Supporting Information
Simulation of Rarefied Gas Flows in Discontinuous
Atmospheric Pressure Interface Mass Spectrometry Systems
Sandilya Garimella1 , Xiaoyu Zhou1 and Zheng Ouyang1*
1
Weldon School of Biomedical Engineering, Purdue University
*Corresponding Author:
Zheng Ouyang
Weldon School of Biomedical Engineering
Purdue University
206 S. Martin Jischke Drive
West Lafayette IN 47907 USA
Email: ouyang@purdue.edu
1
S1. Knudsen Number in MS systems with DAPI and continuous interface
Knudsen Number (Kn) is the ratio of mean free path of flow (
and the length scale of flow (L).
Equation S1
Considering that the gas contained in the MS vacuum chamber is air and treating them as hard
sphere molecules, the diameter of the molecules (
) is 4.1x10-10m. The length scale of flow (L)
is the diameter of the DAPI capillary. In the above equation the other terms used are - kB is
Boltzmann Constant, p is pressure and T is Temperature.
For a conventional atmospheric pressure interface of mass spectrometer, the pressure in
the vacuum chamber (p) is around 1 torr. Considering a capillary of inner diameter 0.5 mm and
at temperature 300K, the value of Knudsen Number of flow in the first stage of mass
spectrometer with a conventional interface is Kn = 0.08. Thus the flow regime is on the border of
continuum regime. If the pressure is lower, transition flow regime occurs.
For a discontinuous atmospheric pressure interface, the base pressure of the vacuum
chamber is 1 mTorr. The pressure rise in the vacuum chamber takes place during the DAPI open
time. Typically the maximum allowed pressure in the vacuum chamber is about 0.5 Torr to
operate within the limits of the turbo molecular pump. Therefore the Knudsen Number varies
from 80 to 0.2. Thus the flow is in the transition regime flow during most of the operation time
of DAPI.
2
S2. Gas expansion in MS systems with DAPI and continuous interfaces
Continuous Interface
0.75mm i.d. capillary1
pb
X
0.5 torr (fixed value)
DAPI
DAPI
(0.5ms after DAPI opening) (2.0ms after DAPI opening)
0.1 torr (instantaneous at
0.5ms)
0.2 torr (instantaneous at
2.0ms)
simulation
theory
simulation
theory
simulation
theory
1.91cm
1.95cm
4.1cm
4.4cm
3.5cm
3.1cm
0.94
0.94cm
1.5cm
1.43cm
1.68
1.5cm
axial
extent of
shock
D
Diameter
of mach
disk
Theoretical expression for size of expansion shock structures has been reported previously.1
Equation S2
Here X is the axial spread of shock expansion and dmach is the diameter of mach disk, , p0 is the
ambient pressure, pb is the vacuum chamber back pressure, dcapillary is the diameter of the
capillary.
3
S3. Boundary conditions in DAPI simulations:
The DAPI process introduces ions from ambient pressure conditions directly into the analyzer
region at low vacuum pressure conditions. The flow dynamics inside the capillary leads to a
pressure drop and velocity increase along the axis of the capillary towards the vacuum chamber.
The boundary conditions for DAPI simulation have been set based on the exit pressure and
velocity of DAPI capillary as inlet into the MS vacuum chamber.
Pressure at exit of DAPI capillary:
The value of pressure at DAPI capillary is calculated using the expression below (Equation
S3.1). Here the pressure is in units of torr, the radius of capillary r and the length of the capillary
L are in centimeters, µ is viscosity of air in poise, T is temperature in kelvin, M is the molecular
weight of air in g/mol.
Equation S3
Equation S4
This formulation is used to obtain the boundary conditions for pressure/number density for the
simulation. Simulating the complete transport process through the capillary is computationally
intensive using the DSMC method as the flow inside the capillary is in the continuum regime.
Only the flow from exit of the capillary into the vacuum chamber suddenly changes and
experiences a steep pressure drop and hence is in the transition regime.
4
Velocity at the exit of the DAPI capillary:
Since the flow in the capillary is choked2, the axial velocity of gas inside the capillary gradually
increases till it reached Mach 1 speed at the exit of the capillary.2-4 For the gas to obey the
principle of continuity, speeds greater than Mach 1 cannot occur inside the capillary. The speed
of sound in air is given by the expression
ratio
. For air with the value of specific heat
, molecular weight M = 28g/mol, T = 300K, R = 8.314J/molK; the value of c can be
calculated to be 353m/s. This value is used as the inlet boundary condition in the DAPI
simulations as the gas exhausts into the vacuum chamber from the DAPI capillary exit. The gas
velocity is assumed to be axially directed and radial velocity is assumed zero just at the exit of
the capillary.
Boundary Conditions:
1. For simulation data shown in Figure 1
Condition
0.6mm i.d. capillary
Pressure at capillary exit
380 torr
Velocity at capillary exit
316 m/s
Vacuum pressure
2 torr
2. For simulation data shown in Figure 2,
Condition
0.50 mm i.d. capillary
5
Number density at capillary exit
1.1e25 m-3
Velocity at capillary exit
350 m/s
Number density in vacuum
3.2e19 m-3
3. For simulation data shown in Figure 3,
Condition
0.25mm i.d. capillary
1.0mm i.d. capillary
Number density at capillary 1.1e25 m-3
exit
Velocity at capillary exit
350 m/s
1.1e25 m-3
3.2e19 m-3
3.2e19 m-3
Number density in vacuum
350 m/s
4. For simulation data shown in Figure 4,
Condition
0.5mm i.d. capillary
Number density at capillary 1.1e25 m-3
exit
Velocity at capillary exit
350 m/s
Number density in vacuum
3.2e19 m-3
5. For simulation data shown in Figure 5,
Condition
0.5mm i.d. capillary DAPI-1
Number density at capillary 1.1e25 m-3
exit
Velocity at capillary exit
350 m/s
0.25 mm i.d. capillary DAPI-2
4.0e24 m-3
350 m/s
6
Number density in vacuum
3.2e19 m-3
3.2e19 m-3
7
S5. Diameter Characterization:
Figure S1
b)
a)
0.25mm i.d. capillary
1.00mm i.d. capillary
1.0
10
100
N/m2
1000
10000
N/m2
Figure S1 a) Contours of neutral density with 1.00m i.d capillary and b) 0.25mm i.d. capillary
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LIST OF REFERENCES
(1)
(2)
(3)
(4)
Jugroot, M.; Groth, C. P. T.; Thomson, B. A.; Baranov, V.; Collings, B. A. Journal of Physics D:
Applied Physics 2004, 37, 1289-1300.
Lin, B. W.; Sunner, J. Journal of the American Society for Mass Spectrometry 1994, 5, 873-885.
Anderson, J. D. Modern Compressible Flow, 2nd Edition, ed., 1990.
Robert W. Fox , A. T. M., Philip J. Pritchard Introduction to Fluid Mechanics 2006.
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