CROSS REFERENCE TO RELATED APPLICATIONS 
   This application claims the benefit of U.S. provisional patent application 60/901,200, filed on Feb. 12, 2007, entitled “High Quality Factor Resonators for Liquid Immersion Biological and Chemical Sensors”, and hereby incorporated by reference in its entirety. 


   FIELD OF THE INVENTION 
   This invention relates to sensors suitable for liquid immersion applications. 
   BACKGROUND 
   Resonant mechanical structures are commonly employed as sensor elements for detecting the presence of biological or chemical analytes. Such detection is typically based on functionalizing the resonant mechanical structure such that the analyte or analytes of interest can bind to the mechanical resonator, if they are present. The binding of analytes to the mechanical resonator alters the resonant frequency of the mechanical resonator due to the mass of the bound analytes. Measurement of the resonant frequency of the mechanical resonator can thereby provide a sensitive indication as to the presence of the analytes. 
   In order for such sensors to provide high sensitivity, it is important for the mechanical resonator to have low mechanical loss, which is frequently expressed in terms of the resonator having a high quality factor (i.e., high Q). High Q results in a reduction of measurement noise, thereby improving sensitivity. However, it is challenging to provide high-Q mechanical resonators for use in liquid immersion applications, because liquid loading of the mechanical resonator due to immersion tends to significantly and undesirably decrease resonator Q. 
   In U.S. Pat. No. 6,906,450, resonator Q in immersion is electronically enhanced by providing electronic feedback control of the mechanical resonator. However, imposing a requirement on the sensor control electronics to provide appropriate Q-enhancing feedback may conflict with other sensor design considerations. Accordingly, it would be an advance in the art to provide mechanically resonant sensors having intrinsically high Q in fluid immersion. 
   SUMMARY 
   A mechanical resonator capable of providing an intrinsically high mechanical quality factor in immersion is provided. The resonator includes a membrane attached at its perimeter to a frame, such that a front side of the membrane is in contact with the liquid, and the back side of the membrane is not in contact with the liquid or the frame. The membrane can act as a mechanical resonator. The quality factor of this resonator is enhanced by providing a pressure release boundary region on the frame in proximity to the membrane and in contact with the liquid. The pressure release boundary region provides a soft boundary condition, in the sense that a mechanical impedance on the solid side of the solid-liquid interface is less than the liquid mechanical impedance. Providing such a soft boundary condition reduces the mechanical energy loss due to excitation of waves in the liquid, thereby improving resonator quality factor. Such high-Q resonators are particularly useful for sensor applications. 


   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1   a  shows a top view of an embodiment of the invention. 
       FIGS. 1   b - c  show side views of two embodiments of the invention. 
       FIG. 1   d  shows a sensor according to an embodiment of the invention. 
       FIG. 2  shows average displacement vs. frequency for an immersed mechanical resonator surrounded by regions providing two different boundary conditions. 
       FIGS. 3   a - b  show two ways of providing soft boundary conditions. 
       FIGS. 4   a - b  show an embodiment of the invention having soft boundary conditions provided by passive mechanical resonators. 
       FIG. 5  shows average displacement vs. frequency for an immersed mechanical resonator surrounded by various arrangements of passive mechanical resonators. 
       FIG. 6  shows a 1-D array of soft boundary condition regions centered on a sensor membrane. 
       FIG. 7   a  shows a top view of a 1-D array of sensor membranes, each sensor membrane surrounded by a corresponding soft boundary condition region. 
       FIG. 7   b  shows a side view of the example of  FIG. 7   a.    
       FIG. 7   c  shows a microfluidic device including the example of  FIGS. 7   a - b.    
       FIG. 8   a  shows a 2-D array of soft boundary condition regions centered on a sensor membrane. 
       FIG. 8   b  shows electrodes in contact with array elements of the example of  FIG. 8   a.    
   


   DETAILED DESCRIPTION 
     FIGS. 1   a - b  show top and side views respectively of an embodiment of the invention. This embodiment is a sensor subassembly including a membrane  106 , a frame  102  attached to the perimeter of membrane  106 , and a pressure release boundary region  104  disposed on frame  102  in proximity to membrane  106 . The pressure release boundary region is a key aspect of the invention that is described in detail below. However, it is convenient to first consider  FIG. 1   d , which shows a sensor including the sensor subassembly of  FIGS. 1   a - b , prior to describing the significance of the pressure release boundary region.  FIGS. 1   b - c  shows side views along line  108  of  FIG. 1   a.    
   During operation of the sensor of  FIG. 1   d , a sensor surface  112  of membrane  106  is in contact with a fluid (typically a liquid), and a back surface  114  of membrane  106  is not in contact with the fluid. In other words, a liquid-free space  110  is formed behind membrane  106 . Membrane  106  is driven to oscillate by energizing circuit  118 , and a resonant frequency of membrane  106  is measured by sensing circuit  120 . The presence of analytes  116  bound to sensor surface  112  of membrane  106  can be detected by measuring the shift in resonant frequency due to the mass of the bound analytes, typically by means of a displacement measurement. 
   As will become apparent below, practice of the invention does not depend critically on details of the membrane geometry, or on the means employed to measure membrane resonant frequency. Circular membranes are shown in the examples herein, but membranes according to embodiments of the invention can have any shape. Typical membranes have a diameter from tens of microns to hundreds of microns and a thickness on the order of microns. Decreasing membrane size tends to improve detection sensitivity, while increasing membrane size tends to increase Q, so detailed sensor design can consider a trade off of these two tendencies. 
   Displacement of membrane  106  can be measured by any technique, including but not limited to: optically, capacitively, magnetically, and piezoelectrically. For example, an optical interferometer can measure membrane displacement. Capacitance of a capacitor having an electrode on the membrane as one of its plates can be measured to provide a membrane displacement sensor. Motion of a metal coil on the membrane can be magnetically sensed to provide membrane displacement. Motion of a piezoelectric film affixed to the membrane can be electrically sensed to provide membrane displacement information. Atomic tip displacement sensing can also be employed. For example, variation in a tunneling current across a gap between membrane  106  and a reference electrode can be measured according to principles of tunneling microscopy. The atomic tip for such an approach can be disposed on membrane  106  or on the reference electrode. The reference electrode can be in the form of a cantilever. 
   Pressure release boundary region  104  is in contact with the fluid during sensor operation, and provides what is convenient to refer to as a “soft boundary condition” at the interface between the pressure release boundary region and the fluid. More precisely, such a soft boundary condition is defined as providing a mechanical impedance at the solid side of the solid-fluid interface that is less than the mechanical impedance of the fluid at this interface. Details of the shape or arrangement of the pressure release boundary region are not critical in practicing the invention. For example, pressure release boundary region  104  can be fully embedded in frame  102  as shown on  FIG. 1   b , or it can be disposed on top of frame  102  as shown on  FIG. 1   c , or any intermediate degree of embedding in frame  102  can be employed. More generally, any structure or device which provides a soft boundary condition as defined above can be regarded as a pressure release boundary region for practicing embodiments of the invention. Several exemplary implementations of pressure release boundary regions are described below. 
   The importance of the boundary conditions provided near an immersed mechanical oscillator can be appreciated by considering the idealized displacement vs. frequency modeling results shown on  FIG. 2 . In this example, a harmonic pressure load is applied to a circular membrane fully supported at its perimeter and facing a liquid half space. Curve  204  (Q=9.3) is the result when the boundary condition around the membrane is idealized to be perfectly “hard” (i.e., no displacement of the solid). Curve  202  (Q=496.0) is the corresponding result when the boundary condition around the membrane is idealized to be perfectly “soft” (i.e., no pressure at the boundary, solid displacement follows displacement of the liquid). 
   As is evident from curves  202  and  204 , perfectly soft boundary conditions make the resonator have much higher Q (i.e., sharper and higher peak) than perfectly hard boundary conditions. The physical reason for this dependence on boundary conditions near (but not on) the resonator itself is that such boundary conditions affect the efficiency with which acoustic waves are generated in the liquid by the oscillating membrane. Since such acoustic waves take energy away from the resonator, they provide a loss mechanism that decreases resonator Q. Such radiative energy loss is hindered by providing soft boundary conditions near the oscillating membrane. 
   Special measures are required to provide the desirable soft boundary conditions identified above, because typical materials for sensor construction (e.g., silicon, tungsten, aluminum) tend to have substantially higher mechanical impedances than typical liquids of interest, such as water. There are various approaches for providing soft boundary conditions in practice. 
   One approach is shown in the side view of  FIG. 3   a . In this example, pressure release boundary region  104  of  FIG. 1   b  is implemented by providing a thin annular secondary membrane  302  around membrane  106  and separating the fluid from a fluid free region  304 . Membrane  302  preferably has a sufficiently low stiffness that the mechanical impedance provided by this structure at the solid side of the fluid-solid interface is substantially less than the fluid mechanical impedance at this interface. Since the back surface of secondary membrane  302  is not in contact with either the fluid or with frame  102 , this arrangement can provide sufficiently low mechanical impedance. 
   Another approach for providing pressure release boundary region  104  of  FIG. 1   b  is shown in the side view of  FIG. 3   b . In this example, the pressure release boundary region is implemented by providing soft solid  306  as shown. Suitable compositions for solid  306  include, but are not limited to: silicone rubber (e.g., polydimethylsiloxane (PDMS)), room temperature vulcanizing (RTV) silicone rubber, RTV or PDMS like materials, polymers including air bubbles, silica aerogels, glass bubbles in an epoxy or RTV binder, and sealed balsa wood. Solid  306  can be provided as one or more layers. In cases where multiple layers are employed, the layers can be arranged to provide a passive mechanical resonator in analogy with the following membrane resonator examples. 
   Another approach for providing soft boundary conditions is shown in the example of  FIGS. 4   a - b , where  FIG. 4   b  is a side view along line  406  of  FIG. 4   a . In this example, the pressure release boundary regions are implemented as passive membrane resonators. Membrane  402  is disposed over fluid-free space  408 , and membrane  404  is disposed over fluid-free space  410 . A mechanical resonator provides a low (ideally zero) mechanical impedance at its resonant frequency. Therefore, by designing the passive resonators to have substantially the same resonant frequency as the active sensor membrane  106 , suitable soft boundary conditions can be provided. In this context, it is helpful to define passive resonators as any resonators present in a sensor structure for which a displacement measurement is not performed to provide sensing, and active resonators as any resonators for which a displacement measurement is performed to provide sensing. 
   Mechanical impedances can be frequency-dependent. The above-stated requirement that the pressure release boundary region provide a lower mechanical impedance than the fluid is understood to apply to frequencies at or near the resonant frequency of the active resonator (e.g., the resonator formed by sensor membrane  106  in the preceding examples). It is not necessary to provide soft boundary conditions at frequencies well away from the resonant frequency of the active resonator, although some approaches (e.g., the examples of  FIGS. 3   a - b ) tend to provide such broad-band softness. 
   Although changes of the resonant frequency of the active resonator occur during sensor operation, such changes tend to be very small fractional frequency changes. Therefore, any particular sensor will have a well-defined nominal resonant frequency of the active resonator which the passive resonators can be matched to. 
     FIG. 5  shows modeling results for three different passive resonator configurations of the kind shown on  FIGS. 4   a - b . In all cases, the active membrane has a 1 micron thickness and a 20 micron radius. For case  1 , the passive resonators are concentric 40 micron wide, 1 micron thick annuli separated by idealized 0 micron wide solid pillars. For case  2 , the passive resonators are concentric 44 micron wide, 1 micron thick annuli separated by idealized 0 micron wide solid pillars. For case  3 , the passive resonators are concentric 41 micron wide, 1 micron thick annuli separated by 3 micron wide solid pillars. For all three cases, the number of concentric passive resonators was increased to a point where the boundary conditions assumed beyond the outermost passive resonator has no significant effect on the calculated results. Curves  502 ,  504  and  506  on  FIG. 5  correspond to cases  1 ,  2 , and  3  above, respectively. In all three cases, high Q (˜350) in liquid immersion is obtained. 
   Passive resonators can also be disposed in a 1-D or 2-D array centered on the active resonator. For example,  FIG. 6  shows membrane  604  (i.e., the active resonator) centered in a 1-D array formed by passive resonators  606 ,  608 ,  610 , and  612  on frame  602 . In cases where multiple passive resonators are employed to provide soft boundary conditions, the passive resonators can be mechanically independent of each other, or they can be mechanically coupled such that they act as a system of coupled mechanical oscillators. 
   In some embodiments of the invention, a 1-D or 2-D array of sensor elements is provided.  FIG. 7   a  shows a top view of a 1-D array of sensor membranes on frame  702 , each sensor membrane surrounded by a corresponding soft boundary condition region.  FIG. 7   b  shows a side view of the example of  FIG. 7   a . One of the sensor membranes is referenced as  704 , and its corresponding pressure release boundary region is referenced as  706 . In cases where multiple sensor membranes are employed, the sensor membranes can be mechanically isolated from each other, or they can be mechanically coupled to act as a system of coupled mechanical resonators. In either case, increasing sensor Q by providing soft boundary conditions in accordance with principles of the invention can be helpful for improving sensor sensitivity. Practice of the invention is not critically dependent on geometrical details of the pressure release boundary regions. The example of  FIG. 7   a  shows a pressure release boundary region around each active sensor membrane. It is also possible for the pressure release boundary region to surround the entire set of active sensor membranes (e.g., a single pressure release boundary region around the array of 4 sensors of the example of  FIG. 7   a ). 
     FIG. 7   c  shows a microfluidic device including the example of  FIGS. 7   a - b . In this example, a cap  708  is attached to frame  702  to form a microfluidic channel  710  through which a liquid containing analytes of interest can flow. Ports  712  and  714  enable the flow of liquid through this sensor. 
     FIG. 8   a  shows a 2-D array of soft boundary condition regions centered on a sensor membrane. Here active membrane  802  is surrounded by passive resonators  804  arranged as a 2-D array on frame  805 .  FIG. 8   b  shows electrodes in contact with array elements of the example of  FIG. 8   a . Electrode  808  is in contact with active resonator  802 , and electrode  806  is in contact with passive resonators  804 . Such electrodes can be helpful for matching the resonant frequency of passive resonators  804  to the resonant frequency of active resonator  802 . For example, a DC bias is typically applied to active resonator  802  to make its oscillation more nearly sinusoidal. Such a DC bias shifts the resonant frequency of active resonator  802 . In cases where active resonator  802  and passive resonators  804  have substantially the same mechanical construction, the same DC bias can be applied to passive resonators  804  in order to match the active and passive resonant frequencies. More generally, providing a DC bias to the passive resonators provides a helpful capability for adjusting and optimizing the soft boundary conditions provided by the passive resonators. In some cases, it may be preferred to have individual control of the DC bias at each passive resonator (by providing individual traces to each passive resonator), as opposed to common electrode  806  of the example of  FIG. 8   b.
A mechanical resonator capable of providing an intrinsically high mechanical quality factor in immersion is provided. The resonator includes a membrane attached at its perimeter to a frame, such that a front side of the membrane is in contact with the liquid, and the back side of the membrane is not in contact with the liquid or the frame. The membrane can act as a mechanical resonator. The quality factor of this resonator is enhanced by providing a pressure release boundary region on the frame in proximity to the membrane and in contact with the liquid. The pressure release boundary region provides a soft boundary condition, in the sense that a mechanical impedance on the solid side of the solid-liquid interface is less than the liquid mechanical impedance. Providing such a soft boundary condition reduces the mechanical energy loss due to excitation of waves in the liquid, thereby improving resonator quality factor. Such high-Q resonators are particularly useful for sensor applications.