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Ion Funnel
Introduction
An electrodynamic ion funnel provides an efficient means of transferring ions from regions of high pressure to high vacuum. The ion funnel can couple devices which generally operate at pressures of different orders of magnitude, such as ion mobility spectrometers and mass spectrometers, allowing mixtures of ionized gases to be separated and analyzed while minimizing losses. For this reason, ion funnels can be used to improve instrument sensitivity in a wide variety of applications, such as the analysis of complex biological molecules or the detection of explosives.
This example uses the Charged Particle Tracing interface to model the movement of ions through an electrodynamic ion funnel. The Electric Force feature is used to apply DC and RF potentials to guide ions through the funnel, while the Elastic Collision Force feature is used to model collisions with background gas molecules.
Note: This application requires the Particle Tracing Module.
Model Definition
The ion funnel is a converging series of insulated ring-shaped electrodes, each subjected to an RF potential, with adjacent electrodes out of phase. The RF potential confines ions radially while a DC bias guides the ions toward successively narrower electrodes. The superposition of the DC and RF fields focuses the ions and offsets the effects of thermal dispersion and Coulombic repulsion.
The ion funnel contains a buffer gas of neutral argon at 1 Torr, which is assumed to follow a Maxwellian velocity distribution:
where
•
vi (SI unit: m/s) is the ith velocity component,
•
mp (SI unit: kg) is the molecular mass,
•
T0 (SI unit: K) is the temperature, and
•
k = 1.3806488 × 10−23 J/K is the Boltzmann constant.
In this example, the background gas molecules have molar mass 0.04 kg/mol, for a molecular mass of 6.6422 × 10−26 kg. The temperature is assumed to be 293.15 K.
The number density of the gas can be computed using the Ideal Gas Law,
where R = 8.3144621 J/(mol·K) is the universal gas constant and NA = 6.02214129 × 1023 1/mol is the Avogadro constant. For a pressure of 1 Torr this yields n = 3.294 × 1022 atoms/m3.
The interaction of ions with the background gas is modeled using the Collisions node with an Elastic subnode. At each time step taken by the solver, for each model particle a background gas particle is sampled at random from the Maxwellian distribution. The frequency of elastic collisions is then computed from the collision cross-section, background gas number density, and the relative velocity of the model particle with respect to the randomly sampled background gas molecule:
where the collision cross section σ (SI unit: m2) is usually a function of the particle kinetic energy. The collision probability is then computed as a function of the collision frequency and the time step size:
The model uses three physics interfaces: Electrostatics, Electric Currents, and Charged Particle Tracing. The Electrostatics and Electric Currents interfaces are used to compute the DC and AC fields, respectively. These fields are then coupled to the Charged Particle Tracing interface, which models the motion of the ions due to the electric fields and interaction with neutral particles in the background gas. Interactions between the ions are neglected. In order to accurately model collisions of ions with the background gas, the average time between elastic collisions should be significantly greater than the maximum time step taken by the solver. Strict or manual time stepping is recommended.
Results and Discussion
The electric potential in the ion funnel is plotted in Figure 1. The gradual DC bias guides positive ions from the larger end of the funnel to the smaller end. The AC voltage, which is out of phase between adjacent electrodes, causes the gradient of the electric potential to become very large close to the electrodes, keeping the ions confined within the funnel.
Figure 1: The combined DC and AC potential is plotted in the ion funnel at time t=0.
The ion trajectories are plotted in Figure 2. Because the ions are confined to an area of reduced size, they can be transported to another device, such as a mass spectrometer, more efficiently.
The x- and y-coordinates of the particles at the narrow end of the funnel are plotted in Figure 3. Although the ions are released along the positive x-axis, they are uniformly distributed around the z-axis by the time they reach the end of the funnel. Because the Collisions node uses random numbers to determine whether a collision takes place at each time step, the results may be slightly different from those shown in Figure 2 and Figure 3.
Figure 2: The trajectories of positive ions in the funnel.
Figure 3: Phase portrait showing the x- and y-coordinates of the ions at the narrow end of the funnel. The color expression is red for particles that have exited the funnel and blue otherwise.
Reference
1. A. V. Phelps, “The application of scattering cross sections to ion flux models in discharge sheaths,” J. Appl. Phys. vol. 76, pp. 747-753, 1994.
Application Library path: ACDC_Module/Particle_Tracing/ion_funnel
Modeling Instructions
From the File menu, choose New.
New
In the New window, click Model Wizard.
Model Wizard
1
In the Model Wizard window, click 2D Axisymmetric.
2
In the Select Physics tree, select AC/DC>Electric Fields and Currents>Electrostatics (es).
3
Click Add.
4
In the Select Physics tree, select AC/DC>Electric Fields and Currents>Electric Currents (ec).
5
Click Add.
6
In the Select Physics tree, select AC/DC>Particle Tracing>Charged Particle Tracing (cpt).
7
Click Add.
8
Click Study.
9
In the Select Study tree, select Preset Studies for Some Physics Interfaces>Stationary.
10
Click Done.
Global Definitions
To save time, load the parameters from a file.
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
Click Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file ion_funnel_parameters.txt.
Geometry 1
1
In the Model Builder window, under Component 1 (comp1) click Geometry 1.
2
In the Settings window for Geometry, locate the Units section.
3
From the Length unit list, choose mm.
Rectangle 1 (r1)
1
In the Geometry toolbar, click Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type welec.
4
In the Height text field, type telec.
5
Locate the Position section. In the r text field, type rmax.
Rectangle 2 (r2)
1
In the Geometry toolbar, click Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type welec.
4
In the Height text field, type telec.
5
Locate the Position section. In the r text field, type rmax.
6
In the z text field, type telec+tgap.
Create arrays containing the electrodes which are in phase with each other. This setup allows the number of electrodes to be changed at any time by changing the corresponding parameters.
Array 1 (arr1)
1
In the Geometry toolbar, click Transforms and choose Array.
2
Select the object r1 only.
3
In the Settings window for Array, locate the Size section.
4
From the Array type list, choose Linear.
5
In the Size text field, type (Nstraight+1)/2.
6
Locate the Displacement section. In the z text field, type 2*(telec+tgap).
7
Locate the Selections of Resulting Entities section. Select the Resulting objects selection check box.
8
From the Show in physics list, choose Boundary selection.
9
Click Build Selected.
Array 2 (arr2)
1
In the Geometry toolbar, click Transforms and choose Array.
2
Select the object r2 only.
3
In the Settings window for Array, locate the Size section.
4
From the Array type list, choose Linear.
5
In the Size text field, type (Nstraight-1)/2.
6
Locate the Displacement section. In the z text field, type 2*(telec+tgap).
7
Locate the Selections of Resulting Entities section. Select the Resulting objects selection check box.
8
From the Show in physics list, choose Boundary selection.
9
Click Build Selected.
Rectangle 3 (r3)
1
In the Geometry toolbar, click Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type welec.
4
In the Height text field, type telec.
5
Locate the Position section. In the r text field, type rmin.
6
In the z text field, type hfunnel-telec.
Rectangle 4 (r4)
1
In the Geometry toolbar, click Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type welec.
4
In the Height text field, type telec.
5
Locate the Position section. In the r text field, type rmin+(rmax-rmin)/Ninclined.
6
In the z text field, type hfunnel-(2*telec+tgap).
7
Click the Zoom Extents button in the Graphics toolbar.
Array 3 (arr3)
1
In the Geometry toolbar, click Transforms and choose Array.
2
Select the object r3 only.
3
In the Settings window for Array, locate the Size section.
4
From the Array type list, choose Linear.
5
In the Size text field, type (Ninclined+1)/2.
6
Locate the Displacement section. In the r text field, type 2*(rmax-rmin)/Ninclined.
7
In the z text field, type -2*(telec+tgap).
8
Locate the Selections of Resulting Entities section. Select the Resulting objects selection check box.
9
From the Show in physics list, choose Boundary selection.
Array 4 (arr4)
1
In the Geometry toolbar, click Transforms and choose Array.
2
Select the object r4 only.
3
In the Settings window for Array, locate the Size section.
4
From the Array type list, choose Linear.
5
In the Size text field, type (Ninclined-1)/2.
6
Locate the Displacement section. In the r text field, type 2*(rmax-rmin)/Ninclined.
7
In the z text field, type -2*(telec+tgap).
8
Locate the Selections of Resulting Entities section. Select the Resulting objects selection check box.
9
From the Show in physics list, choose Boundary selection.
Add a rectangle to enclose the modeling domain. Note that the rectangle is positioned a small distance away from the axis of symmetry. This makes the particle tracing model more robust, since the centrifugal force acting on the particles approaches infinity as the radial coordinate approaches zero.
Rectangle 5 (r5)
1
In the Geometry toolbar, click Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type rmax+welec+10[mm].
4
In the Height text field, type hfunnel+20[mm].
5
Locate the Position section. In the r text field, type 0.2[mm].
6
In the z text field, type -10[mm].
7
Click Build All Objects.
8
Click the Zoom Extents button in the Graphics toolbar.
Definitions
Union 1
1
In the Definitions toolbar, click Union.
2
In the Settings window for Union, type Odd electrodes in the Label text field.
3
Locate the Geometric Entity Level section. From the Level list, choose Boundary.
4
Locate the Input Entities section. Under Selections to add, click Add.
5
In the Add dialog box, in the Selections to add list, choose Array 1 and Array 4.
6
Click OK.
Union 2
1
In the Definitions toolbar, click Union.
2
In the Settings window for Union, type Even electrodes in the Label text field.
3
Locate the Geometric Entity Level section. From the Level list, choose Boundary.
4
Locate the Input Entities section. Under Selections to add, click Add.
5
In the Add dialog box, in the Selections to add list, choose Array 2 and Array 3.
6
Click OK.
Union 3
1
In the Definitions toolbar, click Union.
2
In the Settings window for Union, type All electrodes in the Label text field.
3
Locate the Geometric Entity Level section. From the Level list, choose Boundary.
4
Locate the Input Entities section. Under Selections to add, click Add.
5
In the Add dialog box, in the Selections to add list, choose Odd electrodes and Even electrodes.
6
Click OK.
Explicit 1
1
In the Definitions toolbar, click Explicit.
2
In the Settings window for Explicit, type Exterior boundaries in the Label text field.
3
Locate the Input Entities section. From the Geometric entity level list, choose Boundary.
4
Select Boundaries 2, 3, and 144 only.
Union 4
1
In the Definitions toolbar, click Union.
2
In the Settings window for Union, type All boundaries in the Label text field.
3
Locate the Geometric Entity Level section. From the Level list, choose Boundary.
4
Locate the Input Entities section. Under Selections to add, click Add.
5
In the Add dialog box, in the Selections to add list, choose All electrodes and Exterior boundaries.
6
Click OK.
Materials
Material 1 (mat1)
1
In the Model Builder window, under Component 1 (comp1) right-click Materials and choose Blank Material.
2
In the Settings window for Material, locate the Material Contents section.
3
In the table, enter the following settings:
Property
Variable
Value
Unit
Property group
Relative permittivity
epsilonr
1
1
Basic
Electrical conductivity
sigma
0
S/m
Basic
Electrostatics (es)
1
In the Model Builder window, under Component 1 (comp1) click Electrostatics (es).
2
In the Settings window for Electrostatics, locate the Domain Selection section.
3
Click Clear Selection.
4
Select Domain 1 only.
Electric Potential 1
1
In the Physics toolbar, click Boundaries and choose Electric Potential.
2
In the Settings window for Electric Potential, locate the Boundary Selection section.
3
From the Selection list, choose All boundaries.
4
Locate the Electric Potential section. In the V0 text field, type EDC*z.
Electric Currents (ec)
1
In the Model Builder window, under Component 1 (comp1) click Electric Currents (ec).
2
In the Settings window for Electric Currents, locate the Domain Selection section.
3
Click Clear Selection.
4
Select Domain 1 only.
Terminal 1
1
In the Physics toolbar, click Boundaries and choose Terminal.
2
In the Settings window for Terminal, locate the Boundary Selection section.
3
From the Selection list, choose Odd electrodes.
4
Locate the Terminal section. From the Terminal type list, choose Voltage.
5
In the V0 text field, type Vpp.
Terminal 2
1
In the Physics toolbar, click Boundaries and choose Terminal.
2
In the Settings window for Terminal, locate the Boundary Selection section.
3
From the Selection list, choose Even electrodes.
4
Locate the Terminal section. From the Terminal type list, choose Voltage.
5
In the V0 text field, type -Vpp.
Charged Particle Tracing (cpt)
1
In the Model Builder window, under Component 1 (comp1) click Charged Particle Tracing (cpt).
2
In the Settings window for Charged Particle Tracing, locate the Domain Selection section.
3
Click Clear Selection.
4
Select Domain 1 only.
5
Locate the Additional Variables section. Select the Include out-of-plane degrees of freedom check box.
Particle Properties 1
1
In the Model Builder window, under Component 1 (comp1)>Charged Particle Tracing (cpt) click Particle Properties 1.
2
In the Settings window for Particle Properties, locate the Particle Mass section.
3
In the mp text field, type 0.146[kg/mol]/N_A_const.
4
Locate the Charge Number section. In the Z text field, type 1.
Wall 2
1
In the Physics toolbar, click Boundaries and choose Wall.
2
Select Boundary 1 only.
3
In the Settings window for Wall, locate the Wall Condition section.
4
From the Wall condition list, choose Bounce.
Release from Grid 1
1
In the Physics toolbar, click Global and choose Release from Grid.
2
In the Settings window for Release from Grid, locate the Initial Coordinates section.
3
Click R Range.
4
In the Range dialog box, type 2 in the Start text field.
5
In the Step text field, type 1.
6
In the Stop text field, type 20.
7
Click Replace.
8
In the Settings window for Release from Grid, locate the Initial Velocity section.
9
From the Initial velocity list, choose Maxwellian.
10
In the Nv text field, type 5.
Electric Force 1
1
In the Physics toolbar, click Domains and choose Electric Force.
2
Select Domain 1 only.
3
In the Settings window for Electric Force, locate the Electric Force section.
4
From the E list, choose Electric field (es/ccn1).
5
Locate the Advanced Settings section. Select the Use piecewise polynomial recovery on field check box.
Electric Force 2
1
In the Physics toolbar, click Domains and choose Electric Force.
2
Select Domain 1 only.
3
In the Settings window for Electric Force, locate the Electric Force section.
4
From the E list, choose Electric field (ec/cucn1).
5
Locate the Advanced Settings section. From the Time dependence of field list, choose Time harmonic.
6
Select the Use piecewise polynomial recovery on field check box.
Now define the cross sections used for the collision model.
Definitions
Enter the analytic approximation for momentum cross section for elastic scattering between SF6+ ions and neutral Ar atoms from Ref. 1, which depends on the kinetic energy of the particles.
Analytic 1 (an1)
1
In the Home toolbar, click Functions and choose Global>Analytic.
2
In the Settings window for Analytic, type Qm in the Function name text field.
3
Locate the Definition section. In the Expression text field, type 1.15e-18*x^(-0.1)*(1+0.015/x)^0.6.
4
Locate the Units section. In the Arguments text field, type eV.
5
In the Function text field, type m^2.
Enter the analytic approximation for isotropic elastic collision between SF6+ ions and neutral Ar atoms from Ref. 1, which depends on the kinetic energy of the particles.
Analytic 2 (an2)
1
In the Home toolbar, click Functions and choose Global>Analytic.
2
In the Settings window for Analytic, type Qi in the Function name text field.
3
Locate the Definition section. In the Expression text field, type 2e-19/(x^(0.5)*(1+x))+3e-19*x/(1+x/3)^(2.3).
4
Locate the Units section. In the Arguments text field, type eV.
5
In the Function text field, type m^2.
Charged Particle Tracing (cpt)
Collisions 1
1
In the Physics toolbar, click Domains and choose Collisions.
2
Select Domain 1 only.
3
In the Settings window for Collisions, locate the Fluid Properties section.
4
In the Nd text field, type ND.
5
Locate the Collision Statistics section. Select the Count all collisions check box.
Elastic 1
1
In the Physics toolbar, click Attributes and choose Elastic.
2
In the Settings window for Elastic, locate the Collision Frequency section.
3
In the σ text field, type Qi(cpt.Ep).
4
Locate the Collision Statistics section. Select the Count collisions check box.
Collisions 1
In the Model Builder window, click Collisions 1.
Resonant Charge Exchange 1
1
In the Physics toolbar, click Attributes and choose Resonant Charge Exchange.
2
In the Settings window for Resonant Charge Exchange, locate the Collision Frequency section.
3
In the σ text field, type (Qm(cpt.Ep)-Qi(cpt.Ep))/2.
4
Locate the Collision Statistics section. Select the Count collisions check box.
Study 1
Step 1: Stationary
1
In the Model Builder window, under Study 1 click Step 1: Stationary.
2
In the Settings window for Stationary, locate the Physics and Variables Selection section.
3
In the table, clear the Solve for check box for Electric Currents and Charged Particle Tracing.
Frequency Domain
1
In the Study toolbar, click Study Steps and choose Frequency Domain>Frequency Domain.
2
In the Settings window for Frequency Domain, locate the Study Settings section.
3
In the Frequencies text field, type f0.
4
Locate the Physics and Variables Selection section. In the table, clear the Solve for check box for Electrostatics and Charged Particle Tracing.
5
Click to expand the Values of Dependent Variables section. Find the Values of variables not solved for subsection. From the Settings list, choose User controlled.
6
From the Method list, choose Solution.
7
From the Study list, choose Study 1, Stationary.
Step 1: Stationary
1
In the Model Builder window, click Step 1: Stationary.
2
In the Settings window for Stationary, click to expand the Values of Dependent Variables section.
3
In the Study toolbar, click Compute.
Results
Electric Potential (es)
Create a contour plot of the electric potential when t = 0.
2D Plot Group 5
1
In the Home toolbar, click Add Plot Group and choose 2D Plot Group.
2
In the Settings window for 2D Plot Group, type Sum of AC and DC Potentials in the Label text field.
Contour 1
1
Right-click Sum of AC and DC Potentials and choose Contour.
2
In the Settings window for Contour, locate the Expression section.
3
In the Expression text field, type V+V2.
4
Locate the Levels section. In the Total levels text field, type 30.
5
Locate the Coloring and Style section. From the Contour type list, choose Filled.
6
Clear the Color legend check box.
7
In the Sum of AC and DC Potentials toolbar, click Plot.
8
Click the Zoom Extents button in the Graphics toolbar. Compare this plot to Figure 1. The plot indicates that the potential gradient is steep in the area surrounding the electrodes. The large electric field magnitude in this area confines ions within the funnel.
Add Study
1
In the Home toolbar, click Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select General Studies>Time Dependent.
4
Find the Physics interfaces in study subsection. In the table, clear the Solve check box for Electrostatics (es) and Electric Currents (ec).
5
Click Add Study in the window toolbar.
6
In the Home toolbar, click Add Study to close the Add Study window.
Study 2
Step 1: Time Dependent
1
In the Settings window for Time Dependent, locate the Study Settings section.
2
Click Range.
3
In the Range dialog box, type 1e-5 in the Step text field.
4
In the Stop text field, type 15e-4.
5
Click Replace.
6
In the Settings window for Time Dependent, locate the Study Settings section.
7
From the Tolerance list, choose User controlled.
8
In the Relative tolerance text field, type 1e-3.
9
Click to expand the Values of Dependent Variables section. Find the Values of variables not solved for subsection. From the Settings list, choose User controlled.
10
From the Method list, choose Solution.
11
From the Study list, choose Study 1, Frequency Domain.
Solution 3 (sol3)
1
In the Study toolbar, click Show Default Solver.
2
In the Model Builder window, expand the Solution 3 (sol3) node, then click Time-Dependent Solver 1.
3
In the Settings window for Time-Dependent Solver, click to expand the Absolute Tolerance section.
4
From the Tolerance method list, choose Manual.
5
In the Absolute tolerance text field, type 1e-4.
6
Click to expand the Time Stepping section. From the Steps taken by solver list, choose Strict.
7
From the Maximum step constraint list, choose Constant.
8
In the Maximum step text field, type 1e-8.
9
Click to expand the Output section. Clear the Store reaction forces check box.
10
Clear the Store time derivatives check box.
11
In the Model Builder window, expand the Study 2>Solver Configurations>Solution 3 (sol3)>Time-Dependent Solver 1 node, then click Advanced.
12
In the Settings window for Advanced, locate the General section.
13
From the Solver log list, choose Minimal.
14
In the Study toolbar, click Compute.
Results
Particle Trajectories 1
1
In the Model Builder window, expand the Particle Trajectories (cpt) node, then click Particle Trajectories 1.
2
In the Settings window for Particle Trajectories, locate the Coloring and Style section.
3
Find the Line style subsection. From the Type list, choose Line.
4
In the Particle Trajectories (cpt) toolbar, click Plot. Compare the result to Figure 2.
2D Plot Group 7
1
In the Home toolbar, click Add Plot Group and choose 2D Plot Group.
2
In the Settings window for 2D Plot Group, type Transverse Particle Positions in the Label text field.
3
Locate the Data section. From the Dataset list, choose Particle 1.
4
From the Time (s) list, choose 0.0015.
5
Locate the Plot Settings section. Clear the Plot dataset edges check box.
Phase Portrait 1
1
In the Transverse Particle Positions toolbar, click More Plots and choose Phase Portrait.
2
In the Settings window for Phase Portrait, locate the Expression section.
3
From the x-axis list, choose Manual.
4
In the Expression text field, type qr*cos(qphi).
5
From the y-axis list, choose Manual.
6
In the Expression text field, type qr*sin(qphi).
Color Expression 1
1
Right-click Phase Portrait 1 and choose Color Expression.
2
In the Settings window for Color Expression, locate the Expression section.
3
In the Expression text field, type qz>120[mm].
4
In the Transverse Particle Positions toolbar, click Plot.
5
Click the Zoom Extents button in the Graphics toolbar. Compare the resulting plot to Figure 3.
Export
Create an animation showing the x- and y-components of the particle positions over time.
Animation 1
1
In the Results toolbar, click Animation and choose File.
2
In the Settings window for Animation, locate the Target section.
3
From the Target list, choose Player.
4
Locate the Scene section. From the Subject list, choose Transverse Particle Positions.
5
Right-click Animation 1 and choose Play.
Finally, create a 1D plot showing the average radial position over time. Set the color expression to be the total radial force, averaged over all particles.
1D Plot Group 8
1
In the Results toolbar, click 1D Plot Group.
2
In the Settings window for 1D Plot Group, type Average Radial Particle Position in the Label text field.
3
Locate the Data section. From the Dataset list, choose Particle 1.
Particle 1
1
In the Average Radial Particle Position toolbar, click More Plots and choose Particle.
2
In the Settings window for Particle, click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Model>Component 1>Charged Particle Tracing>Particle position>qr - Particle position, r component - m.
3
Locate the Data Series Operation section. From the Operation list, choose Average.
Color Expression 1
1
Right-click Particle 1 and choose Color Expression.
2
In the Settings window for Color Expression, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Model>Component 1>Charged Particle Tracing>Forces>Total force - N>cpt.Ftr - Total force, r component.
3
In the Average Radial Particle Position toolbar, click Plot.
4
Click the Zoom Extents button in the Graphics toolbar.