is at unacceptable levels when the compression peak corresponds to the location where the shape of the glass becomes fixed. Again, the modeling method of the invention allows the elimination of this cooling sequence without having to actually perform the cooling of a sample. In keeping with the benefits of these embodiments of the invention, this reduces waste and improves yield in manufacture. The third cooling sequence, represented by curve 303, also has three cooling segments, namely a first segment 311, a second segment 312 , and a third segment 313. This third cooling sequence results in stress in the glass shown in stress curve 314. In this exemplary embodiment, the first segment 311 has a comparatively increased slope compared to curve 301, as well as to the other segments, 312 and 313. The first cooling segment 311 results in some compression as shown. The second and third cooling segments 312 and 313, respectively, have a comparatively reduced slope compared to that of the first segment 311, and to the slope of the curve 301. This second cooling segment 312 results in a rather pronounced upward and positive stress peak 315. This cooling profile provides compensation for the effects of the non-linear coefficient of thermal expansion in the glass transition temperature region, thus yielding a desired tensile thermal stress as the glass traverses the glass transition region. As can be appreciated from graph 300, a variety of stress curves can be generated by the selective alteration of the slopes of the cooling segments. Moreover, the lengths of the segments can be altered as well. Each of these groups of cooling segments results in a substantially unique stress curve for the particular glass material being processed. As such, a variety of resultant stress values can be realized. For example, a family of curves can be determined and are characterized by a faster to slower cooling of the glass in the range of approximately 780°C to approximately 720°C, which is the region of the lower end of the glass transition region for the sample of glass of this embodiment. From this family of curves, a particular curve may be chosen to meet the particular desired end result in the glass sheet. Similarly, a family of curves can be determined which is characterized by a slower to faster cooling of the glass in and/or around the glass transition temperature range of the particular glass being manufactured. From this alternate family, a particular curve may be chosen to meet specific curtain warp requirements. Along the same lines, combinations of faster-to-slower and slower-to-faster patterns can be used, if desired, to achieve specific flatness criteria. Quantitatively, in a cooling sequence of an exemplary embodiment described in keeping with third curve 303, in order to substantially eliminate curtain warp a cooling segment (not shown), which is before the first segment 311, has a relatively high cooling rate at temperatures above the glass transition region. Illustratively, this rate is in a range of approximately 6 °C/in. to approximately 15 °C/in. This occurs in a region that is approximately 25 inches to approximately 40 inches from the root. Next, at the upper portion of the glass transition region, and along the first segment 311 the rate of cooling is reduced to in the range of approximately 4 °C/in. to approximately 10 °C/in. This slower cooling rate is continued for the next approximately 10 inches to approximately 15 inches. In the example, during the second and third cooling segments 312 and 313, the rate of cooling is changed at the lower end of the glass transition region to a cooling rate in the range of approximately 2 °C/in. to approximately 5 °C/in. This rate of cooling is effected for the next approximately 15 inches to approximately 25 inches. After the glass has traversed the glass transition region, the cooling rate is maintained at a substantially constant level, for example as at segment 308. Accordingly, the exemplary embodiment includes a non-linear temperature profile during the cooling process, which significantly reduces, or substantially eliminates curtain warp caused by compression stress in the glass by providing desirable tension stress in the glass. Figure 6 illustrates a further discovery based on the modeling, namely, the substantial independence of the stress pattern in the critical glass transition region from the cooling rate at points further down the draw. As shown in this figure, three slopes in the temperature range from 650°C to 300°C yield the same stress pattern at the higher temperature range of 650-850°C. This substantial independence has the benefit that it makes controlling curtain warp easier since it allows the focus to be on obtaining an efficacious cooling pattern for just the GTTR as opposed to over the entire temperature range of the draw. To summarize, from the above it can be seen that the correlation of sheet stress and glass cooling rates in the glass transition temperature range (GTTR) is the cornerstone for curtain warp control. By means of this correlation, tension stress can be generated to keep the sheet flat as the sheet goes from the relatively soft glass of the transition zone to a stiff solid below that zone. In particular, a tension band at the lower end of the GTTR corresponding to the beginning stage of solid sheet provides a final product with reduced curtain warp compared to a product produced without such a tension band. Figure 7 is a flow chart of an illustrative method 200 for implementing the control of curtain warp in a glass manufacturing process. The example embodiment of Figure 7 can, for example, be applied to the fabrication of substantially flat glass panels using a fabrication apparatus such as described above in connection with Figure 2. At step 201 of Figure 7, uniform or linear cooling data for a particular glass sample formed into sheet glass is obtained by computer-based mathematical modeling. As discussed above, the computer modeling may be effected using one or more of a variety of well-known mathematical modeling techniques. As these are known, details are omitted so as to not obscure the description of example embodiments. These cooling data includes the stress (normally in p.s.i.) for the glass sample as it cools during fabrication. For example, these data may be the stress versus temperature and time (or distance from the root) when the glass cools from its molten state upon leaving the isopipe to its final state at room temperature. Alternatively, the data can be for just a portion of the cooling process from just above the glass transition temperature range to the point where individual glass sheets would be separated from a continuous glass ribbon. As discussed above, uniform or linear cooling of the glass results in curtain warping due to compression of the glass in the glass transition temperature region over which the glass transforms from a fluid ₍to a glass-like material. This glass transition temperature region occurs in many glasses over temperature ranges between approximately 850°C and approximately 650°C. For example, this uniform cooling could be the cooling of the glass to an ambient or a somewhat above ambient temperature as it traverses the fabrication assembly. The compression stress (negative stress) in the glass transition region results from nonlinear thermal expansion in the glass material as it cools uniformly through the glass transition temperature. In the glass transition temperature region, the coefficient of thermal expansion varies non-linearly with a linear change in the temperature. Accordingly, the expansion of the glass with temperature is also non-linear, and thus not uniform. If unchecked, this will result in a compression in the glass, which will cause curtain warping in the glass. Upon obtaining the linear or uniform cooling stress data of the glass at step 201 , the method includes modifying the cooling sequence, particularly through the glass transition temperature region at step 202. One objective of step 202 of the present embodiment is to optimize the glass stress so that over the glass transition temperature region, the glass has either substantially zero stress or a positive stress, or tension. At the end of the glass transition temperature region, the glass will be substantially flat because of the positive thermal stress induced by varying the cooling rate selectively over the region, thereby reducing the likelihood of the deleterious curtain warping. It is noted that the control of the cooling rate may be effected by heating/cooling with external heating/cooling devices to enable cooling at a rate that is slower/faster than that realized using unaided radiation of heat and convection. Such heating/cooling devices within the purview of one of ordinary skill in the art of glass sheet manufacture may be used to realize this controlled cooling rate. It is noted that the modification of the cooling sequence from the linear cooling sequence of step 201 beneficially optimizes the cooling of the glass in the glass transition region to substantially eliminate the compression and thus the curtain warping that can result. However, as discussed above, the elimination of the compression in the glass can be effected by various cooling sequences. Some of these sequences will result in the glass have a net positive stress at the end of the glass transition region, and other sequences will result in the glass having substantially no stress at the end of the glass transition region. In an exemplary embodiment, in order to substantially eliminate curtain warp the glass is cooled at a relatively high rate at temperatures above the glass transition region, illustratively in a range of approximately 6 °C/in. to approximately 15 °C/in. This region can be approximately 25 inches to 40 inches from the root. It is noted that the cooling rate in this region and other regions after the root are controlled by adjusting heating/cooling power to the glass surface. Next, at the upper portion of the glass transition region, the rate of cooling can be reduced to a rate in the range of approximately 4 °C/in. to approximately 10 °C/in. This slower cooling rate can be continued for the next approximately 10 inches to approximately 15 inches. For the next 15-25 inches, the rate of cooling can be changed at the lower end of the glass transition region to a cooling rate in the range of approximately 2 °C/in. to approximately 5 °C/in. After the glass has traversed the glass transition region, the cooling rate can be maintained at a constant level, thereby creating a near-linear temperature profile. It is noted that the slowing of the cooling rate at the beginning of the glass transition region enables the glass molecules to rearrange in structure thereby reaching a reduced energy state, and less compaction or compression. The further reduction in the cooling rate at the lower end of the glass transition region creates a tensile stress band in the horizontal width of the glass sheet. The tensile stress band is created, for example, within the zone where there is a reduction in slope of the cooling curve. Such tension in the glass is useful to foster a stretching of the glass as the glass transforms from a glassy/semi-liquid state to its solid state. This is exceedingly beneficial in preventing curtain warping of the glass due to compression. Finally, the control of the cooling rate below the glass transition region is less rigorous, because it does not strongly affect the stress in the curtain warp- prone glass transition region. Stated differently, the stress in the glass does not have a significant effect during the cooling to room temperature from the end of the glass transition region. However, it is noted that the cooling rate below the glass transition region preferably can be controlled to create a curve that is substantially free from sudden changes in the slope (i.e., a smooth temperature curve) in order to minimize the development of a temporary bow in the glass, which may propagate to the glass still within the glass transition zone, causing additional shape defects in the glass sheet. Each of the cooling regions of the exemplary embodiments can have a substantially constant slope on a temperature versus distance curve. The rate of change of the cooling (with respect to distance) can be decided based on the stress curve determined from the linear cooling (temperature) simulation. For example, when the temperature is reached that corresponds to a peak (relative maximum) internal compression stress, a change in the rate of cooling can be employed. In practice, this may be implemented at a selected position from the root or when the glass reaches this temperature during the cooling process. More specifically, in the example embodiment described above, the cooling rates across the glass transition region are a series of two or more controlled cooling segments, where each segment provides linear cooling at a particular rate and each linear cooling segment begins with a nearly instantaneous transition in the cooling rate. However, it noted that this is merely an illustrative embodiment, and other embodiments may be used to realize the elimination of the compression of the glass in the glass transition region. For example, it is noted that linear cooling rates other than those referenced above may be used to effect the desired elimination of compression. Moreover, depending on the glass material used, other linear cooling rates may be required in order to meet this desired end. Furthermore, it is noted that additional cooling segments may be used to meet this desired end for the same or different glass materials. Finally, it is noted that the cooling sequence across the glass transition does not necessarily need to be a plurality of linear cooling segments. Rather, one or more non-linear cooling segments may be chosen, with the slope of the cooling rate varying to effect the desired reduction in compression, or the creation of tension during cooling, or both. Step 203 of the illustrative method of Figure 7 is an optional step, and includes repeating the modifying of the cooling sequence by selecting different cooling rates across the glass transition region. This may be done as desired to optimize the cooling sequence to achieve a particular resultant stress level or a lower curtain warp. Finally, step 204 of Figure 7 is the implementation of the chosen cooling sequence into a production process. The exemplary methods of fabricating the glass described above foster a significant reduction in the complexity of manufacturing glass with reduced curtain warp. As can be appreciated, these methods are carried out using modeling techniques in order to determine the optimal cooling sequences and not using empirical trial-and-error techniques. As such, a significant reduction in downtime and waste is realized by the example embodiments. To this end, by virtue of the example embodiments, new glass materials, or curtain warp requirements, or both, can be effected in production rather quickly by the selection of the desired cooling sequences, rather than through inefficient and costly trial-and-error techniques. These and other benefits of the example embodiments will be apparent to one skilled in the glass processing arts. Without intending to limit it in any manner, the present invention will be more fully described by the following examples. Example 1 This example illustrates the application of the invention to the selection of a cooling pattern for a new installation of glass forming equipment, specifically, glass forming equipment designed to process more pounds of glass per hour than existing equipment in order to produce glass sheets having larger dimensions. Initially, a cooling pattern that had worked successfully with the prior equipment was implemented on the new equipment. The cooling pattern produced unacceptable curtain warp in the 1000-1200 micron range. A trial-and- error approach was then undertaken to find a cooling pattern that would produce curtain warp levels in the range of 200 microns. The unnumbered curves of Figure 8 show representative profiles that were tested without success. The modeling procedures described above were then employed and resulted in curve 402 of Figure 8. That cooling curve reduced the glass' curtain warp from the 1000-1200 micron range down to 250-300 microns. As can be seen in Figure 8, cooling curve 402 has a transition from faster-to-slower cooling at 720-780°C, which is at the lower end of the glass transition temperature range for the glass being produced (i.e., Corning Incorporated's Code 1737 glass). Curve 402 required a high level of cooling capacity at distances in the range of 40-60 inches from the root of the isopipe. To lessen the demand on cooling capacity, further modeling was performed and it was found that a smaller upward inflection at around 760-780°C would also reduce the level of curtain warp in the final glass. Curve 400 shows the cooling curve that resulted from this further modeling. The 760-780°C temperature range is at the upper end of the 720-780°C range, but still within the lower end of the glass transition temperature range for Code 1737 glass. Cooling curve 400 was found to consistently achieve curtain warp levels in the 200 micron range without the need to increase the cooling capacity of the new equipment. Example 2 This example illustrates the application of the invention to the selection of a cooling pattern for a change in the type of glass being processed, specifically, a change from Corning Incorporated's Code 1737 glass to Corning Incorporated's Code Eagle 2000 glass. Curve 408 in Figure 9 shows a cooling pattern which had been found to produce glass sheets having low levels of curtain warp when processing 1737 glass on a particular fusion glass forming machine. The same equipment was then used to process Eagle 2000 glass using a cooling pattern having substantially the same shape as the successful 1737 pattern. Curve 404 shows the Eagle 2000 pattern. As can be seen, it is parallel to the 1737 pattern but at slightly higher temperatures because the Eagle 2000 glass was processed at slightly higher temperatures. Surprisingly, the 404 pattern resulted in unacceptable curtain warp levels for the Eagle 2000 glass. The above modeling techniques were then used to explain and address this difference in the behavior of the two glasses. Figure 10 is a plot of estimated expansion curves for 1737 (curve 410) and Eagle 2000 (curve 412). As shown in this figure, Eagle 2000 exhibits less expansion than 1737. The lower end of the GTTR for these glasses can also be seen in this figure, i.e., it is generally the region between the peak of the expansion curve at about 780°C and the beginning of the linear portion of the curve at about 720°C. Although a person of ordinary skill in the art would think that Eagle 2000 would be less susceptible to curtain warp problems because of its lower expansion, in fact using the modeling approach of the present invention, it was found that the lower expansion of Eagle 2000 actually required a more aggressive cooling pattern at the lower end of the GTTR to control curtain warp than that needed for 1737. As discussed above, using the modeling techniques of the invention it has been found that curtain warp is related to thermal stress caused by temperature gradients as a glass is cooled through its GTTR. The levels and signs of the stress are, in turn, related to the glass' non-linear CTE in the GTTR. Comparison of the CTE curves of 1737 and Eagle 2000 of Figure 10 reveals that, while similar in general, Eagle 2000's CTE (curve 412) has a smaller temperature dependence than that of 1737 (curve 410). The lower CTE slope of the Eagle 2000 glass means its stress is less sensitive to temperature variations in the GTTR. Therefore, to create a similar stress pattern in this critical zone, a steeper temperature profile is needed to compensate for the lower CTE slope of the Eagle 2000 glass. Curve 406 in Figure 9 shows a cooling pattern that was tested for Eagle 2000 glass based on the above analysis. As can be seen, this curve has a steeper temperature profile than curve 404 which, as discussed above, had poor curtain warp performance for Eagle 2000 glass. As predicted by the modeling, curve 406 with its steeper temperature profile was found to produce glass which had low levels of curtain warp. Thus, as illustrated by this example, when switching glass compositions, CTE curves need to be examined and cooling profiles adjusted based on those curves to achieve low curtain warp levels. From the foregoing disclosure, it can be seen that the most sensitive zone for curtain warp formation/control is at temperatures of approximately 650-850°C for display glasses. The reasons this zone is so important are: first, the nonlinear dependence of CTE on temperature in this region results in uneven contraction as glass sheet cools through this temperature range. Secondly, sheet in this region is soft enough to deform and hard enough for the deformation to stay, therefore resulting in permanent sheet deformation, i.e., permanent curtain warp. By modeling the effects of the non-linear CTE on stress and adjusting the cooling rate to take account of these effects, glass sheets having low levels of curtain warp can be achieved. Although specific embodiments of the invention have been described and illustrated, it is to be understood that modifications can be made without departing from the invention's spirit and scope. For example, a non-linear cooling profile comprised of linear cooling segments, such as the embodiments described above, is merely illustrative of the invention. The method may be comprised of more than three cooling segments or fewer than three cooling segments. Moreover, one or more of the cooling segments may be non-linear. Finally, a combination of linear and non-linear cooling segments may be used to realize the non-linear cooling sequence across the glass transition region temperature range. While exemplary embodiments having been described in detail, it is clear that modifications of the invention will be apparent to one having ordinary skill in the art having had the benefit of the present disclosure. Such modifications and variations are included in the scope of the appended claims. APPENDIX A EQUATIONS FOR THERMAL STRESS

The stresses and strains in the sheet must satisfy the following sets of field equations.

Compatibility __| dydz dx dx dy dz d²€. yy d d yz dzdx dy \ dy dz dx

Q"€χ O e_x v €yy ^J dxdy dy² dx² U~€y_Z O~£yy <J~6_ZZ dydz dz² dy* _n d²e_xz _ d²e_zz d²e_ dzdx dx² dz²

where ε is the total strain, subscripts denote components in the conventional manner, and x,y,z are rectangular coordinates. See, for example, Sokolnikoff, I.S., 1956, Mathematical Theory of Elasticity, Robert E. Krieger Publishing Company, Malabar, Florida. The compatibility equations express that the displacement field be continuous. That is, they express that holes do not form in the body, and that the same space is ^"not occupied by more than one part of the body. In an elastic model, the total strain is the sum of the elastic and thermal strains. The elastic strains ε are: £ffi!κ ^"= ^ffla; cal ^€yy = €yy - - <xT €zz = €zz - - o (2) €χy = €_y ^€ z = € z XZ — -2 where α is the coefficient of thermal expansion, taken to be isotropic here, and T is the temperature difference from the base temperature at which thermal strains are zero. Note that Tcan be a function of spatial location.

Constitutive Law 1 € x ^{== "}S (&XX ~ ^{v σ}yy + ^σzz)) 1 €μy — ^■""=^" {^σVV ~ v σ _ + r_zz)) z = -_p (<?zz - V (σ_x_ + σ_yy)) E ^CE5/ -σ. ay (3)

where σ_y is the stress, E the Young's modulus, and v the Poisson's ratio. E and v can be functions of temperature. This set of equations describes the stress-strain behavior of the material, which is taken to be linear elastic, although it could be viscoelastic, if desired.

Equilibrium dσ_xx dσ_xy dσ_xz _ dx dy dz ΞL + ^dσyy -(- ^dav^z = o (4) dx dy dz dσ_xz dσ_{yz (} dσ_zz _ _n dx dy dz

To find the thermally induced stresses, the preceding sets of coupled equations are solved, along with the boundary conditions and the given temperature distribution. In the present curtain warp model, the boundary conditions are that the sheet is free of external forces. If the thermal strains satisfy compatibility by themselves, the total strain can simply be the thermal strain, and there will be no stress. For example, if the thermal strains are uniform, or if they have a uniform gradient, they satisfy compatibility by themselves, so there will be no stresses. When the thermal strains do not satisfy compatibility, elastic (or viscoelastic) strains enter the picture, such that the total strain satisfies compatibility. The warp model is implemented using the ANSYS finite element software. The preceding sets of equations, and procedures to solve them, are built into ANSYS. The geometry, material properties (E, v, α) as functions of temperature, and the temperature distribution are specified to the ANSYS software.

Claims

What is claimed is: 1. A method of fabricating a glass sheet, the method comprising providing a thermal stress that is a tensile stress, or substantially zero in at least a portion of a specific temperature range that the glass sheet passes through during fabrication, said glass sheet, after fabrication, exhibiting no more than a preselected level of curtain warp, wherein for at least some temperatures within the specific temperature range, the glass of the glass sheet (i) has a coefficient of thermal expansion which varies non-linearly with temperature and (ii) undergoes a transformation from substantially a fluid to substantially a glass-like material.

2. A method as recited in claim 1, wherein the specific temperature range is all or part of a glass transition temperature range of the glass.

3. A method as recited in claim 2, wherein the specific temperature range is approximately 60°C.

4. A method as recited in claim 2, wherein the specific temperature range is approximately 20°C.

5. A method as recited in claim 2, wherein the glass transition temperature range of the glass is from approximately 650°C to approximately 850°C.

6. A method as recited in claim 2, wherein the glass transition temperature range of the glass is from approximately 700°C to approximately 850°C.

7. A method as recited in claim 6, wherein the specific temperature range is from approximately 720°C to approximately 780°C.

8. A method as recited in claim 6, wherein the specific temperature range is from approximately 760°C to approximately 780°C.

9. A method as recited in claim 1, wherein the providing of the thermal stress is effected by cooling the glass sheet in a non-uniform manner over at least a portion of the specific temperature range.

10. A method as recited in claim 9, wherein the non-uniform manner of cooling the glass sheet is determined using a computer model for the glass sheet which incorporates the non-linear variation with temperature of the glass' coefficient of thermal expansion.

11. A method as recited in claim 9, wherein the cooling comprises cooling the glass sheet over at least a first cooling segment having a first slope and a second contiguous cooling segment having a second slope, the first slope being different from the second slope.

12. A method as recited in claim 11, wherein the first cooling segment precedes the second cooling segment, and the first slope is greater than the second slope.

13. A method as recited in claim 11, wherein the first cooling segment precedes the second cooling segment, and the first slope is less than the second slope.

14. A method as recited in claim 11, wherein the slope of at least one of the cooling segments is non-linear.

15. A method as recited in claim 1 wherein the pre-selected level of curtain warp is approximately 250 microns.

16. A method of fabricating glass sheets comprising providing one or more substantially non-linear cooling sequences over a glass transition temperature range of the glass of the glass sheet and using the one or more sequences to obtain modeled stress data for a glass sample that represents at least a portion of the sheet.

17. A method as recited in claim 16, wherein the method comprises selecting one of the one or more substantially non-linear cooling sequences which results in substantially no compression stress in the glass sample or results in tension stress in the glass sample in a desired zone.

18. A method as recited in claim 17, wherein the selected substantially non-linear cooling sequence comprises a first cooling segment having a first slope and a second contiguous cooling segment having a second slope, the first slope being different from the second slope.

19. A method as recited in claim 18, wherein the first cooling segment precedes the second cooling segment, and the first slope is greater than the second slope.

20. A method as recited in claim 18, wherein the first cooling segment precedes the second cooling segment, and the first slope is less than the second slope.

21. A method as recited in claim 18, wherein the slope of at least one of the cooling segments is non-linear.

22. A method as recited in claim 17, wherein the method comprises employing the selected substantially non-linear cooling sequence in fabricating glass sheets.

23. A method as recited in claim 22, wherein the fabricated glass sheets exhibit no more than a pre-selected level of curtain warp.

24. A method as recited in claim 16 wherein a plurality of substantially non-linear cooling sequences is provided iteratively.

25. A method of fabricating glass sheets using a fusion forming apparatus that has a root, said glass having a glass transition temperature range (GTTR) and said method comprising applying a cooling pattern to the glass as a function of distance from the root and insuring that the cooling pattern includes at least one non-linearity in the glass' GTTR which is sufficient to result in glass sheets whose curtain warp is below a designated level.

26. A method as recited in claim 25, wherein the at least one non- linearity includes a non-linearity which produces a tension band or a band of substantially zero compression in the region of the lower temperature end of the GTTR.

27. A method as recited in claim 25, wherein a slope of the cooling is variable within the cooling pattern.

28. A method as recited in claim 27, wherein the cooling pattern comprises a plurality of cooling segments.

29. A method as recited in claim 28, wherein each of the plurality of cooling segments has a linear slope and at least one the slopes is different from at least one other slope.

30. A method as recited in claim 28, wherein at least one of the plurality of cooling segments has a non-linear slope.