Patent Abstract:
A method for glass-blowing on a microscopic level includes the steps of defining a plurality of microholes in a wafer, disposing a sheet of thermally formable material onto the wafer covering the microholes, heating the sheet of thermally formable material until a predetermined degree of plasticity is achieved, applying self-induced fluidic pressure by expansion of the heated trapped gas in the microholes to the sheet of thermally formable material, while the sheet is still plastic, and simultaneously forming a plurality of blown micro-objects in the sheet on the wafer by means of continued application of pressure for a predetermined time.

Full Description:
RELATED APPLICATIONS 
     The present application is related to U.S. Provisional Patent Application Ser. No. 60/915,904, filed on May 3, 2007, which is incorporated herein by reference and to which priority is claimed pursuant to 35 USC 119. 
    
    
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The invention relates to a method for shaping glass on a microscopic scale utilizing self-inflation. 
     2. Description of the Prior Art 
     Glass blowing is an art that dates back over 2000 years. Today, glass blowing is used in a wide array of applications, including scientific glassware, optical components, consumer glass containers, and visual arts. Although blow-molding techniques are used in the glass industry to automate the fabrication of bottles and other containers, many fine glass products are still shaped one at a time by glass blowers. 
     The property that enables the successful shaping of glass is that its viscosity is highly dependent on the temperature. In order to shape glass it needs to be heated above its softening point, i.e., the temperature at which glass has a viscosity of 10 6.6  Pascal-seconds (Pa-s) (about 800° C. for borosilicate glass). In conventional glass blowing, a gob of glass is first heated inside a furnace. The gob is then removed from the furnace and blown into desired shapes. Often the heating and blowing steps are repeated multiple times. Once the glass is shaped, it is usually annealed to remove stresses that developed during the blowing. The original implementation of micro-glass blowing was a direct adaptation of conventional glass blowing techniques on a microscale, i.e., to bond a glass wafer to a through-etched silicon wafer, heat the bonded wafers, and directly apply fluidic pressure through the etched holes in order to blow spheres—described in US Patent Application Publication 2007/0071922. 
     Microspheres have been fabricated in the past using different fabrication methods. For example, see: R. Cook, “Creating Microsphere Targets for Inertial Confinement Fusion Experiments”, Energy &amp; Technology Review, pp. 1-9, April 1995; R. Dagani, “Microspheres Play Role in Medical, Sensor, Energy, Space Technologies”, Chemical and Engineering News, pp. 33-35, December 1994. However, previously fabricated microspheres are not attached to a substrate and can only be filled with certain light gases (e.g. hydrogen) through diffusion. 
     BRIEF SUMMARY OF THE INVENTION 
     Glass blowing techniques can normally only be used on a macroscopic scale, and the glass products have to be shaped one at a time. We here disclose and demonstrate how multiple micro-glass-spheres can be formed simultaneously on a silicon substrate. A thin sheet of glass is first bonded to an etched wafer. The sample is then heated inside a furnace above the softening point of glass, and due to the expansion of the trapped gas the glass is blown into spherical shapes. Other alternative ways of shaping the glass are also included with the scope and spirit of the invention. The capability to blow glass on a wafer level enables several applications, e.g. micro-lenses and small gas confinement chambers. Potentially this technology can also be used for drug delivery and diagnostic devices, as well as other biomedical applications. It must be understood that the term “wafer” is or can be used interchangeably with the term “chip” throughout this specification as appropriate. In general, a wafer may include a multiplicity of chips or be diced into separate chips. A chip may also include a plurality of spheres and need not be a considered as restricted to carrying a single sphere or micro-object included on it. Additionally, while spherical shapes are considered for illustrative purposes, non-spherical shapes can also be fabricated by applying blowmolding techniques. Furthermore, cylindrical (in-plane) micro-glass channels can be achieved by defining narrow etched trenches in the silicon wafer. 
     Thus, the illustrated embodiment is particularly directed to glass blowing on a microscopic level, glass blowing compatible with microfabrication technologies, glass blowing on a wafer level, a method for fabricating microspheres or other micro-glass shapes, simultaneous manufacturing of numerous micro-structures on a chip, an ability to simultaneously fill multiple glass shells with gases and other substances 
     This disclosure introduces fabrication processes where glass is blown on a wafer level allowing thousands of glass parts to be built simultaneously. The presented micro glass blowing also opens opportunities for integration with electrical and mechanical components on a chip using conventional microfabrication techniques. The illustrated embodiment of the fabrication process was developed for a micromachined implementation of a nuclear magnetic resonance gyroscope (NMRG), where a spherical gas confinement chamber is preferred in order to minimize the self-magnetization of the atoms. Although no previous micro-NMRGs have been reported, large NMRGs built around traditionally blown glass spheres have been demonstrated in the past. 
     Many other novel applications may be enabled by this new fabrication technique, including microscopic spherical gas confinement chambers, complex three-dimensional microfluidic networks for gas analyzers or miniature drug delivery systems, spacers and hermetic enclosures for wafer-level packaging, micro discharge lamps and plasma light sources, and micro-optical components (e.g. mass-produced microscopic glass lenses). 
     The illustrated embodiments thus include a method for glass-blowing on a microscopic level comprising the steps of defining a plurality of blind microholes in a wafer; disposing a sheet of thermally formable material onto the wafer covering the microholes to trap a gas in the microholes; heating the sheet of thermally formable material until a predetermined degree of plasticity is achieved; applying thermally generated pressure arising from the thermal expansion of the trapped gas in the microholes to the sheet of thermally formable material, while the sheet of glass is plastic; and simultaneously forming a plurality of blown micro-objects in the sheet on the wafer by means of continued application of thermally generated pressure for a predetermined time. 
     The step of defining the microholes comprises etching the microholes using deep-reactive ion etching (DRIE). However, it must be understood that the microholes may be made using any methodology now known or later devised, such as etching by either wet or dry etchants, micromechanical drilling, laser etching, microelectromachining and the like. 
     The step of defining the microholes comprises etching the microholes using wet etchants. 
     The step of defining the microholes comprises etching the microholes using any currently known or future discovered means of etching. 
     The step of disposing a sheet of thermally formable material comprises bonding the thermally formable material to the wafer using anodic bonding to seal the plurality of microholes. 
     The step of disposing a sheet of thermally formable material comprises bonding the thermally formable material to the wafer using any bonding methods currently known or future discovered. 
     The step of disposing a sheet of thermally formable material comprises bonding borosilicate glass to the wafer. 
     The step of disposing a sheet of thermally formable material comprises bonding the thermally formable material inside a controlled pressure environment so that the volume of the corresponding micro-object formed by the thermally generated pressure can be accurately controlled. 
     The illustrated embodiments of the method further comprises the step of fabricating integrated electrical and mechanical components on or into the wafer. 
     The illustrated embodiments of the method further comprises the step of disposing a gas-source material in a solid state in the micro-objects and heating the gas-source material to produce a vapor inside the micro-objects. 
     The illustrated embodiments of the method further comprises the step of sealing the micro-objects by bonding a layer to the backside of the wafer. 
     The step of simultaneously forming a plurality of blown micro-objects in the sheet on the wafer by means of continued application of pressure for a predetermined time comprises blowmolding the micro-objects into a mold. 
     The step of simultaneously forming a plurality of blown micro-objects in the sheet on the wafer by means of continued application of pressure for a predetermined time comprises controlling the pressure of the surrounding environment so that the volume of the corresponding micro-object formed by the thermally generated pressure can be accurately controlled. 
     The step of simultaneously forming a plurality of blown micro-objects in the sheet on the wafer by means of continued application of pressure for a predetermined time comprises forming a hollow substantially spherical micro-object or hemispherical micro-object. 
     The step of simultaneously forming a plurality of blown micro-objects in the sheet on the wafer by means of continued application of pressure for a predetermined time comprises forming hollow substantially cylindrical micro-channels. 
     The step of defining the plurality of blind microholes into the wafer further comprises defining an enlarged volume chamber within each blind microhole so that the corresponding micro-object formed by the thermally generated pressure is increased in volume as compared to the micro-object formed by the thermally generated pressure in the corresponding microhole without the enlarged chamber formed therein. 
     Each microhole has an opening adjacent to the sheet communicating with an interior of the micro-object when the micro-object is formed. The opening has a reduced diameter r 0  compared to remaining portions of the microhole such that sphericity of the micro-object formed in the sheet as determined by the ratio of the height of the micro-object to its radius tends toward 1. The reduced diameter opening is of the order of a few microns. 
     In one embodiment the wafer is comprised of two layers bonded together, namely a top layer having the microholes defined therethrough with the reduced diameter opening and a bottom layer having blind microholes defined therein with a larger diameter than the opening aligned with the microholes in the top layer, and where disposing the sheet of thermally formable material onto the wafer covering the microholes to trap a gas in the microholes comprises disposing the sheet of thermally formable material onto the top layer. 
     The illustrated embodiment includes an apparatus in which some of the foregoing embodiments of the method is practiced. 
     The illustrated embodiment further includes the fabricated micro-objects which are made from some of the foregoing embodiments of the method. 
     While the apparatus and method has or will be described for the sake of grammatical fluidity with functional explanations, it is to be expressly understood that the claims, unless expressly formulated under 35 USC 112, are not to be construed as necessarily limited in any way by the construction of “means” or “steps” limitations, but are to be accorded the full scope of the meaning and equivalents of the definition provided by the claims under the judicial doctrine of equivalents, and in the case where the claims are expressly formulated under 35 USC 112 are to be accorded full statutory equivalents under 35 USC 112. The invention can be better visualized by turning now to the following drawings wherein like elements are referenced by like numerals. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIGS. 1   a - 1   e  is a diagram of the steps of a method wherein micro-objects are made without the assistance of external blowing or pressure, but are thermally self-inflated or blown. 
         FIGS. 2   a - 2   d  is a diagram of the steps of a method to fabricate micro-objects in which larger or highly spherical shapes can be made. 
         FIG. 3  is a cross-sectional diagram of a substantially hemispherical micro-object fabricated with a wafer according to the invention. 
         FIGS. 4   a - 4   h  is a diagram of the steps of a method wherein micro-objects are thermally self-inflated or blown and then later filled with an alkali, gas and/or other substance. 
         FIG. 5  is a graph of the estimated height of a blown structure or micro-object, as a function of the radius of the undeformed membrane, r 0 . 
         FIG. 6  is a graph of the estimated sphericity of a blown structure or micro-object, i.e., the ratio between the height and the diameter of the hollow semisphere, as a function of the radius of the undeformed membrane, r 0 . 
         FIG. 7  is a diagram of an embodiment wherein two wafers are employed to fabricate a substantially spherical micro-object with a size and volume that can be defined independently of the radius of the undeformed membrane, r 0 . 
         FIG. 8  is a graph of microsphere estimated height verse estimated blow up time required to blow uniformly heated hollow glass semispheres at 850° C. 
         FIG. 9  is a diagram illustrating the variables which parameterize wall thickness thinning in a microsphere. 
         FIG. 10  is a microphotograph of a microsphere fabricated according to the invention with scanning electron microscope insets showing wall thickness and structure in two locations with are included with solid outlined boxes. 
     
    
    
     The invention and its various embodiments can now be better understood by turning to the following detailed description of the preferred embodiments which are presented as illustrated examples of the invention defined in the claims. It is expressly understood that the invention as defined by the claims may be broader than the illustrated embodiments described below. 
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     The illustrated embodiments of the invention disclosed below provide a way of simultaneously forming multiple microscopic glass components on a wafer. These glass-structures are orders of magnitude smaller than what can be achieved using traditional glass blowing techniques. In the illustrated embodiment, the glass spheres are attached to a wafer, allowing for integration with traditional micro-fabrication techniques. Furthermore, the glass structures can be filled with gaseous, liquid, and/or solid materials post fabrication. 
     The illustrated embodiment of the invention satisfies the need to implement a microscopic gas confinement chamber. Many specific applications for such a chamber can be considered, e.g. nuclear magnetic resonance gyroscopes, microlamps, and hydrogen capsules for H-vehicles. Other possible applications include micro-lenses, optical switches, laser fusion targets, magnetic shielding when a shielding material is applied on the inside/outside of the sphere, as well as lab-on-a-chip, drug delivery systems, medication capsules, and other biomedical devices. 
     In the preferred fabrication process as depicted in  FIGS. 1   a - 1   e  a silicon wafer  10  is first patterned with a layer of AZ4620 photoresist  12 . Cylindrical cavities  14  are then etched in the wafer using deep-reactive ion etching (DRIE). The photoresist  12  is removed with acetone and a thin sheet of borosilicate glass  16  (e.g. Pyrex 7740) is anodically bonded to the top surface of wafer  10 , covering the openings to etched cavities  14 . Once bonded, the glass  16  may also be ground and polished if thinner cross sections or particular surface finishes are desired. Next the glass covered wafer  10  of  FIG. 1   d  is placed inside a furnace at a temperature above the softening point of the glass  16 . Since the pressure inside the sealed cavities  14  increases when the wafer  10  is heated, the glass  16  will deform into spherical shapes  18 , as illustrated in  FIG. 1   e.    
     Once the glass  16  is shaped, the backside  20  of the wafer  10  can be etched as shown in  FIG. 4   e  described below to allow for filling of various gases or other substances into spheres  18 . The backside  20  can then be resealed using conventional wafer bonding techniques. Etching of the backside  20  is also necessary if the process is used for creating micro-lenses, although the backside would naturally not be resealed in this case since an optical path would be needed between the two sides of the lens. 
     The fabrication method illustrated in  FIGS. 1   a - 1   d  constitute the foundation of the micro glass blowing process and define the shape and size of the glass structures  18 . While these steps are usually included in the fabrication process, additional steps can be added as needed to suit a particular application. For many applications, e.g. micro-lenses and gas confinement chambers, it is necessary to etch the backside of the silicon wafer  10  after the glass spheres  18  have been formed. For example,  FIGS. 4   a - 4   h  illustrate how a gas confinement chamber can be fabricated by etching the backside of the silicon wafer  10  in order to be able to fill the glass sphere  18  with an alkali metal and/or buffer gas. The steps illustrated in  FIGS. 4   a - 4   c  are the same as those in  FIGS. 1   c - 1   e  respectively (the steps in  1   a - 1   b  were omitted from this illustration, but would be included in the actual fabrication). In the step illustrated in  FIG. 4   d , the wafer  10  is placed in a temporary holder  22  in order to protect the glass spheres  18 . The backside  20  is then patterned and etched in the step  FIG. 4   e , using for example DRIE. For illustrative purposes, a filling technique similar to that described in S. Knappe, et al., Opt. Lett. 30 (2005) 2351-2353 is used. It is to be understood that any filing technique now known or later devised could be substituted. BaN 6  and  87 RbCl are placed inside a small glass ampoule  24  with a 5 mm long nozzle  26  of 700 μm diameter. The wafer  10  with the glass-blown cell or sphere  18  is placed inside a vacuum chamber and the ampoule  24  is aligned with the opening  14 . Next, the ampoule  24  is heated in order to react the compounds, as shown in the step  FIG. 4   f  of the fabrication process. Since the vapor pressure of rubidium is higher than that of Ba and Cl, a fairly pure beam of  87 Rb emerges from the ampoule  24  and is deposited into the sphere  18 . The nitrogen produced during the reaction is pumped away. The vacuum chamber inside which the filling is performed (not shown) is then filled with the desired combination of buffer gases, here a mixture consisting of Xe in natural isotopic abundance and N 2 . The backside  20  is then sealed by anodic bonding of a glass wafer or sheet  28  in the step of  FIG. 4   g . Finally, the wafer  10  is taken out from the vacuum chamber and the temporary holder  22  is removed from the wafer  10  as shown in the step of  FIG. 4   h . The materials used to fill sphere  18  have been described here only for the purposes of illustration and it is to be expressly understood that any materials and gases may be substituted as desired. 
     One alternative option is to fill the cavities  14  with the desired substances before the glass  16  is bonded in the step of  FIG. 1   d  (or  FIG. 4   b ). If this is done, the etch, filling, and resealing steps in  FIGS. 4   d - 4   g  would not be required. However, certain light gases may diffuse through the glass  16  when the glass covered wafer  10  is heated inside a furnace. Furthermore, some substances may vaporize and Increase the pressure Inside the etched cavity  14  more than desired. For certain substances, an additional filling step (before the step of  FIG. 1   d ) is a preferred option instead of filling cavities  14  and spheres  18  from the back with substances post-fabrication. 
     Another embodiment of the process is to etch non-cylindrical cavities in the wafer. For example, if a trench of substantial length (not shown) is etched in the wafer and the thermally formable material is subsequently bonded and shaped by the thermally generated pressure, the micro-structure  18 ′ would assume the shape of a cylindrical channel with its axial direction parallel to the wafer plane (or out-of-plane relative the sketch in  FIG. 3 ). By defining a network of connecting trenches in the wafer, a complex three-dimensional network of micro-glass channels can be obtained. 
     Yet another embodiment of the process is to use a mold to shape non-spherical structures  18 ′. For example, a wafer with predefined etched cavities (e.g. cubical molds) can be temporarily attached on top of the glass before the step in  FIG. 1   e . After the blowing in step in  FIG. 1   e , hollow cubical glass structures  18 ′ would now be obtained instead of hollow semispheres (not shown). Many other types of glass shapes can be made by employing this molding principle similar to conventional blow molding at macroscopic scales. 
     Using the processes in the above embodiments, multiple glass structures  18  can be batch fabricated simultaneously. The fabrication process also allows for potential integration of other electrical and mechanical components on the wafer  10  using conventional microfabrication techniques. If the wafer  10  needs to be diced, some care needs to be taken to assure that the glass  18  is not damaged. Several methods can be employed for this purpose, e.g. covering the wafer  10  with wax before the dicing. The wax can then be removed by heating the sample in a water bath. Alternatively the dicing can be performed before the glass is blown (between the steps in  FIGS. 1   d  and  1   e ). As was illustrated in the fabrication processes disclosed above, no external blowing needs to be involved in the fabrication of the glass spheres  18  although it may be included. Instead, the glass components  18  can be formed by themselves due to the increased pressure inside the sealed cavities  14 , which is understood to include the volume within the glass component  18 , when heated. An estimate of the pressure that develops inside the cavity can be obtained from the ideal gas law
 
PV=nRT  (1)
 
     where P is the pressure, V is the volume, n is the number of moles, R is the Boltzmann constant, and T is the temperature. Since n and R are both constants, the ideal gas law can also be written as P 1 V 1 /T 1 =P 2 V 2 /T 2 . 
     An estimate of the pressure that develops inside the cavity  14  before the glass  16  is deformed can be obtained from the ideal gas law (for constant volume): P=P i T f /T i  where T f  is the final temperature, T i  is the initial temperature, and P i  is the initial pressure. For example, inside a container that is initially at room temperature and atmospheric pressure, and is then heated to 1200 K, the pressure will increase to four atmospheres. 
     In order to control the size of the glass shapes, the pressure inside the sealed chambers  14  needs to be controlled. Large shapes may be obtained by either increasing the pressure or increasing the volume of the etched cavity  14 . While it is also possible to increase the temperature at which the glass shapes are formed, the range of usable temperatures is pretty narrow since the temperature needs to be just slightly above the softening point of glass. Thus, the pressure and volume of the etched cavity will affect the size of the shapes to a greater extent than the temperature. If a larger cavity  14  is etched, a larger glass bubble  18  can be blown due to the ideal gas law at an elevated but constant temperature when the glass volume begins to expand: V=V etched  P i /P f  where V is the total volume enclosed by both the etched cavity and the deformed glass, V etched  is the volume of the etched cavity only, P i  is the initial pressure at an elevated temperature (just before the glass starts to deform), and P f  is the final pressure once the bubble has been blown (˜1 atm if the self-inflation of the glass is performed in a furnace at atmospheric pressure). Making wafers  10  thicker and etching deeper cavities  14  is one way of achieving larger glass spheres. Alternatively the wafer  10  may be heated inside a vacuum furnace in order to amplify the pressure difference between the inside and outside of the etched cavity  14  by decreasing P f . Another option is to perform the anodic bonding inside a pressurized chamber, allowing precise control over P i . Yet another option is to fill the cavity  14  with a substance that vaporizes and increases the pressure inside the cavity  14  (i.e. increase P i ). Yet another mode to increase the volume of the etched cavity  14  is to use a 2-step DRIE process as illustrated in  FIGS. 2   a - 2   d . After the initial DRIE etch in  FIG. 2   a , the sidewalls are passivated and coated with a masking material. In  FIG. 2   b  the bottom of the cavities are then etched using either a dry or wet etchant. This will increase the etched volume for a certain depth, and thus enable the blowing of larger structures. Yet another alternative allowing for increased volume of the blown glass is shown in  FIG. 7 . Here two silicon wafers have been bonded (before the glass was bonded). The first wafer now defines the “base” of the sphere, and the second wafer defines the volume (V etched ). 
     The principles of the glass blowing processes described above are based on the free inflation and large deformation of an initially flat glass sheet at elevated temperatures. Thus, the modeling is related to that of biaxial inflation of viscoelastic membranes, commonly used for material characterization in the polymer industry. A few assumptions are made regarding the glass in order to model the fabrication process. At room temperature glass essentially behaves like an elastic solid, responding rapidly to applied stress. However at sufficiently high temperatures, stress is immediately relieved from the material due to the low viscosity of the glass. At high temperatures (and consequently low viscosities) glass can be modeled as a Newtonian fluid. Glass also has a viscoelastic region for viscosities between approximately 10 8  Pa-s and 10 13  Pa-s. In the fabrication processes described in the illustrated embodiments, the glass is shaped at temperatures between 850 and 900° C. The viscosity in this temperature region is less than 10 6  Pa-s for borosilicate glass. It is therefore assumed in the following that the glass can be modeled as an incompressible Newtonian fluid due to the low viscosity at the elevated temperatures. 
     In the illustrated embodiments the glass blowing takes place inside a furnace at atmospheric pressure, although this need not be required in all embodiments. When the wafers  10  are placed inside the furnace, the high temperature will cause the pressure to increase rapidly inside the sealed cavities  14  of the silicon wafer  10 . At the same time the viscosity of the glass  16  decreases and the glass sheet  16  starts to deform. The glass  16  will grow into a spherical shape due to the uniform pressure distribution. After a sufficiently long period of time the pressure inside the glass shells, hemispheres or spheres  18  will be almost equal to the atmospheric pressure inside the furnace and most of the stresses in the glass shells  18  will be relieved. Since the final pressure is approximately equal on the inside and the outside of the hollow semisphere  18  and the cavities  14  were sealed at atmospheric pressure, the ideal gas law yields the following relation between the initial volume V e  of the etched cavity  14 , and the volume of the blown glass shell  18 , 
     
       
         
           
             
               
                 
                   
                     V 
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                       V 
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                             T 
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                         - 
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                   ( 
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     where T f  is the furnace temperature and T s  is the temperature at which the cavities  14  etched in the silicon wafer  10  were sealed by the glass wafer  10 . Note that this equation only holds true in the illustrated embodiment when the etched cavities are sealed at the same pressure as the pressure inside the furnace in which the shaping of the glass is performed (here 1 atm). As was previously discussed, the bonding and/or furnace pressures may alternatively be individually controlled in order to provide better control over the size and volume of the glass structures. In this case the complete ideal gas law has to be considered: P s V e /T s =P f (V e +V g )/T f , where P s  and P f  are the pressure at which the etched cavities are sealed and the pressure at which the glass structures are shaped (e.g. inside a furnace), respectively. 
     From geometry considerations, the radius of curvature of the hollow glass semisphere  18  develops according to 
     
       
         
           
             
               
                 
                   
                     r 
                     g 
                   
                   = 
                   
                     
                       
                         h 
                         g 
                         2 
                       
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                         r 
                         0 
                         2 
                       
                     
                     
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                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         h 
                         g 
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
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     where h g  is the height of the glass semisphere  18  and it is assumed that the undeformed membrane was circular with a radius of r 0 . Note that the height of the glass  16  is measured from the bottom of the undeformed glass sheet  16  to the interior wall of the top of the blown glass shell  18 , as illustrated in  FIG. 3 . 
     By considering the ratio between the volume of the undeformed glass membrane, πr 0   2 δ 0 , and the approximate final volume of the glass shell  18 , 2πr g h g δ, and assuming that the glass  16  is incompressible, the thickness of the hollow semisphere  18  can be estimated as 
     
       
         
           
             
               
                 
                   δ 
                   = 
                   
                     
                       
                         r 
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                   ( 
                   4 
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     where δ 0  is the initial thickness before the deformation. However, in reality the thickness will vary slightly over the surface of the shell  18  with the smallest thickness at the top. 
     In the process that was illustrated in  FIGS. 1   a - 1   e  the etched cavity  14  is cylindrical and the blown glass shell  18  is spherical. Thus, their respective enclosed volumes are 
     
       
         
           
             
               
                 
                   
                     
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                         h 
                         e 
                       
                     
                   
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                   ⁢ 
                   and 
                 
               
               
                 
                   ( 
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                       . 
                     
                   
                 
               
               
                 
                   ( 
                   6 
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     By combining Equations (3) and (6), the final height of the semisphere  18  can be shown to develop as a function of the furnace temperature, the temperature at which the cavity  14  was sealed, and the depth and radius of the etched cavity  14  according to 
     a. 
     
       
         
           
             
               
                 
                   
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                   ) 
                 
               
             
           
         
       
     
     where V g =h e πr 0   2 (T f /T s −1) is obtained from Equations (2) and (5). While it is possible to shape glass over a wide range of temperatures, empirical trials show that if the temperature is lower than 800° C. it will take a long time for the glass spheres  18  to develop. Also, if the temperature is higher than 950° C. the spheres  18  tend to break due to the low viscosity at higher temperatures. The best shapes were obtained at temperatures between 850 and 900° C. when using Pyrex 7740 borosilicate glass. The height of the semisphere  18  as a function of the initial radius of the undeformed glass membrane (equal to the radius of the etched cavity) is plotted in  FIG. 5  for etch depths of 300, 500, 700, and 900 μm. Plots are shown for both 850° C. (solid) and 900° C. (dashed). 
     Note that the variation in height due to furnace temperature is relatively small in the region of 850-900° C. The radius of the etched cavity  14  has the largest influence on the final volume of the glass shell  18  due to the square of r 0  in Equation (5). 
     In certain applications a highly spherical shape is desired. For example in a nuclear magnetic resonance gyroscope, which is the application that the wafer-level glass blowing was initially developed for, a spherical gas confinement chamber reduces the self-magnetization of the confined atoms due to symmetry. Thus, a spherical chamber can potentially improve the performance of the inertial instrument. In order to make the shells as spherical as possible, the base radius at the bottom of the hollow semisphere  18  should be small. Therefore it is advantageous to use thick wafers and etch deep cavities (large h e ) instead of increasing the etched radius. The ratio between the height and the diameter of the blown semispheres  18 , i.e., the sphericity measured in percent, is shown in the graph of  FIG. 6  for different etch depths and as a function of the radius of the undeformed glass membrane. 
     Naturally a narrower opening, r 0 , gives a more spherical shape. But even for a fairly large radius of 200 μm the estimated ratio between the height and the diameter of the semisphere  18  is greater than 90%, as long as the etched cavity  14  is deeper than 500 μm, as can be seen in  FIG. 6 . An alternative process, which potentially allows for larger sphericity, is illustrated in  FIG. 7 . In this process two silicon wafers  10   a  and  10   b  are bonded. The first double-side polished (thin) wafer  10   a  will define the base of the hollow glass semisphere  18  and is etched all the way though with a small radius microhole  14 . In the second wafer  10   b  a large chamber or enlarged microhole  14 ′ is etched. Once etched, the silicon wafers  10   a  and  10   b  are bonded using, for example, a fusion bond process. Next, a thin glass wafer is anodically bonded to wafer  10   a  and the bonded wafers are then placed inside a furnace in order to blow the glass. In this process the radius of the base of the glass shell  18 , r 0 , can be designed to be much smaller than the radius of the cavity etched in wafer  10   b , r e . While it is here assumed that the cavity etched in wafer  10   b  is cylinder-shaped, only the volume matters and wafer  10   b  can readily be etched into any desired shape using either wet or dry etching. The final volume enclosed by the glass shell  18  is determined primarily by the volume of the cavity etched in wafer  10   b  and the sphericity is now independent of the microhole&#39;s  14 ′ radius, r e . By utilizing this process, r 0  can be made as small as a few microns, which in turn gives a ratio between the height and the diameter of the blown hollow semisphere  18  of close to 100%, and thus potentially an almost perfect sphere. 
     Consider the axis symmetric inflation of a thin circular membrane. Force equilibrium conditions lead to the following estimation of the pressure difference between the inside and outside of the resulting thin spherical membrane: 
     
       
         
           
             
               
                 
                   
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     P 
                   
                   = 
                   
                     
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         δ 
                       
                       
                         r 
                         g 
                       
                     
                     ⁢ 
                     σ 
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     where δ is the thickness of the membrane, r g  is the radius of the semisphere, and σ is the stress. A few assumptions were made during the derivations of this equation. First, the shell thickness is assumed to be much thinner than the radius of curvature, so stress gradients across the shell  18  can be ignored. Furthermore, the thickness of the inflated membrane is assumed to be uniform. While this is not quite true for the described glass blowing process, the above spherical shell equation can still be used to get an idea of the approximate blow-up time of the glass spheres  18 . 
     As was previously discussed, the properties of the heated glass  16  depend on the temperature. For low temperatures the glass  16  behaves like an elastic solid, but for higher temperatures viscoelastic models are normally used. At very high temperatures glass is modeled as a Newtonian fluid. The stress can be split into a viscoelastic part and a viscous part. The resistance to fast deformations is determined primarily by the viscous response. Now consider the top of the hollow semisphere  18 , where the flow is purely elongational due to the biaxial stretching of the membrane. For elongational flows of a Newtonian fluid the stress is given by σ=−3ηdε/dt, where η is the viscosity and dε/dt is the strain rate. The strain is ε=ln(δ/δ 0 ), and hence the stress can be written as 
     
       
         
           
             
               
                 
                   σ 
                   = 
                   
                     
                       - 
                       3 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     η 
                     ⁢ 
                     
                       
                         ⅆ 
                         
                             
                         
                       
                       
                         ⅆ 
                         t 
                       
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           ln 
                           ⁢ 
                           
                             δ 
                             
                               δ 
                               0 
                             
                           
                         
                         ) 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     In order to estimate the time required to shape the glass  16 , Equations (8) and (9) are combined. The height of the glass shell, h g , now develops according to 
     
       
         
           
             
               
                 
                   
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     P 
                   
                   = 
                   
                     24 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     η 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       r 
                       0 
                       2 
                     
                     ⁢ 
                     
                       δ 
                       0 
                     
                     ⁢ 
                     
                       
                         h 
                         g 
                         2 
                       
                       
                         
                           ( 
                           
                             
                               r 
                               0 
                               2 
                             
                             + 
                             
                               h 
                               g 
                               2 
                             
                           
                           ) 
                         
                         3 
                       
                     
                     ⁢ 
                     
                       
                         ⅆ 
                         
                           h 
                           g 
                         
                       
                       
                         ⅆ 
                         t 
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     where ΔP=P i −P o  is the pressure difference between the inside, P i , and outside, P o , of the shell  18 . In the fabrication process illustrated in  FIGS. 1   a - 1   e , P o  is equal to the furnace pressure (1 atm). The pressure inside the glass shell  18  depends on the furnace temperature as well as the time-dependent height of the semisphere  18 . The magnitude of this pressure was derived from the ideal gas law and the geometry considerations above as 
     
       
         
           
             
               
                 
                   
                     P 
                     i 
                   
                   = 
                   
                     
                       
                         P 
                         s 
                       
                       ⁢ 
                       
                         
                           T 
                           f 
                         
                         
                           T 
                           s 
                         
                       
                     
                     
                       1 
                       + 
                       
                         
                           
                             h 
                             g 
                           
                           
                             6 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               r 
                               0 
                               2 
                             
                             ⁢ 
                             
                               h 
                               e 
                             
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               h 
                               g 
                               2 
                             
                             + 
                             
                               3 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 r 
                                 0 
                                 2 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     where P s  is the pressure at which the cavities  14  etched in the silicon wafer  10  were sealed by the glass wafer (assumed to be 1 atm). It was also assumed that the glass membrane will not significantly deform until the final temperature has been distributed uniformly throughout the wafer, and thus the ideal gas law can be applied. This assumption was based on the fact that the wafers are small and quickly positioned inside the furnace and should therefore heat fairly uniformly as well as rapidly. While this assumption does not quite hold true in reality, it is sufficient for the rough estimations of the order of magnitude of the blow-up time presented here. 
     As described by equations (10) and (11), the pressure difference, ΔP, increases rapidly to a few atmospheres when the wafers are placed inside the furnace. As the glass shell  18  grows, the pressure inside the shell  18  will decrease until it is almost equal to the pressure inside the furnace (1 atm). After a sufficient period of time, the pressure difference will be close to zero. 
     The plot in  FIG. 8  was obtained from equations (10) and (11). The height of the hollow glass semisphere  18  is shown for etch depths of 300, 500, 700, and 900 μm. It was assumed that the etched radius, r 0 , was 200 μm, the initial glass thickness, δ 0 , was 100 μm, and the viscosity of glass, η, was 10 6  Pa-s (approximate viscosity of borosilicate glass at 850° C.). The blow-up time is on the order of 15 s. Since a few extra seconds need to be added to allow for the heating of the wafers, the time required to fully form the glass spheres  18  inside a furnace is estimated to be on the order of one minute. 
     In the discussion above it was assumed that the thickness of the glass  16  was uniform throughout the surface of the shell  18 . However, due to the viscous nature of the heated glass  16  this is not true. The top of the semisphere  18  will be slightly thinner than the parts closer to the base. An estimate of the nonuniform wall thickness of the shell can be derived 
     
       
         
           
             
               
                 
                   δ 
                   = 
                   
                     
                       
                         δ 
                         0 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               r 
                               0 
                               4 
                             
                             + 
                             
                               
                                 r 
                                 2 
                               
                               ⁢ 
                               
                                 h 
                                 g 
                                 2 
                               
                             
                           
                           
                             
                               r 
                               0 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   r 
                                   0 
                                   2 
                                 
                                 + 
                                 
                                   h 
                                   g 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                     2 
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     where a particle that was initially positioned at radius r before the circular membrane was deformed is considered. As the glass  16  is blown and forms a hollow semisphere  18 , this particle travels to a new position as shown in  FIG. 9 . Note that in the middle of the membrane, and thus the thickness of the top of the glass shell  18  is described by. δ=δ 0 (1+h g   2 /r 0   2 ) −2 . 
     Depending on the particular application of the glass structures, a nonuniform wall thickness may be more or less detrimental. For some applications this property can even be utilized, e.g., to create microlenses. The focal length of a glass shell due to the nonuniform wall thickness can be estimated from the lens makers&#39; equation 
     
       
         
           
             
               
                 
                   
                     1 
                     f 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           n 
                           g 
                         
                         - 
                         
                           n 
                           0 
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           1 
                           
                             R 
                             1 
                           
                         
                         - 
                         
                           1 
                           
                             R 
                             2 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     where R 1  and R 2  are the two different meridional radii of curvature, and n g  and n 0  are the refractive indices of the glass and the surrounding medium, respectively. 
     Consider again an example of an actual fabrication according to the process illustrated in  FIGS. 1   a - 1   e . The fabrication was performed using 2-inch diameter single crystal silicon and Pyrex 7740 wafers  10 . An array of cylindrical cavities  14  was first etched in the silicon wafer using deep reactive ion etching (DRIE). Structures have been successfully fabricated for etched diameters ranging from 100 to 1000 μm. The targeted depth of the etched cavities  14  varied from 300 to 800 μm. 
     Once the cavities were etched in the silicon wafer  10 , a 100 μm thin Pyrex 7740 sheet  16  was anodically bonded to the silicon wafer  10 . The bonding was done at atmospheric pressure on top of a hot plate set to 400° C. and using a voltage of 600 V. Next the wafers  10  were diced using a diamond saw. Optionally the wafer  10  can be diced after the blowing of the glass shells  18  but in order to avoid potential damages to the glass structures the dicing was here performed before the hollow glass semispheres  18  are blown. If the dicing is instead performed as the last fabrication step, some additional care needs to be taken in order to protect the fragile glass shells. 
     The wafers  10  were placed inside a furnace at a temperature of approximately 850° C. for about 3 minutes in order to shape the glass spheres. 
     As was previously discussed, an issue that potentially affects the final shape of the shells  18  in the illustrated embodiments the temperature used during the anodic bonding. The final height (and consequently radius) of the glass shells  18  depends on the temperature at which the etched chambers were sealed, T s . Above it has been assumed that is equal to room temperature. However, in order for this assumption to be valid, a sufficient force must be applied to the top electrode until the anodic bonding is completed to provide a temporary seal between the glass  16  and the silicon of wafer  10 . If the glass  16  and silicon wafers  10  are not perfectly sealed in this manner at room temperature, some air will escape from the etched cavities  14  when heated during the anodic bonding, leading to a higher and thus a smaller final height of the glass shells  18 . The anodic bonding can alternatively be performed inside a pressure chamber. By controlling both the temperature and the pressure during the anodic bonding, the final size of the glass shells  18  can be accurately predicted. Once the hollow glass spheres  18  are fabricated, a few optional fabrication steps may be required depending on the particular application. 
     For example, if the chambers need to be filled with gas or other substances, it may be necessary to open the backside  20  of the wafer  10 . The backside  20  can be patterned and etched using either wet or dry etchants to gain access to the hollow semispheres  18  (assuming the glass shells on the front side are protected). If double-side polished wafers are used, a rim can be maintained on the backside that will allow for resealing of the chamber using anodic bonding techniques. 
     Other additional processing steps may include applying an anti-relaxation coating, etching of the bulk glass to gain access to the silicon, and integration with other electrical and mechanical components. 
     The fabricated shells in the example above were covered with photoresist and diced at the center of the spheres  18  in order to be able to perform metrology. A scanning electron microscope image of the cross-section of one of the hollow semispheres is shown in  FIG. 10 . The shell  18  was fabricated using a 1 mm thick silicon wafer bonded to a 100 μm thin Pyrex 7740 wafer  10 , and was formed at 850° C. The cylinder-shaped etched cavity  14  is 750 μm deep and 500 μm in diameter. Table I shows a comparison between the experimental results and the values predicted by the presented analytical model, calculated using the equations above as specified in the table. 
     
       
         
               
             
               
               
               
             
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE I 
               
             
             
               
                   
               
               
                 COMPARISON BETWEEN THE GlASS BLOWING MODEL AND 
               
               
                 THE EXPERIMENTAL RESULTS 
               
             
          
           
               
                   
                 Chip 1 (h e  = 350 μm, r 0  = 375 μm) 
                 Chip 2 (h e  = 750 μm, r 0  = 250 μm) 
               
             
          
           
               
                 Parameter 
                 Equation 
                 Calculated 
                 Sphere 1 
                 Sphere 2 
                 Sphere 3 
                 Calculated 
                 Sphere 4 
                 Sphere 5 
                 Sphere 6 
               
               
                   
               
               
                 Glass height, h g   
                 (7) 
                 794 μm 
                 806 μm 
                 834 μm 
                 718 μm 
                 860 μm 
                 818 μm 
                 814 μm 
                 803 μm 
               
               
                 Glass radius, r g   
                 (3) 
                 486 μm 
                 520 μm 
                 540 μm 
                 479 μm 
                 466 μm 
                 431 μm 
                 439 μm 
                 436 μm 
               
               
                 Uniform thickness, δ 
                 (4) 
                  18 μm 
                 N/A 
                 N/A 
                 N/A 
                  7.8 μm 
                 N/A 
                 N/A 
                 N/A 
               
               
                 Thickness at top, δ 
                 (12)  
                     3.3 μm 
                  14 μm 
                  13 μm 
                  14 μm 
                     0.6 μm 
                     5.3 μm 
                     5.5 μm 
                     7.2 μm 
               
               
                 Thickness at side, δ 
                 (12)  
                  33 μm 
                  28 μm 
                  22 μm 
                  29 μm 
                  18 μm 
                  11 μm 
                  12 μm 
                  16 μm 
               
               
                   
               
             
          
         
       
     
     Two different chips from wafer  10  were diced and three glass shells  18  were measured on each chip. Chip  1  was fabricated from a 450 μm thick silicon wafer by etching 350 μm deep cavities with a radius of 375 μm. A 1-mm-thick wafer was instead used to fabricate Chip  2 , with an etch depth of 750 μm and a radius of 250 μm. The calculated height and radius agree with the experimental values in Table I. However, both equations (4) and (12) failed to predict the final thickness of the shells  18 . While the thickness was not quite uniform, the thickness variation was overestimated using equation (12). Instead the true glass thickness was somewhere in between the thicknesses predicted by the uniform and the nonuniform models. 
     It should be mentioned that two other variables may have affected the results in Table I. First, while great care was taken to attempt to dice the cross-sections in the middle of the spheres  18 , a slight offset from the center was inevitable. Therefore the actual height and radius of the glass spheres  18  may be slightly larger than the values displayed in Table I. In addition, the specified thickness of the Pyrex 7740 wafer was 100 μm±25 μm. This potential variation of 50 μm naturally leads to some discrepancies in the thickness results. The surface quality of both the inside and the outside of the side of the glass semisphere  18  (dashed area in  FIG. 10 ) was measured using an optical profiler (Hyphenated-Systems NanoScale 150OP). Although both surfaces were still relatively smooth, the surface roughness was greater on the outside surface. The specified initial surface roughness of the Pyrex 7740 wafers was &lt;10 Å. The average surface roughness after the spheres were formed was 2 nm on the inside surface and 9 nm on the outside. It is believed that the reason for this difference in surface roughness is that the inside surface was subjected to a uniform pressure during the blowing of the spheres, while the outside surface was directly exposed to the surrounding nitrogen gas flow and particulates inside the furnace. 
     Many alterations and modifications may be made by those having ordinary skill in the art without departing from the spirit and scope of the invention. Therefore, it must be understood that the illustrated embodiment has been set forth only for the purposes of example and that it should not be taken as limiting the invention as defined by the following invention and its various embodiments. 
     Therefore, it must be understood that the illustrated embodiment has been set forth only for the purposes of example and that it should not be taken as limiting the invention as defined by the following claims. For example, notwithstanding the fact that the elements of a claim are set forth below in a certain combination, it must be expressly understood that the invention includes other combinations of fewer, more or different elements, which are disclosed in above even when not initially claimed in such combinations. A teaching that two elements are combined in a claimed combination is further to be understood as also allowing for a claimed combination in which the two elements are not combined with each other, but may be used alone or combined in other combinations. The excision of any disclosed element of the invention is explicitly contemplated as within the scope of the invention. 
     The words used in this specification to describe the invention and its various embodiments are to be understood not only in the sense of their commonly defined meanings, but to include by special definition in this specification structure, material or acts beyond the scope of the commonly defined meanings. Thus if an element can be understood in the context of this specification as including more than one meaning, then its use in a claim must be understood as being generic to all possible meanings supported by the specification and by the word itself. 
     The definitions of the words or elements of the following claims are, therefore, defined in this specification to include not only the combination of elements which are literally set forth, but all equivalent structure, material or acts for performing substantially the same function in substantially the same way to obtain substantially the same result. In this sense it is therefore contemplated that an equivalent substitution of two or more elements may be made for any one of the elements in the claims below or that a single element may be substituted for two or more elements in a claim. Although elements may be described above as acting in certain combinations and even initially claimed as such, it is to be expressly understood that one or more elements from a claimed combination can in some cases be excised from the combination and that the claimed combination may be directed to a subcombination or variation of a subcombination. 
     Insubstantial changes from the claimed subject matter as viewed by a person with ordinary skill in the art, now known or later devised, are expressly contemplated as being equivalently within the scope of the claims. Therefore, obvious substitutions now or later known to one with ordinary skill in the art are defined to be within the scope of the defined elements. 
     The claims are thus to be understood to include what is specifically illustrated and described above, what is conceptionally equivalent, what can be obviously substituted and also what essentially incorporates the essential idea of the invention.

Technology Classification (CPC): 8