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Numerical study of segmented-electrode planar Orbitraps
International Journal of Mass Spectrometry 417 (2017) 58–68
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Numerical study of segmented-electrode planar Orbitraps Hrishikesh S. Sonalikar a , Atanu K. Mohanty a,b,∗ a b
Department of Instrumentation and Applied Physics, Indian Institute of Science, Bangalore 560012, India Department of Computational and Data Sciences, Indian Institute of Science, Bangalore 560012, India
a r t i c l e
i n f o
Article history: Received 11 November 2016 Received in revised form 2 March 2017 Accepted 15 March 2017 Available online 28 March 2017 Keywords: Orbitrap Planar Orbitrap Segmented-electrode traps Boundary element method Image current Field optimization
a b s t r a c t This study proposes a planar geometry mass analyzer which has fields and performance like that of a reference Orbitrap. The planar geometry taken up for investigation consists of two planar surfaces with concentric ring electrodes on each facing surface. Appropriate potentials are applied to the individual rings for effecting trapping of ions. The study used both single particle and multi-particle simulations. In the multi-particle simulations spatial and energetic distribution as well as space charge effects in the ion ensemble have been incorporated. The performance of this planar geometry has been compared to that of a reference Orbitrap by computing (1) the variation in potential along the principle directions, (2) ion trajectories and (3) induced image currents in the two geometries. The potentials applied to the ring electrodes were optimized to improve the electric field inside the planar geometry. © 2017 Elsevier B.V. All rights reserved.
1. Introduction
quadratic variation in axial (z) direction. The potential at a point (r, z), U(r, z), is expressed as [5],
This study proposes a planar geometry mass analyzer which has fields and performance similar to that of a reference Orbitrap. The Orbitrap geometry was first proposed by Makarov in 1999 [1] and a commercial instrument based on the Orbitrap was reported in 2005 [2–4]. The geometry consists of two axially symmetric electrodes, an inner spindle shaped electrode and an outer barrel shaped (split) electrode. The profile of these electrodes, z1,2 , are given by [5],
z1,2 =
2 R1,2 R r2 2 ln 1,2 − + Rm 2 2 r
(1)
Here, z1 and z2 refer to the profile of the inner and the outer electrodes, respectively; r is a the radial coordinate; R1 and R2 are the radii of the inner and the outer electrodes, respectively; Rm is known as the Orbitrap’s characteristic radius. The operation of the Orbitrap is based on electrostatic orbital trapping of ions [6,7,3]. For its operation, a negative DC potential is applied to the inner electrode and the split outer electrode is kept at ground potential. Due to the special shapes of these electrodes, the potential inside the Orbitrap has a logarithmic variation in radial (r) direction and
∗ Corresponding author at: Department of Instrumentation and Applied Physics, Indian Institute of Science, Bangalore 560012, India. E-mail addresses: [email protected] (H.S. Sonalikar), [email protected] (A.K. Mohanty). http://dx.doi.org/10.1016/j.ijms.2017.03.004 1387-3806/© 2017 Elsevier B.V. All rights reserved.
U(r, z) =
k 2
z2 −
r2 2
−
k (Rm )2 ln 2
R m
r
+C
(2)
Here, k and C are the constants related to the Orbitrap geometry defined later in Eqs. (4) and (5). Ions of an analyte, which are pulsed into the Orbitrap from an external ion source, undergo a (near) circular motion in the r direction and a simple harmonic motion in the z direction (because of the linear field in this direction). The frequency of ion motion in z direction, ωaxial , is related to the mass-to-charge ratio of ions by [5],
ωaxial =
k
q m
(3)
Here, q is the charge and m is the mass of an ion. The mass-tocharge ratios of fragment ions of an analyte gas are determined by first measuring the induced image current waveform across the split outer electrode and then computing its Fourier transform to obtain the frequencies of the ion’s axial motion, similar to what is done in the FT-ICR [5,8]. The Orbitrap mass analyzer is known for its ability to provide accurate mass analysis with high resolution [5]. The use of Orbitrap mass spectrometer was demonstrated in the research areas such as proteomics and metabolomics [9]. Simulation exercises were carried out to study the ion motion inside the Orbitrap under the influence of different excitations on the electrodes [10,11] as well as space charge effects [12].
H.S. Sonalikar, A.K. Mohanty / International Journal of Mass Spectrometry 417 (2017) 58–68
A few studies are available in the literature which have attempted to modify the geometry of the Orbitrap for improving its performance. In the first modification known as high-field Orbitrap, Makarov [13] reduced the gap between the inner and the outer electrodes by increasing the radius of the inner electrode, keeping the radius of the outer electrode unchanged. This geometry was found to be useful for providing higher field strength which resulted in higher resolution as well as lesser effect of space charge on mass shifts. Denisov et al. [14] suggested another design of the Orbitrap in which radii of the inner and the outer electrodes were scaled down by the factors of 1.2 and 1.5, respectively. These modified Orbitrap designs demonstrated very high mass resolution. A recent study which modified the geometry of the Orbitrap was that of the Sonalikar et al. [15] which presented segmented geometries of Orbitraps having simplified, easily machinable electrodes. This last study was motivated by both the initial suggestion of Makarov [16] and a more recent study of segmented electrode geometries for the CIT [17]. Planar geometry mass analyzers are not new in mass spectrometry. Planar ion traps typically consist of one or two planar substrates on which the number of metallic electrodes are printed by micro-fabrication techniques [18]. Planar geometries typically use a large number of segmented electrodes with appropriate potential applied on them in order to satisfy the boundary values of a desired field profile [19]. Planar ion traps have been used for a variety of applications in the literature. Austin et al. [20] presented a geometry consisting of two planar substrates with concentric ring electrodes lithographically printed on them. This geometry was demonstrated to create toroidal trapping volume by applying the optimized values of potentials to the ring electrodes as well as a quadrupolar electric field of the Paul trap [21–24]. Hansen et al. [25] and Zhou et al. [26] presented planar geometries consisting of a combination of grounded, RF and DC potentials applied to the electrodes to create an electric field similar to that of Linear Ion Trap. Chaudhary et al. [27] reported a design of planar ion funnel for miniature ion optics by applying a gradient of electrostatic potentials to the concentric ring electrodes printed on a planar substrate. Such planar geometries of ion traps are suitable for micro-fabrication and hence suitable for miniaturization [19]. Also, planar geometries provide a comparatively large trapping volume as well as an easier access to trapped ions [20]. In the present study, we have taken up for numerical investigation a planar geometry mass analyzer consisting of two planes, each plane having a number of concentric ring electrodes. Appropriate potentials are applied to the individual rings for effecting trapping of ions. This geometry has been named as PORB (‘Planar ORBitrap’). Both single particle and multi-particle simulations are performed. The multi-particle simulations are carried out on an ensemble of 1000 ions. These simulations incorporate spatial and energetic distributions as well as space charge effects. The performance of the PORB will be compared with that of a reference Orbitrap. The following comparative studies will be undertaken: (1) potential variation in the principle directions, (2) ion trajectories and (3) image currents in the PORB and the reference Orbitrap. The following section describes the geometries considered in this study. Section 3 briefly summarizes the numerical methods used in this work. Section 4 presents the results and discussion. A few concluding remarks are presented in Section 5.
2. Geometries considered The geometry and geometry parameters of the reference Orbitrap and the PORB geometry taken up for investigation in this study are presented below.
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Table 1 Geometry parameters of the reference Orbitrap and the PORB. All dimensions are in millimeter. Parameter
Reference Orbitrap
PORB
R1 R2 Zmax g h
7.0 80.0 100.0 – –
7.0 80.1 15.0 0.2 2.47
2.1. Reference Orbitrap The geometry of the reference Orbitrap used in this study is shown in Fig. 1(a). While outer electrode of the practical Orbitrap is symmetrically split into two at z = 0 for ion entry as well as for the measurement of the image current, in our present numerical study, no split has been incorporated. R1 and R2 are the radii of the inner and the outer electrodes of the Orbitrap, respectively. The Orbitrap at z = Zmax . The characteristic radius is truncated in axial direction √ of the Orbitrap is Rm = 2R2 [5]. The dimensions of the geometry parameters used in our study are listed in Table 1 under the column Orbitrap. The values of geometry parameters R1 , R2 and Rm of our reference Orbitrap are used to compute the values of constants k and C by using equations, 2V0
k=
C=
k 2
2 ln R /R (R22 − R12 )/2 − Rm 2 1
R22 2
2 + Rm ln
R m
R2
(4)
(5)
These have been derived from Eq. (2). Here, V0 denotes the potential applied to the inner electrode of the Orbitrap. The value of V0 is generally kept negative for trapping positive ions. The values of k and C are used to compute potentials applied to the different ring electrodes of the PORB geometry. The length of the Orbitrap is arbitrarily truncated at Zmax = 100 mm. This length is sufficient since we have restricted the maximum distance of ion motion in z direction to 10 mm in our simulations. R2 of 80 mm chosen in our reference Orbitrap is much larger than the R2 values in practical traps reported in the literature. The reason for this choice of large R2 will be discussed in Section 4.1. 2.2. PORB The geometry of the PORB consists of two planar surfaces, each consisting of NR concentric metallic ring electrodes. The cross section of the PORB geometry is shown in Fig. 1(b). Fig. 1(e) and (f) shows the top and 3D views of the PORB, respectively. Both the planes of PORB share the same central z axis. The distance between the two planes is 2Zmax . The radius of the outermost ring electrode is denoted by R2 . h denotes the width of each ring electrode and g denotes the width of an air gap separating neighboring electrodes. The values of these geometry parameters are listed in Table 1 under the column PORB. The potential applied to each ring electrode of the PORB is computed by putting the r and z coordinates corresponding to the center of each ring electrode in Eq. (2). The values of constants k, C and Rm used in this computation correspond to the parameters of the reference Orbitrap (Section 2.1). The value of R2 used for the PORB geometry is 80.1 mm. The value of Zmax used for the PORB is 15 mm, which is much smaller than the Zmax in the reference Orbitrap. The reason for this will be discussed in Section 4.1. In addition to this geometry, the PORB
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Fig. 1. (a) Cross section view of the reference Orbitrap. (b) Cross section view of the PORB. (c) Equipotential contours inside the reference Orbitrap. (d) Equipotential contours inside the PORB. (e) Top view of the planar electrode of the PORB. (f) 3D view of both the upper and the lower planar electrodes of the PORB.
geometries with increased values of R2 = 120 mm and 160 mm are also studied. The equipotential curves inside the reference Orbitrap and the PORB are shown in Fig. 1(c) and (d), respectively. These curves are plotted by computing the potential inside the two traps on
the grid of points which ranges from r =−80 mm to +80 mm in the radial direction and z =−14 mm to +14 mm in axial direction. The observation of these figures suggests that the potential and field in the PORB will be similar to that of the reference Orbitrap.
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3. Computational methods The numerical methods used in this study are briefly outlined in this section.
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The frequencies of ion motion in x, y and z direction in the PORB were computed from the ion trajectory by using the ‘harminv’ program [31,32]. 3.2. Image current
3.1. Boundary element method (BEM) and ion trajectory computation The geometries of the Orbitrap and the PORB have axial symmetry around the z axis. The potential and the electric field inside the Orbitrap can by analytically computed using Eq. (2) and its derivatives, respectively. On the other hand, the potential and field inside the PORB is computed by first computing the surface charge distribution on its electrodes. To compute the surface charge, the electrode surfaces of the PORB are divided in to small ring elements. The charge on each ring element is computed by using 2D boundary element method (2D BEM) [28]. Then the potential and the electric field inside the PORB is computed by applying the principle of superposition to elemental charges [29]. Once the electric field is known, the trajectory of an ion can be computed. Both the single particle and multi-particle ion trajectory simulations have been used in this study. The computer program was developed in-house to perform these simulations using RungeKutta 4th order (RK4) method [30]. The space charge effects due to Coulomb forces between the ions as well as loss of ions due to unfavorable initial velocities were taken into account in these simulations. The trajectories of ions were used for computing the contribution of each ion to the induced image current. In order to take into account the space charge effects, the force on ion i due the charge on ion j was computed as qi qj rji
Fij =
40 r 2 ji
(6)
where qi and qj are charges on ions i and j respectively, rji is a vector from position of ion j to the position of ion i with a magnitude of rji , 0 is a permittivity of a free space. The total force on ion is computed by adding Fij and the force due charges on different ring elements computed from 2D BEM. For a single ion simulation, the initial position would be x = r0 , y = 0 and z = z0 , where r0 and z0 are the selected radial and axial position of the ion. In the case of multi-particle simulation, the ions were assumed to have a Gaussian distribution with the mean initial position same as x, y and z (as above) with the standard deviation of x , y and z in the x, y and z directions, respectively. In our computations, x = y = z = 0.5 mm. The Box–Muller method [30] was used to obtain the random numbers with a Gaussian distribution. Similarly, the initial velocities of ions were assumed to have Maxwell–Boltzmann distribution with the mean of vx , vy and vz in the x, y and z directions, respectively. In our simulations, vx and vz were set to zero and vy was determined from the formula
vy =
Iimage =
dQ dt
(9)
Here, Q = Qupper − Qlower
(10)
Qupper and Qlower denote the total charge induced on the upper and lower split outer electrode, respectively. In the case of PORB geometry, both the upper and the lower electrodes consist of number of ring electrodes. Therefore, for PORB geometry, Qupper and Qlower are calculated by summing up the charges induced on individual ring electrodes as, Qupper/lower =
NR
qk
(11)
k=1
where NR is a number of ring electrodes present in the upper or lower planes of the PORB and qk is a charge induced on the kth ring electrode. Computing an induced charge due an ion on an arbitrary electrode normally requires a full 3D numerical solution of a Laplace’s equation for a given boundary conditions at each position of a moving charge. This method is computationally expensive. The induced charge can also be computed using the Green’s reciprocity theorem [5,11,33]. In this method, the induced charge computation requires the use of numerical solution only once and therefore it is computationally very efficient. In this paper, we have used the reciprocity principle to compute induced charge on the electrodes of the reference Orbitrap and the PORB. 3.3. Electric field optimization The potential values applied to the ring electrodes of the PORB were optimized to minimize the error between the electric field of the PORB and the reference Orbitrap. The method proposed in Ref. [15] was used to perform the optimization. In this method, the values of electric field were computed on a grid of points inside the PORB. To minimize the error in the z component of the field, the following objective function GZ was chosen as
N i,j M
Ez zi,j GZ = Ei,a − zi,1 z
(12)
i=1 j=2
2 − r2) kq(Rm 0
2m
(7)
√ where k is the Orbitrap constant obtained from Eq. (4), Rm (= R2 2) is the characteristic radius of the Orbitrap, r0 is the initial radial position of the ion, q/m is the charge-to-mass ratio of the ion. The thermal velocity of ions at 300 K was taken as the standard deviation in velocity of ions in each of the three directions. The thermal velocity of an ion of mass m at temperature T is given by
vth =
In the Orbitrap, the image current induced by ions is measured across the upper and lower part of the split outer electrode. The image current is given by,
kB T m
where kB is the Boltzmann constant.
Here, M and N denote number of points in grid in r and z direction, i,j respectively. Ez and zi,j denotes the value of electric field and z coordinate of point (i, j) on the grid, respectively. As the motion of ion is limited to |z| = 10 mm, the size of the grid at which the electric field is computed is also limited to ±10 mm in the z direction. The correctness of field in the z directions also insures the correctness in the r direction.1 Therefore, in this study, only the field in the z direction was optimized using function GZ . For additional control over the field in the r direction, the option to
(8) 1 This was pointed out by an anonymous reviewer of the PhD thesis of the first author [34].
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Fig. 2. Variation of potential in r direction in PORB from R1 = 7 mm to (a) R2 = 60 mm, (b) R2 = 80 mm at z = 0 mm. The continuous curve corresponds to the potential within the Orbitrap and the curve marked by stars correspond to the potential in PORB.
use the field errors in the r direction is also available as suggested in Ref. [15]. To optimize the electric field, the objective function GZ , which represents the error in the electric field, was minimized by using the Nelder–Mead simplex method [35]. Due to the use of principle of superposition, the use of 2D BEM for surface charge computation during the optimization was required only as many times as the number of ring electrodes in the PORB geometry. This made the optimization procedure computationally efficient. 4. Results and discussion The performance of the PORB is compared with that of the reference Orbitrap by computing (1) the variation in potential along r and z directions, (2) ion trajectories and (3) induced image currents in the two traps. In these simulations, appropriate values of potentials were applied to the ring electrodes of the PORB as discussed in Section 2.2. For comparison of potential as well as the single particle simulations presented in this work, the inner electrode of the Orbitrap was kept at −50 V and the outer electrode was kept at the ground potential. For multi-particle simulations of ion trajectory and image currents, the inner electrode of the Orbitrap was kept at −2000 V while the outer electrode was grounded. 4.1. Potential In this section we present the variation in potential in the Orbitrap and the PORB as a function of distance in the radial and the axial directions. This study will be used to choose the values of geometry parameters for the PORB reported in Section 2. The geometry of the PORB does not have the conventional inner and outer electrodes that the Orbitrap has. As a result, the field inside the PORB will not match with that in the Orbitrap near the innermost as well as the outermost ring electrodes of the PORB. Therefore it is expected that the potential inside the PORB will be significantly different from that of the Orbitrap near the innermost as well as outermost regions of the PORB. Consequently, the geometry parameter R2 of the PORB should be carefully chosen such that the electric field in the PORB matches with that of the Orbitrap for most of the region between the innermost and the outermost ring electrodes. Fig. 2 shows the variation of potential with r for different values of R2 in the PORB and the Orbitrap. The values of R1 and Zmax are 7 mm and 15 mm, respectively for the PORB as well as the Orbitrap. The variation of potential is computed when r is varied from R1 to R2 and z is fixed at 0 mm. The continuous curve indicates the
potential in the Orbitrap and the curve marked by stars indicates the potential in the PORB. Fig. 2(a) shows the variation of potential in the PORB for R2 = 60 mm. For this value of R2 , we observe that the potential in the PORB agrees with that in the reference Orbitrap around r = 40 mm. For other values of r, however, the potential in the PORB differs from that in the Orbitrap. When the value of R2 is set to 80 mm, the potential in the PORB closely matches with that in the Orbitrap when r ranges from 40 mm to 60 mm as can be seen from Fig. 2(b). We note that when the R2 = 80 mm, the potentials of the two traps closely match for the region which extends for 20 mm. Therefore the value of R2 = 80 mm was found to be suitable for the PORB to work as an Orbitrap. Although not shown here, higher values of R2 were seen to give better agreement between the potentials of the two traps for larger ranges of r. On the other hand, the lower values of R2 were seen to show increasing difference between the potentials in the two traps. We have chosen R2 = 80 mm for our reference Orbitrap in the present study. Fig. 3 shows the variation of potential in the PORB and Orbitrap with z for different values of Zmax used in the geometry of the PORB. The values of Zmax used are 15 mm and 20 mm. The values of geometry parameters R1 and R2 were fixed at 7 mm and 80 mm, respectively. The variation of potential was computed by varying z from 0 to Zmax for the fixed value of r = 50 mm. The continuous curve indicates the potential in the Orbitrap and the curve marked by stars indicates the potential in the PORB. As can be noted from Fig. 3, the agreement between the potential in the PORB and the Orbitrap improves as the value of Zmax is reduced from 20 mm to 15 mm. The error in the potential variation in the PORB and the Orbitrap is minimum when Zmax = 15 mm as seen in Fig. 3(a). Therefore, Zmax = 15 mm was chosen as a suitable value for the PORB geometry. As the amplitude of the z motion of ion in the Orbitrap is around 8 mm, the value of Zmax was not reduced further. Although not shown here, the difference between the potential variation in the PORB and the reference Orbitrap increases for the increased values of Zmax . For the reasons given above, we have chosen R2 for the reference Orbitrap to be 80 mm and Zmax for the PORB to be 15 mm. Here, we would like to emphasize that as long as the variation of potential in the r direction has a logarithmic form and that in the z direction has a quadratic form, respectively, the PORB will still function as an orbital trapping device regardless of the values of geometry parameters chosen. A good match between the potential in the reference Orbitrap and the PORB of a fixed geometry parameters can also be obtained by multiplying the potentials applied to the different electrodes of the PORB by a suitable factor.
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Fig. 3. Variation of potential in z direction in (a) PORB with Zmax = 15 mm, (b) PORB with Zmax = 20 mm at r = 50 mm. The continuous curve corresponds to the potential within the Orbitrap and the curve marked by stars correspond to the potential in PORB.
4.2. Ion trajectories In order to use the geometry of the PORB for electrostatic orbital trapping of ions, an ion should have stable motion in both the radial and axial directions inside the PORB. To observe how an ion behaves inside the PORB, the trajectory of ion was computed and compared to the ion motion inside the Orbitrap. In these trajectory simulations, the initial position of ion having a unit charge and a mass of 78 u was set at x = 50 mm, y = 0 mm and z = 10 mm. At this initial position of ion, its radial distance from the center of the trap is close to 50 mm where the field in the PORB is in close agreement with that inside the Orbitrap. The initial velocity components of ion were Vx = 0 m s−1 , Vy = 4769.3 m s−1 and Vz = 0 m s−1 . This initial velocity was chosen such that ion will move in a near circular orbit around the central axis of the trap. The trajectory of ion was computed for 1 ms using 4th order Runge-Kutta method with the time step set to 0.1 s. Fig. 4 shows the trajectory of an ion inside the PORB and the Orbitrap, respectively. These figures show the ion motion in x, y and z directions as well as the orbit of ion in x–y place around the central axis. The frequencies of ion motion in x, y and z direction are listed in Table 2. From Table 2, we note that ion motion’s fx and fy frequencies inside PORB are smaller compared to those inside the Orbitrap. On the other hand, fz in the PORB is larger compared to that in the Orbitrap. From Fig. 4, we note that the ion has a stable motion in both the radial (x–y) as well as axial (z) direction. Also it can be observed that the motion of ion inside the PORB is comparable to that inside the Orbitrap. The amplitude of ion motion inside the Orbitrap can be seen to be constant whereas the amplitude of ion motion inside the PORB seems to be slightly fluctuating with time. The radial spread of ion motion in x–y plane inside the PORB is larger compared to the perfectly circular orbit of an ion inside the Orbitrap as can be observed from Fig. 4(d) and (h). Trajectory simulations were also performed to compute the useful trapping area between the planar electrodes. In these simulations, the initial r position of ion was varied keeping the initial z position of ion fixed at 10 mm. The initial velocity of ion was adjusted so that ion might have a near circular orbit around the central axis. The useful trapping area was determined by observing whether the ion has stable trajectory in both radial and axial directions for a given set of initial conditions. It was observed that the ions have stable trajectories when initial r position of ion lies in the range varying from r = 35 mm to r = 56 mm. Outside this range, the ions were found to be unstable. This simulation shows that ions entering the PORB geometry in this range of r
values will be trapped while others will escape due to their unstable trajectories. 4.3. Variation of axial frequency In the Orbitrap, the frequency of ion motion in z direction (fz ) is used to determine the mass of an ion. Due to the precise shape of the Orbitrap, the frequencies of ion motion in the Orbitrap are independent of the initial position of the ion. The electric field in the PORB, however, is only an approximation to the field in the reference Orbitrap. Due to the errors in the field, the frequencies of ion motion inside the PORB geometry will vary with the initial conditions of the ion. Consequently, it is important to reduce the dependence of fz on the initial z position of the ion to improve the accuracy of mass detection by optimizing the electric field inside the PORB. For optimizing the electric field inside the PORB, the potentials applied to the ring electrodes of the PORB were adjusted to minimize the objective function given in Section 3.3. Once the values of potentials to be applied to the ring electrodes to achieve an optimum electric field were obtained, the frequency of ion motion in z direction was computed for different initial z positions of a single ion. Fig. 5(a) shows the variation of fz with the initial z position of an ion. A curve marked by stars represents variation in fz before optimization. A curve marked by triangles represents variation in fz after optimization. From Fig. 5(a), we observe that before optimization, the variation in fz was 118.0 Hz as z is varied from 1 mm to 10 mm. After optimization the variation in fz was minimized to just 12.9 Hz. This result shows that the optimization procedure used in this work was successful in minimizing the errors in the electric field of the PORB which results in the reduced variation of axial frequency of ion with the z position of an ion. Fig. 5(b) shows the values of potentials applied to the ring electrodes of the PORB before and after optimization. The blue curve indicates potentials before optimizations and the red curve indicates the potentials after optimization. From Fig. 5(b) and (a), we can observe that by slight change in the applied potentials, the errors in the electric field can be significantly reduced. 4.4. Image current The mass-to-charge ratio of ions in the Orbitrap is determined by detecting the image current induced across the split outer electrodes of the Orbitrap. The frequency analysis of the acquired image current transient is done to obtain the accurate mass analysis of trapped ions. In this section, we present the comparison of image currents computed in the Orbitrap and the PORB. A multi-particle simulation involving an ensemble of 1000 ions, each having a unit
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Fig. 4. (a) Displacement of ion in x direction in the PORB. (b) Displacement of ion in y direction in the PORB. (c) Displacement of ion in z direction in the PORB. (d) Motion of ion in x–y plane in the PORB. (e) Displacement of ion in x direction in the Orbitrap. (f) Displacement of ion in y direction in the Orbitrap. (g) Displacement of ion in z direction in the Orbitrap. (h) Motion of ion in x–y plane in the Orbitrap.
charge and a mass of 78 u has been carried out. The image current was computed for the duration of at least 1 ms. The initial positions and velocities of ions were calculated using the Gaussian distribution. The mean radial and axial position of ions were r = 50 mm and z = 10 mm with standard deviation of 0.5 mm in each direction. Ions
were given a mean initial tangential velocity such that they follow a near circular path in the trapping region. The thermal velocity of ions at 300 K was used as a standard deviation in velocity distribution among the ions. The value of applied potential V0 in these simulations was −2000 V.
H.S. Sonalikar, A.K. Mohanty / International Journal of Mass Spectrometry 417 (2017) 58–68 Table 2 Frequencies of ion motion in the Orbitrap and the PORB. Frequency (kHz)
Orbitrap
PORB
fx fy fz
15.18 15.18 10.57
14.35 14.35 11.16
Fig. 6(a) and (b) shows the image current induced in the reference Orbitrap and the PORB, respectively. The image current in the reference Orbitrap has a constant amplitude. The amplitude of image current in the PORB is initially larger compared to that of the reference Orbitrap but this rapidly decays. Fig. 6(c) and (d) shows the frequency spectrum of image current computed in the reference Orbitrap and the PORB, respectively. It can be observed that the width of a principle peak in the image current spectrum of PORB is larger compared to that of the reference Orbitrap, implying that the mass resolution in the PORB is lower than that in the reference Orbitrap. Fig. 6(e) and (f) shows a snapshot of positions of ions in X–Y plane after 1 ms in the reference Orbitrap and the PORB, respectively. The ions were initially bunched together in a volume of sphere having radius of 2.1 mm indicated by circles in these figures. It can be observed that the ions have formed a ring due the space charge effects. Figures 6(g) and (h) shows the snapshot of ion motion in Y–Z plane for the two traps after 1 ms. From these figures, it can be observed that in the reference Orbitrap, the spread of ions in axial direction is from z = 1 mm to z = 4 mm where as the spread of ions in the PORB in the axial direction is from z =−10 mm to z = 10 mm. The relatively large spread of ion in z direction in the PORB is on account of the non-linearities of field in the PORB. This large spread of ions in the PORB results in the rapid decay of image current signal due to out-of-phase motion of ions in the PORB. To characterize the non-linearities in the electric field in the ideal Orbitrap as well as the PORB in the axial direction, the variation of potential in the axial direction in both the traps was represented by a polynomial of degree 6 as shown in Eq. (13): V (z) = C6 z 6 + C5 z 5 + C4 z 4 + C3 z 3 + C2 z 2 + C1 z + C0
(13)
The coefficients of this polynomial were calculated by a least square fit. The values of the even order coefficients are listed in Table 3 for both the Orbitrap and the PORB. The coefficient C2 is the contribution of quadratic potential which represents the linearity of the
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electric field while C0 represents a constant potential which does not contribute to the electric field. The higher order coefficients represent the non-linearities in the field. As can be observed from Table 3, the higher order coefficients for the Orbitrap potential are negligibly small and C2 is an only significant coefficient contributing a linear field. On other hand, the values of coefficients C4 and C6 for the PORB are considerably large compared to those for the Orbitrap indicating that the axial field in the PORB contains non-linear components. The values of odd order coefficients such as C1 , C3 and C5 were less than 2.0e−15 for both the traps due to top-bottom symmetry of their geometries. As an aside, we have also carried out simulations to measure image current in two other PORB geometries. In one, R2 has been increased to 120 mm (compared to R2 value of 80 mm in our study) and in the other, the value of R2 has been further increased to 160 mm. In these simulations, the initial mean radial positions of ions were also increased to 75 mm and 90 mm, respectively. The image currents computed for 2 ms in these two geometries are shown in Fig. 7(a) and 7(b), Fig. 7(a) corresponding to R2 = 120 mm and Fig. 7(b) corresponding to R2 = 160 mm, respectively. A comparison of the image current presented in Fig. 6(b) (for R2 = 80 mm), Fig. 7(a) (for R2 = 120 mm) and Fig. 7(b) (for R2 = 160 mm) indicates an interesting trend. It is expected that as R2 is increased, the non-linearity in the trapping region decreases as the fringing fields at the edges of the trap have reduced effect on the field in the trapping region. This results in the field which is in a better agreement with that of the Orbitrap. This is the reason why there is reduction in the rate of decay with increase in R2 . In Fig. 6(b), the image current has rapidly decayed in 0.3 ms, whereas in the image current depicted in Fig. 7(a), the decay is much slower. In Fig. 7(b), the decay is negligible. From these results, it can be concluded that the field in the PORB geometry with higher values of R2 is better. While this is good, it however involves increasing the size of the trap which may not be desirable. In our search for a better performance of the original PORB (R2 = 80 mm), we have undertaken two additional simulations. In the first, the potentials applied to the ring electrodes of the PORB were optimized by using the method outlined in Section 3.3. The image current computed after the optimization is presented in Fig. 8(a). Fig. 8(a) shows the image current signal in the PORB with R2 = 80 mm and Zmax = 15 mm for the duration of 2 ms. The mean initial z position of 1000 ions in this simulation was 10 mm. It can
Fig. 5. (a) Variation of axial frequency fz with initial z position in PORB. Without optimization, the variation in fz is 118.0 Hz as z is varied from 1 mm to 10 mm. After optimization the variation in fz is minimized to 12.9 Hz. A curve marked by stars represents variation in fz before optimization. A curve marked by triangles represents variation in fz after optimization. (b) Potentials applied to the ring electrodes of the PORB before and after optimization. The blue curve indicates potentials before optimization and the red curve indicates the potentials after optimization. The ring electrodes are numbered from 0 to 29. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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H.S. Sonalikar, A.K. Mohanty / International Journal of Mass Spectrometry 417 (2017) 58–68
Fig. 6. Image current signal in (a) the reference Orbitrap and (b) the PORB. The image current spectrum in (c) the reference Orbitrap and (d) the PORB. Snapshot of ion motion in X–Y plane in (e) the reference Orbitrap and (f) the PORB. Snapshot of ion motion in Y–Z plane in (g) the reference Orbitrap and (h) the PORB taken at t = 1 ms.
be seen from the figure that the optimization has resulted in a better performance with the image current decaying at a slower pace. Although this is better than the earlier performance, it still falls short of the image current obtainable in the Orbitrap. Consequently, in the second experiment, the spacing between the planar electrodes was reduced from 30 mm, which is a spacing used in
the PORB, to 25 mm. The mean initial z position of ions was also reduced to 8 mm. All other parameters were kept same. The potentials applied to the ring electrodes were optimized as was done for obtaining Fig. 8(a). Fig. 8(b) shows the image current computed after applying the optimized values of potentials to the ring electrodes of this modified geometry. It is clear from the figure that
H.S. Sonalikar, A.K. Mohanty / International Journal of Mass Spectrometry 417 (2017) 58–68 Table 3 Values of coefficients of V(z) for the Orbitrap and the PORB. Coefficient
Orbitrap
PORB
C0 C2 C4 C6
−7.2590 0.30171 −2.3321e−14 1.7552e−14
−7.2728 0.30795 1.2757e−2 −1.2770e−2
the decay of the image current for the 2 ms period is negligible in comparison to the image current plotted in Fig. 8(a). This indicates that the reduced spacing has resulted in a further improved performance.
5. Concluding remarks This study investigated a planar geometry mass analyzer which has fields and performance similar to that of a reference Orbitrap. In this study, we have taken up for investigation a geometry referred to as PORB. This geometry consists of two planes, each having 30 ring electrodes. Comparisons of (1) the potential variation in the principle directions, (2) ion trajectories, and (3) the image currents in these two traps have been made. In this numerical study, both single particle and multi-particle simulations have been performed.
67
Our investigations show that it is possible to mimic the quadrologarithmic potential distribution of the reference Orbitrap in our PORB. However, the PORB being a non-linear device, has a performance inferior to the reference Orbitrap. In our simulations, we have considered 30 concentric ring electrodes in each plane of the PORB. The geometry with the gap of 0.5 mm between the ring electrodes was studied and it was found that the field was not significantly affected by an increased gap. It is expected that larger number of rings will improve the field in the trap. Separate studies will have to be carried out to quantify the effect of parameters g and h on the electric field. The optimization of the electric field in the PORB has been carried out and it was shown that an improved performance was obtained after applying the optimized values of potentials to the segmented-electrodes. The performance could be further improved by reducing the spacing between the planar electrodes. An attempt can be made to minimize the amplitude of higher frequency harmonics present in the image current spectrum of the PORB by further optimizing the geometry as well as by super imposing the image currents induced in different electrodes of the PORB by first multiplying them by a suitable factor. Finally, we present some thoughts on how the electrical connections can be made in PORB. A resistive network can be used to provide appropriate DC potentials to the electrodes. Since the same electrodes are used for application of DC potentials for ion
Fig. 7. (a) The image current in the PORB with R2 = 120 mm. (b) The image current in the PORB with R2 = 160 mm.
b Fig. 8. (a) The image current in the optimized PORB with R2 = 80 mm and Zmax = 15 mm. (b) The image current in the PORB with R2 = 80 mm and Zmax = 12.5 mm.
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trapping as well as detection of image current for mass analysis, these DC voltages need to be isolated from the induced image current by using inductors. Also, the induced image current signal needs to be passed through a capacitive coupling to isolate it from the applied DC potentials. The use of the inductive and capacitive coupling should have only a minimum effect on the sensitivity of an instrument as the losses in the inductors and capacitors are negligible. Acknowledgements We thank an anonymous reviewer of the Ph.D. thesis of the first author for suggesting improvements in the optimization procedure used in this study. We thank Prof. A.G. Menon for his valuable suggestions during the preparation of the manuscript. We also thank three reviewers for giving important suggestions for significantly improving the manuscript. References [1] A.A. Makarov, Mass spectrometer, U.S. Patent 5,886,346 (1999). [2] M. Hardman, A.A. Makarov, Interfacing the Orbitrap mass analyzer to an electrospray ion source, Anal. Chem. 75 (2003) 1699–1705. [3] Q. Hu, R.J. Noll, H. Li, A. Makarov, M. Hardman, R.G. Cooks, The Orbitrap: a new mass spectrometer, J. Mass Spectrom. 40 (2005) 430–443. [4] A. Makarov, E. Denisov, A. Kholomeev, W. Balschun, O. Lange, K. Strupat, S. Horning, Performance evaluation of a hybrid linear ion trap/Orbitrap mass spectrometer, Anal. Chem. 78 (2006) 2113–2120. [5] A.A. Makarov, Electrostatic axially harmonic orbital trapping: a high-performance technique of mass analysis, Anal. Chem. 72 (2000) 1156–1162. [6] K.H. Kingdon, A method for the neutralization of electron space charge by positive ionization at very low gas pressures, Phys. Rev. 21 (1922) 408–418. [7] R.D. Knight, Storage of ions from laser produced plasmas, Appl. Phys. Lett. 38 (1981) 221–223. [8] A.G. Marshall, F.R. Verdun, Fourier Transforms in NMR, Optical and Mass Spectrometry – A User’s Handbook, Elsevier, Amsterdam, 1990. [9] R.H. Perry, R.G. Cooks, R.J. Noll, Orbitrap mass spectrometry: instrumentation, ion motion and applications, Mass Spectrom. Rev. 27 (2008) 661–699. [10] Q. Hu, A. Makarov, R.G. Cooks, R.J. Noll, Resonant ac dipolar excitation for ion motion control in the Orbitrap mass analyzer, J. Phys. Chem. A 110 (2006) 2682–2689. [11] G. Wu, R.J. Noll, W.R. Plass, Q. Hu, R. Perry, R.G. Cooks, Ion trajectory simulations of axial ac dipolar excitation in the orbitrap, Int. J. Mass Spectrom. 254 (2006) 53–62. [12] A. Kharchenko, G. Vladimirov, R.M.A. Heeren, E.N. Nikolaev, Performance of orbitrap mass analyzer at various space charge and non-ideal field conditions: simulation approach, J. Am. Soc. Mass Spectrom. 23 (2012) 977–987. [13] A. Makarov, E. Denisov, O. Lange, Performance evaluation of a high field Orbitrap mass analyzer, J. Am. Soc. Mass Spectrom. 20 (2009) 1391–1396.
[14] E. Denisov, E. Damoc, O. Lange, A. Makarov, Orbitrap mass spectrometry with resolving powers above 1,000,000, Int. J. Mass Spectrom. 325–327 (2012) 80–85. [15] H.S. Sonalikar, A.A. Ovhal, A.K. Mohanty, Segmented electrode orbitraps, Int. J. Mass Spectrom. 395 (2016) 36–48. [16] A.A. Makarov, Multi-electrode ion trap, U.S. Patent 7,767,960 B2 (2010). [17] H.S. Sonalikar, A.K. Mohanty, Variable field ion traps, Int. J. Mass Spectrom. 356 (2013) 60–70. [18] Y. Peng, D.E. Austin, New approaches to miniaturizing ion trap mass analyzers, Trends Anal. Chem. 30 (10) (2011) 1560–1567. [19] D.E. Austin, Y. Peng, B.J. Hansen, I.W. Miller, A.L. Rockwood, A.R. Hawkins, S.E. Tolley, Novel ion traps using planar resistive electrodes: implications for miniaturized mass analyzers, J. Am. Soc. Mass Spectrom. 19 (2008) 1435–1441. [20] D. Austin, M. Wang, S.E. Tolley, J.D. Maas, A.R. Hawkins, A.L. Rockwood, H.D. Tolley, E.D. Lee, M.L. Lee, Halo ion trap mass spectrometer, Anal. Chem. 79 (2007) 2927–2932. [21] D. Austin, B. Hansen, Y. Peng, Z. Zhang, Multipole expansion in quadrupolar devices comprised of planar electrode arrays, Int. J. Mass Spectrom. 295 (2010) 153–158. [22] Z. Zhang, Y. Peng, B. Hansen, I. Miller, M. Wang, M. Lee, A. Hawkins, D. Austin, Paul trap mass analyzer consisting of opposing microfabricated electrode plates, Anal. Chem. 81 (2009) 5241–5248. [23] Z. Zhang, H. Quist, Y. Peng, B. Hansen, J. Wang, A. Hawkins, D. Austin, Effects of higher order multipoles on the performance of a two-plate quadrupole ion trap mass analyzer, Int. J. Mass Spectrom. 299 (2011) 151–157. [24] Y. Peng, B.J. Hansen, H. Quist, Z. Zhang, M. Wang, A.R. Hawkins, D.E. Austin, Coaxial ion trap mass spectrometer: concentric toroidal and quadrupolar trapping regions, Anal. Chem. 83 (2011) 5578–5584. [25] B.J. Hansen, R.J. Niemi, A.R. Hawkins, S.A. Lammert, D.E. Austin, A lithographically patterned discrete planar electrode linear ion trap mass spectrometer, J. Microelectromech. Syst. 22 (4) (2013) 876–883. [26] X. Zhou, G. Xu, C. Xiong, C. Chen, H. Qiao, Z. Nie, Characteristics of electrical field and ion motion in surface-electrode ion traps, J. Mass Spectrom. 47 (2012) 286–293. [27] A. Chaudhary, F.H.W. van Amerom, R.T. Short, A novel planar ion funnel design for miniature ion optics, Rev. Sci. Instrum. 85 (2014) 105101. [28] P.K. Tallapragada, A.K. Mohanty, A. Chatterjee, A.G. Menon, Geometry optimization of axially symmetric ion traps, Int. J. Mass Spectrom. 264 (2007) 38–52. [29] D.J. Griffiths, Introduction to Electrodynamics, Prentice Hall, New Delhi, 2008. [30] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in C, Cambridge University Press, New Delhi, 2007. [31] V.A. Mandelshtam, H.S. Taylor, Harmonic inversion of time signals and its applications, J. Chem. Phys. 107 (17) (1997) 6756–6769. [32] S.G. Johnson. http://ab-initio.mit.edu/wiki/index.php/harminv (accessed 23.08.16). [33] J.A. Marto, A.G. Marshall, L. Schweikhard, A two-electrode ion trap for Fourier transform ion cyclotron resonance mass spectrometry, Int. J. Mass Spectrom. Ion Process. 137 (1994) 9–30. [34] H.S. Sonalikar, Numerical Investigation of Segmented Electrode Designs for the Cylindrical Ion Trap and the Orbitrap Mass Analyzers (Unpublished PhD thesis), Department of Instrumentation and Applied Physics, Indian Institute of Science, Bangalore, India, 2016. [35] J.A. Nelder, R. Mead, A simplex method for function minimization, Comput. J. 7 (4) (1965) 303–313.
Contents lists available at ScienceDirect
International Journal of Mass Spectrometry journal homepage: www.elsevier.com/locate/ijms
Numerical study of segmented-electrode planar Orbitraps Hrishikesh S. Sonalikar a , Atanu K. Mohanty a,b,∗ a b
Department of Instrumentation and Applied Physics, Indian Institute of Science, Bangalore 560012, India Department of Computational and Data Sciences, Indian Institute of Science, Bangalore 560012, India
a r t i c l e
i n f o
Article history: Received 11 November 2016 Received in revised form 2 March 2017 Accepted 15 March 2017 Available online 28 March 2017 Keywords: Orbitrap Planar Orbitrap Segmented-electrode traps Boundary element method Image current Field optimization
a b s t r a c t This study proposes a planar geometry mass analyzer which has fields and performance like that of a reference Orbitrap. The planar geometry taken up for investigation consists of two planar surfaces with concentric ring electrodes on each facing surface. Appropriate potentials are applied to the individual rings for effecting trapping of ions. The study used both single particle and multi-particle simulations. In the multi-particle simulations spatial and energetic distribution as well as space charge effects in the ion ensemble have been incorporated. The performance of this planar geometry has been compared to that of a reference Orbitrap by computing (1) the variation in potential along the principle directions, (2) ion trajectories and (3) induced image currents in the two geometries. The potentials applied to the ring electrodes were optimized to improve the electric field inside the planar geometry. © 2017 Elsevier B.V. All rights reserved.
1. Introduction
quadratic variation in axial (z) direction. The potential at a point (r, z), U(r, z), is expressed as [5],
This study proposes a planar geometry mass analyzer which has fields and performance similar to that of a reference Orbitrap. The Orbitrap geometry was first proposed by Makarov in 1999 [1] and a commercial instrument based on the Orbitrap was reported in 2005 [2–4]. The geometry consists of two axially symmetric electrodes, an inner spindle shaped electrode and an outer barrel shaped (split) electrode. The profile of these electrodes, z1,2 , are given by [5],
z1,2 =
2 R1,2 R r2 2 ln 1,2 − + Rm 2 2 r
(1)
Here, z1 and z2 refer to the profile of the inner and the outer electrodes, respectively; r is a the radial coordinate; R1 and R2 are the radii of the inner and the outer electrodes, respectively; Rm is known as the Orbitrap’s characteristic radius. The operation of the Orbitrap is based on electrostatic orbital trapping of ions [6,7,3]. For its operation, a negative DC potential is applied to the inner electrode and the split outer electrode is kept at ground potential. Due to the special shapes of these electrodes, the potential inside the Orbitrap has a logarithmic variation in radial (r) direction and
∗ Corresponding author at: Department of Instrumentation and Applied Physics, Indian Institute of Science, Bangalore 560012, India. E-mail addresses: [email protected] (H.S. Sonalikar), [email protected] (A.K. Mohanty). http://dx.doi.org/10.1016/j.ijms.2017.03.004 1387-3806/© 2017 Elsevier B.V. All rights reserved.
U(r, z) =
k 2
z2 −
r2 2
−
k (Rm )2 ln 2
R m
r
+C
(2)
Here, k and C are the constants related to the Orbitrap geometry defined later in Eqs. (4) and (5). Ions of an analyte, which are pulsed into the Orbitrap from an external ion source, undergo a (near) circular motion in the r direction and a simple harmonic motion in the z direction (because of the linear field in this direction). The frequency of ion motion in z direction, ωaxial , is related to the mass-to-charge ratio of ions by [5],
ωaxial =
k
q m
(3)
Here, q is the charge and m is the mass of an ion. The mass-tocharge ratios of fragment ions of an analyte gas are determined by first measuring the induced image current waveform across the split outer electrode and then computing its Fourier transform to obtain the frequencies of the ion’s axial motion, similar to what is done in the FT-ICR [5,8]. The Orbitrap mass analyzer is known for its ability to provide accurate mass analysis with high resolution [5]. The use of Orbitrap mass spectrometer was demonstrated in the research areas such as proteomics and metabolomics [9]. Simulation exercises were carried out to study the ion motion inside the Orbitrap under the influence of different excitations on the electrodes [10,11] as well as space charge effects [12].
H.S. Sonalikar, A.K. Mohanty / International Journal of Mass Spectrometry 417 (2017) 58–68
A few studies are available in the literature which have attempted to modify the geometry of the Orbitrap for improving its performance. In the first modification known as high-field Orbitrap, Makarov [13] reduced the gap between the inner and the outer electrodes by increasing the radius of the inner electrode, keeping the radius of the outer electrode unchanged. This geometry was found to be useful for providing higher field strength which resulted in higher resolution as well as lesser effect of space charge on mass shifts. Denisov et al. [14] suggested another design of the Orbitrap in which radii of the inner and the outer electrodes were scaled down by the factors of 1.2 and 1.5, respectively. These modified Orbitrap designs demonstrated very high mass resolution. A recent study which modified the geometry of the Orbitrap was that of the Sonalikar et al. [15] which presented segmented geometries of Orbitraps having simplified, easily machinable electrodes. This last study was motivated by both the initial suggestion of Makarov [16] and a more recent study of segmented electrode geometries for the CIT [17]. Planar geometry mass analyzers are not new in mass spectrometry. Planar ion traps typically consist of one or two planar substrates on which the number of metallic electrodes are printed by micro-fabrication techniques [18]. Planar geometries typically use a large number of segmented electrodes with appropriate potential applied on them in order to satisfy the boundary values of a desired field profile [19]. Planar ion traps have been used for a variety of applications in the literature. Austin et al. [20] presented a geometry consisting of two planar substrates with concentric ring electrodes lithographically printed on them. This geometry was demonstrated to create toroidal trapping volume by applying the optimized values of potentials to the ring electrodes as well as a quadrupolar electric field of the Paul trap [21–24]. Hansen et al. [25] and Zhou et al. [26] presented planar geometries consisting of a combination of grounded, RF and DC potentials applied to the electrodes to create an electric field similar to that of Linear Ion Trap. Chaudhary et al. [27] reported a design of planar ion funnel for miniature ion optics by applying a gradient of electrostatic potentials to the concentric ring electrodes printed on a planar substrate. Such planar geometries of ion traps are suitable for micro-fabrication and hence suitable for miniaturization [19]. Also, planar geometries provide a comparatively large trapping volume as well as an easier access to trapped ions [20]. In the present study, we have taken up for numerical investigation a planar geometry mass analyzer consisting of two planes, each plane having a number of concentric ring electrodes. Appropriate potentials are applied to the individual rings for effecting trapping of ions. This geometry has been named as PORB (‘Planar ORBitrap’). Both single particle and multi-particle simulations are performed. The multi-particle simulations are carried out on an ensemble of 1000 ions. These simulations incorporate spatial and energetic distributions as well as space charge effects. The performance of the PORB will be compared with that of a reference Orbitrap. The following comparative studies will be undertaken: (1) potential variation in the principle directions, (2) ion trajectories and (3) image currents in the PORB and the reference Orbitrap. The following section describes the geometries considered in this study. Section 3 briefly summarizes the numerical methods used in this work. Section 4 presents the results and discussion. A few concluding remarks are presented in Section 5.
2. Geometries considered The geometry and geometry parameters of the reference Orbitrap and the PORB geometry taken up for investigation in this study are presented below.
59
Table 1 Geometry parameters of the reference Orbitrap and the PORB. All dimensions are in millimeter. Parameter
Reference Orbitrap
PORB
R1 R2 Zmax g h
7.0 80.0 100.0 – –
7.0 80.1 15.0 0.2 2.47
2.1. Reference Orbitrap The geometry of the reference Orbitrap used in this study is shown in Fig. 1(a). While outer electrode of the practical Orbitrap is symmetrically split into two at z = 0 for ion entry as well as for the measurement of the image current, in our present numerical study, no split has been incorporated. R1 and R2 are the radii of the inner and the outer electrodes of the Orbitrap, respectively. The Orbitrap at z = Zmax . The characteristic radius is truncated in axial direction √ of the Orbitrap is Rm = 2R2 [5]. The dimensions of the geometry parameters used in our study are listed in Table 1 under the column Orbitrap. The values of geometry parameters R1 , R2 and Rm of our reference Orbitrap are used to compute the values of constants k and C by using equations, 2V0
k=
C=
k 2
2 ln R /R (R22 − R12 )/2 − Rm 2 1
R22 2
2 + Rm ln
R m
R2
(4)
(5)
These have been derived from Eq. (2). Here, V0 denotes the potential applied to the inner electrode of the Orbitrap. The value of V0 is generally kept negative for trapping positive ions. The values of k and C are used to compute potentials applied to the different ring electrodes of the PORB geometry. The length of the Orbitrap is arbitrarily truncated at Zmax = 100 mm. This length is sufficient since we have restricted the maximum distance of ion motion in z direction to 10 mm in our simulations. R2 of 80 mm chosen in our reference Orbitrap is much larger than the R2 values in practical traps reported in the literature. The reason for this choice of large R2 will be discussed in Section 4.1. 2.2. PORB The geometry of the PORB consists of two planar surfaces, each consisting of NR concentric metallic ring electrodes. The cross section of the PORB geometry is shown in Fig. 1(b). Fig. 1(e) and (f) shows the top and 3D views of the PORB, respectively. Both the planes of PORB share the same central z axis. The distance between the two planes is 2Zmax . The radius of the outermost ring electrode is denoted by R2 . h denotes the width of each ring electrode and g denotes the width of an air gap separating neighboring electrodes. The values of these geometry parameters are listed in Table 1 under the column PORB. The potential applied to each ring electrode of the PORB is computed by putting the r and z coordinates corresponding to the center of each ring electrode in Eq. (2). The values of constants k, C and Rm used in this computation correspond to the parameters of the reference Orbitrap (Section 2.1). The value of R2 used for the PORB geometry is 80.1 mm. The value of Zmax used for the PORB is 15 mm, which is much smaller than the Zmax in the reference Orbitrap. The reason for this will be discussed in Section 4.1. In addition to this geometry, the PORB
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Fig. 1. (a) Cross section view of the reference Orbitrap. (b) Cross section view of the PORB. (c) Equipotential contours inside the reference Orbitrap. (d) Equipotential contours inside the PORB. (e) Top view of the planar electrode of the PORB. (f) 3D view of both the upper and the lower planar electrodes of the PORB.
geometries with increased values of R2 = 120 mm and 160 mm are also studied. The equipotential curves inside the reference Orbitrap and the PORB are shown in Fig. 1(c) and (d), respectively. These curves are plotted by computing the potential inside the two traps on
the grid of points which ranges from r =−80 mm to +80 mm in the radial direction and z =−14 mm to +14 mm in axial direction. The observation of these figures suggests that the potential and field in the PORB will be similar to that of the reference Orbitrap.
H.S. Sonalikar, A.K. Mohanty / International Journal of Mass Spectrometry 417 (2017) 58–68
3. Computational methods The numerical methods used in this study are briefly outlined in this section.
61
The frequencies of ion motion in x, y and z direction in the PORB were computed from the ion trajectory by using the ‘harminv’ program [31,32]. 3.2. Image current
3.1. Boundary element method (BEM) and ion trajectory computation The geometries of the Orbitrap and the PORB have axial symmetry around the z axis. The potential and the electric field inside the Orbitrap can by analytically computed using Eq. (2) and its derivatives, respectively. On the other hand, the potential and field inside the PORB is computed by first computing the surface charge distribution on its electrodes. To compute the surface charge, the electrode surfaces of the PORB are divided in to small ring elements. The charge on each ring element is computed by using 2D boundary element method (2D BEM) [28]. Then the potential and the electric field inside the PORB is computed by applying the principle of superposition to elemental charges [29]. Once the electric field is known, the trajectory of an ion can be computed. Both the single particle and multi-particle ion trajectory simulations have been used in this study. The computer program was developed in-house to perform these simulations using RungeKutta 4th order (RK4) method [30]. The space charge effects due to Coulomb forces between the ions as well as loss of ions due to unfavorable initial velocities were taken into account in these simulations. The trajectories of ions were used for computing the contribution of each ion to the induced image current. In order to take into account the space charge effects, the force on ion i due the charge on ion j was computed as qi qj rji
Fij =
40 r 2 ji
(6)
where qi and qj are charges on ions i and j respectively, rji is a vector from position of ion j to the position of ion i with a magnitude of rji , 0 is a permittivity of a free space. The total force on ion is computed by adding Fij and the force due charges on different ring elements computed from 2D BEM. For a single ion simulation, the initial position would be x = r0 , y = 0 and z = z0 , where r0 and z0 are the selected radial and axial position of the ion. In the case of multi-particle simulation, the ions were assumed to have a Gaussian distribution with the mean initial position same as x, y and z (as above) with the standard deviation of x , y and z in the x, y and z directions, respectively. In our computations, x = y = z = 0.5 mm. The Box–Muller method [30] was used to obtain the random numbers with a Gaussian distribution. Similarly, the initial velocities of ions were assumed to have Maxwell–Boltzmann distribution with the mean of vx , vy and vz in the x, y and z directions, respectively. In our simulations, vx and vz were set to zero and vy was determined from the formula
vy =
Iimage =
dQ dt
(9)
Here, Q = Qupper − Qlower
(10)
Qupper and Qlower denote the total charge induced on the upper and lower split outer electrode, respectively. In the case of PORB geometry, both the upper and the lower electrodes consist of number of ring electrodes. Therefore, for PORB geometry, Qupper and Qlower are calculated by summing up the charges induced on individual ring electrodes as, Qupper/lower =
NR
qk
(11)
k=1
where NR is a number of ring electrodes present in the upper or lower planes of the PORB and qk is a charge induced on the kth ring electrode. Computing an induced charge due an ion on an arbitrary electrode normally requires a full 3D numerical solution of a Laplace’s equation for a given boundary conditions at each position of a moving charge. This method is computationally expensive. The induced charge can also be computed using the Green’s reciprocity theorem [5,11,33]. In this method, the induced charge computation requires the use of numerical solution only once and therefore it is computationally very efficient. In this paper, we have used the reciprocity principle to compute induced charge on the electrodes of the reference Orbitrap and the PORB. 3.3. Electric field optimization The potential values applied to the ring electrodes of the PORB were optimized to minimize the error between the electric field of the PORB and the reference Orbitrap. The method proposed in Ref. [15] was used to perform the optimization. In this method, the values of electric field were computed on a grid of points inside the PORB. To minimize the error in the z component of the field, the following objective function GZ was chosen as
N i,j M
Ez zi,j GZ = Ei,a − zi,1 z
(12)
i=1 j=2
2 − r2) kq(Rm 0
2m
(7)
√ where k is the Orbitrap constant obtained from Eq. (4), Rm (= R2 2) is the characteristic radius of the Orbitrap, r0 is the initial radial position of the ion, q/m is the charge-to-mass ratio of the ion. The thermal velocity of ions at 300 K was taken as the standard deviation in velocity of ions in each of the three directions. The thermal velocity of an ion of mass m at temperature T is given by
vth =
In the Orbitrap, the image current induced by ions is measured across the upper and lower part of the split outer electrode. The image current is given by,
kB T m
where kB is the Boltzmann constant.
Here, M and N denote number of points in grid in r and z direction, i,j respectively. Ez and zi,j denotes the value of electric field and z coordinate of point (i, j) on the grid, respectively. As the motion of ion is limited to |z| = 10 mm, the size of the grid at which the electric field is computed is also limited to ±10 mm in the z direction. The correctness of field in the z directions also insures the correctness in the r direction.1 Therefore, in this study, only the field in the z direction was optimized using function GZ . For additional control over the field in the r direction, the option to
(8) 1 This was pointed out by an anonymous reviewer of the PhD thesis of the first author [34].
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Fig. 2. Variation of potential in r direction in PORB from R1 = 7 mm to (a) R2 = 60 mm, (b) R2 = 80 mm at z = 0 mm. The continuous curve corresponds to the potential within the Orbitrap and the curve marked by stars correspond to the potential in PORB.
use the field errors in the r direction is also available as suggested in Ref. [15]. To optimize the electric field, the objective function GZ , which represents the error in the electric field, was minimized by using the Nelder–Mead simplex method [35]. Due to the use of principle of superposition, the use of 2D BEM for surface charge computation during the optimization was required only as many times as the number of ring electrodes in the PORB geometry. This made the optimization procedure computationally efficient. 4. Results and discussion The performance of the PORB is compared with that of the reference Orbitrap by computing (1) the variation in potential along r and z directions, (2) ion trajectories and (3) induced image currents in the two traps. In these simulations, appropriate values of potentials were applied to the ring electrodes of the PORB as discussed in Section 2.2. For comparison of potential as well as the single particle simulations presented in this work, the inner electrode of the Orbitrap was kept at −50 V and the outer electrode was kept at the ground potential. For multi-particle simulations of ion trajectory and image currents, the inner electrode of the Orbitrap was kept at −2000 V while the outer electrode was grounded. 4.1. Potential In this section we present the variation in potential in the Orbitrap and the PORB as a function of distance in the radial and the axial directions. This study will be used to choose the values of geometry parameters for the PORB reported in Section 2. The geometry of the PORB does not have the conventional inner and outer electrodes that the Orbitrap has. As a result, the field inside the PORB will not match with that in the Orbitrap near the innermost as well as the outermost ring electrodes of the PORB. Therefore it is expected that the potential inside the PORB will be significantly different from that of the Orbitrap near the innermost as well as outermost regions of the PORB. Consequently, the geometry parameter R2 of the PORB should be carefully chosen such that the electric field in the PORB matches with that of the Orbitrap for most of the region between the innermost and the outermost ring electrodes. Fig. 2 shows the variation of potential with r for different values of R2 in the PORB and the Orbitrap. The values of R1 and Zmax are 7 mm and 15 mm, respectively for the PORB as well as the Orbitrap. The variation of potential is computed when r is varied from R1 to R2 and z is fixed at 0 mm. The continuous curve indicates the
potential in the Orbitrap and the curve marked by stars indicates the potential in the PORB. Fig. 2(a) shows the variation of potential in the PORB for R2 = 60 mm. For this value of R2 , we observe that the potential in the PORB agrees with that in the reference Orbitrap around r = 40 mm. For other values of r, however, the potential in the PORB differs from that in the Orbitrap. When the value of R2 is set to 80 mm, the potential in the PORB closely matches with that in the Orbitrap when r ranges from 40 mm to 60 mm as can be seen from Fig. 2(b). We note that when the R2 = 80 mm, the potentials of the two traps closely match for the region which extends for 20 mm. Therefore the value of R2 = 80 mm was found to be suitable for the PORB to work as an Orbitrap. Although not shown here, higher values of R2 were seen to give better agreement between the potentials of the two traps for larger ranges of r. On the other hand, the lower values of R2 were seen to show increasing difference between the potentials in the two traps. We have chosen R2 = 80 mm for our reference Orbitrap in the present study. Fig. 3 shows the variation of potential in the PORB and Orbitrap with z for different values of Zmax used in the geometry of the PORB. The values of Zmax used are 15 mm and 20 mm. The values of geometry parameters R1 and R2 were fixed at 7 mm and 80 mm, respectively. The variation of potential was computed by varying z from 0 to Zmax for the fixed value of r = 50 mm. The continuous curve indicates the potential in the Orbitrap and the curve marked by stars indicates the potential in the PORB. As can be noted from Fig. 3, the agreement between the potential in the PORB and the Orbitrap improves as the value of Zmax is reduced from 20 mm to 15 mm. The error in the potential variation in the PORB and the Orbitrap is minimum when Zmax = 15 mm as seen in Fig. 3(a). Therefore, Zmax = 15 mm was chosen as a suitable value for the PORB geometry. As the amplitude of the z motion of ion in the Orbitrap is around 8 mm, the value of Zmax was not reduced further. Although not shown here, the difference between the potential variation in the PORB and the reference Orbitrap increases for the increased values of Zmax . For the reasons given above, we have chosen R2 for the reference Orbitrap to be 80 mm and Zmax for the PORB to be 15 mm. Here, we would like to emphasize that as long as the variation of potential in the r direction has a logarithmic form and that in the z direction has a quadratic form, respectively, the PORB will still function as an orbital trapping device regardless of the values of geometry parameters chosen. A good match between the potential in the reference Orbitrap and the PORB of a fixed geometry parameters can also be obtained by multiplying the potentials applied to the different electrodes of the PORB by a suitable factor.
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Fig. 3. Variation of potential in z direction in (a) PORB with Zmax = 15 mm, (b) PORB with Zmax = 20 mm at r = 50 mm. The continuous curve corresponds to the potential within the Orbitrap and the curve marked by stars correspond to the potential in PORB.
4.2. Ion trajectories In order to use the geometry of the PORB for electrostatic orbital trapping of ions, an ion should have stable motion in both the radial and axial directions inside the PORB. To observe how an ion behaves inside the PORB, the trajectory of ion was computed and compared to the ion motion inside the Orbitrap. In these trajectory simulations, the initial position of ion having a unit charge and a mass of 78 u was set at x = 50 mm, y = 0 mm and z = 10 mm. At this initial position of ion, its radial distance from the center of the trap is close to 50 mm where the field in the PORB is in close agreement with that inside the Orbitrap. The initial velocity components of ion were Vx = 0 m s−1 , Vy = 4769.3 m s−1 and Vz = 0 m s−1 . This initial velocity was chosen such that ion will move in a near circular orbit around the central axis of the trap. The trajectory of ion was computed for 1 ms using 4th order Runge-Kutta method with the time step set to 0.1 s. Fig. 4 shows the trajectory of an ion inside the PORB and the Orbitrap, respectively. These figures show the ion motion in x, y and z directions as well as the orbit of ion in x–y place around the central axis. The frequencies of ion motion in x, y and z direction are listed in Table 2. From Table 2, we note that ion motion’s fx and fy frequencies inside PORB are smaller compared to those inside the Orbitrap. On the other hand, fz in the PORB is larger compared to that in the Orbitrap. From Fig. 4, we note that the ion has a stable motion in both the radial (x–y) as well as axial (z) direction. Also it can be observed that the motion of ion inside the PORB is comparable to that inside the Orbitrap. The amplitude of ion motion inside the Orbitrap can be seen to be constant whereas the amplitude of ion motion inside the PORB seems to be slightly fluctuating with time. The radial spread of ion motion in x–y plane inside the PORB is larger compared to the perfectly circular orbit of an ion inside the Orbitrap as can be observed from Fig. 4(d) and (h). Trajectory simulations were also performed to compute the useful trapping area between the planar electrodes. In these simulations, the initial r position of ion was varied keeping the initial z position of ion fixed at 10 mm. The initial velocity of ion was adjusted so that ion might have a near circular orbit around the central axis. The useful trapping area was determined by observing whether the ion has stable trajectory in both radial and axial directions for a given set of initial conditions. It was observed that the ions have stable trajectories when initial r position of ion lies in the range varying from r = 35 mm to r = 56 mm. Outside this range, the ions were found to be unstable. This simulation shows that ions entering the PORB geometry in this range of r
values will be trapped while others will escape due to their unstable trajectories. 4.3. Variation of axial frequency In the Orbitrap, the frequency of ion motion in z direction (fz ) is used to determine the mass of an ion. Due to the precise shape of the Orbitrap, the frequencies of ion motion in the Orbitrap are independent of the initial position of the ion. The electric field in the PORB, however, is only an approximation to the field in the reference Orbitrap. Due to the errors in the field, the frequencies of ion motion inside the PORB geometry will vary with the initial conditions of the ion. Consequently, it is important to reduce the dependence of fz on the initial z position of the ion to improve the accuracy of mass detection by optimizing the electric field inside the PORB. For optimizing the electric field inside the PORB, the potentials applied to the ring electrodes of the PORB were adjusted to minimize the objective function given in Section 3.3. Once the values of potentials to be applied to the ring electrodes to achieve an optimum electric field were obtained, the frequency of ion motion in z direction was computed for different initial z positions of a single ion. Fig. 5(a) shows the variation of fz with the initial z position of an ion. A curve marked by stars represents variation in fz before optimization. A curve marked by triangles represents variation in fz after optimization. From Fig. 5(a), we observe that before optimization, the variation in fz was 118.0 Hz as z is varied from 1 mm to 10 mm. After optimization the variation in fz was minimized to just 12.9 Hz. This result shows that the optimization procedure used in this work was successful in minimizing the errors in the electric field of the PORB which results in the reduced variation of axial frequency of ion with the z position of an ion. Fig. 5(b) shows the values of potentials applied to the ring electrodes of the PORB before and after optimization. The blue curve indicates potentials before optimizations and the red curve indicates the potentials after optimization. From Fig. 5(b) and (a), we can observe that by slight change in the applied potentials, the errors in the electric field can be significantly reduced. 4.4. Image current The mass-to-charge ratio of ions in the Orbitrap is determined by detecting the image current induced across the split outer electrodes of the Orbitrap. The frequency analysis of the acquired image current transient is done to obtain the accurate mass analysis of trapped ions. In this section, we present the comparison of image currents computed in the Orbitrap and the PORB. A multi-particle simulation involving an ensemble of 1000 ions, each having a unit
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Fig. 4. (a) Displacement of ion in x direction in the PORB. (b) Displacement of ion in y direction in the PORB. (c) Displacement of ion in z direction in the PORB. (d) Motion of ion in x–y plane in the PORB. (e) Displacement of ion in x direction in the Orbitrap. (f) Displacement of ion in y direction in the Orbitrap. (g) Displacement of ion in z direction in the Orbitrap. (h) Motion of ion in x–y plane in the Orbitrap.
charge and a mass of 78 u has been carried out. The image current was computed for the duration of at least 1 ms. The initial positions and velocities of ions were calculated using the Gaussian distribution. The mean radial and axial position of ions were r = 50 mm and z = 10 mm with standard deviation of 0.5 mm in each direction. Ions
were given a mean initial tangential velocity such that they follow a near circular path in the trapping region. The thermal velocity of ions at 300 K was used as a standard deviation in velocity distribution among the ions. The value of applied potential V0 in these simulations was −2000 V.
H.S. Sonalikar, A.K. Mohanty / International Journal of Mass Spectrometry 417 (2017) 58–68 Table 2 Frequencies of ion motion in the Orbitrap and the PORB. Frequency (kHz)
Orbitrap
PORB
fx fy fz
15.18 15.18 10.57
14.35 14.35 11.16
Fig. 6(a) and (b) shows the image current induced in the reference Orbitrap and the PORB, respectively. The image current in the reference Orbitrap has a constant amplitude. The amplitude of image current in the PORB is initially larger compared to that of the reference Orbitrap but this rapidly decays. Fig. 6(c) and (d) shows the frequency spectrum of image current computed in the reference Orbitrap and the PORB, respectively. It can be observed that the width of a principle peak in the image current spectrum of PORB is larger compared to that of the reference Orbitrap, implying that the mass resolution in the PORB is lower than that in the reference Orbitrap. Fig. 6(e) and (f) shows a snapshot of positions of ions in X–Y plane after 1 ms in the reference Orbitrap and the PORB, respectively. The ions were initially bunched together in a volume of sphere having radius of 2.1 mm indicated by circles in these figures. It can be observed that the ions have formed a ring due the space charge effects. Figures 6(g) and (h) shows the snapshot of ion motion in Y–Z plane for the two traps after 1 ms. From these figures, it can be observed that in the reference Orbitrap, the spread of ions in axial direction is from z = 1 mm to z = 4 mm where as the spread of ions in the PORB in the axial direction is from z =−10 mm to z = 10 mm. The relatively large spread of ion in z direction in the PORB is on account of the non-linearities of field in the PORB. This large spread of ions in the PORB results in the rapid decay of image current signal due to out-of-phase motion of ions in the PORB. To characterize the non-linearities in the electric field in the ideal Orbitrap as well as the PORB in the axial direction, the variation of potential in the axial direction in both the traps was represented by a polynomial of degree 6 as shown in Eq. (13): V (z) = C6 z 6 + C5 z 5 + C4 z 4 + C3 z 3 + C2 z 2 + C1 z + C0
(13)
The coefficients of this polynomial were calculated by a least square fit. The values of the even order coefficients are listed in Table 3 for both the Orbitrap and the PORB. The coefficient C2 is the contribution of quadratic potential which represents the linearity of the
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electric field while C0 represents a constant potential which does not contribute to the electric field. The higher order coefficients represent the non-linearities in the field. As can be observed from Table 3, the higher order coefficients for the Orbitrap potential are negligibly small and C2 is an only significant coefficient contributing a linear field. On other hand, the values of coefficients C4 and C6 for the PORB are considerably large compared to those for the Orbitrap indicating that the axial field in the PORB contains non-linear components. The values of odd order coefficients such as C1 , C3 and C5 were less than 2.0e−15 for both the traps due to top-bottom symmetry of their geometries. As an aside, we have also carried out simulations to measure image current in two other PORB geometries. In one, R2 has been increased to 120 mm (compared to R2 value of 80 mm in our study) and in the other, the value of R2 has been further increased to 160 mm. In these simulations, the initial mean radial positions of ions were also increased to 75 mm and 90 mm, respectively. The image currents computed for 2 ms in these two geometries are shown in Fig. 7(a) and 7(b), Fig. 7(a) corresponding to R2 = 120 mm and Fig. 7(b) corresponding to R2 = 160 mm, respectively. A comparison of the image current presented in Fig. 6(b) (for R2 = 80 mm), Fig. 7(a) (for R2 = 120 mm) and Fig. 7(b) (for R2 = 160 mm) indicates an interesting trend. It is expected that as R2 is increased, the non-linearity in the trapping region decreases as the fringing fields at the edges of the trap have reduced effect on the field in the trapping region. This results in the field which is in a better agreement with that of the Orbitrap. This is the reason why there is reduction in the rate of decay with increase in R2 . In Fig. 6(b), the image current has rapidly decayed in 0.3 ms, whereas in the image current depicted in Fig. 7(a), the decay is much slower. In Fig. 7(b), the decay is negligible. From these results, it can be concluded that the field in the PORB geometry with higher values of R2 is better. While this is good, it however involves increasing the size of the trap which may not be desirable. In our search for a better performance of the original PORB (R2 = 80 mm), we have undertaken two additional simulations. In the first, the potentials applied to the ring electrodes of the PORB were optimized by using the method outlined in Section 3.3. The image current computed after the optimization is presented in Fig. 8(a). Fig. 8(a) shows the image current signal in the PORB with R2 = 80 mm and Zmax = 15 mm for the duration of 2 ms. The mean initial z position of 1000 ions in this simulation was 10 mm. It can
Fig. 5. (a) Variation of axial frequency fz with initial z position in PORB. Without optimization, the variation in fz is 118.0 Hz as z is varied from 1 mm to 10 mm. After optimization the variation in fz is minimized to 12.9 Hz. A curve marked by stars represents variation in fz before optimization. A curve marked by triangles represents variation in fz after optimization. (b) Potentials applied to the ring electrodes of the PORB before and after optimization. The blue curve indicates potentials before optimization and the red curve indicates the potentials after optimization. The ring electrodes are numbered from 0 to 29. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 6. Image current signal in (a) the reference Orbitrap and (b) the PORB. The image current spectrum in (c) the reference Orbitrap and (d) the PORB. Snapshot of ion motion in X–Y plane in (e) the reference Orbitrap and (f) the PORB. Snapshot of ion motion in Y–Z plane in (g) the reference Orbitrap and (h) the PORB taken at t = 1 ms.
be seen from the figure that the optimization has resulted in a better performance with the image current decaying at a slower pace. Although this is better than the earlier performance, it still falls short of the image current obtainable in the Orbitrap. Consequently, in the second experiment, the spacing between the planar electrodes was reduced from 30 mm, which is a spacing used in
the PORB, to 25 mm. The mean initial z position of ions was also reduced to 8 mm. All other parameters were kept same. The potentials applied to the ring electrodes were optimized as was done for obtaining Fig. 8(a). Fig. 8(b) shows the image current computed after applying the optimized values of potentials to the ring electrodes of this modified geometry. It is clear from the figure that
H.S. Sonalikar, A.K. Mohanty / International Journal of Mass Spectrometry 417 (2017) 58–68 Table 3 Values of coefficients of V(z) for the Orbitrap and the PORB. Coefficient
Orbitrap
PORB
C0 C2 C4 C6
−7.2590 0.30171 −2.3321e−14 1.7552e−14
−7.2728 0.30795 1.2757e−2 −1.2770e−2
the decay of the image current for the 2 ms period is negligible in comparison to the image current plotted in Fig. 8(a). This indicates that the reduced spacing has resulted in a further improved performance.
5. Concluding remarks This study investigated a planar geometry mass analyzer which has fields and performance similar to that of a reference Orbitrap. In this study, we have taken up for investigation a geometry referred to as PORB. This geometry consists of two planes, each having 30 ring electrodes. Comparisons of (1) the potential variation in the principle directions, (2) ion trajectories, and (3) the image currents in these two traps have been made. In this numerical study, both single particle and multi-particle simulations have been performed.
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Our investigations show that it is possible to mimic the quadrologarithmic potential distribution of the reference Orbitrap in our PORB. However, the PORB being a non-linear device, has a performance inferior to the reference Orbitrap. In our simulations, we have considered 30 concentric ring electrodes in each plane of the PORB. The geometry with the gap of 0.5 mm between the ring electrodes was studied and it was found that the field was not significantly affected by an increased gap. It is expected that larger number of rings will improve the field in the trap. Separate studies will have to be carried out to quantify the effect of parameters g and h on the electric field. The optimization of the electric field in the PORB has been carried out and it was shown that an improved performance was obtained after applying the optimized values of potentials to the segmented-electrodes. The performance could be further improved by reducing the spacing between the planar electrodes. An attempt can be made to minimize the amplitude of higher frequency harmonics present in the image current spectrum of the PORB by further optimizing the geometry as well as by super imposing the image currents induced in different electrodes of the PORB by first multiplying them by a suitable factor. Finally, we present some thoughts on how the electrical connections can be made in PORB. A resistive network can be used to provide appropriate DC potentials to the electrodes. Since the same electrodes are used for application of DC potentials for ion
Fig. 7. (a) The image current in the PORB with R2 = 120 mm. (b) The image current in the PORB with R2 = 160 mm.
b Fig. 8. (a) The image current in the optimized PORB with R2 = 80 mm and Zmax = 15 mm. (b) The image current in the PORB with R2 = 80 mm and Zmax = 12.5 mm.
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trapping as well as detection of image current for mass analysis, these DC voltages need to be isolated from the induced image current by using inductors. Also, the induced image current signal needs to be passed through a capacitive coupling to isolate it from the applied DC potentials. The use of the inductive and capacitive coupling should have only a minimum effect on the sensitivity of an instrument as the losses in the inductors and capacitors are negligible. Acknowledgements We thank an anonymous reviewer of the Ph.D. thesis of the first author for suggesting improvements in the optimization procedure used in this study. We thank Prof. A.G. Menon for his valuable suggestions during the preparation of the manuscript. We also thank three reviewers for giving important suggestions for significantly improving the manuscript. References [1] A.A. Makarov, Mass spectrometer, U.S. Patent 5,886,346 (1999). [2] M. Hardman, A.A. Makarov, Interfacing the Orbitrap mass analyzer to an electrospray ion source, Anal. Chem. 75 (2003) 1699–1705. [3] Q. Hu, R.J. Noll, H. Li, A. Makarov, M. Hardman, R.G. Cooks, The Orbitrap: a new mass spectrometer, J. Mass Spectrom. 40 (2005) 430–443. [4] A. Makarov, E. Denisov, A. Kholomeev, W. Balschun, O. Lange, K. Strupat, S. Horning, Performance evaluation of a hybrid linear ion trap/Orbitrap mass spectrometer, Anal. Chem. 78 (2006) 2113–2120. [5] A.A. Makarov, Electrostatic axially harmonic orbital trapping: a high-performance technique of mass analysis, Anal. Chem. 72 (2000) 1156–1162. [6] K.H. Kingdon, A method for the neutralization of electron space charge by positive ionization at very low gas pressures, Phys. Rev. 21 (1922) 408–418. [7] R.D. Knight, Storage of ions from laser produced plasmas, Appl. Phys. Lett. 38 (1981) 221–223. [8] A.G. Marshall, F.R. Verdun, Fourier Transforms in NMR, Optical and Mass Spectrometry – A User’s Handbook, Elsevier, Amsterdam, 1990. [9] R.H. Perry, R.G. Cooks, R.J. Noll, Orbitrap mass spectrometry: instrumentation, ion motion and applications, Mass Spectrom. Rev. 27 (2008) 661–699. [10] Q. Hu, A. Makarov, R.G. Cooks, R.J. Noll, Resonant ac dipolar excitation for ion motion control in the Orbitrap mass analyzer, J. Phys. Chem. A 110 (2006) 2682–2689. [11] G. Wu, R.J. Noll, W.R. Plass, Q. Hu, R. Perry, R.G. Cooks, Ion trajectory simulations of axial ac dipolar excitation in the orbitrap, Int. J. Mass Spectrom. 254 (2006) 53–62. [12] A. Kharchenko, G. Vladimirov, R.M.A. Heeren, E.N. Nikolaev, Performance of orbitrap mass analyzer at various space charge and non-ideal field conditions: simulation approach, J. Am. Soc. Mass Spectrom. 23 (2012) 977–987. [13] A. Makarov, E. Denisov, O. Lange, Performance evaluation of a high field Orbitrap mass analyzer, J. Am. Soc. Mass Spectrom. 20 (2009) 1391–1396.
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