Theoretical Study of Dual-Direction Dipolar Excitation of Ions in Linear Ion Traps

Abstract

The ion enhanced activation and collision-induced dissociation (CID) by simultaneous dipolar excitation of ions in the two radial directions of linear ion trap (LIT) have been recently developed and tested by experiment. In this work, its detailed properties were further studied by theoretical simulation. The effects of some experimental parameters such as the buffer gas pressure, the dipolar excitation signal phases, power amplitudes, and frequencies on the ion trajectory and energy were carefully investigated. The results show that the ion activation energy can be significantly increased by dual-direction excitation using two identical dipolar excitation signals because of the addition of an excitation dimension and the fact that the ion motion radius related to ion kinetic energy can be greater than the field radius. The effects of higher-order field components, such as dodecapole field on the performance of this method are also revealed. They mainly cause ion motion frequency shift as ion motion amplitude increases. Because of the frequency shift, there are different optimized excitation frequencies in different LITs. At the optimized frequency, ion average energy is improved significantly with relatively few ions lost. The results show that this method can be used in different kinds of LITs such as LIT with 4-fold symmetric stretch, linear quadrupole ion trap, and standard hyperbolic LIT, which can significantly increase the ion activation energy and CID efficiency, compared with the conventional method.

Introduction

Collision-induced dissociation (CID) is the most commonly employed method for structural elucidation of various compounds ranging from small organic molecules to large biomolecules in mass spectrometry [1, 2]. CID is usually accomplished by ion activation with electric fields and then ion energetic collisions with a neutral gas followed by chemical bond dissociation and ion fragmentation [3]. Because of the ease of implementation, multiple tandem mass spectrometers with CID have been developed, including magnetic-sector mass spectrometer [4], time-of-flight/time-of-flight mass spectrometer [5], triple quadrupole mass spectrometer [6], Fourier transform-ion cyclotron resonance mass spectrometer [7], Orbitrap-based instrument [8], and quadrupole ion trap (QIT) mass spectrometer [9]. The different ion activation process in different mass spectrometry platforms will result in different collision energy and thus different activation time scale spanning from the order of a few microseconds to hundreds of milliseconds, which gives rise to different behavior, such as dissociation channel and efficiency [2]. For example, the ions in QIT can be trapped for a long time, so multistage tandem mass analysis can be realized in a single chamber, and the access to very slow dissociation channel and very high dissociation efficiency (up to 100%) can be obtained [2, 10]. These are also the unique advantages for QIT in performing CID and tandem mass spectrometry analysis.

CID in a QIT is usually implemented by applying a supplementary direct current (DC) potential [1113] or alternating current (AC) potential [9, 14, 15] between a pair of trap electrodes and an auxiliary electric field is produced by the above potential. The mass-selected ions are excited by absorbing energy from the electric field and thus move close to the electrodes; therefore, the ion kinetic energy is increased [16]. Collisions of the high kinetic energy ion with the buffer gas will convert the kinetic energy into ion internal energy. The ion could be fragmented if the deposited internal energy is greater than the bond dissociation energy. As single collision can only deposit a small amount of internal energy into the ion, tens to hundreds of collisions are required to eventually overcome the bond energy barriers for ion dissociation. Moreover, there is competition between ion activation and deactivation, for the reason that collisions can also lead to the dissipation of the internal energy and kinetic energy, particularly in the case where the ion with high internal energy undergoes relatively low kinetic energy collisions [17]. As a result, the internal energy deposited into the ions is limited and it usually takes a relatively long time to perform CID. Additionally, a very significant shortage of CID in QIT is the low mass cut-off (LMCO) effect, which is referred to as the fact that product ions below a certain mass-to-charge ratio determined by the Mathieu parameter q u [where u = x, y for linear ion traps (LITs) and u = z for three-dimensional ion traps (ITs)] value of parent ions cannot be detected [18]. Generally, the smaller the q u value, the lower the mass-to-charge ions that can be trapped. However, the maximum ion kinetic energy will decrease with the decreasing Mathieu parameter q u value, which cannot exceed the potential well of the trap. As a consequence, at small q u value, the internal energy is always insufficient to break chemical bond for dissociation [19].

To address the above issues, several methods for increasing the ion internal energy in ITs have been investigated and reported over the past decades [2024]. The first method is to increase the ion kinetic energy, which is achieved by employing a high-amplitude AC signal at relatively high q u value for ion activation with a short time followed by an ion trapping period at a low q u value, such as pulsed Q dissociation (PQD) [20], high amplitude short time excitation (HASTE) [21] and pulsed q dynamic CID [22]. The second is to increase the ion initial kinetic energy by heating the buffer gas, termed as thermally assisted collision-induced dissociation (TA-CID) [23]. The third is to improve the conversion efficiency of ion kinetic energy to the ion internal energy, which is accomplished by increasing the mass of the buffer gas, e.g., the use of argon as buffer gas [24].

In the conventional CID process in a LIT, ions are generally excited in only one direction by applying one dipolar AC signal to one pair of electrodes. Wells [25] described a rotating excitation method in linear ion processing apparatus, such as LIT, in which two mutually orthogonal dipoles could be applied in phase quadrature (90° out of phase). The ion circulates about central axis of the trapping field because of the rotating supplemental resonant field and the ion kinetic energy can be increased by this method. Dang et al. [26] recently introduced a method that can enhance the ion activation and CID in a LIT by employing two identical dipolar AC signals. In this method, ions are excited in two radial coordinate directions simultaneously and mainly move along the diagonal direction of the trapping region. It has been found that more energy could be deposited into the mass-selected precursor ions, and therefore the ion dissociation rate constant and fragmentation efficiency were all increased in the experiment using triangular-electrode linear ion trap (TeLIT) by this method. In addition, the LMCO was redeemed and some lower mass fragment ions could be detected.

In this paper, detailed simulations were performed to further investigate the dual-direction dipolar excitation properties by using different shape LITs and experimental conditions.

Simulations

Theoretical Method

In an ideal quadrupole field in the absence of collisions, ion motion in the x- or y-direction are uncoupled, which can be described by the Mathieu equation:

$$ \frac{d^2u}{d{\varepsilon}^2}+\left(a-2q \cos 2\varepsilon \right)u=0 $$
(1)

where u = x, y.

Baranov [27] implemented an analytical method of the solution of the above equation in conjunction with algebraic presentation of Mathieu functions. By this analytical approach, the time and phase averaged radial ion energy can be described [28]:

$$ {E}_r=\frac{m{w}_2}{8}{r}_i^2{v}_{21}^2\left(0,{q}_u\right)+\frac{m}{2{\gamma}_{\perp}^2}{v}_{22}^2\left(0,{q}_u\right) $$
(2)
$$ {\mathrm{r}}_i^2={x}_i^2+{y}_i^2 $$
(3)

where m is the ion mass, w 2 is the main radio frequency (rf) power angular frequency, r i is the initial radial position, \( \frac{1}{\gamma_{\perp }} \) is the most probable ion thermal speed in the radial direction, and \( {v}_{21}^2\left(0,{q}_u\right) \) and \( {v}_{22}^2\left(0,{q}_u\right) \) are the dimensionless parameters that depend only on the field properties. The second term is very small and can usually be neglected.

As previously reported [16], the major contributor to the increase in effective temperature during resonance excitation is the power absorption from the main rf trapping field rather than from the dipolar excitation signal. The main role of the dipolar excitation signal is to move ions into regions of the trap where the rf electric field is stronger than the trap center [16]. Therefore, Equation 2 can approximately represent the ion energy with excitation by auxiliary AC or DC signals.

It has been recognized that the ion moving radius (i.e., the r i value) by dual-direction excitation method is greater than that by conventional single-direction excitation method [26]. This also can be seen from Supplementary Figure S1 (see Supplemental Information), which shows the comparison of ion trajectory with the above two methods by direct numerical simulations (the simulation method is described in the following section). As a consequence, the ion average energy is improved (the average energy with single-direction and dual-direction excitation method are 1.7eV and 2.8 eV, respectively). Because the well depth in the xy plane is greater than that in either x- or y-direction [26], ions can still be trapped.

Numerical Simulation

It is well known that the details of ion motion can be calculated directly at any time with great accuracy by numerical simulations, and the effects of several significant factors, such as collisions with the buffer gas, higher-order fields can be included. Here, simulations of ion motion were performed using AXSIM software [29].The model LITs, which were 2D slices for the radial cross section in the center of the trap, were prepared using SIMION 8.1 (Scientific Instrument Services, Ringoes, NJ, USA) with a potential grid unit of 0.025 mm. A hard sphere collision model was used at a pressure of 1 mTorr with helium as buffer gas and a temperature of 300 K, unless otherwise noted. The main rf voltage with a frequency of 768 kHz was applied in the quadrupolar fashion. The AC and rf signals are defined by the following equation:

$$ V={V}_{0-p} \cos \left(2\pi ft+\upvarphi \right) $$
(4)

where \( {V}_{0-p} \) is the zero-to-peak amplitude, f is the frequency, and φ is the initial phase.

The multipole expansion coefficients were calculated using the previously reported method [30], and were seen as the Fourier components of potential distribution at a circle with the radius of 5 mm in different LITs with quadrupole supply. The normalization radius for all LITs is 5 mm.

It has been shown that there is a competition between activation and deactivation for resonance excitation under conditions of multiple ion-neutral collisions in the trap, which leads to a steady-state internal energy. The internal energy of ions in the trap can be characterized by a Boltzmann distribution and effective ion temperature [31]. Here the following equation was used to calculate the effective temperature [31]:

$$ {\mathrm{T}}_{eff}=\frac{m_r}{m_g}T+\frac{m_r}{m_i}\frac{2<{K}_i>}{3k} $$
(5)

where m r, m g, and m i are the reduced mass, the buffer gas mass, and the ion mass, respectively; <K i> is the average laboratory frame ion kinetic energy.

Ion Trap Model

Four different IT models, including pure LIT, LIT with standard hyperbolic geometry, LIT with 4-fold symmetric stretch (4FSS), and linear quadrupole ion trap (LQIT) were used in this simulation. The field radius of all ITs is 5 mm and they all have no slots on the electrodes. The trapping field in the pure LIT is a pure quadrupole field calculated according to Equation

$$ \Phi \left(\mathrm{x},\mathrm{y},\mathrm{t}\right)=\left(\frac{x^2-{y}^2}{25}\right){V}_t $$
(6)

while the AC auxiliary fields were calculated using the following standard hyperbolic LIT model. The point (x,y) = (0,0) is the center of the LIT model and \( {V}_t \) is the applied rf signal. In the standard hyperbolic LIT, four hyperbolic electrodes were truncated at distances of 2r 0 (r 0 is the field radius) from the field center. The multipole expansion coefficients in this trap are as follows: A0 = 0.0000, A2 = 1.0015, A6 = –0.0006, A10 = –0.0000, A14 = 0.0001, A18 = –0.0001. The LITs with 4FSS have been widely used in commercial instruments [32].They have some significant advantages over the LITs with 2-fold symmetric stretch, such as zero axial potential at the field center, and thereby allow efficient ion injection at a wide range of q u value [33]. Here the LIT with 4FSS is composed of four identical electrodes and the cross section of each electrode is an equilateral hyperbola with a semi-major axis of 4.2 mm. They were also truncated at distances of 2r0.The multipole expansion coefficients in this trap are as follows: A0 = 0.0000, A2 = 0.9901, A6 = 0.0094, A10 = 0.00098, A14 = 0.0001, A18 = –0.0000. In LQIT, the ratio of the rod radius to the field radius is about 1.126 [34, 35].The multipole expansion coefficients in this trap are as follows: A0 = 0.0000, A2 = 1.0027, A6 = 0.0010, A10 = –0.0024, A14 = –0.0002, A18 = –0.0001.

Initial Conditions

All simulations were performed using ions of m/z 556 with a cross section of 160 A2, which correspond to the leucine enkephalin molecule ions [36]. Ion coordinates and energy were randomly chosen from a Gaussian distribution around the center of the trap with a 0.1 mm standard deviation and around 0 eV with a 0.07 eV deviation, respectively. Then they were subjected to a collision cooling period of 10 ms. The initial phases of the dipolar AC signals and the rf signals were set at zero.

Results and Discussion

Ion Trajectory Analysis

It is well known that ion motion in the trap is composed of a low-frequency secular motion and a superimposed high-frequency rf-ripple or micromotion [18, 37]. Figure 1 shows the ion motion in response to dual-direction dipolar excitation without and with random collisions in the pure quadrupole field. As can be seen from Figure 1a, even though the ion initial conditions in the x- and y-directions and two AC dipolar signals are the same, ion motion between x- and y-directions is different and this difference becomes more obvious as the amplitude of the ion motion increases (Figure 1b), which results in the ion lost after 5.3 ms excitation. This is because the ion displacement difference between the two directions should be smaller to ensure ions cannot strike the electrodes, when the ion moving radius increases (Supplementary Figure S2). Since the amplitude, phase, and frequency of the secular motion in the two directions are almost the same, this difference mainly arises from the difference in micromotion between the two directions (Figure 1a). At 5.3 ms, the amplitude in each direction is more than 7.0 mm and the displacement difference is about 1.8 mm (Figure 1a). If the ion initial conditions in the x- and y-directions are different, for example, x = 0.2 mm, y = –0.3 mm, E x = 0.01eV, E y = –0.02 eV (negative signal represents –x-direction), the frequency and phase of the secular motion in x- and y-directions are significantly different at the beginning but converge quickly, whereas the amplitude in the y-direction is always slightly greater than that in the x-direction (Supplementary Figure S3a and b).

Figure 1
figure1

Simulation of dual-direction dipolar excitation of m/z 556 ion with the following initial condition: x = 0.2 mm, y = 0.2 mm, E x = 0.01 eV, E y = 0.01 eV at q u = 0.25 in the pure quadrupole field. The excitation amplitude is 50 mV0-p, and the excitation frequency is 68.85 kHz. (a) Temporal variation of ion displacement in the x- and y-directions during 0–0.3 ms and 5.0–5.3 ms; and (b) ion trajectory over a period of 5 ms, no buffer gas. (c) Temporal variation of ion displacement in the x- and y-directions during 0–0.3 ms and 5.0–5.3 ms; and (d) ion trajectory over a period of 5 ms, in the presence of 1 mTorr helium buffer gas

The above is the result under condition of no buffer gas. Figure 1c and d show the ion motion in the presence of helium gas at 1 mTorr. The amplitude of oscillation initially increases with time, just like the situation without buffer gas. But a steady state is finally reached because of collisions of the ion with buffer gas. The two amplitudes of the secular motion in the x- and y-directions are different and this difference that changes randomly (but always small) is mainly caused by the random collisions, whereas the frequency and phase in the two directions are almost the same. Compared with the multiple collisions, the effect of the initial conditions on the ion motion can be neglected (Supplementary Figure S3). In particular, ions always experience a period of cooling stage before ion excitation.

The maximum average energy of the ion in Figure 1 during the initial 5 ms excitation period that can be reached by the dual-direction method with buffer gas is about 21.34 eV (excitation amplitude 220 mV0-p), compared with about 5.97 eV (excitation amplitude 95 mV0-p) by the conventional method under the same condition. At this point, the maximum moving radius for dual-direction excitation is 7.36 mm (X max = 6.16 mm, Y max = 5.79 mm), whereas it is 4.98 mm (X max = 4.98 mm, Y max = 0.62 mm) for the conventional single-direction method. From Figure 1a, it can be seen that the maximum displacement in either x-direction or y-direction is about 7.5 mm and the moving radius is about 9.61 mm in the absence of gas. They are much greater than that in the presence of buffer gas. This is consistent with the above analysis of the ion trajectory. That is to say, collisions enlarge the difference of the amplitudes of ion motion in the x- and y-direction and lead to the decrease of ion maximum moving radius. From the above we can see that the increment of the ion average energy is 257%, more than once (the addition of one excitation dimension), compared with the conventional method. So, larger moving radius in each direction (greater than the radius of the LIT or the maximum excursion with excitation only in one direction) makes a great contribution to the improvement of the ion energy by the dual-direction excitation method.

Two Different AC Signals

The dual-direction excitation can be accomplished with two identical dipolar AC signals, which were produced by only one AC power. It can also be performed with two separate power supplies. So the two AC excitation signals can have different phases, amplitudes, and frequencies. To further investigate the properties of dual-direction excitation method, the ion motion using two different AC excitation signals with different phase, amplitude, and frequency was simulated. The pure LIT was used in the simulation.

Figures 2a, 3, and Supplementary Figure S4 show the simulation results of ion motion and energy using two AC signals with same amplitude and frequency but different phase. In Figure 2a, it is observed that the phase difference of the ion secular motion between x- and y-directions is around \( \frac{\uppi}{2} \) (small variation is caused by random collisions), which is equal to that of the two AC signals. The two amplitudes of the secular motion at the stable state in the x- and y-directions are very close and the small difference should be attributed to the random collisions. The two secular frequencies in the two directions are the same, 68.85 kHz. One circle of the ion trajectory is like an ellipse. Generally these conclusions are also valid for the two AC signals with other phase differences in the pure LIT.

Figure 2
figure2

Temporal variation of the displacement (during 5.0–5.15 ms and 9.0–9.15 ms) and ion trajectory [during 5.0–5.05 ms (green) and 9.0–9.05 ms (red)] of m/z 556 ion (initial condition: x = 0.2 mm, y = 0.2 mm, E x = 0.01 eV, E y = 0.01 eV) in the x- and y-directions with dual-direction dipolar excitation at q u = 0.25 at helium gas pressure of 1 mTorr using two different AC signals in the pure quadrupole field. (a) different phase: \( {\upvarphi}_x={\upvarphi}_{RF}=0,\;{\upvarphi}_y=\frac{\uppi}{2} \), AC amplitude V x=V y=50 mV0-p, AC frequency f x = f y = 68.85 kHz; (b) different amplitude: V x = 60 mV0-p, V y = 40 mV0-p, \( {\upvarphi}_x={\upvarphi}_y={\upvarphi}_{RF}=0, \) f x = f y = 68.85 kHz; (c) different frequency: f x = 68.9 kHz, f y = 68.7 kHz, \( {\upvarphi}_x={\upvarphi}_y={\upvarphi}_{RF}=0, \) V x=V y=50 mV0-p

Figure 3
figure3

Simulations in the pure quadrupole field by single-direction excitation and dual-direction excitations with phases of 0, 45, 90, 135, and 180°, respectively. (a) The effective ion temperature which corresponds to 200 survival ions at m/z 556. (b) Ion survival number as a function of the AC amplitude over a excitation period of 30 ms at q u = 0.25 (secular frequency 68.85 kHz) in the presence of helium gas at 1 mTorr

The comparisons of the ion effective temperature and the corresponding ion survival number of 200 ions among five different phase differences are illustrated in Figure 3. It can be seen that the effective temperature by dual-direction excitation method is always higher than that by single-direction excitation method, whatever the phase difference between the two AC signals is. However, the highest AC amplitude without significant ion loss is dramatically different at different phase differences. The amplitude under conditions of the phase differences of 0 and 180° can even reach 110 mV0-p, whereas it is only about 60 mV0-p for 90°. As a result, the attainable effective temperature at phase differences of 0 and 180° is much higher than that at other phase differences. This is due to the fact that the phase difference of two AC excitation signals does affect the ion trajectories in the IT region (Supplementary Figure S4). At 0° and 180°, ions approximately move along the diagonal direction, which allows more room for ion excitation. In the case of two AC signals with a phase difference of 90°, ion trajectory is close to a circle. The maximum moving radius in either x- or y-direction is the same as that with excitation in one direction. Under the same conditions, the maximum excitation amplitude using two different AC signals with 90° phase difference is close to that with excitation in one direction, whereas it is much less than that using two identical excitation signals. Owing to the excitation in another dimension, the average kinetic energy can be increased, but the increment is much smaller. Moreover, as the room for ion motion gets smaller, ions are easier to be lost. But at the same AC amplitude, the phase difference will not obviously affect the ion average kinetic energy (Figure 3). It should be noted that the variation of the effective temperature and ion survival number for the two pairs: 0 and 180, 45 and 135 is almost the same. As can be seen from Supplementary Figure S4, the main difference of the ion trajectory between these pairs is the direction and the shapes are identical. This situation also exists in the pairs: 90 and 270, 45 and 225, 225 and 315.

Figure 2b shows the ion motion using two AC signals with different amplitudes. The amplitude of the secular motion in the y-direction is smaller, which corresponds to the smaller amplitude of dipolar AC signal, whereas there is no difference of the phase and frequency of the secular motion in the two directions. It is apparent from Supplementary Figure S5 that compared with single-direction excitation, dual-direction excitation at all voltage ratios can enhance the ion activation energy. However, different voltage ratios show different performance and the available effective ion temperature at the voltage ratio of 1.0 is the highest. It is understandable that if two identical AC signals are employed, ions mainly move along the line y = x and in this instance there is enough space for ions to avoid the neutralization on the rods. For the ratios of 1.2 and 1.5, although the ion trajectory is still approximately along a line, the line is biased towards the electrodes (Figure 2b). That is why ions are easier to be lost as the amplitude increases.

For two AC signals with different frequencies, the relation between the secular motion in x- and y-directions is more complicated (Figure 2c). At the steady state, the frequency of the secular motion in each direction is equal to that of the dipolar AC signal applied in the corresponding direction. Owing to different frequencies in the two directions, the phase difference between the two directions changes over time. It has been reported that the slow ion motion in the IT in response to dipolar AC excitation consists of a frequency component at the excitation frequency (f ac) and at the secular frequency (f 0) under off-resonance excitation (f acf 0) [38]. Under the damping effect of the buffer gas, the motion at the secular frequency is decayed, whereas the ion motion at the excitation frequency is stable. Therefore, it can be seen that ion motion amplitude in the y-direction at the beginning of the excitation is much greater than that at steady state (the inset in Figure 2c). Xu et al. [38] introduced the expression of the ion motion amplitude at the excitation frequency:

$$ {A}_{\omega_{ac}}=\frac{P}{\sqrt{c^2{\omega}_{ac}^2+{\left({\omega}_{ac}^2-{\omega}_0^2\right)}^2}} $$
(7)

where \( P=a\left({U}_{ac}/{u}_0\left)\right(e/m\right) \) with U ac defined as the amplitude of the applied AC signal, u 0 as the ion trap dimension in the direction of the dipolar AC excitation, and a is a constant. The smaller the difference between the AC frequency and the secular frequency is, the bigger the amplitude. The frequency 69.0 kHz is much closer to the secular frequency 68.85 kHz. This is why the amplitude of the secular motion in x-direction at the steady state is much greater than that in y-direction. The ion trajectory (Figure 2c) always changes dramatically over time because of the variation of the phase and amplitude. It can be inferred that the performance cannot be better than that using identical AC signals.

In conclusion, if two different AC signals are employed, ion average energy or effective temperature can be improved because of the addition of another excitation dimension compared with single-direction excitation method. However, its performance is worse than that using identical AC signals because the ion motion amplitude is smaller and ions are easier to strike the rods.

The Effects of Higher-Order Fields

In practical LITs, higher-order fields are inevitable for many reasons, such as the truncation of the electrodes, the use of simplified electrodes, and the existence of the ion ejection slots [30, 3335, 3942]. Moreover, it is well known that ITs with some positive even-order fields might improve performance [33, 3945]. For example, the addition of octopole [4045] or dodecapole fields [30, 33, 39] can compensate for the detrimental field caused by the slot and, therefore, the mass resolution and mass accuracy are enhanced.

In this paper, the effects of the dodecapole field component on the dual-direction excitation were investigated. In some conventional LITs (i.e., LQIT and LIT with 4FSS), which are rotationally symmetric with respect to 90°, the dodecapole components play a significant role and the amplitudes of other higher-order terms are usually relatively small. In the simulation, The LIT with 4FSS (A6/A2 = 0.95%, A10/A2 = 0.1%) was used as an example.

Frequency shift is an important phenomenon induced by the higher-order fields. Supplementary Figure S6 shows the secular frequency as a function of the ion amplitude. The ion amplitude was calculated by direct simulations in the LIT with 4FSS (no buffer gas) using an m/z 556 ion with zero velocity and different initial positions. For the ion with amplitude of 0.2 mm in the x-direction, the amplitude in the y-direction is equal to the initial y-position. For the ion with the same x- and y-position, the ion amplitudes in the two directions are equivalent and usually larger than the initial position. It can be seen that there is a shift to higher frequency in the y-direction, which is proportional to the ion amplitude in the y-direction when the amplitude in the x-direction is very small (i.e. 0.2 mm). This is consistent with the previous result that positive even-order fields cause a blue frequency shift [46, 47]. However, in spite of no variation of the amplitude in x-direction, there is a shift to lower frequency in the x-direction. A similar phenomenon was observed when the amplitudes in the x- and y-directions were equivalent.

In the previous study, because ions are excited in only one direction (e.g., y-direction), the amplitude in the x-direction is very small and can be neglected. Thus, the cross term xnym is neglected. It will not be reasonable when the amplitude in the x-direction is large. The potential of a LIT with an added dodecapole field and no other multipoles can be described as:

$$ \phi ={A}_2\left(\frac{x^2-{y}^2}{r_0^2}\right)+{A}_6\left(\frac{x^6-15{x}^4{y}^2+15{x}^2{y}^4-{y}^6}{r_0^6}\right) $$
(8)

Equation 8 can be rewritten as

$$ {\phi}_y=-\left\{\left({A}_2+15{A}_6\frac{x^4}{r_0^4}\right)\left(\frac{y^2}{r_0^2}\right)-\left(15{A}_6\frac{x^2}{r_0^2}\right)\left(\frac{y^4}{r_0^4}\right)+{A}_6\left(\frac{y^6}{r_0^6}\right)-{A}_2\left(\frac{x^2}{r_0^2}\right)\right\} $$
(9)

The multipole coefficients can be defined as

$$ {A}_0^{\prime }=-{A}_2\left(\frac{x^2}{r_0^2}\right) $$
(10)
$$ {A}_2^{\prime }={A}_2+15{A}_6\frac{x^4}{r_0^4} $$
(11)
$$ {A}_4^{\prime }=-\left(15{A}_6\frac{x^2}{r_0^2}\right) $$
(12)
$$ {A}_6^{\prime }={A}_6 $$
(13)

All the above coefficients except the dodecapole term vary with the x value. To estimate approximately their magnitude, pseudo-potential well model (q < 0.4) is employed here and the x-motion is regarded as a simple harmonic motion:

$$ {x}_t={x}_0 sin{\omega}_xt $$
(14)

where ω x /2 \( \uppi \) is the so-called secular frequency in the units of hertz and x 0 is the ion motion amplitude. The multipole coefficients averaged over a period of the secular motion in y-direction is

$$ \left\langle {A}_n\right\rangle =\frac{1}{2\uppi /{\omega}_y}{\displaystyle \underset{0}{\overset{2\pi /{\omega}_y}{\int }}}{A}_ndt $$
(15)

Making the assumption that \( {\omega}_x\approx {\omega}_y \), substituting for x from Equation 14 and integration of Equation 15 gives

$$ {A}_0^{\prime }=-0.5{A}_2\kern0.14em \frac{x_0^2}{r_0^2} $$
(16)
$$ {A}_2^{\prime }={A}_2+5.625{A}_6\kern0.14em \frac{x_0^4}{r_0^4} $$
(17)
$$ {A}_4^{\prime }=-7.5{A}_6\kern0.14em \frac{x_0^2}{r_0^2} $$
(18)

Taking x 0/r 0 = 1.5/5 for example, the multipole coefficients are as follows: \( {A}_2^{\prime }=0.9905 \), \( {A}_4^{\prime }=-0.0063 \), \( {A}_6^{\prime }=0.0094 \). The magnitude of the negative octopole term is comparable to that of positive dodecapole. If x 0/r 0 increases to 4/5, the multipole coefficients: \( {A}_2^{\prime }=1.0118 \), \( {A}_4^{\prime }=\hbox{--} 0.04512 \), \( {A}_6^{\prime }=0.0094 \). It can be concluded that it is analogous to adding a negative octopole component in the y-direction for excitation in the x-direction. This is why a shift to lower frequency is observed (Supplementary Figure S6). Similarly, if there are some negative dodecapole components, a shift to higher frequency will occur, contrary to the excitation in one direction. In the truncated standard hyperbolic LIT, the higher-order field components are negative, and the secular frequency as a function of the ion amplitude is shown in the Supplementary Figure S7.

Frequency shift greatly affects the performance of dual-direction excitation method in the nonlinear IT. In Figure 4a, although there is no difference between the two AC signals applied in the x- and y-directions, an obvious phase shift and amplitude difference occur after 1.9 ms excitation, which causes the ion to be lost after 2.05 ms excitation. As a result, the maximum displacement (determine the energy) in the two directions during 0–2.05 ms is 4.5 mm, which is relatively small compared with that of 7.5 mm in the pure LIT. This phenomenon can be regarded as the effect of the frequency shift on the beat motion. It can be seen from the inset to Figure 4a the ion beat motion is formed in both directions. As previously reported [38], the beat motion is mainly caused by the off-resonance excitation and the beat frequency is the difference between the excitation and secular frequency (f acf 0). Initially, the beat motion between x- and y-direction is almost the same but collisions will induce the two amplitudes in the two directions to be different. As the secular frequency depends on the amplitude of oscillation, the amplitude difference will produce the frequency difference of the beat motion. This frequency difference will exacerbate the amplitude difference. As the ion motion in the x- and y- directions fall out of step, ion is easy to strike the electrode under action of relatively high AC amplitude. In order to prove the above explanation, the ion trajectory without collisions was simulated (Figure 4c). The ion follows a stable beat-like oscillation. In addition, as mentioned previously [38], the decay of the secular motion attributable to the damping effect of the buffer gas will suppress the beat motion. High pressure will help to rapidly decay the motion. Therefore, the beat motion disappears quickly and the ion motion remains stable when the pressure is increased to 10 mTorr (Figure 4b).

Figure 4
figure4

Temporal variation of the displacement of m/z 556 ion (initial condition: x = 0.2 mm, y = 0.2 mm, Ex = 0.01 eV, Ey = 0.01 eV) in the x- and y-directions with dual-direction dipolar excitation during 1.7–2.1 ms at q u = 0.25 at helium gas pressure of 1 mTorr (a), 10 mTorr (b), and 0 mTorr (c) in the LIT with 4FSS (A6/A2 = 0.95%, A10/A2 = 0.1%), excitation amplitude1100 mV0-p, excitation frequency 69.0 kHz

Figure 5a shows the relationship between the effective temperature and the AC amplitude at different frequencies. It can be seen that there is a sudden jump of the effective temperature when the amplitude reaches 400 mV at 67.0 kHz. Then the temperature increases smoothly with increasing AC voltage amplitudes. A similar situation also prevails at 65.0 and 69.0 kHz (single-direction excitation). The corresponding ion displacement also soars around the critical point (i.e., 400 mV0-p for 67.0 kHz). For example, the maximum excursions of the m/z 556 ion in x- and y- directions (x = 0.2 mm, y = 0.2 mm, E x = 0.01 eV, E y = 0.01 eV) are 2.73 and 2.77 mm at amplitude of 400 mV0-p, respectively, while they increase to 5.06 and 4.88 mm at 410 mV0-p. This also can be explained by the frequency shift. Because of the decay of the secular motion, the amplitude of the ion motion at the steady state is mainly determined by the amplitude of the motion at the excitation frequency. From Equation 7, the amplitude at the excitation frequency is dependent upon the difference between the secular frequency and excitation frequency. Supplementary Figure S6 shows that the secular frequency decreases with increasing ion motion amplitude in the dual-direction excitation method. Hence as the excitation frequency is lower than the secular frequency and the amplitude of the ion motion increases, the frequency difference reduces. This will lead to a further increase of the amplitude. Higher amplitude in turn further narrows the frequency gap. This cycling effect significantly promotes the increase of the ion motion amplitude and therefore the ion energy (i.e., effective temperature), which leads to the sudden jump at certain excitation amplitude. However, the ion motion amplitude cannot infinitely increase, because when the amplitude continues to increase, the secular frequency is less than the excitation frequency and becomes smaller and smaller, which enlarges the frequency difference and, consequently, leads the decrease of the amplitude. As a result, the ion finally reaches a steady state. When the excitation amplitude is over the critical point, increasing the AC amplitude will increase the P value but also enlarge the frequency difference (Equation 7). Hence, the ion energy increases slowly with increasing the AC amplitude. From Figure 5a, another phenomenon is that the smaller the excitation frequency (for example, 65.0 kHz) is, the higher the effective temperature after the jumping point is. This is due to the fact that larger frequency shift requires higher ion motion amplitude. The maximum excursion of the above m/z 556 ion in the x- and y-directions is about 6.0 mm at the excitation amplitude of 1200 mV0-p, compared with 5.06 mm at 67.0 kHz. However, as the maximum excursion is larger at smaller frequency after the jumping point, the amplitude range during which ions are not significantly lost becomes narrower.

Figure 5
figure5

Simulations in the LIT with 4FSS showing the effective ion temperature at different frequencies (a) and the corresponding ion survival number of 200 m/z 556 ions (b) as a function of the AC amplitude over a excitation period of 30 ms at q u = 0.25 (secular frequency 68.85 kHz) in the presence of helium gas at 1 mTorr. Circles, upright triangles, and inverted triangles represent the effective temperature with the dual-direction excitation frequencies of 69.0, 67.0, and 65.0 kHz, respectively. Squares represent the effective temperature with the single-direction excitation frequency of 69.0 kHz

For excitation only in one direction, there is a shift to higher frequency and a sudden jump is observed at 69.0 kHz (during the amplitude from 30 to 40 mV0-p), which is consistent with the previous study [48]. An interesting phenomenon is that at the same frequency (i.e., 69.0 kHz) by dual-direction method, there is a rapid rise of the effective temperature as the amplitude increases from 50 to 150 mV0-p. This is because ions are greatly excited in one-direction while de-excited in the other direction, like the single-direction excitation, but it only occurs for a short period during the excitation time (Supplementary Figure S8a). It almost disappears when the amplitude reaches 150 mV0-p (Supplementary Figure S8b).

Compared with the single-direction excitation method, there is no obvious increase in the effective temperature at 69.0 kHz by dual-direction excitation method. The main reason is that ions are easily lost as the amplitude increases (Figure 5b), which has been explained before (Figure 4). At 67.0 and 65.0 kHz, the effective temperature is dramatically improved and the ion survival number is large (Figure 5b). This may be due to the fact that the secular frequency at the steady state is lower than the excitation frequency. If the amplitude in one direction decreases, the secular frequency in this direction increases, which reduces the frequency difference between the secular frequency and excitation frequency. This will increase the amplitude in turn. Similarly, if the amplitude in one direction increases, the secular frequency will decrease and thus the frequency gap enlarges, which decrease the amplitude in turn. As a consequence, the amplitude difference between x- and y-directions is relatively small and ions follow a stable motion.

It can be concluded that the excitation frequency is critical for dual-direction excitation method. The optimized frequency is dependent upon the field distribution. For the LIT with positive dodecapole component, the frequency less than the secular frequency without consideration of the higher-order fields components (unperturbed frequency) is preferred.

The Dual-Direction Excitation Applied in Different Types of LIT

As we discussed before, several kinds of ITs were developed and used commercially, such as LIT with standard hyperbolic geometry, LIT with 4FSS, and LQIT and so on. All of them can be applied with dual-direction excitation method. Their performance was simulated and Figure 6 shows the comparison of the effective temperature and ion survival number by single-direction excitation method and dual-direction method in LQIT and standard hyperbolic LIT. In LQIT, the effective temperature can reach 900°, which is about 1.6 times greater than that by single-direction method (550°), while the ion survival number is about 140 out of 200 ions. In standard hyperbolic LIT, with the survival number of 100, the effective temperature is about 1050° by dual-direction excitation method, compared with 600° by single-direction excitation method. The unperturbed secular frequency at q u = 0.25 in standard hyperbolic LIT is about 68.85 kHz. The excitation frequency employed in the dual-direction method is a little more than 68.85 because of the negative dodecapole field components.

Figure 6
figure6

Simulations showing the effective ion temperature and the corresponding ion survival number of 200 m/z 556 ions as a function of the AC amplitude over a excitation period of 30 ms at q u = 0.25, using LQIT by single-direction excitation method (a) with excitation frequency of 69.0 kHz, and dual-direction method (b) with excitation frequency of 67.0 kHz, and standard hyperbolic LIT by single-direction excitation method (c) with excitation frequency of 69.0 kHz, and dual-direction method (d) with excitation frequency of 69.1 kHz

Conclusion

Simulations have been performed to investigate the ion motion in response to dual-direction dipolar excitation in LITs. Compared with the conventional one-direction excitation, ion average energy can be increased significantly, which arises from the following two factors: the addition of an excitation dimension and larger ion motion amplitude in either x- or y-direction. The basic effect of higher-order field components is the frequency shift. Contrary to the excitation in one-direction, positive dodecapole components bring about a red frequency shift. In the LIT with such higher-order components, the excitation frequency less than the unperturbed frequency of the ion motion is preferred, where the effective temperature is dramatically increased while the ions have relatively less to lose. The dual-direction dipolar excitation can be applied in most LQITs and standard hyperbolic LITs, and the CID efficiency can be obviously improved.

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Dang, Q., Xu, F., Wang, L. et al. Theoretical Study of Dual-Direction Dipolar Excitation of Ions in Linear Ion Traps. J. Am. Soc. Mass Spectrom. 27, 596–606 (2016). https://doi.org/10.1007/s13361-015-1317-5

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Keywords

  • Linear ion trap
  • Ion activation
  • Dual-direction dipolar excitation
  • Two identical excitation signals
  • Higher-order fields