Patent Abstract:
An electrospinning system using a spinneret and a counter electrode is first operated for a fixed amount of time at known system and operational parameters to generate a fiber mat having a measured fiber mat width associated therewith. Next, acceleration of the fiberizable material at the spinneret is modeled to determine values of mass, drag, and surface tension associated with the fiberizable material at the spinneret output. The model is then applied in an inversion process to generate predicted values of an electric charge at the spinneret output and an electric field between the spinneret and electrode required to fabricate a selected fiber mat design. The electric charge and electric field are indicative of design values for system and operational parameters needed to fabricate the selected fiber mat design.

Full Description:
ORIGIN OF THE INVENTION 
     This invention was made by an employee of the United States Government and may be manufactured and used by or for the Government of the United States of America for governmental purposes without the payment of any royalties thereon or therefor. Pursuant to 35 U.S.C. §119, the benefit of priority from provisional application 60/990,673, with a filing date of Nov. 28, 2007, is claimed for this non-provisional application, and the specification thereof is incorporated in its entirety herein by reference. 
    
    
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     This invention relates to electrospinning. More specifically, the invention is a method of predicting as well as optimizing various parameters for an electrospinning system using a single exemplary test run of the system. 
     2. Description of the Related Art 
     Electrospinning is a polymer manufacturing process that has been revived over the past decade in order to produce micro and nano-fibers as well as resulting fiber groups (or mats as they are known) with properties that can be tailored to specific applications by controlling fiber diameter and mat porosity. The individual fibers are formed by applying a high electrostatic field to a polymer solution that carries a charge sufficient to attract the solution to a grounded source. The polymer solution is ejected as a stream from a spinneret. The stream is directed towards a collector where it forms a fiber thereon. Parameters that determine fiber formation include physical system parameters defining the spinneret, the collector, and the distance between the spinneret and collector, as well as material parameters such as polymer solution viscosity, polymer/solvent interaction, surface tension, applied voltage, and the conductivity of the solution. 
     Typically, only non-woven mats can be produced during this process due to splaying of the fibers and jet instability of the polymer expelled from the spinneret. These non-woven mats are used as scaffolds for tissue engineering, wound dressings, clothing, filters and membranes. While non-woven mats have proven to be useful for a variety of applications, controlling fiber alignment in the mat is a desirable characteristic to expand the applications of electrospun materials. Particularly for the case of tissue engineering scaffolds, the control of fiber distribution, fiber alignment, and porosity of the scaffold are crucial for the success of any scaffold. Current manufacturing techniques are limited by erratic polymer whipping that often produces dense nano-fiber mats, which cannot support cell infiltration or cell alignment. 
     An improved system for aligning fibers in an electrospinning process was recently disclosed in U.S. patent application Ser. No. 12/131,420, filed Jun. 2, 2008. Briefly, this new system and technique direct a jet of a fiberizable material towards an uncharged collector from a dispensing location that is spaced apart from the collector. While the fiberizable material is directed towards the collector, an elliptical (the term “elliptical” including elliptical and all dipole field-like shapes, including both symmetric and unsymmetric, and including both spherical and ovoid) electric field is generated. The electric field spans between the dispensing location and a control location that is within line-of-sight of the dispensing location such that the electric field impinges upon at least a portion of the collector. The generation of the elliptical electric field and placement of the uncharged collector therein provide for fiber alignment when the fiberizable material is deposited on the collector. However, development of a particular fiber mat design requires a lengthy trial-and-error process to establish the various system parameters. 
     SUMMARY OF THE INVENTION 
     Accordingly, it is an object of the present invention to provide a method of selecting or predicting a number of system parameters for an electrospinning system. 
     Another object of the present invention is to provide a method of optimizing system parameters for an electrospinning system without requiring a lengthy trial-and-error process. 
     Other objects and advantages of the present invention will become more obvious hereinafter in the specification and drawings. 
     In accordance with the present invention, a method is provided for optimizing electrode parameters for an electrospinning configuration. The system for fabricating an aligned-fiber mat includes: a conductive, semi-conductive or non-conductive collector; an electrically-conductive spinneret having an output facing the collector and maintained in a spaced-apart relationship therewith; an electrode having a tip positioned at a control location that is spaced apart from the collector, with the collector being substantially disposed between the output and tip while they remain in line-of-sight of one another and aligned along a defined x-axis; the output and the tip having substantially the same geometric shape, the application of voltages of opposing polarity to the spinneret and electrode; and the pumping of a fiberizable material through the spinneret. The system is first operated for a fixed amount of time at known values of i) the voltages, ii) a distance between the spinneret output and the electrode tip, iii) length of the spinneret, iv) length of the electrode, v) radius of the spinneret, and vi) radius of the electrode. As a result, a fiber mat is deposited on the collector. The fiber mat has a measured fiber mat width associated therewith. Next, acceleration of the fiberizable material at the spinneret output is modeled to determine values of mass, drag, and surface tension associated with the fiberizable material at the spinneret output. Modeling is repeated until the values are in correspondence with the measured fiber mat width. The model used to determine the values of mass, drag, and surface tension is then applied in an inversion process to generate predicted values of an electric charge at the spinneret output and an electric field between the spinneret and electrode corresponding to a selected fiber mat design. More specifically, the inversion modeling uses the earlier-determined particular width and values for mass, drag, and surface tension to generate the predicted values of electric charge and electric field. The electric charge and field are indicative of design values for i) the voltages, ii) the distance between the spinneret output and electrode tip, iii) length of the spinneret, iv) length of the electrode, v) radius of the spinneret, and vi) radius of the electrode. The design values are used as the system parameters when fabricating the selected fiber mat design. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a schematic view of a system for producing aligned electrospun fibers; 
         FIG. 2  is a side view of a portion of the system in  FIG. 1  taken along line 2-2 thereof and illustrating positions for the fiberizable material dispenser and the electrode in accordance with an embodiment of the system, and 
         FIG. 3  is a diagrammatic representation of the fiberizable material dispenser, collector, and electrode illustrating various system parameter relationships. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Prior to describing the method of the present invention, an exemplary electrospinning system will be described. This electrospinning system is one that can benefit from the novel system parameter optimization scheme of the present invention. The electrospinning system shown and described herein has been previously disclosed in the afore cited U.S. patent application Ser. No. 12/131,420, filed Jun. 2, 2008. 
     Referring now to the drawings and more particularly to  FIG. 1 , the exemplary electrospinning system for fabricating a mat of aligned fibers is shown and is referenced generally by numeral  10 . For simplicity of discussion, system  10  will be described for its use in producing a single-ply mat with aligned single fibers or fiber bundles that are substantially parallel to one another. However, as will be explained further below, the system can also be used to produce a multiple-ply mat where fiber orientation between adjacent plies is different to thereby create a porous multi-ply mat. Such multi-ply porous mats could be used in a variety of industries/applications, as would be understood by one of ordinary skill in the art. 
     In general, system  10  includes a dispenser  12  capable of discharging a fiberizable material  14  therefrom in jet stream form (as indicated by arrow  14 A) that will be deposited as a single fiber or fiber bundles (not shown) on a collector  16 . Dispenser  12  is typically a spinneret through which fiberizable material  14  is pumped, as is well known in the art of electrospinning. The type and construction of dispenser  12  will dictate whether a single fiber or fiber bundles are deposited on collector  16 . Fiberizable material  14  is any viscous solution that will form a fiber after being discharged from dispenser  12  and deposited on collector  16 . Typically, material  14  includes a polymeric material and can include disparate material fillers mixed therein to give the resulting fiber desired properties. Collector  16  can be a static plate, a wire mesh, a moving-conveyor-type collector, or a rotating drum fabricated in a variety of shapes and configurations, the choice of which is not a limitation of the present invention. For the illustrated example, collector  16  will be rotated about its longitudinal axis  16 A as indicated by rotational arrow  16 B. Collector  16  is maintained in an electrical uncharged state (e.g., floating or coupled to an electric ground potential  18  as illustrated). The fiber deposition surface of collector  16  can be electrically conductive, semi-conductive, or non-conductive. 
     Dispenser  12  is positioned such that its dispensing aperture  12 A faces collector  16  a short distance therefrom as would be understood in the electrospinning art. For example, if dispenser  12  is a spinneret, aperture  12 A represents the exit opening of the spinneret. In the present invention, the portion of dispenser  12  defining aperture  12 A should be electrically conductive. Typically, dispenser  12  is a “needle electrode.” As is known in the art, a needle electrode is essentially a hollow tube made from an electrically conductive material. A voltage source  20  is coupled to dispenser  12  such that an electric charge is generated at the portion of dispenser  12  defining aperture  12 A. 
     Positioned near collector  16  and within the line-of-sight of aperture  12 A is an electrode  22 . More specifically, a tip  22 A of electrode  22  is positioned within line-of-sight of aperture  12 A as is readily seen in  FIG. 2  where dashed line  24  indicates the line-of-sight communication between aperture  12 A and electrode tip  22 A. A voltage source  26  is coupled to electrode  22  such that an electric charge is generated at electrode tip  22 A. The charge is opposite in polarity to that of the charge on the portion of dispenser  12  defining aperture  12 A. That is, if the charge is positive at aperture  12 A (as indicated), the charge should be negative at electrode tip  22 A (as illustrated) Similarly, if the charge is negative at aperture  12 A, the charge should be positive at electrode tip  22 A. The magnitude of the voltages applied to dispenser  12  and electrode  22  can be the same or different, although they are typically the same. 
     The opposite-polarity charges at dispenser aperture  12 A and electrode tip  22 A cause an elliptical electric field to be generated therebetween as represented by dashed lines  30 . Typically, aperture  12 A and electrode tip  22 A will be circular, and they can be the same or different in terms of their size. Since aperture  12 A and electrode tip  22 A are in line-of-sight of one another, some portion of electric field  30  will impinge upon the surface of collector  16 . This will be true whether electrode tip  22 A is positioned centrally with respect to collector  16  (as illustrated), or at any position along collector  16 . For purpose of an illustrated example, dispenser  12  is a cylindrical needle electrode while electrode  22  is a cylindrical electrode having the same outer dimensions as dispenser  12 . Further, aperture  12 A and electrode tip  22 A are aligned along an axis referenced by line-of-sight communication line  24 . 
     In operation, dispenser  12  and electrode  22  are positioned with respect to collector  16  as described above. Opposite-polarity voltages are applied to dispenser  12  and electrode  22  in order to establish electric field  30  with at least a portion of collector  16  being disposed in electric field  30 . Fiberizable material  14  is plumped from dispenser  12  such that a jet stream  14 A thereof is subject to electric field  30 . A pulsed electric field, generated for example by pulsing the voltages applied to dispenser  12  and electrode  22 , may also be used. 
     As mentioned above, the present invention is a method of predicting and optimizing the various physical system parameters for an electrospinning system such as the one described herein. A diagrammatic representation of dispenser  12  (e.g., a cylindrical needle electrode), collector  16  (e.g., a rotating drum), and electrode  22  (e.g., a cylindrical electrode), is illustrated in  FIG. 3  with various system parameters being denoted. It is to be understood that relative sizes of and distances between dispenser  12 , collector  16 , and electrode  22  are not to scale as they are merely sized and positioned to facilitate a description of the present invention. The line-of-sight communication axis  24  forms the x-axis for the relationships discussed below. The y-axis denotes the reference direction for the width of the fiber mat (not shown) that gets deposited on collector  16  during the electrospinning process. 
     The external dimensions of dispenser  12  and electrode  22  are the same for the following explanation where the length of cylindrical dispenser  12  and cylindrical electrode  22  is “L”, and the distance between dispenser aperture  12 A and electrode tip  22 A is “D”. These parameters are illustrated along the x-axis and are referenced to an origin defined at dispenser aperture  12 A. Points in a spatial region of free-space between dispenser aperture  12 A and electrode tip  22 A are referenced by coordinate (x′,y′) The charge density on dispenser  12  due to an applied voltage is “ρ”, and the charge density on electrode  22  due to an equal and opposite applied voltage is “−ρ”. The external radius of dispenser  12  and electrode  22  is “R”, 
     Using an electrospinning system as described above, the present invention first requires an exemplary test run of the system in order to generate a sample fiber mat where the width dimension thereof is used in the predicting/optimizing scheme. Briefly and with simultaneous reference to  FIGS. 1-3 , system  10  is operated for some short and fixed period of time (e.g., on the order of seconds) with the various system parameters being known. That is, system  10  is set up such that voltage sources  20  and  26  apply equal and opposite voltages to dispenser  12  and electrode  22 , respectively. Further, distance D is known, length L is known (and the same for dispenser  12  and electrode  22  in this example), and the radius R of dispenser  12  and electrode  22  is known (and the same in this example). As a result of this operation, a sample fiber mat (not shown) will be deposited on collector  16 . The width of the fiber mat along the axial length of collector  16  (i.e., perpendicular to axis  24 ) is measured and is designated herein as “y N ”. 
     In the remaining steps of the present invention, well known electric field/potential relationships (as they apply to electrospinning) and a novel particle acceleration model are used to predict and optimize various system parameters when a particular fiber mat design is to be fabricated. The development of the model will now be explained. 
     The electric field generated between dispenser aperture  12 A and electrode tip  22  is the negative gradient of the electric potential, given by the well known relationship
 
 E=−∇V    (1)
 
     where E is the electric field and V is the electric potential that can be calculated for points in the free-space region between dispenser aperture  12 A and electrode tip  22 A in accordance with 
     
       
         
           
             
               
                 
                   
                     V 
                     ⁡ 
                     
                       ( 
                       
                         x 
                         , 
                         y 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       
                         ɛ 
                         0 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             q 
                             1 
                           
                           
                             r 
                             1 
                           
                         
                         + 
                         
                           
                             q 
                             2 
                           
                           
                             r 
                             2 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     where q 1  is the charge on dispenser  12  for a given applied voltage, 
     q 2  is the charge on electrode  22  for a given applied voltage, 
     r 1  is the distance from the charge at dispenser  12  to the location (x,y) in the free-space region, 
     r 2  is the distance from the charge at electrode  22  to the location (x,y) in the free-space region, and 
               ɛ   o     =     8.8541878176   ×     10     -   12       ⁢       C   2       J   ·   m               
is the permittivity of free space.
 
     For the exemplary arrangement at some point (x′,y′) in the free-space region, 
     
       
         
           
             
               
                 
                   
                     V 
                     ⁡ 
                     
                       ( 
                       
                         
                           x 
                           ′ 
                         
                         , 
                         
                           y 
                           ′ 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ρ 
                         
                           ɛ 
                           0 
                         
                       
                       ⁢ 
                       
                         
                           ∫ 
                           
                             - 
                             L 
                           
                           0 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             ⅆ 
                             x 
                           
                           
                             
                               ( 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         x 
                                         ′ 
                                       
                                       - 
                                       x 
                                     
                                     ) 
                                   
                                   2 
                                 
                                 + 
                                 
                                   y 
                                   ′2 
                                 
                               
                               ) 
                             
                             
                               1 
                               / 
                               2 
                             
                           
                         
                       
                     
                     + 
                     
                       
                         - 
                         ρ 
                       
                       ⁢ 
                       
                         
                           ∫ 
                           D 
                           
                             D 
                             + 
                             L 
                           
                         
                         ⁢ 
                         
                           
                             
                                 
                             
                             ⁢ 
                             
                               ⅆ 
                               x 
                             
                           
                           
                             
                               ( 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         x 
                                         ′ 
                                       
                                       - 
                                       x 
                                     
                                     ) 
                                   
                                   2 
                                 
                                 + 
                                 
                                   y 
                                   ′2 
                                 
                               
                               ) 
                             
                             
                               1 
                               / 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     where the charge density ρ is calculated based upon the required voltage to bring the potential on dispenser  12  and electrode  22  to the operating voltage V O . The charge density is given by 
     
       
         
           
             
               
                 
                   ρ 
                   = 
                   
                     
                       ± 
                       
                         V 
                         O 
                       
                     
                     ⁢ 
                     
                       
                         ɛ 
                         0 
                       
                       / 
                       
                         
                           ∫ 
                           
                             
                               - 
                               L 
                             
                             / 
                             2 
                           
                           
                             L 
                             / 
                             2 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             ⅆ 
                             x 
                           
                           / 
                           
                             
                               ( 
                               
                                 
                                   x 
                                   2 
                                 
                                 + 
                                 
                                   R 
                                   2 
                                 
                               
                               ) 
                             
                             
                               1 
                               / 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     In these equations for the exemplary arrangement, D is the distance between dispenser aperture  12 A and electrode tip  22 A, L is the length of dispenser  12  and electrode  22 , R is the radius of dispenser  12  and electrode  22 , and 
               ɛ   o     =     8.8541878176   ×     10     -   12       ⁢       C   2       J   ·   m               
is the permittivity of free space.
 
     By assuming that the charge q 0  on a droplet of polymer at dispenser aperture  12 A is that required to bring the surface potential to the operating voltage, all parameters needed to calculate the electrostatic force “F” throughout the above-defined free-space region can be defined. The acceleration vector “A” for the polymer droplet can be written in accordance with the well known relationship 
     
       
         
           
             
               
                 
                   A 
                   = 
                   
                     
                       F 
                       m 
                     
                     = 
                     
                       
                         
                           q 
                           0 
                         
                         ⁢ 
                         E 
                       
                       m 
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     where “m” is the mass of the polymer particle. 
     In addition to the electrostatic forces, the polymer kinetics are dependent upon drag and the surface tension of the polymer as it exits dispenser  12 . In the exemplary system described above, these effects can be modeled as additional forces on the polymer droplet. Drag “μ” is modeled as a force proportional to the square of the velocity “v” of the droplet in the opposite direction of the droplet&#39;s velocity vector “v”. Surface tension “σ” is modeled as a force inversely proportional to the cube of the distance “d” between dispenser aperture  12 A and the droplet along the vector “d” from the droplet to dispenser aperture  12 A. Thus, the novel acceleration model applied in the present invention models the kinetics of the polymer during electrospinning as follows 
     
       
         
           
             
               
                 
                   
                     
                       A 
                       i 
                     
                     = 
                     
                       
                         1 
                         m 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               q 
                               0 
                             
                             ⁢ 
                             E 
                           
                           - 
                           
                             μ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               v 
                               i 
                               2 
                             
                             ⁢ 
                             
                               
                                 v 
                                 i 
                               
                               
                                  
                                 
                                   v 
                                   i 
                                 
                                  
                               
                             
                           
                           - 
                           
                             
                               σ 
                               
                                 d 
                                 i 
                                 3 
                               
                             
                             ⁢ 
                             
                               
                                 d 
                                 i 
                               
                               
                                  
                                 
                                   d 
                                   i 
                                 
                                  
                               
                             
                           
                         
                         ) 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     6 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     a 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       v 
                       
                         i 
                         + 
                         1 
                       
                     
                     = 
                     
                       
                         
                           A 
                           i 
                         
                         ⁢ 
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         t 
                       
                       + 
                       
                         v 
                         i 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     6 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     b 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       d 
                       
                         i 
                         + 
                         1 
                       
                     
                     = 
                     
                       
                         
                           A 
                           i 
                         
                         ⁢ 
                         
                           
                             
                               ( 
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 t 
                               
                               ) 
                             
                             2 
                           
                           2 
                         
                       
                       + 
                       
                         
                           v 
                           1 
                         
                         ⁡ 
                         
                           ( 
                           
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             t 
                           
                           ) 
                         
                       
                       + 
                       
                         d 
                         i 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     6 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     c 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     d 
                     n 
                   
                   = 
                   
                     
                       
                         x 
                         n 
                       
                       ⁢ 
                       x 
                     
                     + 
                     
                       
                         y 
                         n 
                       
                       ⁢ 
                       y 
                     
                   
                 
               
               
                 
                   ( 
                   
                     6 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     d 
                   
                   ) 
                 
               
             
           
         
       
     
     where q 0  is the charge on the droplet exiting dispenser aperture  12 A, 
     E is an electric field between dispenser  12 A and electrode  22 , 
     v i  is the velocity of the droplet at an instant (Δt*i) in a fixed amount of system operating time, 
     v i  is the velocity vector at the i-th instant, 
     d i  is a distance from dispenser aperture  12 A to the droplet at the i-th instant, 
     d i  is the distance vector associated with the distance d i , 
     x is a unit vector aligned with the x-axis defined by line-of-sight axis  24 , 
     y is a unit vector perpendicular to the x-axis, 
     x n  is equal to the distance D, and 
     y n  is equal to the width of the fiber mat deposited on collector  16  during the fixed amount of system operating time. 
     In accordance with the present invention, the particle acceleration model presented in equations (6a)-(6d) is first used in an iteration process. Specifically, the model is iterated over the amount of time used to create the sample fiber mat in order to generate values for mass m, drag μ, and surface tension σ that will yield, at the n-th time step, a calculated fiber mat width y n  that is equal to (or within an acceptable tolerance) of the sample fiber mat width y M . As would be understood by one of ordinary skill in the art, the iteration process begins with some selected initial values for mass, drag, and surface tension. 
     Following the iteration process, the determined values for mass, drag, and surface tension are used in an inversion application of the particle acceleration model that yields optimized predictions of system parameters. More specifically, the inversion application solves the particle acceleration model using a combination of (i) a value for y n  that is set equal to a desired fiber mat width, and (ii) the determined values of mass, drag, and surface tension. Solving the model with these given parameter values yields both the required charge and the electric field. The above-described equations (1)-(4) are then used in a straight-forward fashion to define the operating voltages V O , distance D, length L, and radius R. 
     The present invention is further described in Carnell, Lisa S.; Wincheski, Russell A.; Siochi, Emilie, J.; Holloway, Nancy M.; and Clark, Robert L., “Electric Field Effects on Fiber Alignment Using an Auxiliary Electrode during Electrospinning,” 2007 Materials Research Society (MRS) Fall Meeting, 29 Nov. 2007, Boston, Mass., the contents of which are hereby incorporated by reference in their entirety. 
     The advantages of the present invention are numerous. Parameter prediction and optimization for a recently-developed electrospinning technique will enhance the value thereof. The results of a single sample run for the electrospinning system in combination with a novel particle acceleration model will allow system parameters to be defined without time-consuming trial-and-error processing. 
     Although the invention has been described relative to a specific embodiment thereof, there are numerous variations and modifications that will be readily apparent to those skilled in the art in light of the above teachings. The present invention can be readily extended to electrospinning systems using a dispenser and electrode of differing length and/or radius dimensions. For example, if the lengths are different, the first integral in equation (3) is bounded on one side by −L 1 , and the second integral in equation (3) is bounded on one side by D+L 2 , where L 1  is the length of dispenser  12  and L 2  is the length of electrode  22 . If the radius dimensions are different, equation (4) is calculated twice, i.e., one time to generate a charge density for dispenser  12  using the radius thereof and the potential applied thereto, and a second time to generate a charge density for electrode  22  using the radius thereof and the potentials applied thereto. The “dispenser” charge density would then be used for the first term in equation (3), while the “electrode” charge density would then be used for the second term of equation (3). It is therefore to be understood that, within the scope of the appended claims, the invention may be practiced other than as specifically described.

Technology Classification (CPC): 3