Abstract:
A thermoelastic device comprising an expansive element is disclosed. The expansive element is formed from a material, which is preselected on the basis that it has one or more of the following properties: a resistivity between 0.1 μΩm and 10.0 μΩm; chemically inert in air; chemically inert in the chosen ink; and depositable by CVD, sputtering or other thin film deposition technique.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
       [0001]    This is a Continuation of U.S. application Ser. No. 11/768875 filed on Jun. 26, 2007, which is a Continuation of U.S. application Ser. No. 10/854,269 filed on May 27, 2004, now issued U.S. Pat. No. 7,270,475 which is a Continuation of U.S. application Ser. No. 10/636,259 filed on Aug. 8, 2003 now issued Pat. No. 6,793,974, which is a Continuation of U.S. application Ser. No. 09/693,079 filed on Oct. 20, 2000, now issued U.S. Pat. No. 7,095,309 all of which are herein incorporated by reference. 
     
    
     CO-PENDING APPLICATIONS 
       [0002]    Various methods, systems and apparatus relating to the present invention are disclosed in the following co-pending applications filed by the applicant or assignee of the present invention on May 23, 2000: 
         [0000]                                                6,428,133   6,526,658   6,315,399   6,338,548   6,540,319       6,328,431   6,328,425   6,991,320   6,383,833   6,464,332       6,390,591   7,018,016   6,328,417   6,322,194   6,382,779       6,629,745   09/575,197   7,079,712   6,825,945   7,330,974       6,813,039   6,987,506   7,038,797   6,980,318   6,816,274       7,102,772   7,350,236   6,681,045   6,728,000   7,173,722       7,088,459   09/575,181   7,068,382   7,062,651   6,789,194       6,789,191   6,644,642   6,502,614   6,622,999   6,669,385       6,549,935   6,987,573   6,727,996   6,591,884   6,439,706       6,760,119   7,295,332   6,290,349   6,428,155   6,785,016       6,870,966   6,822,639   6,737,591   7,055,739   7,233,320       6,830,196   6,832,717   6,957,768   7,456,820   7,170,499       7,106,888   7,123,239   6,409,323   6,281,912   6,604,810       6,318,920   6,488,422   6,795,215   7,154,638   6,859,289       6,924,907   6,712,452   6,416,160   6,238,043   6,958,826       6,812,972   6,553,459   6,967,741   6,956,669   6,903,766       6,804,026   7,259,889   6,975,429                    
The disclosures of these co-pending applications are incorporated herein by cross reference.
 
       FIELD OF THE INVENTION 
       [0003]    The present invention relates to materials potentially suitable for use as the expansive element in thermoelastic design and to methods for ranking the potential relative suitabilities of those materials. 
         [0004]    The invention as developed originally as a means of identifying and ranking a range of materials that potentially may exhibit superior properties for use in the manufacture of microscopic thermal bend actuators for use in micro-electro mechanical systems (MEMS), and will be described hereinafter with reference to this field. However, it will be appreciated that the invention is not limited to this particular use and is equally applicable to macroscopic design even though the overall design considerations are vastly different and certainly less complex. 
       BACKGROUND OF THE INVENTION 
       [0005]    It is important to clarify that thermoelastic actuation is characterized using force, deflection and temperature as opposed to switching, which is characterized using deflection and temperature rise alone. Macroscopic thermoelastic actuators are typically used as switches that activate other more energy efficient actuation systems, however, microscopic thermoelastic actuators are an attractive actuation mechanism for a number of reasons. This includes the down scaling of certain physical phenomena. For example, it is possible to fabricate very thin films that decrease the thermal mass and minimize efficiency losses. Opposing gravitational and inertial forces become negligible on the microscopic scale. Other advantages include ease of fabrication (although more complex than simple electrostatic actuators) and the possibility of low voltage operation. Disadvantages include a low operational bandwidth determined by the thermal conductivities of substrate materials—this is more of an advantage for the current application allowing for rapid firing. 
         [0006]    A relatively diverse range of output force and deflection values can be obtained by altering actuator geometry. However, the fundamental operation of actuation is directly related to the mechanical and thermal properties of the component materials. Correct material selection in association with effective design can result in either a smaller or a more efficient actuator. Such an actuator increases wafer yield and is thus more commercially viable. A more efficient actuator may be battery powered increasing operation simplicity and negating the requirement for expensive voltage transformers. An increase in thermal efficiency improves the operational firing frequency, and decreases the possibility of thermal crosstalk. This is especially relevant for arrays of thermal actuators in a micro-cilia device. 
         [0007]    However, material selection for MEMS application is not straightforward. Firstly, published thin film properties can vary greatly due to different fabrication methods and difficulties associated with experimentally quantifying material properties on the microscopic scale. Secondly, certain thin films can only be fabricated with certain layer thicknesses because inherent stress can shatter or curl the substrate wafer. Thirdly, only certain materials can be used in the fabrication process at most fabs as the introduction of a new material can contaminate machinery. 
       Progress to Date 
       [0008]    Until recently, the only materials regularly used or considered for use in such applications were polysilicon, single crystal silicon. However, the applicant just previously made the surprising discovery that titanium nitride and titanium boride/diboride exhibited excellent properties relevant to this application. 
         [0009]    Realizing the breakthrough this surprising discovery signified, the applicant sought to try and identify possible alternatives in order to provide designers of thermoelastic systems with more choice and flexibility. However, given the lack of available data on their film properties for various materials and the fact that empirical testing with MEMS would be prohibitively expensive, there was clearly a need, or it was at least highly desirable to be able to determine a method of evaluating materials for this use based solely on the commonly available macro material properties. 
       SUMMARY OF THE INVENTION 
       [0010]    It is therefore an ultimate object of one aspect of this invention to identify a range of alternative materials that will potentially exhibit superior properties for use in thermoelastic design and of another aspect to provide a means of ranking the potential suitability of a given range of materials for this same use. 
         [0011]    According to an aspect of the invention there is provided a method for selecting a material for a thermal bend actuator of a printhead nozzlel, the method including the steps of: 
         [0012]    selecting the material from the group including suicides and carbides of titanium; borides, suicides, carbides and nitrides of tantalum, molybdenum, niobium, chromium, tungsten, vanadium, and zirconium; and 
         [0013]    selecting the material on the basis of ε, wherein ε is a dimensionless constant for the material in accordance with the formula 
         [0000]    
       
         
           
             ɛ 
             = 
             
               
                 E 
                  
                 
                     
                 
                  
                 
                   γ 
                   2 
                 
                  
                 T 
               
               
                 ρ 
                  
                 
                     
                 
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                 C 
               
             
           
         
       
     
         [0014]    wherein E is Young&#39;s module of the material; γ is the coefficient of thermal expansion; T is the maximum operating temperature, ρ is the density and C is the specific heat capacity. 
         [0015]    Other aspects are also disclosed. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0016]    Derivation of the dimensionless constant ε of the first aspect of the invention, together with sample applications and examples of derived values of this constant and other properties for a range of materials, will now be described in detail with reference to the accompanying drawings in which: 
           [0017]      FIG. 1  shows a schematic representation of a thermoelastic actuator; 
           [0018]      FIG. 2  shows a plot of longitudinal work versus heat energy for single material clamped/free titanium beam (length 20 μm, thickness 1 μm, width 5 μm); 
           [0019]      FIG. 3  shows a plot derived from  FIG. 2  of expansion efficiency versus temperature efficiency for a clamped/free titanium beam; and 
           [0020]      FIG. 4  shows a comparison of mechanical work versus the heat energy of thermoelastic actuator fabricated from Titanium and Silicon. 
       
    
    
     DETAILED DESCRIPTION 
       [0021]    A non-dimensionalized material actuation efficiency is presented that assesses the potential application of a material to thermoelastic design. The method is based on the material thermal and mechanical properties and assists in a structured approach of material selection for effective design. 
       The Material Actuation Efficiency 
       [0022]    Actuators are characterized by a combination of deflection, force and operation temperature in contrast to switches that are characterized by operation temperature and deflection alone. Fundamental thermoelastic design is characterized by the differential longitudinal expansion of two bonded layers. Thus, the expansion of isolated unbonded layers directly relates to global behavior. A single material beam is used here to illustrate the material actuation efficiency. The approach is straightforward and relates to general thermoelastic design. The derivation assumes that material properties are constant across the thermal range. 
         [0023]    Equations 1 to 3 are fundamental thermomechanical equations describing the behavior of simple single material beam subjected to a quantity of heat, Q as illustrated in  FIG. 1 . Equation 1 describes the extension, δL, of a free/free beam and equation 2 describes the reaction force, F, of a clamped/clamped beam. 
         [0000]      δL=γL 0 T  (EQ 1) 
         [0000]    Where: δL=extension of beam, L 0 =original length of beam, T=operation temperature (temperature rise), and γ=coefficient of thermal expansion of beam. 
         [0000]      F=AEγT  (EQ2) 
         [0000]    F=force exerted by beam expansion, A=cross sectional area of beam, E=Young&#39;s Modulus. 
         [0000]      Q=V ρCt   (EQ3) 
         [0000]    Where: Q=heat energy input, V=volume of beam, ρ=density, and C=specific heat capacity of beam. 
         [0024]    Potential mechanical work is given by equation 4 and is defined as the product of the clamped beam force, F, and free beam deflection, δL. The quadratic relationship between the heat input and output mechanical work for the simple monolithic beam is shown in  FIG. 18 . 
         [0000]      W=FδL  (EQ4) 
         [0000]    Where: W=mechanical work 
         [0025]    Equation 5 describes the non-dimensional thermoelastic actuation efficiency and is formulated as the quotient of the mechanical work and heat energy as described by equations 3 and 4. The efficiency is independent of geometry and is a primary indication of a material&#39;s potential application to thermoelastic design. The linear relationship between the actuation efficiency and material temperature for the simple beam is shown in  FIG. 3 . The graph indicates that high temperature operation is desirable for maximum efficiency. The plot is limited by the applicable operation temperature and therefore, different material plots are of different lengths. The assumption used in this text is that the operation temperature is the material melting point because it is indicative of the operable thermal range. Thus, the material actuation efficiency, ε, is defined as the actuation efficiency at the maximum operable temperature, T, of that material. The slope of the efficiency curve is a constant, m ε  and is defined in equation 6. The combination of ε and m ε  fully characterize a materials actuation characteristics non graphically. 
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       Material Selection 
       [0026]    Different thin film materials including materials with extreme properties (PTFE—high g, Diamond—high E) and compounds from all the major CVD groups including borides, silicides, nitrides and carbides is shown in Table 2. The efficiency values are scaled according to silicon efficiency values because the inclusion of scaled values greatly simplifies design equations described in the following text. The scaling or comparison of a material with respect to a reference material is an integral step in the material selection process. In addition, scaling also results in a more readable index as illustrated by the following comparisons. Silicon is chosen as the reference material because of its predominance in lithographic fabrication. 
         [0027]    Preliminary candidates for thermoelastic actuation can be selected according to efficiencies and slopes, however, it is important to note that two materials that have identical ε but differing m ε  will output different amounts of work for any constant geometry (see Comparison 1 below, different amounts of heat energy are also required). Three important design parameters are defined here as heat input, work output and volume. A design matrix can be constructed by varying each parameter and can then be used to select suitable materials. The following comparisons are used to assemble the design matrix. 
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 2 
               
             
             
               
                   
               
               
                 Material Properties. 
               
             
          
           
               
                   
                 g 
                   
                   
                 C 
                   
                   
                   
                   
                   
                   
                   
               
               
                   
                 10 −6 / 
                 E 
                 r 
                 J/ 
                 m e /m r,e   
                 O.T 
                 M.P. 
                 MN 
                 MN 
                 KXX 
                 R 
               
               
                 Material 
                 ° C. 
                 GPa 
                 kg/m 3   
                 kg ° C. 
                 ° C. −1   
                 ° C. 
                 ° C. 
                 O.T. 
                 M.P. 
                 W/m · K 
                 mWm 
               
               
                   
               
             
          
           
               
                 Aluminum 
                 23.1 
                 68.9 
                 2700 
                 897 
                 17.12 
                   
                 657 
                   
                 7.98 
                 231 
                 0.027 
               
               
                 Boron Carbide 
                 4.5 
                 454 
                 2520 
                 955 
                 4.31 
                   
                 2450 
                   
                 7.49 
                 35 
                 5e4 
               
               
                 Chromium diBoride 
                 11.1 
                 540 
                 5600 
                 690 
                 19.42 
                 1000 
                 2150 
                 13.78 
                 29.62 
                 32 
                 0.18 
               
               
                 Chromium diSilicide 
                 5.9 
                   
                 5600 
                   
                   
                 1150 
                 1560 
                   
                   
                   
                 0.8 
               
               
                 Chromium Carbide 
                 9.9 
                 385 
                 6680 
                 530 
                 12.02 
                 1100 
                 1895 
                 9.38 
                 16.16 
                 19 
                 0.75 
               
               
                 Chromium Oxide 
                 9.0 
                 102 
                 5210 
                 730 
                 2.45 
                 1000 
                 2603 
                 1.74 
                 4.52 
                 30 
                 13 
               
               
                 Copper 
                 16.5 
                 110 
                 8940 
                 386 
                 9.79 
                   
                 1085 
                   
                 7.53 
                 398 
                 0.017 
               
               
                 Gold 
                 14.2 
                 80 
                 19300 
                 129 
                 7.31 
                   
                 1064 
                   
                 5.52 
                 315 
                 0.023 
               
               
                 Hafnium Carbide 
                 6.3 
                 410 
                 12670 
                 190 
                 7.63 
                 600 
                 3930 
                 3.24 
                 21.25 
                 13 
                 0.4-0.6 
               
               
                 Hafnium diBoride 
                 7.6 
                   
                 11200 
                 300 
                   
                 1500 
                 3250 
                   
                   
                 51 
                 0.1 
               
               
                 Hafnium diSilicide 
                   
                   
                 8030 
                   
                   
                 1100 
                 1700 
               
               
                 Hafnium 
                 6.5 
                 424 
                 11940 
                   
                   
                   
                 3890 
                   
                   
                 8 
                 0.5 
               
               
                 Monocarbide 
               
               
                 Hafnium Nitride 
                 6.5 
                   
                 13,940 
                   
                   
                 500 
                 3300 
                   
                   
                 17 
                 32 
               
               
                 Molybdenum 
                 4.8 
                 343 
                 10200 
                 251 
                 3.48 
                   
                 2623 
                   
                 6.48 
                 138 
               
               
                 Molybdenum Boride 
                 5 
                 685 
                 7480 
                 530 
                 4.87 
                 1000 
                 2140 
                 3.46 
                 7.40 
                 27 
                 0.18 
               
               
                 Molybdenum 
                 6.7 
                 530 
                 9120 
                 315 
                 9.34 
                 500 
                 2500 
                 3.31 
                 16.56 
                 22 
                 0.57 
               
               
                 Carbide 
               
               
                 Molybdenum 
                 8.4 
                 450 
                 6240 
                 550 
                 10.44 
                 1700 
                 2050 
                 12.58 
                 15.17 
                 49 
                 0.7 
               
               
                 diSilicide 
               
               
                 Nickel 
                 13.4 
                 200 
                 8900 
                 444 
                 10.25 
                   
                 1455 
                   
                 10.58 
                 90.7 
               
               
                 Niobium diBoride 
                 8.6 
                 650 
                 7210 
                 420 
                 17.91 
                 850 
                 3000 
                 10.80 
                 38.10 
                 0.12 
                 17 
               
               
                 Niobium diSilicide 
                 8.5 
                   
                 5690 
                   
                   
                 900 
                 2050 
                   
                   
                   
                 0.5 
               
               
                 Niobium Carbide 
                 7.4 
                 450 
                 7820 
                 290 
                 12.26 
                 650 
                 3500 
                 5.65 
                 30.42 
                 14 
                 0.19 
               
               
                 PTFE 
                 220 
                 1.3 
                 2130 
                 1024 
                 32.54 
                   
                 200 
                   
                 4.62 
                 140 
                 10e22 
               
               
                 Silicon 
                 3.0 
                 162 
                 2330 
                 705 
                 1.00 
                 1410 
                 1410 
                 1 
                 1 
                 149 
                 2300 
               
               
                 Silicon Carbide 
                 4.7 
                 304 
                 3440 
                 669 
                 3.29 
                   
                 2700 
                   
                 6.30 
                 90 
                 0.5 
               
               
                 Tantalum Carbide 
                 6.7 
                 510 
                 14500 
                 190 
                 9.37 
                 650 
                 3900 
                 4.32 
                 25.93 
                 23 
                 0.35 
               
               
                 Tantalum diBoride 
                 8.5 
                 250 
                 12600 
                 250 
                 6.47 
                 850 
                 3090 
                 3.90 
                 14.17 
                 16 
                 0.14 
               
               
                 Tantalum diSilicide 
                 9.5 
                   
                 9080 
                 360 
                   
                 800 
                 2670 
                   
                   
                   
                 0.46 
               
               
                 Titanium Carbide 
                 7.4 
                 462 
                 4920 
                 480 
                 12.08 
                 700 
                 3160 
                 6.00 
                 27.08 
                 17.2 
                 1.55 
               
               
                 Titanium diBoride 
                 8.2 
                 575 
                 4450 
                 632 
                 15.51 
                 1400 
                 3253 
                 15.40 
                 35.78 
                 26.4 
                 0.13 
               
               
                 Titanium diSilicide 
                 10.7 
                 270 
                 4100 
                 480 
                 17.72 
                 1300 
                 1540 
                 16.34 
                 19.35 
                 46 
                 0.145 
               
               
                 Titanium Nitride 
                 9.4 
                 600 
                 5450 
                 636 
                 17.25 
                 500 
                 2950 
                 6.12 
                 36.10 
                 30 
                 1.35 
               
               
                 Tungsten Boride 
                 5.0 
                 790 
                 13100 
                 460 
                 3.70 
                 1000 
                 2365 
                 2.62 
                 6.20 
                 52 
                 0.19 
               
               
                 Tungsten Carbide 
                 5.2 
                 690 
                 15800 
                 200 
                 6.66 
                 500 
                 2780 
                 2.36 
                 13.13 
                 29 
                 0.2 
               
               
                 Tungsten diSilicide 
                 7.0 
                 300 
                 9750 
                 330 
                 5.15 
                 1200 
                 2165 
                 4.39 
                 7.91 
                 48 
                 33e10 
               
               
                 Vanadium diBoride 
                 7.6 
                 260 
                 5100 
                 670 
                 4.96 
                 600 
                 2430 
                 2.11 
                 8.54 
                 42 
                 0.13 
               
               
                 Vanadium Carbide 
                 6.7 
                 420 
                 5480 
                 530 
                 7.32 
                 600 
                 2730 
                 3.12 
                 14.18 
                 10 
                 0.59 
               
               
                 Vanadium diSilicide 
                 11.2 
                   
                 5100 
                   
                   
                 1000 
                 1700 
                   
                   
                 25 
                 0.66 
               
               
                 Vanadium Nitride 
                 8.1 
                 460 
                 6080 
                 630 
                 8.89 
                 450 
                 2170 
                 2.84 
                 13.68 
                 5.2 
                 0.85 
               
               
                 Zirconium Carbide 
                 6.3 
                 410 
                 6560 
                 250 
                 11.19 
                 600 
                 3440 
                 4.76 
                 27.31 
                 22 
                 0.42 
               
               
                 Zirconium diBoride 
                 5.9 
                 340 
                 6170 
                   
                   
                 1300 
                 3245 
                   
                   
                 58 
                 0.15 
               
               
                 Zirconium diSilicide 
                 8.7 
                 270 
                 4900 
                   
                   
                 1150 
                 1600 
                   
                   
                 15 
                 0.76 
               
               
                 Zirconium Nitride 
                 5.9 
                 500 
                 7350 
                 400 
                 6.68 
                 500 
                 2950 
                 2.37 
                 13.97 
                 10 
                 0.2-0.3 
               
               
                   
               
               
                 Where: 
               
               
                 γ = Coefficient of thermal expansion. 
               
               
                 E = Young&#39;s Modulus, 
               
               
                 ρ = density, 
               
               
                 C = specific heat capacity, 
               
               
                 O.T. = Oxidizing temperature, 
               
               
                 M.T. = Melting Temperature, 
               
               
                 m ε  = Efficiency Slope (normalized to Silicon m ε  value, normalized Silicon value m (r,ε)  = 0.8865e−06), 
               
               
                 ε c  = Material Index (normalized to Silicon ε value, normalized Silicon ε r  = 1.25e−03), 
               
               
                 KXX = thermal conductivity, and 
               
               
                 R = resistivity. 
               
             
          
         
       
     
       Comparison 1  
       [0028]    The mechanical work and heat input between a material and silicon for a constant beam volume is compared. Thus, Comparison 1 calculates the maximum possible relative work and associated relative heat input required due to a direct material substitution. Details of the comparison for different materials are included in Table 3 which shows that CVD ceramics are far superior actuator materials than silicon (Table 3 is formulated using melting point and Table 4 is formulated using oxidation temperature). Titanium nitride can output 159.3 times more the amount of mechanical work than silicon with only 4.41 times the amount of heat input. The factor in equation 8 and the scaled material efficiency ratio (as included in Table 2) repeatedly occur in the following comparisons illustrating the versatility of the method. 
         [0000]    
       
         
           
             
               
                 
                   
                     
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                       c 
                     
                     
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                               ρ 
                               c 
                             
                              
                             
                               C 
                               c 
                             
                              
                             
                               T 
                               c 
                             
                           
                           
                             
                               ρ 
                               r 
                             
                              
                             
                               C 
                               r 
                             
                              
                             
                               T 
                               r 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   EQ7 
                   ) 
                 
               
             
           
         
       
     
         [0029]    The r subscript denotes the reference material which is silicon in this case. The c subscript denotes the compared material. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Q 
                       c 
                     
                     
                       Q 
                       r 
                     
                   
                   = 
                   
                     ( 
                     
                       
                         
                           ρ 
                           c 
                         
                          
                         
                           C 
                           c 
                         
                          
                         
                           T 
                           c 
                         
                       
                       
                         
                           ρ 
                           r 
                         
                          
                         
                           C 
                           r 
                         
                          
                         
                           T 
                           r 
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   EQ8 
                   ) 
                 
               
             
           
         
       
     
       Comparison 2  
       [0030]    Different materials increase in temperature by different amounts when subjected to the same quantity of heat energy for a constant volume. The material volume is scaled relative to the silicon volume according to the constraints that the same amount of silicon heat energy is input to both actuators and the compared material attains its operational temperature. Thus, the actuation efficiency value remains unchanged because it is not a function of volume and the operable temperature is reached (as equation 5 shows). Comparison 2 represents the design case where heat and volume are critical factors. 
         [0031]    The scaled volume and output mechanical work are calculated using equations 9 and 10. The volume change is typically implemented by modifying one geometric dimension, i.e. length, width or thickness. Titanium nitride is capable of 36.1 times the amount of work that silicon is capable with the same heat energy input but with only 0.23 times the volume. Equation 9 is the inverse of equation 8 and equation 10 is simply the scaled efficiency number as included in Table 2. 
         [0000]    
       
         
           
             
               
                 
                   
                     Q 
                     r 
                   
                   = 
                   
                     
                       
                         V 
                         r 
                       
                        
                       
                         ρ 
                         r 
                       
                        
                       
                         C 
                         r 
                       
                        
                       
                         T 
                         r 
                       
                     
                     = 
                     
                       
                         Q 
                         c 
                       
                       = 
                       
                         
                           
                             
                               V 
                               c 
                             
                              
                             
                               ρ 
                               c 
                             
                              
                             
                               C 
                               c 
                             
                              
                             
                               T 
                               c 
                             
                           
                           ⇒ 
                           
                             
                               V 
                               
                                 ( 
                                 
                                   c 
                                   , 
                                   Qr 
                                 
                                 ) 
                               
                             
                             
                               V 
                               r 
                             
                           
                         
                         = 
                         
                           
                             
                               ρ 
                               r 
                             
                              
                             
                               C 
                               r 
                             
                              
                             
                               T 
                               r 
                             
                           
                           
                             
                               ρ 
                               c 
                             
                              
                             
                               C 
                               c 
                             
                              
                             
                               T 
                               c 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     EQ 
                      
                     
                         
                     
                      
                     9 
                   
                   ) 
                 
               
             
           
         
       
     
         [0032]    The first entry of the bracketed subscript in these equations refers to the material that the beam is constructed from. The second entry refers to the constraining variable for the described parameter. For example—W (c,Vc) =Mechanical work output from beam constructed of compared material with a volume of V c . 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       W 
                       
                         ( 
                         
                           c 
                           , 
                           Vc 
                         
                         ) 
                       
                     
                     
                       W 
                       
                         ( 
                         
                           r 
                           , 
                           Vr 
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           ɛ 
                           c 
                         
                          
                         
                           Q 
                           r 
                         
                       
                       
                         
                           ɛ 
                           r 
                         
                          
                         
                           Q 
                           r 
                         
                       
                     
                     = 
                     
                       
                         ɛ 
                         c 
                       
                       
                         ɛ 
                         r 
                       
                     
                   
                 
               
               
                 
                   ( 
                   EQ10 
                   ) 
                 
               
             
           
         
       
     
       Comparison 3  
       [0033]    The output mechanical work resulting from silicon heat energy for constant volume beams is compared. The operation temperature and efficiency value for the compared material changes. However, the new efficiency is easily calculated using a multiplicative ratio of the new and old operation temperatures because of the linear relationship between temperature and efficiency (as shown in  FIG. 3 ). The new operation temperature and work are given by equations 11 and 12. This comparison represents the design case where heat is a critical parameter. 
         [0034]    PTFE will melt when subjected to the input silicon heat value. Titanium disilicide outperforms titanium nitride mainly because of the higher computed operating temperature (Table 3). 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           Q 
                           r 
                         
                         = 
                           
                          
                         
                           
                             V 
                             r 
                           
                            
                           
                             ρ 
                             r 
                           
                            
                           
                             C 
                             r 
                           
                            
                           
                             T 
                             r 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           Q 
                           c 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               V 
                               c 
                             
                              
                             
                               ρ 
                               c 
                             
                              
                             
                               C 
                               c 
                             
                              
                             
                               T 
                               
                                 ( 
                                 
                                   c 
                                   , 
                                   Qr 
                                 
                                 ) 
                               
                             
                           
                           ⇒ 
                           
                             T 
                             
                               ( 
                               
                                 c 
                                 , 
                                 Qr 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             T 
                             r 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   ρ 
                                   r 
                                 
                                  
                                 
                                   C 
                                   r 
                                 
                               
                               
                                 
                                   ρ 
                                   c 
                                 
                                  
                                 
                                   C 
                                   c 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     EQ 
                      
                     
                         
                     
                      
                     11 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       W 
                       
                         ( 
                         
                           c 
                           , 
                           Vc 
                         
                         ) 
                       
                     
                     
                       W 
                       
                         ( 
                         
                           r 
                           , 
                           Qr 
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           ɛ 
                           
                             ( 
                             
                               c 
                               , 
                               Qr 
                             
                             ) 
                           
                         
                          
                         
                           Q 
                           r 
                         
                       
                       
                         
                           ɛ 
                           r 
                         
                          
                         
                           Q 
                           r 
                         
                       
                     
                     = 
                     
                       
                         
                           
                             T 
                             
                               ( 
                               
                                 c 
                                 , 
                                 Qr 
                               
                               ) 
                             
                           
                            
                           
                             ɛ 
                             2 
                           
                         
                         
                           
                             T 
                             c 
                           
                            
                           
                             ɛ 
                             r 
                           
                         
                       
                       = 
                       
                         
                           ( 
                           
                             
                               
                                 ρ 
                                 r 
                               
                                
                               
                                 C 
                                 r 
                               
                                
                               
                                 T 
                                 r 
                               
                             
                             
                               
                                 ρ 
                                 c 
                               
                                
                               
                                 C 
                                 c 
                               
                                
                               
                                 T 
                                 c 
                               
                             
                           
                           ) 
                         
                          
                         
                           
                             ɛ 
                             c 
                           
                           
                             ɛ 
                             r 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     EQ 
                      
                     
                         
                     
                      
                     12 
                   
                   ) 
                 
               
             
           
         
       
     
       Comparison 4  
       [0035]    The material volume is scaled with respect to the silicon volume according to the constraint that the compared material operation temperature and silicon work are maintained. Thus, if the silicon work value is less then the original work then the volume is scaled down. Otherwise the volume is increased as is the case for PTFE or amorphous Silicon Dioxide. The material actuation efficiency reoccurs in the calculations as an inverse as shown in equation 14 
         [0036]    Titanium nitride can output the same amount of work as silicon but with a volume that is less than two orders of magnitude smaller with an input heat energy that is less than an order smaller. 
         [0000]    
       
         
           
             
               ? 
             
              
             
               ? 
             
           
         
       
       
         
           
             
               ? 
             
              
             
               indicates text missing or illegible when filed 
             
           
         
       
     
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Q 
                       
                         ( 
                         
                           c 
                           , 
                           Vc 
                         
                         ) 
                       
                     
                     
                       Q 
                       
                         ( 
                         
                           r 
                           , 
                           Vr 
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           ɛ 
                           r 
                         
                          
                         
                           W 
                           r 
                         
                       
                       
                         
                           ɛ 
                           c 
                         
                          
                         
                           W 
                           r 
                         
                       
                     
                     = 
                     
                       
                         ɛ 
                         r 
                       
                       
                         ɛ 
                         c 
                       
                     
                   
                 
               
               
                 
                   ( 
                   EQ14 
                   ) 
                 
               
             
           
         
       
     
       Comparison 5  
       [0037]    The input heat energy required to output silicon mechanical work for constant volume beams is compared. The operation temperature and thus efficiency value for the compared material changes. The new efficiency can be calculated in an identical fashion to that described in comparison 3. The operational temperature and heat input values are calculated using equations 15 and 16. 
         [0038]    The table shows that titanium disilicide slightly outperforms titanium nitride whereas both PTFE and silicon dioxide will melt. The CVD ceramics are again shown to have the best performance. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           W 
                           r 
                         
                         = 
                           
                          
                         
                           
                             V 
                             r 
                           
                            
                           
                             E 
                             r 
                           
                            
                           
                             γ 
                             r 
                             2 
                           
                            
                           
                             T 
                             r 
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           W 
                           c 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               V 
                               c 
                             
                              
                             
                               E 
                               c 
                             
                              
                             
                               γ 
                               c 
                               2 
                             
                              
                             
                               T 
                               c 
                               2 
                             
                           
                           ⇒ 
                           
                             T 
                             
                               ( 
                               
                                 c 
                                 , 
                                 Wr 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             ( 
                             
                               
                                 γ 
                                 r 
                               
                               
                                 γ 
                                 c 
                               
                             
                             ) 
                           
                            
                           
                             
                               
                                 E 
                                 r 
                               
                               
                                 E 
                                 c 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   EQ15 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       Q 
                       
                         ( 
                         
                           c 
                           , 
                           Wr 
                         
                         ) 
                       
                     
                     
                       Q 
                       
                         ( 
                         
                           r 
                           , 
                           Wr 
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           ɛ 
                           r 
                         
                          
                         
                           W 
                           r 
                         
                       
                       
                         
                           ɛ 
                           
                             ( 
                             
                               c 
                               , 
                               Qr 
                             
                             ) 
                           
                         
                          
                         
                           W 
                           r 
                         
                       
                     
                     = 
                     
                       
                         
                           
                             ɛ 
                             r 
                           
                            
                           
                             T 
                             c 
                           
                         
                         
                           
                             ɛ 
                             c 
                           
                            
                           
                             T 
                             
                               ( 
                               
                                 c 
                                 , 
                                 Qr 
                               
                               ) 
                             
                           
                         
                       
                       = 
                       
                         
                           
                             
                               ɛ 
                               r 
                             
                              
                             
                               T 
                               c 
                             
                              
                             
                               γ 
                               c 
                             
                           
                           
                             
                               ɛ 
                               c 
                             
                              
                             
                               T 
                               r 
                             
                              
                             
                               γ 
                               r 
                             
                           
                         
                          
                         
                           
                             
                               E 
                               c 
                             
                             
                               E 
                               r 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     EQ 
                      
                     16 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 3 
               
             
             
               
                   
               
               
                 Design comparisons for materials included in Table 2. Comparisons are done using melting point temperature 
               
             
          
           
               
                   
                 Comparison 1 
                 Comparison 2 
                 Comparison 3 
                 Comparison 4 
                 Comparison 5 
               
             
          
           
               
                   
                 Constant 
               
             
          
           
               
                   
                 V 
                 Q 
                 V, Q 
                 W 
                 V, W 
               
             
          
           
               
                   
                   
                 W c / 
                 V (c,Qr) / 
                 W (c,Vc) / 
                   
                 W (c,Qr) / 
                 V (c,Wr) / 
                 Q (c,Vc) / 
                   
                 Q (c,Wr) / 
               
               
                   
                 Q c /Q r   
                 W r   
                 V (r,Qr)   
                 W (r,Vr)   
                 T (c,Qr)   
                 W (r,Qr)   
                 V (r,Vr)   
                 Q (r,Vr)   
                 T (c,Wr)   
                 Q (r,Wr)   
               
               
                   
                   
               
             
          
           
               
                 Aluminum 
                 0.69 
                 5.48 
                 1.46 
                 7.98 
                 &gt;Tmelt 
                   
                 0.183 
                 0.125 
                 280.79 
                 0.29 
               
               
                 Boron Carbide 
                 2.55 
                 19.06 
                 0.39 
                 7.49 
                 962.41 
                 2.94 
                 0.053 
                 0.133 
                 561.51 
                 0.58 
               
               
                 Chromium diBoride 
                 3.59 
                 106.23 
                 0.28 
                 29.62 
                 599.41 
                 8.26 
                 0.009 
                 0.0330 
                 208.73 
                 0.35 
               
               
                 Chromium Carbide 
                 2.90 
                 46.80 
                 0.35 
                 16.16 
                 654.20 
                 5.58 
                 0.021 
                 0.062 
                 277.16 
                 0.42 
               
               
                 Chromium Oxide 
                 4.27 
                 19.34 
                 0.23 
                 4.52 
                 608.98 
                 1.06 
                 0.052 
                 0.221 
                 592.32 
                 0.97 
               
               
                 Copper 
                 1.62 
                 12.18 
                 0.62 
                 7.53 
                 671.18 
                 4.66 
                 0.082 
                 0.132 
                 311.11 
                 0.46 
               
               
                 Gold 
                 1.14 
                 6.31 
                 0.87 
                 5.52 
                 930.29 
                 4.82 
                 0.159 
                 0.181 
                 423.90 
                 0.46 
               
               
                 Hafnium Carbide 
                 4.08 
                 86.81 
                 0.24 
                 21.25 
                 962.13 
                 5.20 
                 0.012 
                 0.047 
                 422.05 
                 0.44 
               
               
                 Molybdenum 
                 2.90 
                 18.78 
                 0.34 
                 6.48 
                 904.67 
                 2.23 
                 0.053 
                 0.154 
                 605.63 
                 0.67 
               
               
                 Molybdenum Boride 
                 3.66 
                 27.09 
                 0.27 
                 7.40 
                 584.23 
                 2.02 
                 0.037 
                 0.135 
                 411.42 
                 0.70 
               
               
                 Molybdenum Carbide 
                 3.10 
                 51.36 
                 0.32 
                 16.56 
                 806.23 
                 5.34 
                 0.019 
                 0.061 
                 349.05 
                 0.43 
               
               
                 Molybdenum diSilicide 
                 3.04 
                 46.09 
                 0.33 
                 15.17 
                 674.86 
                 4.99 
                 0.022 
                 0.066 
                 302.14 
                 0.45 
               
               
                 Nickel 
                 2.48 
                 26.26 
                 0.40 
                 10.58 
                 586.13 
                 4.26 
                 0.038 
                 0.095 
                 284.10 
                 0.48 
               
               
                 Niobium diBoride 
                 3.92 
                 149.44 
                 0.25 
                 38.10 
                 764.86 
                 9.71 
                 0.007 
                 0.026 
                 245.55 
                 0.32 
               
               
                 Niobium Carbide 
                 3.43 
                 104.26 
                 0.29 
                 30.42 
                 1021.31 
                 8.88 
                 0.010 
                 0.032 
                 342.97 
                 0.34 
               
               
                 PTFE 
                 0.19 
                 0.87 
                 5.31 
                 4.62 
                 &gt;Tmelt 
                   
                 1.152 
                 0.216 
                 &gt;Tmelt 
               
               
                 Silicon 
                 1.00 
                 1.00 
                 1.00 
                 1 
                 1410.00 
                 1.00 
                 1.000 
                 1 
                 1410.00 
                 1.00 
               
               
                 Silicon Carbide 
                 2.68 
                 16.91 
                 0.37 
                 6.30 
                 1006.42 
                 2.35 
                 0.059 
                 0.158 
                 657.00 
                 0.65 
               
               
                 Tantalum Carbide 
                 4.64 
                 120.27 
                 0.22 
                 25.93 
                 840.70 
                 5.59 
                 0.008 
                 0.038 
                 355.83 
                 0.42 
               
               
                 Tantalum diBoride 
                 4.20 
                 59.57 
                 0.24 
                 14.17 
                 735.28 
                 3.37 
                 0.017 
                 0.071 
                 400.60 
                 0.54 
               
               
                 Titanium 
                 1.70 
                 7.27 
                 0.59 
                 4.28 
                 984.12 
                 2.52 
                 0.138 
                 0.234 
                 619.87 
                 0.63 
               
               
                 Titanium diBoride 
                 3.95 
                 141.32 
                 0.25 
                 35.78 
                 823.54 
                 9.06 
                 0.007 
                 0.028 
                 273.81 
                 0.33 
               
               
                 Titanium diSilicide 
                 1.31 
                 25..32 
                 0.76 
                 19.35 
                 1176.90 
                 14.79 
                 0.040 
                 0.0517 
                 306.22 
                 0.26 
               
               
                 Titanium Nitride 
                 4.41 
                 159.36 
                 0.23 
                 36.10 
                 668.21 
                 8.18 
                 0.006 
                 0.0277 
                 233.83 
                 0.35 
               
               
                 Tungsten Boride 
                 6.15 
                 38.16 
                 0.16 
                 6.20 
                 384.36 
                 1.01 
                 0.026 
                 0.161 
                 383.10 
                 1.00 
               
               
                 Tungsten Carbide 
                 3.79 
                 49.80 
                 0.26 
                 13.13 
                 732.95 
                 3.46 
                 0.020 
                 0.076 
                 394.10 
                 0.54 
               
               
                 Tungsten diSilicide 
                 3.01 
                 23.80 
                 0.33 
                 7.91 
                 719.86 
                 2.63 
                 0.042 
                 0.126 
                 444.06 
                 0.62 
               
               
                 Vanadium diBoride 
                 3.58 
                 30.63 
                 0.28 
                 8.54 
                 677.83 
                 2.38 
                 0.033 
                 0.117 
                 439.34 
                 0.65 
               
               
                 Vanadium Carbide 
                 3.42 
                 48.53 
                 0.29 
                 14.18 
                 797.46 
                 4.14 
                 0.021 
                 0.071 
                 392.10 
                 0.49 
               
               
                 Vanadium Nitride 
                 3.59 
                 49.09 
                 0.28 
                 13.68 
                 604.67 
                 3.81 
                 0.020 
                 0.0731 
                 309.91 
                 0.51 
               
               
                 Zirconium Carbide 
                 2.44 
                 66.51 
                 0.41 
                 27.31 
                 1412.28 
                 11.21 
                 0.015 
                 0.0366 
                 422.05 
                 0.30 
               
               
                 Zirconium Nitride 
                 3.74 
                 52.32 
                 0.27 
                 13.97 
                 787.80 
                 3.73 
                 0.019 
                 0.0716 
                 408.09 
                 0.52 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 4 
               
             
             
               
                   
               
               
                 Design comparisons for material included in Table 2. Comparisons are done using oxidation temperature 
               
             
          
           
               
                   
                 Comparison 1 
                 Comparison 2 
                 Comparison 3 
                 Comparison 4 
                 Comparison 5 
               
             
          
           
               
                   
                 Constant 
               
             
          
           
               
                   
                 Q 
                 V, Q 
                 W 
                 V, W 
               
             
          
           
               
                   
                 V 
                 V (c,Qr) / 
                 W (c,Vc) / 
                   
                 W (c,Qr) / 
                 V (c,Wr) / 
                 Q (c,Vc) / 
                   
                 Q (c,Wr) / 
               
             
          
           
               
                   
                 Q c /Q r   
                 W c /W r   
                 V (r,Qr)   
                 W (r,Vr)   
                 T (c,Qr)   
                 W (r,Qr)   
                 V (r,Vr)   
                 Q (r,Vr)   
                 T (c,Wr)   
                 Q (r,Wr)   
               
               
                   
                   
               
             
          
           
               
                 Vanadium diBoride 
                 0.885 
                 1.864 
                 1.13 
                 2.10 
                 &gt;T oxid. 
                   
                 0.326 
                 0.475 
                 439.337 
                 0.648 
               
               
                 Vanadium Carbide 
                 0.752 
                 2.341 
                 1.33 
                 3.11 
                 &gt;T oxid. 
                   
                 0.26 
                 0.32 
                 392.1 
                 0.49 
               
               
                 Vanadium Nitride 
                 0.74 
                 2.1 
                 1.34 
                 2.83 
                 &gt;T oxid. 
                   
                 0.289 
                 0.353 
                 309.9 
                 0.513 
               
               
                 Zirconium Carbide 
                 0.425 
                 2.02 
                 2.35 
                 4.75 
                 &gt;T oxid. 
                   
                 0.301 
                 0.21 
                 422.05 
                 0.299 
               
               
                 Zirconium Nitride 
                 0.64 
                 1.5 
                 1.57 
                 2.36 
                 &gt;T oxid. 
                   
                 0.405 
                 0.423 
                 408.1 
                 0.518 
               
               
                   
               
             
          
         
       
     
       A Thermoelastic Actuator 
       [0039]    A hot arm/cold arm actuator is presented in  FIG. 1  to illustrate the results contained in Table 3. Only the steady state solution for a quantity of heat input to the heater is analyzed. The device comprises two identical material layers separated by air and connected to each other at the ends by a thermally non-conductive block. The force/deflection characteristics of the output mechanical power can be tuned by altering the separation between the two layers. A greater separation increases the transverse force but decreases deflection. 
         [0040]    Two actuators constructed from titanium and silicon are compared using graphed energy results in  FIG. 4 . Five design comparisons for Titanium are plotted according to the results contained in Table 3. The relationship between volumes, mechanical work and heat energy are identical to those included in Table 3. Titanium volumes are scaled using length for Comparisons 2 and 4. 
       Discussion 
       [0041]    The combination of five separate material properties is important in assessing a material&#39;s potential for thermoelastic design and materials with one predominant property have been shown to not necessarily be the best candidate. This is evident in both Table 3 for PTFE (high g) and diamond (high E). Both gold and copper have high g values but are hindered as good candidates by low E and high r values. Silicon is very inefficient compared to certain other materials, however, amorphous silicon dioxide is possibly the most inefficient material of all. 
         [0042]    Output mechanical work, input heat energy and actuator volume are three essential characterizing parameters for thermoelastic design. The design method described incorporates these parameters using only material properties and provides a structured approach for material selection. The method is versatile because the approach assesses the potential of a material using easily calculated comparison ratios. It is important to note that the approach is a measure of a materials potential and must be used as a tool in conjunction with other appropriate design criteria. For example, criteria such as force/deflection characteristics of the output work, material resistivity, environmental ruggedness and material availability may be important. The operable temperature range is assumed to be from 0 degrees to the melting point on the Centigrade scale because it is indicative of the material thermal range. However, the maximum operable temperature could be different due to oxidation of the material or other thermal design constraints. Titanium nitride has close to the highest actuation efficiency value when melting point is used as a criteria. However, Titanium diSilicide is potentially a better candidate for use when oxidation temperature is used. Titanium nitride is a practical candidate because it is well established as a CMOS barrier material. The oxidation temperature of TiN can be raised from 500° C. to 900° C. by alloying with aluminum. The alloyed material has a symbol (Ti,Al)N. 
         [0043]    The actuation efficiency of a simple thermoelastic titanium beam is low compared to other actuation mechanisms (less than 1 percent). It is theoretically possible to get a thermoelastic actuation efficiency of about 4.5 percent for a simple titanium nitride beam, however, this value typically decreases when the material is implemented in a MEMS device due to associated operational losses (for example—thermal conduction into the substrate). 
         [0044]    The invention has been described herein by way of example only. Skilled workers in this field will readily recognize many variations and modifications which do depart from the spirit and scope of the broad inventive concept.