Abstract:
An apparatus and method for electrochemical fluid analysis comprises a chamber ( 1202 ) having a depth dimension for accommodating a volume of a fluid under test, first and second electrodes (A1) disposed within the chamber and extending along the depth dimension in spaced relation with each other, and a soluble solid, such as an annealed polymer, e.g. EUDRAGIT occupying a lateral gap between the first and second electrodes. The rate of dissolution as monitored by electrochemical impedance spectroscopy (EIS) of the soluble solid within the fluid depends on the chemical concentration of a corresponding analyte present in solution in the fluid. In one embodiment a silicon-based integrated circuit device defining an upper margin includes an array of electrodes disposed along said upper margin to permit direct exposure of the electrode array to the fluid under test. The device is constructed using CMOS technology.

Description:
FIELD OF THE INVENTION 
     The present invention relates to a system, device and method for measuring the concentration of chemical components present within a fluid and, more particularly, to systems, devices and methods for measuring analyte concentration electrochemically. 
     BACKGROUND OF THE INVENTION 
     Chemical concentration of an analyte in a fluid can be measured by transducing presence of the analyte into measurable physical parameters. For example, the concentration of an analyte solution can be determined via such techniques as spectroscopy, chromatography, calorimetry, or optical fluorescence. 
     Further concentration measurement techniques involve probing the electrical characteristics of the analyte solution. Some such techniques involve coultometry. Others involve amperometric, voltametric, and/or potentiametric titration. Many such techniques are capable of a high degree of accuracy, speed (e.g., throughput), and efficiency. Unfortunately, the equipment required to implement such techniques can tend to be both large and bulky. As a result, the use of such equipment is typically limited to a laboratory setting, and technicians in the field who seek to make concentration determinations via measurement of electrical characteristics are often left with few attractive options. 
     Despite efforts to date, a need remains for effective methods and systems for electrochemical measurement of analyte concentration, particularly for applications in which a high premium is placed on portability and field use availability. These and other needs are satisfied by the methods and systems disclosed herein. 
     SUMMARY OF THE INVENTION 
     Apparatus and methods for electrochemical analysis of fluids are provided according to the present disclosure. 
     In an exemplary embodiment of the present disclosure, an apparatus is provided that includes a chamber having a depth dimension for accommodating a volume of a fluid under test, a first electrode disposed within the chamber and extending therewithin along the depth dimension, a second electrode disposed within the chamber and extending therewithin along the depth dimension in laterally spaced relation with the first electrode, and a soluble solid disposed within the chamber between the first and second electrodes so as to substantially completely occupy a lateral gap therebetween to an extent of at least a portion of the depth dimension. A rate of dissolution of the soluble solid within the fluid is at least partially dependent on a chemical concentration of a corresponding analyte present in solution in the fluid. 
     A method for electrochemical analysis of fluids is also provided. In exemplary embodiments of the present disclosure, the method includes exposing a soluble solid to a fluid, measuring a rate of dissolution of the soluble solid in the fluid, and determining a chemical concentration of a corresponding analyte present in solution in the fluid based on the measured rate of dissolution. 
     Additional advantageous features, functions and applications of the disclosed apparatus and methods for electrochemical analysis of fluids will be apparent from the description which follows, particularly when read in conjunction with the appended figures. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       To assist those of skill in the art in making and using the disclosed systems and methods, reference is made to the accompanying figures, wherein: 
         FIG. 1  is a schematic representation of an embodiment of an analyte concentration measurement tool in accordance with the present disclosure; 
         FIG. 2  is a downward perspective view of a CMOS die useable to fabricate the  FIG. 1  measurement tool in accordance with the present disclosure; 
         FIG. 3  is a top plan view of the  FIG. 2  CMOS die after modification via formation on an upper margin thereof of a metallic contact pattern in accordance with the present disclosure; 
         FIG. 4  is a section view of the  FIG. 3  modified CMOS die taken along section line  4 - 4  shown in  FIG. 3 ; 
         FIG. 5  is a downward perspective view of the  FIG. 3  modified CMOS die; 
         FIG. 6  is a top plan view of the  FIG. 3  modified CMOS die after further modification via formation atop the metallic contact pattern thereof of an array of paired electrodes in accordance with the present disclosure; 
         FIG. 7  is a section view of the  FIG. 6  modified CMOS die taken along section line  7 - 7  shown in  FIG. 6 ; 
         FIG. 8  is a downward perspective view of the  FIG. 6  modified CMOS die; 
         FIG. 9  is a top plan view of the  FIG. 6  modified CMOS die after further modification via formation atop the array of paired electrodes thereof of a dielectric material layer in accordance with the present disclosure; 
         FIG. 10  is a section view of the  FIG. 9  modified CMOS die taken along section line  10 - 10  shown in  FIG. 9 ; 
         FIG. 11  is a downward perspective view of the  FIG. 9  modified CMOS die; 
         FIG. 12  is a top plan view of the  FIG. 9  modified CMOS die after filling of the cylindrical chambers thereof with polymeric materials and associated annealing to form an embodiment of the analyte concentration measurement tool of  FIG. 1  in accordance with the present disclosure; 
         FIG. 13  is a section view of the  FIG. 12  analyte concentration measurement tool; 
         FIG. 14  is a section view of a fluid-polymer filled cylinder of the  FIG. 12  analyte concentration measurement tool in accordance with the present disclosure; 
         FIG. 15  is a section view of a fluid filled cylinder of the  FIG. 12  analyte concentration measurement tool in accordance with the present disclosure; 
         FIG. 16  is a schematic diagram of an exemplary electrical circuit corresponding to the fluid-polymer filled cylinder of  FIG. 14 ; and 
         FIG. 17  is a schematic diagram of an exemplary electrical circuit corresponding to the fluid-filled cylinder of  FIG. 15 . 
         FIG. 18  is a schematic diagram of an exemplary electrical circuit. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     An apparatus for electrochemical analysis of fluids is provided that can be adapted to be compact in size, economic to manufacture, and convenient to deploy. Exemplary embodiments of apparatus for electrochemical analysis of fluids include a chamber having a depth dimension for accommodating a volume of a fluid under test, and a pair of electrodes disposed within the chamber and extending along the depth dimension thereof in laterally spaced relation to each other. A soluble solid is disposed within the chamber between the electrodes, occupying a lateral gap therebetween to an extent of at least a portion of the depth dimension of the chamber. A rate of dissolution of the soluble solid within the fluid is at least partially dependent on a chemical concentration of a corresponding analyte present in solution in the fluid To the extent that the soluble solid dissolves in the fluid, the fluid fills the void generated by the dissolving solid. Because the soluble solid is a poorer conductor compared to the fluid, dissolution of the soluble solid leads to an increase of conductance between the electrodes. The rate of conductance change further depends on the properties of the dissolving solid and the actual analyte concentration in solution in the fluid. 
     Materials suitable for use with respect to the soluble solid according to the present disclosure include commercially available materials that exhibit respective solubilities dependent on the concentration in solution of a chemical component or active species of interest, e.g., H+ concentration (i.e., pH), proteins, amino acids, glucose, enzymes and other analytes of interest. Exemplary materials for use with respect to the soluble solid according to the present disclosure include polymers that exhibit a pH-dependent dissolution rate, such as EUDRAGIT acrylic polymers manufactured by Degussa GmbH, and polymers that exhibit dissolution rates that are dependent on the presence of colon enzyme, such as azo polymers used by Alizyme plc (Cambridge, United Kingdom). 
     Apparatus and methods for electrochemical analysis of fluids in accordance with the present disclosure may be used to measure the concentration for a large number of chemical components present within a fluid under test. In embodiments of the present disclosure, such apparatus and methods rely on polymers with specific solubility depending on concentration of compounds mixed within the fluid, and include an electronic device that allows an accurate measurement of the solubility based upon complex conductance measurements. The lifetime of the electronic device may be limited in accordance with embodiments of the present disclosure, and controlled by processing parameters of the device. 
     In accordance with some embodiments of the present disclosure, a small, simple, energy efficient ‘lab-on-a-chip’ solution is provided having a response time in the field at least comparable to, if not superior to, many larger, more bulky systems commonly limited to use within a laboratory. Such an apparatus can be implemented through the use of an integrated circuit (IC) electronic device combined with an array of confined micro-cylinders fabricated via MEMS processes at the surface of a die associated with the IC electronic device, and filled with polymers having known etching rate versus chemical concentration of active species in solution in the fluid under test. 
     For illustration purposes, the disclosed apparatus and methods are described in greater detail herein with reference to a tool for measuring analyte concentration in solution in a fluid under test. However, the disclosed systems and methods have wide ranging applicability, as will be readily apparent to persons skilled in the art, including implementations directed to a variety of analytes. Thus, in one exemplary embodiment of the present disclosure, the apparatus includes a soluble solid in the form of a polymer that does not dissolve until the pH is above a threshold value and, as a result, the conductance between the electrodes does not increase unless the fluid under test has a pH above this threshold. If the pH of the fluid under test is above the applicable threshold, the conductance between electrodes will advantageously increase proportionally to the difference between the actual pH value of the fluid under test and the lower threshold pH of the soluble polymer. 
     Additionally, in one exemplary embodiment of the present disclosure, the apparatus includes a soluble solid in the form of a polymer that does not dissolve unless the pH of the fluid under test is below a threshold value and, as a result, the conductance between the electrodes does not increase unless the fluid under test has a pH below this threshold. If the pH of the fluid under test is below the applicable threshold, the conductance between electrodes will advantageously increase proportionally to the difference between the actual pH value of the fluid under test and the higher threshold pH of the soluble polymer. Therefore, by monitoring a rate of change in conductance between the electrodes between which the soluble solid is disposed, either or both of a pH limit value and an actual pH value can be derived. 
     When an apparatus for electrochemically analyzing a fluid based on this principle of concentration-dependent solubility is used, conductance between each pair of electrodes may be measured as a function of time, and the rate of conductance change may be used to derive the concentration value of the analyte present in solution in the fluid under test. One unique advantage of such an apparatus for electrochemically analyzing a fluid is that the apparatus can be operated without absolute calibration. Variation in manufacturing process and environmental conditions, such as overall conductivity of the fluids under test, can cause variation in absolute conductance between electrodes. These variations, however, do not interfere with derivation of the concentration value of an analyte present in solution in a fluid under test because the concentration value is determined by the change rate of conductance, not by the absolute value of conductance. Of course, such an apparatus can be used in conjunction with a reference electrode to account for environmental changes in the rate of conductance. 
     An apparatus  100  for measuring analyte concentration in solution in a fluid under test in accordance with embodiments of the present disclosure is shown in  FIG. 1 . The apparatus  100  may include a silicon-based integrated circuit (IC)  102 . The IC  102  may incorporate an input/output (IO) data block  104 , a data processor and control unit (DPCU)  106 , an amplitude and frequency control unit (AFCU)  108 , a complex admittance measurement unit (CAMU)  110 , and an electrode selector (ES)  112 . The apparatus  100  may further include an electrode array (EA)  114 . 
     The IO  104  may be an interface of the circuit with respect to external devices. The EA  114  is a matrix of electrodes present at an upper margin or surface of the IC  102 . Each of the electrodes of the EA  114  can be connected through the ES  112  block to corresponding measurement ports of the CAMU  110 . All other electrodes of the EA  114  may be grounded. The ES  112  may be an array of analog switches which allows the selection of a single electrode out of the EA  114 . The CAMU  110  can measure the complex admittance of the circuit connected at the selected electrode from the EA  114 . The frequency and amplitude of the test signal can be controlled and/or imposed by the AFCU  108 . The DPCU  106  may receive analog signals provided by the CAMU  110  and convert the same to digital values. The DPCU  106  may further store and/or process such digital values, take decisions regarding the frequencies and amplitudes of operations from the AFCU  108  and operate the ES  112  accordingly. The DPCU  106  may further be employed to transfer to the IO  104  measurement results with respect to concentration(s) of one or more analytes present in solution in the fluid under test. 
     In accordance with embodiments of the present disclosure, the CMOS die  200  shown in  FIG. 2  can embody the  FIG. 1  IC  102 . For example, the CMOS die  200  may include an upper margin  202  featuring an array of peripherally-disposed contacts  204  associated with an input/output interface of the circuit (e.g., associated with the  FIG. 1  IO  104 ). 
     A MEMS process may be utilized to modify and/or convert the CMOS die  200  of  FIG. 2  to form an embodiment of the  FIG. 1  apparatus  100  in accordance with the present disclosure. An example of such a process is shown and described below with reference to  FIGS. 3-13 . 
     Referring now to  FIGS. 3 ,  4  and  5 , a modified CMOS die  300  can be formed by modifying the  FIG. 2  CMOS die  102  via conventional metal deposition process and associated appropriate patterning to form a contact pattern  302  on an upper margin  304  of the die  300  operative to permit electrical interconnection between an IC (e.g.,  FIG. 1  IC  102 ) and an electrode array (e.g.,  FIG. 1  EA  114 ) in accordance with the present invention. 
     Referring now to  FIGS. 6 ,  7  and  8 , a modified CMOS die  600  can be formed by further modifying the  FIG. 3  modified CMOS die  300  via an appropriate aluminum-silicon deposition and etch processes (e.g., with a highly selective RIE) to form a mask  602 . The mask  602  may include an array of chambers  604  for accommodating small volumes of a liquid under test, each of which may include a cylindrically shaped microbarrel  606  connected to ground and a column shaped central electrode  608 . The central electrodes  608  may be disposed within the microbarrels  606 , and, further may be coaxial with, and/or coextensive (e.g., depthwise) therewith. 
     Referring now to  FIGS. 9 ,  10  and  11 , a modified CMOS die  900  can be formed by further modifying the  FIG. 6  modified CMOS die  600  via an appropriate material layer deposition and etch process, e.g., to form a dielectric material layer  902  atop the  FIG. 6  mask  602 . For example, the dielectric material layer  902  may be a SiO 2 -Si 3 N 4  layer. One or more of the  FIG. 6  chambers  604  may be masked during this step so as to prevent the dielectric material layer  902  from extending thereto. For example, four such chambers (e.g., chambers lacking a layer  902  of a dielectric material) may be positioned on respective corners  904 ,  906 ,  908 ,  910  of the die  900 , and/or may be used to measure air admittance (e.g., as part of a measurement control function). 
     Referring now to  FIGS. 12 and 13 , an analyte concentration measurement tool  1200  is shown. For example, the tool  1200  may be implemented to embody the analyte concentration measurement tool  100  of  FIG. 1 . The tool  1200  can be formed by further modifying the  FIG. 9  modified CMOS die  900  via filling one or more of the  FIG. 6  chambers  604  with similar and/or different polymers and executing an appropriate annealing process to form fluid-polymer filled cylinders or chambers  1202  (described further hereinbelow), wherein the dissolution rate of each such polymer may be specific to one or more of the same or different chemical compounds in solution in a fluid under test. The  FIG. 9  corner-disposed chambers  904 ,  906 ,  908 ,  910  may be left unfilled with polymer for purposes of measuring air admittance as part of a measurement control function. Likewise, an entire row  1204  of chambers  1206 , also referred to herein as fluid filled cylinders or chambers  1206 , may be left unfilled with polymer for purposes of measuring an admittance of the fluid under test as described more fully below. 
     The tool  1200  includes an IC  1208  which can embody, for example, the  FIG. 1  IC  102 , and a MEMS region  1210  which can embody the  FIG. 1  EA  114 . The MEMS region  1210  can be configured to be exposed to the fluid under test, while the IC  1208  can be configured such that its internally-disposed electrical circuitry and/or functions are secured from damage from the fluid under test. 
     Referring now to  FIG. 14 , when exposed to fluid  1400  from the fluid under test, the soluble solid  1402  (e.g., polymer) within the chamber  1202  is dissolved by the analyte present in solution in the fluid under test.  FIG. 14  gives a vertical plane cross-section through a fluid-polymer filled cylinder (FPC)  1202  after the soluble solid  1402  contained therein was etched to an etch value equal to h fluid .  FIG. 15  gives a vertical plane cross-section through a fluid filled cylinder (FC)  1206 . When exposed to the fluid under test, the fluid-filled cylinders  1206 , containing no soluble solid (e.g., containing no polymer), are filled with fluid  1500 . The  FIG. 12  tool  1200  measures the complex admittances for all of the FPCs  1202  and FCs  1206  of the MEMS region  1210  at specific time intervals and estimates the electrical properties of polymers within the FPCs  1202  and the fluid within the FCs  1206  and the etch rate for every type of polymer from every FPC  1202 . Based on the etching rate, one may determine the concentrations of active chemical species within the fluid under test. The capacity to make such determinations may last as long as the polymers within the FPCs  1202  are not completely dissolved within the fluid under test. 
     Exemplary electric equivalent circuits  1600 ,  1700  for FPCs  1202  and FCs  1206  are given, respectively, in  FIGS. 16 and 17 . 
     The below-described algorithm is executable by a processor (e.g., by the  FIG. 1  DPCU  106 ) in accordance with the present disclosure to allow the determination at a time tj of the concentration of species present in solution in the fluid under test, given the FPC  1202  are filled with different polymers with etching rates specific to every chemical concentration of the species. 
     V probe  is a voltage source V probe =V 0 e i(ωt+φ)  with i=√−1, V 0 : amplitude, ω: angular frequency, φ: phase, t: time. The complex impedeance of the circuit from  FIG. 8  (seen by the voltage source voltage source V probe  is: 
                       Z   g     =             (       X     Cfe   ⁢   _   ⁢   i       +     X     Cfe   ⁢   _   ⁢   o       +         X   Cfluid     ⁢     R   fluid           X   Cfluid     ⁢     R   fluid           )               (       X     Cpe   ⁢   _   ⁢   i       +     X     Cpe   ⁢   _   ⁢   o       +         X   Cpolymer     ⁢     R   polymer           X   Cpolymer     +     R   polymer           )                     X     Cfe   ⁢   _   ⁢   i       +       X     Cfe   ⁢   _   ⁢   o       ⁢                       X   Cfluid     ⁢     R   fluid           X   Cfluid     +     R   fluid         +         X   Cpolymer     ⁢     R   polymer           X   Cpolymer     +     R   polymer                     ⁢     
     ⁢     where   ⁢     :               (   1   )                   X     Cfe   i       =     1     ⅈ   ⁢           ⁢   ω   ⁢           ⁢     C     fe   ⁢   _   ⁢   i             ,       X     Cfe   ⁢   _   ⁢   o       =     1     ⅈ   ⁢           ⁢   ω   ⁢           ⁢     C     fe   ⁢   _   ⁢   o             ,     
     ⁢       X     Cpe   ⁢   _   ⁢   o       =     1     ⅈ   ⁢           ⁢   ω   ⁢           ⁢     C     pe   ⁢   _   ⁢   o             ,       X     Cpe   ⁢   _   ⁢   i       =     1     ⅈω   ⁢           ⁢     C     pe   ⁢   _   ⁢   i             ,     
     ⁢       X   Cpolymer     =     1     ⅈω   ⁢           ⁢     C   polymer           ,       X   Cfluid     =     1     ⅈω   ⁢           ⁢     C   fluid                   (   2   )               
And the resistors are not frequency dependent (the voltage source frequency is smaller than the minimum ionic rotation frequency within the fluid or polymer).
 
     Capacitances and resistances are calculated with the coaxial model. As a reminder:
         the resistance measured between the inner electrode with radius R i  and output electrode with radius R o  of a coaxial cable of length l filled with a material of resistivity ρ is:       

     
       
         
           
             
               
                 
                   
                     R 
                     ⁡ 
                     
                       ( 
                       l 
                       ) 
                     
                   
                   = 
                   
                     
                       ρln 
                       ⁢ 
                       
                         
                           R 
                           o 
                         
                         
                           R 
                           i 
                         
                       
                     
                     
                       2 
                       ⁢ 
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       l 
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
         
         
           
             the capacitance measured between the inner electrode with radius R i  and output electrode with radius R o  of a coaxial cable of length l filled with a material with a relative dielectric constant ε dielectric  is: 
           
         
       
    
                     C   ⁡     (   l   )       =       2   ⁢     πɛ   dielectric     ⁢     ɛ   0     ⁢   l       ln   ⁢       R   o       R   i                   (   4   )               
Interestingly enough, the time constant associated to this coax cable is independent of any geometrical parameters of the cable:
 
τ= R ( l ) C ( l )=ε dielectric ε 0 ρ  (5)
 
We will use the following notations:
 
                       h   electrodes     =     h   conductor       ⁢     
     ⁢         ɛ   dielectric   ′     =     2   ⁢     πɛ   dielectric     ⁢     ɛ   0         ,       ɛ   fluid   ′     =     2   ⁢     πɛ   fluid     ⁢     ɛ   0         ,     
     ⁢       ɛ   polymer   ′     =     2   ⁢     πɛ   polymer     ⁢     ɛ   0           ⁢     
     ⁢         ρ   fluid   ′     =       ρ   fluid       2   ⁢   π         ,       ρ   polymer   ′     =       ρ   polymer       2   ⁢   π           ⁢     
     ⁢         τ   polymer     =         ɛ   0     ⁢     ρ   polymer     ⁢     ɛ   polymer       =       ɛ   polymer   ′     ⁢     ρ   polymer   ′           ,     
     ⁢       τ   fluid     =         ɛ   0     ⁢     ρ   fluid       =       ɛ   fluid   ′     ⁢     ρ   fluid   ′             ⁢     
     ⁢     ξ   =       h   fluid       h   electrodes                 (   6   )               
The expressions for the capacitances instantiated by equations (2) are:
 
     
       
         
           
             
               
                 
                   
                     
                       C 
                       
                         e 
                         ⁢ 
                         _ 
                         ⁢ 
                         i 
                       
                     
                     = 
                     
                       
                         
                           ɛ 
                           dielectric 
                           ′ 
                         
                         
                           ln 
                           ⁡ 
                           
                             ( 
                             
                               1 
                               + 
                               
                                 
                                   t 
                                   dielectric 
                                 
                                 
                                   R 
                                   i 
                                 
                               
                             
                             ) 
                           
                         
                       
                       ⁢ 
                       
                         h 
                         electrodes 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       C 
                       
                         e 
                         ⁢ 
                         _ 
                         ⁢ 
                         o 
                       
                     
                     = 
                     
                       
                         
                           ɛ 
                           dielectric 
                           ′ 
                         
                         
                           ln 
                           ⁡ 
                           
                             ( 
                             
                               1 
                               + 
                               
                                 
                                   t 
                                   dielectric 
                                 
                                 
                                   
                                     R 
                                     i 
                                   
                                   + 
                                   
                                     t 
                                     dielectric 
                                   
                                   + 
                                   
                                     d 
                                     electrodes 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                       ⁢ 
                       
                         h 
                         electrodes 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         
                           
                             
                               C 
                               
                                 fe 
                                 ⁢ 
                                 _ 
                                 ⁢ 
                                 i 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 h 
                                 fluid 
                               
                               ) 
                             
                           
                           = 
                           
                             
                               
                                 ɛ 
                                 dielectric 
                                 ′ 
                               
                               
                                 ln 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       
                                         t 
                                         dielectric 
                                       
                                       
                                         R 
                                         i 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ⁢ 
                             
                               h 
                               fluid 
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                           
                             
                               ξ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   C 
                                   
                                     fe 
                                     ⁢ 
                                     _ 
                                     ⁢ 
                                     i 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     h 
                                     electorodes 
                                   
                                   ) 
                                 
                               
                             
                             = 
                             
                               ξ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 C 
                                 
                                   e 
                                   ⁢ 
                                   _ 
                                   ⁢ 
                                   i 
                                 
                               
                             
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         
                           
                             
                               C 
                               
                                 fe 
                                 ⁢ 
                                 _ 
                                 ⁢ 
                                 o 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 h 
                                 fluid 
                               
                               ) 
                             
                           
                           = 
                           
                             
                               
                                 ɛ 
                                 dielectric 
                                 ′ 
                               
                               
                                 ln 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       
                                         t 
                                         dielectric 
                                       
                                       
                                         
                                           R 
                                           i 
                                         
                                         + 
                                         
                                           t 
                                           dielectric 
                                         
                                         + 
                                         
                                           d 
                                           electrodes 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ⁢ 
                             
                               h 
                               fluid 
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                           
                             ξ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 C 
                                 
                                   fe 
                                   ⁢ 
                                   _ 
                                   ⁢ 
                                   o 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   h 
                                   electrodes 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                           
                             ξ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               C 
                               
                                 e 
                                 ⁢ 
                                 _ 
                                 ⁢ 
                                 o 
                               
                             
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         
                           
                             
                               C 
                               
                                 pe 
                                 ⁢ 
                                 _ 
                                 ⁢ 
                                 i 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 h 
                                 fluid 
                               
                               ) 
                             
                           
                           = 
                           
                             
                               
                                 ɛ 
                                 dielectric 
                                 ′ 
                               
                               
                                 ln 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       
                                         t 
                                         dielectric 
                                       
                                       
                                         R 
                                         i 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   h 
                                   electrodes 
                                 
                                 - 
                                 
                                   h 
                                   fluid 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                           
                             
                               ( 
                               
                                 1 
                                 - 
                                 ξ 
                               
                               ) 
                             
                             ⁢ 
                             
                               
                                 C 
                                 
                                   pe 
                                   ⁢ 
                                   _ 
                                   ⁢ 
                                   i 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   h 
                                   electrodes 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                           
                             
                               ( 
                               
                                 1 
                                 - 
                                 ξ 
                               
                               ) 
                             
                             ⁢ 
                             
                               C 
                               
                                 e 
                                 ⁢ 
                                 _ 
                                 ⁢ 
                                 i 
                               
                             
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         
                           
                             
                               C 
                               
                                 pe 
                                 ⁢ 
                                 _ 
                                 ⁢ 
                                 i 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 h 
                                 fluid 
                               
                               ) 
                             
                           
                           = 
                           
                             
                               
                                 ɛ 
                                 dielectric 
                                 ′ 
                               
                               
                                 ln 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       
                                         t 
                                         dielectric 
                                       
                                       
                                         
                                           R 
                                           i 
                                         
                                         + 
                                         
                                           t 
                                           dielectric 
                                         
                                         + 
                                         
                                           d 
                                           electrodes 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   h 
                                   electrodes 
                                 
                                 - 
                                 
                                   h 
                                   fluid 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                           
                             
                               ( 
                               
                                 1 
                                 - 
                                 ξ 
                               
                               ) 
                             
                             ⁢ 
                             
                               
                                 C 
                                 
                                   pe 
                                   ⁢ 
                                   _ 
                                   ⁢ 
                                   o 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   h 
                                   electrodes 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                           
                             
                               ( 
                               
                                 1 
                                 - 
                                 ξ 
                               
                               ) 
                             
                             ⁢ 
                             
                               C 
                               
                                 e 
                                 ⁢ 
                                 _ 
                                 ⁢ 
                                 o 
                               
                             
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         
                           
                             
                               C 
                               fluid 
                             
                             ⁡ 
                             
                               ( 
                               
                                 h 
                                 fluid 
                               
                               ) 
                             
                           
                           = 
                           
                             
                               
                                 ɛ 
                                 fluid 
                                 ′ 
                               
                               
                                 ln 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       
                                         d 
                                         electrodes 
                                       
                                       
                                         
                                           R 
                                           i 
                                         
                                         + 
                                         
                                           t 
                                           dielectric 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ⁢ 
                             
                               h 
                               fluid 
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                           
                             ξ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 C 
                                 fluid 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   h 
                                   electrodes 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                           
                             ξ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               C 
                               
                                 FC 
                                 , 
                                 fluid 
                               
                             
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         
                           
                             
                               C 
                               polymer 
                             
                             ⁡ 
                             
                               ( 
                               
                                 h 
                                 fluid 
                               
                               ) 
                             
                           
                           = 
                           
                             
                               
                                 ɛ 
                                 fluid 
                                 ′ 
                               
                               
                                 ln 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       
                                         d 
                                         electrodes 
                                       
                                       
                                         
                                           R 
                                           i 
                                         
                                         + 
                                         
                                           t 
                                           dielectric 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   h 
                                   electrodes 
                                 
                                 - 
                                 
                                   h 
                                   fluid 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                           
                             
                               ( 
                               
                                 1 
                                 - 
                                 ξ 
                               
                               ) 
                             
                             ⁢ 
                             
                               
                                 C 
                                 polymer 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   h 
                                   electrodes 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                           
                             
                               ( 
                               
                                 1 
                                 - 
                                 ξ 
                               
                               ) 
                             
                             ⁢ 
                             
                               C 
                               
                                 FPC 
                                 , 
                                 polymer 
                                 , 
                                 0 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     All capacitances are linear functions of h fluid    
     The expressions for the resistances instantiated by equations (2) are: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               R 
                               fluid 
                             
                             ⁡ 
                             
                               ( 
                               
                                 h 
                                 fluid 
                               
                               ) 
                             
                           
                           = 
                           
                             
                               
                                 ρ 
                                 fluid 
                                 ′ 
                               
                               ⁢ 
                               
                                 ln 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       
                                         d 
                                         electrodes 
                                       
                                       
                                         
                                           R 
                                           i 
                                         
                                         + 
                                         
                                           t 
                                           dielectric 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             
                               h 
                               fluid 
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                           
                             
                               
                                 R 
                                 fluid 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   h 
                                   electrodes 
                                 
                                 ) 
                               
                             
                             ξ 
                           
                         
                       
                     
                     
                       
                         
                           = 
                           
                             
                               R 
                               
                                 FC 
                                 , 
                                 fluid 
                               
                             
                             ξ 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         
                           
                             
                               R 
                               polymer 
                             
                             ⁡ 
                             
                               ( 
                               
                                 h 
                                 fluid 
                               
                               ) 
                             
                           
                           = 
                           
                             
                               
                                 ρ 
                                 polymer 
                                 ′ 
                               
                               ⁢ 
                               
                                 ln 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       
                                         d 
                                         electrodes 
                                       
                                       
                                         
                                           R 
                                           i 
                                         
                                         + 
                                         
                                           t 
                                           dielectric 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             
                               
                                 h 
                                 electrodes 
                               
                               - 
                               
                                 h 
                                 fluid 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                           
                             
                               
                                 R 
                                 polymer 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   h 
                                   electrodes 
                                 
                                 ) 
                               
                             
                             
                               1 
                               - 
                               ξ 
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                           
                             
                               R 
                               
                                 FPC 
                                 , 
                                 polymer 
                                 , 
                                 0 
                               
                             
                             
                               1 
                               - 
                               ξ 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     All conductances are linear functions of h fluid    
     Step 1: Device in Air 
     FPC filled with polymer only 
     FC filled with air 
     Test cylinders filled with air. 
     Measuring the admittance {tilde over (Y)} C,test,air  of the test cylinders in the air (ε air =1) allows the experimental determination of the thickness of the dielectrics within the FPC and FC: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             Y 
                             ~ 
                           
                           
                             C 
                             , 
                             test 
                             , 
                             air 
                           
                         
                         = 
                         
                           ⅈω 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             C 
                             test 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             ⅈω 
                             ⁢ 
                             
                               
                                 2 
                                 ⁢ 
                                 
                                   πɛ 
                                   0 
                                 
                               
                               
                                 ln 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       
                                         
                                           2 
                                           ⁢ 
                                           
                                             t 
                                             dielectric 
                                           
                                         
                                         + 
                                         
                                           d 
                                           electrodes 
                                         
                                       
                                       
                                         R 
                                         i 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ⁢ 
                             
                               h 
                               electrodes 
                             
                           
                           ⇔ 
                           
                             t 
                             dielectric 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             1 
                             2 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   R 
                                   i 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       ⅇ 
                                       
                                         
                                           2 
                                           ⁢ 
                                           
                                             πɛ 
                                             0 
                                           
                                           ⁢ 
                                           
                                             wh 
                                             electrodes 
                                           
                                         
                                         
                                           ?? 
                                           ⁡ 
                                           
                                             ( 
                                             
                                               
                                                 Y 
                                                 _ 
                                               
                                               
                                                 C 
                                                 , 
                                                 test 
                                                 , 
                                                 air 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 d 
                                 electrodes 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     The admittance of the FC filled with air is (ρ air ≈∞, ε air =1) 
                             ⅈω   ⁢           ⁢     h   electrodes           Y   ~       C   ,   test   ,   air         =       ⁢       (         1     ɛ   dielectric   ′       ⁢     ln   ⁡     (     1   +       t   dielectric       R   i         )         +       1     ɛ   air   ′       ⁢     ln   ⁡     (     1   +       d   electrodes         R   i     +     t   dielectric           )         +       1     ɛ   dielectric   ′       ⁢     ln   ⁡     (     1   +       t   dielectric         R   i     +     t   dielectric     +     d   electrodes           )           )     ⇔                   ɛ   dielectric   ′     =       ⁢       ln   ⁡     (       (     1   +       t   dielectric       R   i         )     ⁢     (     1   +       t   dielectric         R   i     +     t   dielectric     +     d   electrodes           )       )             ω   ⁢           ⁢     h   electrodes         ??   ⁡     (       Y   ~       C   ,   test   ,   air       )         -       1     ɛ   air   ′       ⁢     ln   ⁡     (     1   +       d   electrodes         R   i     +     t   dielectric           )                           (   10   )               
The measurements performed in air allowed the determinations of two critical values: ε dielectric  and t dielectric  
 
     Let us measure the FPC in air (h fluid =0): 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         
                           
                             
                               Z 
                               ~ 
                             
                             
                               g 
                               , 
                               FPC 
                               , 
                               air 
                             
                           
                           = 
                             
                           ⁢ 
                           
                             
                               
                                 X 
                                 
                                   Cpe 
                                   ⁢ 
                                   _ 
                                   ⁢ 
                                   i 
                                 
                               
                               + 
                               
                                 X 
                                 
                                   Cpe 
                                   ⁢ 
                                   _ 
                                   ⁢ 
                                   o 
                                 
                               
                               + 
                               
                                 
                                   
                                     X 
                                     Cpolymer 
                                   
                                   ⁢ 
                                   
                                     R 
                                     polymer 
                                   
                                 
                                 
                                   
                                     X 
                                     Cpolymer 
                                   
                                   + 
                                   
                                     R 
                                     polymer 
                                   
                                 
                               
                             
                             ⇔ 
                           
                         
                       
                     
                     
                       
                         
                             
                           ⁢ 
                           
                             
                               
                                 Z 
                                 ~ 
                               
                               
                                 g 
                                 , 
                                 FPC 
                                 , 
                                 air 
                               
                             
                             - 
                             
                               ( 
                               
                                 
                                   X 
                                   
                                     Cpe 
                                     ⁢ 
                                     _ 
                                     ⁢ 
                                     i 
                                   
                                 
                                 + 
                                 
                                   X 
                                   
                                     Cpe 
                                     ⁢ 
                                     _ 
                                     ⁢ 
                                     o 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                           ⁢ 
                           
                             
                               
                                 R 
                                 polymer 
                               
                               ⁡ 
                               
                                 ( 
                                 0 
                                 ) 
                               
                             
                             
                               1 
                               + 
                               
                                 ⅈω 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     R 
                                     polymer 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     0 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     C 
                                     polymer 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     0 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     Replace the values for X Cpe     —     i  and X Cpe     —     o  with the expression given in (7) for h fluid =0: 
                         Z   ~       g   ,   FPC   ,   air       -       1   ⅈω     ⁢     (       1       C     pe   ⁢   _   ⁢   i       ⁡     (   0   )         +     1     C     pe   ⁢   _   ⁢   o           )         =             R   polymer     ⁡     (   0   )         1   +     ⅈω   ⁢           ⁢       R   polymer     ⁡     (   0   )       ⁢       C   polymer     ⁡     (   0   )             ⇔     
     ⁢       ℜ   ⁡     (       Z   ~       g   ,   FPC   ,   air       )       +     ⅈ??   ⁡     (       Z   ~       g   ,   FPC   ,   air       )       +       ⅈ   ω     ⁢     (       1       C     pe   ⁢   _   ⁢   i       ⁡     (   0   )         +     1       C     pe   ⁢   _   ⁢   o       ⁡     (   0   )           )           =           (     1   -     ⅈωτ   polymer       )     ⁢       R   polymer     ⁡     (   0   )           1   +       ω   2     ⁢     τ   polymer   2           ⇔     
     ⁢     {               ℜ   ⁡     (       Z   ~       g   ,   FPC   ,   air       )       =         R   polymer     ⁡     (   0   )         1   +       ω   2     ⁢     τ   polymer   2                           ??   ⁡     (       Z   ~       g   ,   FPC   ,   air       )       +       1   ω     ⁢     (       1       C     pe   ⁢   _   ⁢   i       ⁡     (   0   )         +     1       C     pe   ⁢   _   ⁢   o       ⁡     (   0   )           )         =           -     ωτ   polymer       ⁢       R   polymer     ⁡     (   0   )           1   +       ω   2     ⁢     τ   polymer   2           ⇔             ⁢     {             ℜ   ⁡     (       Z   ~       g   ,   FPC   ,   air       )       =         R   polymer     ⁡     (   0   )         1   +       ω   2     ⁢     τ   polymer   2                           ??   ⁡     (       Z   ~       g   ,   FPC   ,   air       )       +       1   ω     ⁢     (       1       C     pe   ⁢   _   ⁢   i       ⁡     (   0   )         +     1       C     pe   ⁢   _   ⁢   o       ⁡     (   0   )           )         =         -     ωτ   polymer       ⁢     ℜ   ⁡     (       Z   ~       g   ,   FPC   ,   air       )         ⇔                               (   12   )                       ⁢     {             τ   polymer     =           1       C     pe   ⁢   _   ⁢   i       ⁡     (   0   )         +     1       C     pe   ⁢   _   ⁢   o       ⁡     (   0   )         +     ω??   ⁡     (       Z   ~       g   ,   FPC   ,   air       )             ω   2     ⁢     ℜ   ⁡     (     Z     g   ,   FPC   ,   air       )           ⇔                     R   polymer     ⁡     (   0   )       =       (     1   +       ω   2     ⁢     τ   polymer   2         )     ⁢     ℜ   ⁡     (       Z   ~       g   ,   FPC   ,   air       )                                           
Step 2: Measure the Fluid and Polymer Admittance at any Time after the Device has been Immersed in a Fluid.
 
     Assume the fluid etched the polymer, and got to the coordinate h fluid  of the FPC. 
     The FC complex admittance yields: 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     { 
                     
                       
                         
                           
                             
                               τ 
                               fluid 
                             
                             = 
                             
                               
                                 - 
                                 
                                   
                                     
                                       1 
                                       
                                         
                                           C 
                                           
                                             pe 
                                             ⁢ 
                                             _ 
                                             ⁢ 
                                             i 
                                           
                                         
                                         ⁡ 
                                         
                                           ( 
                                           0 
                                           ) 
                                         
                                       
                                     
                                     + 
                                     
                                       1 
                                       
                                         
                                           C 
                                           
                                             pe 
                                             ⁢ 
                                             _ 
                                             ⁢ 
                                             o 
                                           
                                         
                                         ⁡ 
                                         
                                           ( 
                                           0 
                                           ) 
                                         
                                       
                                     
                                     + 
                                     
                                       ω?? 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             Z 
                                             ~ 
                                           
                                           
                                             g 
                                             , 
                                             FC 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   
                                     
                                       ω 
                                       2 
                                     
                                     ⁢ 
                                     
                                       ℜ 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           Z 
                                           
                                             g 
                                             , 
                                             FC 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                               ⇔ 
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 R 
                                 fluid 
                               
                               ⁡ 
                               
                                 ( 
                                 0 
                                 ) 
                               
                             
                             = 
                             
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     
                                       ω 
                                       2 
                                     
                                     ⁢ 
                                     
                                       τ 
                                       fluid 
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 ℜ 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       Z 
                                       ~ 
                                     
                                     
                                       g 
                                       , 
                                       FC 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     The FPC complex admittance yields: 
     
       
         
           
             
               
                 
                   
                     
                       
                         X 
                         ⁡ 
                         
                           ( 
                           
                             h 
                             fluid 
                           
                           ) 
                         
                       
                       + 
                       
                         X 
                         ⁡ 
                         
                           ( 
                           
                             h 
                             fluid 
                           
                           ) 
                         
                       
                       + 
                       
                         
                           
                             
                               X 
                               Cfluid 
                             
                             ⁡ 
                             
                               ( 
                               
                                 h 
                                 fluid 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               R 
                               fluid 
                             
                             ⁡ 
                             
                               ( 
                               
                                 h 
                                 fluid 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               X 
                               Cfluid 
                             
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                           + 
                           
                             
                               R 
                               fluid 
                             
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                         
                       
                     
                     = 
                     
                       
                         
                           1 
                           ξ 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               X 
                               
                                 C 
                                 , 
                                 FC 
                                 , 
                                 
                                   fe 
                                   ⁢ 
                                   _ 
                                   ⁢ 
                                   i 
                                 
                               
                             
                             + 
                             
                               X 
                               
                                 C 
                                 , 
                                 FC 
                                 , 
                                 
                                   fe 
                                   ⁢ 
                                   _ 
                                   ⁢ 
                                   o 
                                 
                               
                             
                             + 
                             
                               
                                 
                                   X 
                                   
                                     C 
                                     , 
                                     FC 
                                     , 
                                     fluid 
                                   
                                 
                                 ⁢ 
                                 
                                   R 
                                   
                                     FC 
                                     , 
                                     fluid 
                                   
                                 
                               
                               
                                 
                                   X 
                                   
                                     C 
                                     , 
                                     FC 
                                     , 
                                     fluid 
                                   
                                 
                                 + 
                                 
                                   R 
                                   
                                     FC 
                                     , 
                                     fluid 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                       = 
                       
                         
                           1 
                           ξ 
                         
                         ⁢ 
                         
                           
                             Z 
                             ~ 
                           
                           
                             g 
                             , 
                             FC 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   and 
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             X 
                             
                               Cpe 
                               ⁢ 
                               _ 
                               ⁢ 
                               i 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               h 
                               fluid 
                             
                             ) 
                           
                         
                         + 
                         
                           
                             X 
                             
                               Cpe 
                               ⁢ 
                               _ 
                               ⁢ 
                               o 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               h 
                               fluid 
                             
                             ) 
                           
                         
                         + 
                         
                           
                             
                               
                                 X 
                                 Cfluid 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   h 
                                   fluid 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               
                                 R 
                                 fluid 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   h 
                                   fluid 
                                 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 X 
                                 Cfluid 
                               
                               ⁡ 
                               
                                 ( 
                                 x 
                                 ) 
                               
                             
                             + 
                             
                               
                                 R 
                                 fluid 
                               
                               ⁡ 
                               
                                 ( 
                                 x 
                                 ) 
                               
                             
                           
                         
                       
                       = 
                       
                         
                           1 
                           
                             1 
                             - 
                             ξ 
                           
                         
                         ⁢ 
                         
                           
                             Z 
                             ~ 
                           
                           
                             g 
                             , 
                             FPC 
                             , 
                             0 
                           
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       Then 
                       ⁢ 
                       
                         : 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         Z 
                         ~ 
                       
                       
                         g 
                         , 
                         FC 
                       
                     
                     = 
                     
                       
                         
                           
                             
                               1 
                               ξ 
                             
                             ⁢ 
                             
                               
                                 Z 
                                 ~ 
                               
                               
                                 g 
                                 , 
                                 FC 
                               
                             
                             ⁢ 
                             
                               1 
                               
                                 1 
                                 - 
                                 ξ 
                               
                             
                             ⁢ 
                             
                               
                                 Z 
                                 ~ 
                               
                               
                                 g 
                                 , 
                                 FPC 
                                 , 
                                 0 
                               
                             
                           
                           
                             
                               
                                 1 
                                 ξ 
                               
                               ⁢ 
                               
                                 
                                   Z 
                                   ~ 
                                 
                                 
                                   g 
                                   , 
                                   FC 
                                 
                               
                             
                             + 
                             
                               
                                 1 
                                 
                                   1 
                                   - 
                                   ξ 
                                 
                               
                               ⁢ 
                               
                                 
                                   Z 
                                   ~ 
                                 
                                 
                                   g 
                                   , 
                                   FPC 
                                   , 
                                   0 
                                 
                               
                             
                           
                         
                         ⇔ 
                         
                           
 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           h 
                           fluid 
                         
                       
                       = 
                       
                         
                           h 
                           electrodes 
                         
                         ⁢ 
                         
                           
                             
                               
                                 Z 
                                 ~ 
                               
                               
                                 g 
                                 , 
                                 FC 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     Z 
                                     ~ 
                                   
                                   
                                     g 
                                     , 
                                     FPC 
                                     , 
                                     0 
                                   
                                 
                                 - 
                                 
                                   
                                     Z 
                                     ~ 
                                   
                                   
                                     g 
                                     , 
                                     FPC 
                                   
                                 
                               
                               ) 
                             
                           
                           
                             
                               
                                 Z 
                                 ~ 
                               
                               
                                 g 
                                 , 
                                 FPC 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     Z 
                                     ~ 
                                   
                                   
                                     g 
                                     , 
                                     FPC 
                                     , 
                                     0 
                                   
                                 
                                 - 
                                 
                                   
                                     Z 
                                     ~ 
                                   
                                   
                                     g 
                                     , 
                                     FC 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     With this complex impedance measurement approach, the etching of the polymer with a FPC h fluid  is calculated based on the initial values of the impedance of the FPC (FPC in air) and the adjacent FPC measurement. This does not solve the variations of the polymer electrical parameters vs. time. 
     Let us look at a frequency swipe method: for the same h fluid  the frequency of the measurement of the AMA is changed within limits larger than the poles and zeros of the complex admittance. 
     For an FC: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                                 
                             
                             ⁢ 
                             
                               
                                 
                                   
                                     
                                       
                                         Z 
                                         ~ 
                                       
                                       
                                         g 
                                         , 
                                         FC 
                                       
                                     
                                     = 
                                     
                                       
                                         
                                           1 
                                           ⅈω 
                                         
                                         ⁢ 
                                         
                                           ( 
                                           
                                             
                                               1 
                                               
                                                 C 
                                                 
                                                   FC 
                                                   , 
                                                   
                                                     fe 
                                                     i 
                                                   
                                                 
                                               
                                             
                                             + 
                                             
                                               1 
                                               
                                                 C 
                                                 
                                                   FC 
                                                   , 
                                                   
                                                     fe 
                                                     o 
                                                   
                                                 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                       + 
                                       
                                         
                                           R 
                                           
                                             FC 
                                             , 
                                             fluid 
                                           
                                         
                                         
                                           1 
                                           + 
                                           
                                             ⅈωτ 
                                             fluid 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           1 
                                           
                                             C 
                                             
                                               FC 
                                               , 
                                               
                                                 fe 
                                                 i 
                                               
                                             
                                           
                                         
                                         + 
                                         
                                           1 
                                           
                                             C 
                                             
                                               FC 
                                               , 
                                               
                                                 fe 
                                                 o 
                                               
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                       = 
                                       Δ 
                                     
                                     ⁢ 
                                     
                                       1 
                                       
                                         C 
                                         
                                           FC 
                                           , 
                                           fe 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             } 
                           
                           ⇒ 
                           
                             
 
                           
                           ⁢ 
                           
                             
                               
                                 
                                   
                                     
                                       Z 
                                       ~ 
                                     
                                     
                                       g 
                                       , 
                                       FC 
                                     
                                   
                                   = 
                                   
                                     
                                       
                                         1 
                                         ⅈω 
                                       
                                       ⁢ 
                                       
                                         1 
                                         
                                           C 
                                           
                                             FC 
                                             , 
                                             fe 
                                           
                                         
                                       
                                     
                                     + 
                                     
                                       
                                         R 
                                         
                                           FC 
                                           , 
                                           fluid 
                                         
                                       
                                       
                                         1 
                                         + 
                                         
                                           ⅈωτ 
                                           fluid 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       R 
                                       
                                         FC 
                                         , 
                                         fluid 
                                       
                                     
                                     ⁢ 
                                     
                                       C 
                                       
                                         FC 
                                         , 
                                         fe 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     = 
                                     Δ 
                                   
                                   ⁢ 
                                   
                                     τ 
                                     df 
                                   
                                 
                               
                             
                           
                         
                         } 
                       
                       ⇒ 
                       
                          
                         
                           
                             Z 
                             ~ 
                           
                           
                             g 
                             , 
                             FC 
                           
                         
                          
                       
                     
                     = 
                     
                       
                         1 
                         
                           C 
                           
                             FC 
                             , 
                             fe 
                           
                         
                       
                       ⁢ 
                       
                         
                           
                             
                               
                                 ω 
                                 2 
                               
                               ⁢ 
                               
                                 τ 
                                 fd 
                                 2 
                               
                             
                             + 
                             
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     
                                       ω 
                                       2 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           τ 
                                           fluid 
                                           2 
                                         
                                         + 
                                         
                                           
                                             τ 
                                             fluid 
                                           
                                           ⁢ 
                                           
                                             τ 
                                             df 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                         
                           ω 
                           ⁡ 
                           
                             ( 
                             
                               1 
                               + 
                               
                                 
                                   ω 
                                   2 
                                 
                                 ⁢ 
                                 
                                   τ 
                                   fluid 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     The 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     measured 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     absolute 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     value 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     of 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     the 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     impedance 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     of 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     the 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     FC 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     has 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     the 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     poles 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     and 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     zeros 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     placed 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     at 
                     ⁢ 
                     
                       : 
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           ω 
                           pole 
                         
                         ∈ 
                         
                           { 
                           
                             0 
                             , 
                             
                               - 
                               
                                 i 
                                 
                                   τ 
                                   fluid 
                                 
                               
                             
                             , 
                             
                               i 
                               
                                 τ 
                                 fluid 
                               
                             
                           
                           } 
                         
                       
                       , 
                       
                         
                           ω 
                           zero 
                         
                         ∈ 
                         
                           { 
                           
                             
                               - 
                               
                                 i 
                                 
                                   
                                     τ 
                                     fluid 
                                   
                                   + 
                                   
                                     τ 
                                     df 
                                   
                                 
                               
                             
                             , 
                             
                               i 
                               
                                 
                                   τ 
                                   fluid 
                                 
                                 + 
                                 
                                   τ 
                                   df 
                                 
                               
                             
                           
                           } 
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       with 
                       ⁢ 
                       
                         : 
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           R 
                           
                             FC 
                             , 
                             fluid 
                           
                         
                         ⁢ 
                         
                           C 
                           
                             FC 
                             , 
                             fe 
                           
                         
                       
                       ⁢ 
                       
                         = 
                         Δ 
                       
                       ⁢ 
                       
                         τ 
                         df 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       τ 
                       df 
                     
                     = 
                     
                       
                         ɛ 
                         dielectric 
                         ′ 
                       
                       ⁢ 
                       
                         ρ 
                         fluid 
                         ′ 
                       
                       ⁢ 
                       
                         
                           ln 
                           ⁡ 
                           
                             ( 
                             
                               1 
                               + 
                               
                                 
                                   d 
                                   electrodes 
                                 
                                 
                                   
                                     R 
                                     i 
                                   
                                   + 
                                   
                                     t 
                                     dielectric 
                                   
                                 
                               
                             
                             ) 
                           
                         
                         
                           
                             
                               
                                 
                                   ln 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       1 
                                       + 
                                       
                                         
                                           t 
                                           dielectric 
                                         
                                         
                                           R 
                                           i 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   ln 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       1 
                                       + 
                                       
                                         
                                           t 
                                           dielectric 
                                         
                                         
                                           
                                             R 
                                             i 
                                           
                                           + 
                                           
                                             t 
                                             dielectric 
                                           
                                           + 
                                           
                                             d 
                                             electrodes 
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 ln 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       ( 
                                       
                                         1 
                                         + 
                                         
                                           
                                             t 
                                             dielectric 
                                           
                                           
                                             
                                               R 
                                               i 
                                             
                                             + 
                                             
                                               t 
                                               dielectric 
                                             
                                             + 
                                             
                                               d 
                                               electrodes 
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         1 
                                         + 
                                         
                                           
                                             t 
                                             dielectric 
                                           
                                           
                                             R 
                                             i 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     From the bode plot of the absolute value of the complex impedance of an FC, one can extract τ fluid +τ df  and τ fluid  which is equivalent to resolving ρ fluid  and ε fluid . 
     We will use the same procedure for extracting the parameters of the polymer within an FPC: 
                         Y   ~       g   ,   FPC       =           1     Z     g   ,   FPC   ,   polymer         +     1     Z     g   ,   FPC   ,   fluid           ⇔     
     ⁢       Y   ~       g   ,   FPC         =           1       1   ξ     ⁢       Z   ~       g   ,   FC           +     1       1     1   -   ξ       ⁢       Z   ~       g   ,   FPC   ,   fluid             ⇔     
     ⁢     Y     g   ,   FPC         =         C     FC   ,   ee       ⁢   ω             [       ωτ   df     -     ⅈ   ⁡     (     1   +       ω   2     ⁢       τ   fluid     ⁡     (       τ   fluid     +     τ   df       )           )         ]               [       ωτ   dp     -     ⅈ   ⁡     (     1   +       ω   2     ⁢       τ   polymer     ⁡     (       τ   polymer     +     τ   dp       )           )         ]                   ⁢     
     ⁢     {               ξ   ⁡     [       ωτ   dp     -     ⅈ   ⁡     (     1   +       ω   2     ⁢       τ   polymer     ⁡     (       τ   polymer     +     τ   dp       )           )         ]       ⁢     (     1   +       ω   2     ⁢     τ   fluid   2         )       +                   (     1   -   ξ     )     ⁡     [       ωτ   df     -     ⅈ   ⁡     (     1   +       ω   2     ⁢       τ   fluid     ⁡     (       τ   fluid     +     τ   df       )           )         ]       ⁢     (     1   +       ω   2     ⁢     τ   polymer   2         )             }             (   19   )               
The poles and zeros for {tilde over (Y)} g,FPC  can be estimated from the Bode plot ∥{tilde over (Y)} g,FPC ∥, therefore one can extract ρ polymer  and ε polymer  at any time during the measurement of the fluid.
 
     Determination of Chemical Active Species Concentration 
     Let us consider an Admittance Measurement Array (AMA) as described above, with the following characteristics: 
     
         
         
           
             Confines a (not necessarily) square array of N cylinders ×N cylinders  cylinders. 
             4 cylinders of the AMA are not covered with Si 3 N 4 /SiO 2  dielectrics and are used to measure (in air) the specifics of the dielectric covering the A1 electrodes (ε dielectric  t dielectric ). 
             N cylinder =N FC,cylinder  cylinders (different from the 4 above) are not filled with any polymers. 
             N polymer =N cylinders ×N cylinders −N FC,cylinder −4 are filled with N polymer  different types of polymers. 
             Every polymer is chemically sensitive to a specific chemical species within the fluid under test. For example, assume the fluid under test confines N specie  active species with concentrations C specie . The polymer in cylinder k ∈≦{1, . . . , N polymer }⊂N reacts with the active species within the fluid. As a result of the reactions, the polymer will become soluble within the solution. The rate of solubility is related to the etching rate of the polymer with a function isomorphic with:
 
 r   ks   =m   ks ( C   s   −C   ks,threshold )θ( C   s   −C   ks,threshold ),∀ k={ 1, . . . , N   polymer   },∀s={ 1, . . . , N   species }  (20)
 
where:
 
             r ks  Etching rate of polymer k reacting with specie s 
             θ(C s −C ks,threshold ) Heaviside function of difference between active specie concentration Cs and concentration threshold Cks,threshold above which polymer k reacts with specie s 
             M ks (C s −C ks,threshold ) Increasingly monotonic function describing the etching rate of polymer k by specie s of concentration C s . 
           
         
       
    
     The total etch rate for polymer k while reacting to all species N specie  within the fluid is the sum of the etch rate of the polymer per specie: 
                           ⁢         r   k     =         ∑     i   =   1       N   species       ⁢           ⁢     r   ks       =       ∑     i   =   1       N   species       ⁢           ⁢       M   ki     ⁡     (       C   i     -     C     ki   ,   threshold         )             ,     
     ⁢           ⁢       ∀   k     =     {     1   ,   …   ⁢           ,     N   polymer       }       ,       ∀   i     =     {     1   ,   …   ⁢           ,     N   species       }                 (   21   )               
with M id (C i −C id,threshold )=m ks (C s −C ks,threshold )θ(C s −C ks,threshold ).
 
The AMA structure is measured in air and the specifics of all Npolymer are stored as (ε k , ρ k , τ k ), ∀k={1, . . . , N polymer }.
 
     The AMA structure immersed in the solution under test. We will assume at any time there is no concentration gradient of any active species at the surface of the AMA, therefore all cylinders “see” at the same time the same value of the concentration value for any species. 
     The admittance measurement circuit measures for every cylinder (Fluid Polymer filled Cylinders (FPF) and Fluid-filled Cylinders (FC)) at a sample rate S all complex admittances and calculates: 
     For every FC 
     
       
         
           
             
                 
             
             ⁢ 
             
               
                 ( 
                 
                   
                     ɛ 
                     
                       FC 
                       , 
                       fluid 
                       , 
                       
                         e 
                         ′ 
                       
                     
                     
                       ( 
                       
                         t 
                         j 
                       
                       ) 
                     
                   
                   , 
                   
                     ρ 
                     
                       FC 
                       , 
                       fluid 
                       , 
                       
                         e 
                         ′ 
                       
                     
                     
                       ( 
                       
                         t 
                         j 
                       
                       ) 
                     
                   
                   , 
                   
                     τ 
                     
                       FC 
                       , 
                       fluid 
                       , 
                       e 
                     
                     
                       ( 
                       
                         t 
                         j 
                       
                       ) 
                     
                   
                 
                 ) 
               
               , 
               
                 
                   ∀ 
                   c 
                 
                 = 
                 
                   { 
                   
                     1 
                     , 
                     … 
                     ⁢ 
                     
                         
                     
                     , 
                     
                       N 
                       
                         FC 
                         , 
                         cylinder 
                       
                     
                   
                   } 
                 
               
               , 
               
                 
 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 
                   ∀ 
                   
                     t 
                     j 
                   
                 
                 = 
                 
                   { 
                   
                     
                       1 
                       S 
                     
                     , 
                     … 
                     ⁢ 
                     
                         
                     
                     , 
                     
                       N 
                       S 
                     
                   
                   } 
                 
               
               , 
               
                 N 
                 ∈ 
                 N 
               
             
           
         
       
     
     The average values of (ε FC,fluid   (t     j     ) ,ρ FC,fluid   (t     j     ) ,τ FC,fluid   (t     j     ) ) at time t j : 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         ɛ 
                         
                           FC 
                           , 
                           fluid 
                         
                         
                           ( 
                           
                             t 
                             j 
                           
                           ) 
                         
                       
                       = 
                       
                         
                           1 
                           
                             N 
                             
                               FC 
                               , 
                               cylinder 
                             
                           
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             
                               N 
                               
                                 FC 
                                 , 
                                 cylinder 
                               
                             
                           
                           ⁢ 
                           
                             ɛ 
                             
                               FC 
                               , 
                               fluid 
                               , 
                               i 
                             
                             
                               ( 
                               
                                 t 
                                 j 
                               
                               ) 
                             
                           
                         
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ρ 
                         
                           FC 
                           , 
                           fluid 
                         
                         
                           ( 
                           
                             t 
                             j 
                           
                           ) 
                         
                       
                       = 
                       
                         
                           1 
                           
                             N 
                             
                               FC 
                               , 
                               cylinder 
                             
                           
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             
                               N 
                               
                                 FC 
                                 , 
                                 cylinder 
                               
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ρ 
                             
                               FC 
                               , 
                               fluid 
                               , 
                               i 
                             
                             
                               ( 
                               
                                 t 
                                 j 
                               
                               ) 
                             
                           
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         τ 
                         
                           FC 
                           , 
                           fluid 
                         
                         
                           ( 
                           
                             t 
                             j 
                           
                           ) 
                         
                       
                       = 
                       
                         
                           1 
                           
                             N 
                             
                               FC 
                               , 
                               cylinder 
                             
                           
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             
                               N 
                               
                                 FC 
                                 , 
                                 cylinder 
                               
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             τ 
                             
                               FC 
                               , 
                               fluid 
                               , 
                               i 
                             
                             
                               ( 
                               
                                 t 
                                 j 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
         
         
           
             Based upon the values obtained above, assuming the fluid parameters within adjacent FC and PFC are the same (no concentration gradient at the surface of the AMA), calculate for every FPC 
           
         
       
    
     
       
         
           
             
                 
             
             ⁢ 
             
               
                 ( 
                 
                   
                     ɛ 
                     
                       polymer 
                       , 
                       k 
                     
                     
                       ( 
                       
                         t 
                         j 
                       
                       ) 
                     
                   
                   , 
                   
                     ρ 
                     
                       polymer 
                       , 
                       k 
                     
                     
                       ( 
                       
                         t 
                         j 
                       
                       ) 
                     
                   
                   , 
                   
                     τ 
                     
                       polymer 
                       , 
                       k 
                     
                     
                       ( 
                       
                         t 
                         j 
                       
                       ) 
                     
                   
                   , 
                   
                     h 
                     
                       fluid 
                       , 
                       k 
                     
                     
                       ( 
                       
                         t 
                         j 
                       
                       ) 
                     
                   
                 
                 ) 
               
               , 
               
                 
                   ∀ 
                   k 
                 
                 = 
                 
                   { 
                   
                     1 
                     , 
                     … 
                     ⁢ 
                     
                         
                     
                     , 
                     
                       N 
                       polymer 
                     
                   
                   } 
                 
               
               , 
               
                 
 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 
                   ∀ 
                   
                     t 
                     j 
                   
                 
                 = 
                 
                   { 
                   
                     
                       1 
                       S 
                     
                     , 
                     … 
                     ⁢ 
                     
                         
                     
                     , 
                     
                       N 
                       S 
                     
                   
                   } 
                 
               
               , 
               
                 N 
                 ∈ 
                 
                   ℕ 
                   . 
                 
               
             
           
         
       
     
     We can approximate the etching rate at time t j  as: 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         
                           
                             r 
                             ~ 
                           
                           k 
                           
                             ( 
                             
                               t 
                               j 
                             
                             ) 
                           
                         
                         ≈ 
                         
                           
                             
                               h 
                               
                                 fluid 
                                 , 
                                 k 
                               
                               
                                 ( 
                                 
                                   t 
                                   j 
                                 
                                 ) 
                               
                             
                             - 
                             
                               h 
                               
                                 fluid 
                                 , 
                                 k 
                               
                               
                                 ( 
                                 
                                   t 
                                   
                                     j 
                                     - 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             
                               t 
                               j 
                             
                             - 
                             
                               t 
                               
                                 j 
                                 - 
                                 1 
                               
                             
                           
                         
                       
                       = 
                       
                         
                           ( 
                           
                             
                               h 
                               
                                 fluid 
                                 , 
                                 k 
                               
                               
                                 ( 
                                 
                                   t 
                                   j 
                                 
                                 ) 
                               
                             
                             - 
                             
                               h 
                               
                                 fluid 
                                 , 
                                 k 
                               
                               
                                 ( 
                                 
                                   t 
                                   
                                     j 
                                     - 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         S 
                       
                     
                     , 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ∀ 
                         k 
                       
                       = 
                       
                         { 
                         
                           1 
                           , 
                           … 
                           ⁢ 
                           
                               
                           
                           , 
                           
                             N 
                             polymer 
                           
                         
                         } 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
     Therefore for any time t j  we get: 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         
                           r 
                           ~ 
                         
                         k 
                         
                           ( 
                           
                             t 
                             j 
                           
                           ) 
                         
                       
                       = 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           
                             N 
                             species 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             M 
                             ki 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 C 
                                 i 
                               
                               - 
                               
                                 C 
                                 
                                   ki 
                                   , 
                                   threshold 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     , 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ∀ 
                         k 
                       
                       = 
                       
                         { 
                         
                           1 
                           , 
                           … 
                           ⁢ 
                           
                               
                           
                           , 
                           
                             N 
                             polymer 
                           
                         
                         } 
                       
                     
                     , 
                     
                       
                         ∀ 
                         i 
                       
                       = 
                       
                         { 
                         
                           1 
                           , 
                           … 
                           ⁢ 
                           
                               
                           
                           , 
                           
                             N 
                             species 
                           
                         
                         } 
                       
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
     We can express the relation above in matrix form: 
     {tilde over (r)} [N     polymer     ,1]   (t     j     ) =           {tilde over (r)} k   (t     j     )           , measured etching rate matrix, at time t j  
     {tilde over (C)} [1,N     specie     ]   (t     j     ) =           {tilde over (c)} i   (t     j     )             , concentration matrix at time t   j , to be determined
     C TH[N     polymer     ,N     specie     ] =           C ki,threshold           , concentrating threshold for polymer k reacting with specie I, known (25)
     M [N     polymer     ,N     specie     ] (C)=           M ik (C ik )         , etching rate function matrix for polymer k, reacting with specie i, known
     C H[N     polymer     ,N     specie     ]   )t     j     ) =1 [N     polymer     ,t] {tilde over (C)} [1,N     specie     ]   (t     j     ) −C TH[N     polymer     ,N     specie     ] , normalized concentration matrix 
     The systems of equations from (24) can be written as:
 
 {tilde over (r)}   [N     polymer     ,t]   (t     j     )   =M   [N     polymer     ,N     specie     ] ( C   H[N     polymer     ,N     specie     ]   (t     j     ) )1 [N     polymer     ,1]   (26)
 
     For the case of when the matrix M [N     polymer     ,N     specie     ]  is has an inverse, i.e., N polymer =N species =N e , the equation reads:
 
 {tilde over (r)}   [N     e     t]   (t     j     )   =M   [N     e     ,N     e     ] ( C   H[N     e     ,N     e     ]   (t     j     ) )1 [N     e     ,1]   (27)
 
     And after some linear transformations becomes:
 
 M   [N     e     ,N     e     ] (1 [N     e     ,1]   {tilde over (c)}   [1,N     e     ]   (t     j     )   −C   TH[N     e     ,N     e     ] ) {tilde over (r)}   [N     e     ,1]   (t     j     ) =1 [N     e     ,1]   (28)
 
     This represents a system of N e  equations N e  unknowns {tilde over (C)} [1,N     e     ]   (t     j     ) , which all have a unique solution, as long as the matrix M [N     e     ,N     e     ]  is inverseable. 
     Although the invention is illustrated and described herein with reference to specific embodiments, the invention is not intended to be limited to the details shown. Rather, various modifications may be made in the details within the scope and range of equivalents of the claims and without departing from the invention.