Optimum Waveforms for Differential Ion Mobility Spectrometry (FAIMS)

Abstract

Differential mobility spectrometry or field asymmetric waveform ion mobility spectrometry (FAIMS) is a new tool for separation and identification of gas-phase ions, particularly in conjunction with mass-spectrometry. In FAIMS, ions are filtered by the difference between mobilities in gases (K) at high and low electric field intensity (E) using asymmetric waveforms. An infinite number of possible waveform profiles make maximizing the performance within engineering constraints a major issue for FAIMS technology refinement. Earlier optimizations assumed the non-constant component of mobility to scale as E², producing the same result for all ions. Here we show that the optimum profiles are defined by the full series expansion of K(E) that includes terms beyond the 1^st that is proportional to E². For many ion/gas pairs, the first two terms have different signs, and the optimum profiles at sufficiently high E in FAIMS may differ substantially from those previously reported, improving the resolving power by up to 2.2 times. This situation arises for some ions in all FAIMS systems, but becomes more common in recent miniaturized devices that employ higher E. With realistic K(E) dependences, the maximum waveform amplitude is not necessarily optimum and reducing it by up to ∼20 – 30% is beneficial in some cases. The present findings are particularly relevant to targeted analyses where separation depends on the difference between K(E) functions for specific ions.

Introduction

Differential ion mobility spectrometry (DMS) is becoming a powerful method of broad utility for analysis of gas-phase ions and separation of their mixtures.^1–5 The introduction of commercial DMS instruments and particularly their integration with mass spectrometry (MS) and/or liquid or gas chromatography since 2003 has enabled rapid growth of the number and diversity of applications that include environmental analyses,^6,7 food and water quality assurance,^8–10 bacterial typing,^11,12 forensic investigations,¹³ proteomics and metabolomics,^14–17 pharmaceutical studies,^18–20 and protein folding research.^21–25 Since its earliest days, DMS has been employed to detect explosives, drugs, and chemical warfare agents, and its role in defense, security, and law enforcement settings continues expanding.^26–31

As captured in the name, DMS separates ions based on the difference between their mobilities (K) at high and relatively low electric field intensity (E).^1,5 The mobility of any ion depends on E and the gas number density (N), and we can expand K(E/N) in a series:^7,30,32,33

$$K(E/N)=K(0)[1+a(E/N)]=K(0)[1+{\displaystyle \sum _{n=1}^{\infty }{a}_n{(E/N)}^{2n}}]$$ (1)

where a is the relative deviation of K from its low-field limit K(0). The a_n coefficients are functions of the ion - gas molecule potential³² and can produce a >0 or a <0, depending on their values and E/N. In principle, one can deduce a for any ion from measurements of K at different E/N using drift tube ion mobility spectrometry (DT IMS).^32,34–36 However, that approach does not permit separating ion mixtures based on the difference, and thus is of limited analytical utility. Also, for larger polyatomic and biomolecular ions that are of most interest, the a(E/N) dependence is usually weak. For E/N allowed by the electrical breakdown limitations of gases at standard temperature and pressure (STP), typical |a| are ∼10⁻² (except for the smallest ions).³⁷ That is close to the accuracy of existing DT IMS systems, and K(E/N) for sizable ions such as peptides appear flat.³⁸

In DMS, a(E/N) is elicited directly using a periodic time-dependent electric field E(t) that comprises short segments E₊(t) with high E and longer segments E₋(t) with lower E of opposite polarity such that mean E over the period t_c is null (the zero-offset condition):^33,39–43

$$\overline{E}=\frac{1}{t_c}{\displaystyle \underset{0}{\overset{t_c}{\int }}E(t)dt}=0$$ (2)

but absolute E̅₊(t) and E̅₋(t differ. It is convenient to normalize E(t) as

$$E(t)=E_{D}F(t)$$ (3)

where E_D is the peak absolute amplitude (“dispersion field”) and F(t) defines the functional form. The condition of F(t) asymmetry is:

$$\langle F_{2n+1}\rangle =\frac{1}{t_c}{\displaystyle \underset{0}{\overset{t_c}{\int }}{F}^{2n+1}(t)dt}\ne 0$$ (4)

for at least one n ≥ 1. In earlier treatises,^33,39–43 this inequality was stipulated for n = 1 or all n ≥ 1. Either condition is sufficient though not necessary, as expression (4) may equal 0 for n = 1 but not some greater n. Current DMS methods mainly utilize n = 1, but higher-order separations based on n ≥ 2 are feasible.⁴⁴ The quantity 〈F₃〉 characterizing the waveform is known as the “form-factor”,⁴⁵ 〈F_2n+1〉 and may be viewed as form-factors of various orders.

The asymmetry of E(t) gave raise to the other name for DMS - field asymmetric waveform IMS or FAIMS. Ions with a = 0 would oscillate in such field without separation. In reality, the displacements during E₊(t) and E₋(t) do not cancel fully: ions drift in the direction of E₊(t) segment when a(high E) > a(low E) and E₋(t) otherwise. The net displacement over the cycle is:

$$d={\displaystyle \underset{0}{\overset{t_c}{\int }}K(E/N)E(t)dt}$$ (5)

To employ this mechanism for spatial dispersion of ions based on a(E/N), one needs a field of >∼60 Td (or ∼15 kV/cm at STP) over large distances. That being impractical, FAIMS is implemented as a filtering method using a constant weak “compensation field” E_C superposed on E(t). A certain E_C of approximately:

$$E_C=d/(K{t}_c)$$ (6)

offsets the net drift due to E(t) for a particular species, while others with different a(E/N) still migrate along the E_C axis. The {E(t) + E_C} field is maintained in a gap between two electrodes carrying rf and dc voltages. This allows the species with correct E_C to stay balanced and pass the gap to be detected, while others move toward an electrode and are neutralized. As with other filtering techniques, such as quadrupole MS, one can fix E_C to monitor selected ions or scan E_C to reveal the spectrum of species present.

Numerous asymmetric F(t) comply with eq (2); one comprises two rectangles:^{33,39,40,45,46}

$$F=1\text{for}t=[0;t_c/(f+1)[;F=-1/f\text{for}t=[t_c/(f+1);t_c[$$ (7)

where f >1 (Fig. 1 a, Fig. 2). The number of possible F(t) is infinite even within eq (7), but not limited to it. For example, two right non-isosceles triangles (Fig. 1 b) would do. As the integral of a sum equals the sum of integrals, any sequence of F(t) satisfying eqs (2, 5) that remains asymmetric will also work, e.g., a trapezoidal (Fig. 1 c) built from rectangles and triangles.

Fig. 1.

Asymmetric waveforms: rectangular (a), triangular (b), and trapezoidal (c).

Fig. 2.

Rectangular waveforms with fixed peak-to-peak amplitude have greater peak amplitudes at higher f: profiles with f = 2 (a), 4 (b), and 6 (c).

To find the best F(t) for FAIMS analyses, we need to define the optimization criterion. In general, the electric field in FAIMS may not just separate different ions but also focus them to the gap median, reducing losses to electrodes.^5,46 Focusing requires inhomogeneous field created in gaps of curved (e.g., cylindrical or spherical) shape. In planar geometries, homogeneous field permits no focusing. While focusing improves ion transmission through FAIMS, it introduces discrimination based on a(E/N) and limits the resolving power R by rendering ions with multiple E_C stable in the gap. For low ion currents, the disadvantages outweigh gains and the overall performance (quantified via the resolution/sensitivity diagrams) maximizes for planar gaps.⁴ This study formally addresses planar FAIMS, but the conclusions should extend to all geometries. In the absence of focusing, the electric field effects separation only and the F(t) providing best separation is optimum. In global analyses, that means the maximum of R normally defined as the absolute separation parameter (here E_C) divided by the full peak width at half maximum, w_1/2:

$$R=\left|E_C\right|/w_{1/2}$$ (8)

In targeted analyses, the resolution of specific features (e.g., X and Y) is characterized by

$$r=2\left[\left|E_C(X)-E_C(Y)\right|\right]/\left[w_{1/2}(X)+w_{1/2}(Y)\right]$$ (9)

This metric may be extended to three or more species.

Previous efforts to optimize FAIMS waveforms^33,39,40,45 sought to maximize |E_C| rather than R, i.e., a constant w_1/2 was implied. While the choice of F(t) affects the average E/N in FAIMS, and thus the average diffusion that determines the peak width,^32,47 the effect on E_C is much stronger and fixing w_1/2 is a fair approximation that we follow in this work. By eqs (5, 6), maximizing |E_C| means maximizing |d|. Introducing the reduced mobility K₀ = KN/N₀ (where N₀ is N at STP) and combining eqs (1 – 5), one obtains:

$$d=K_0(0)(E_D/N)t_c{\displaystyle \sum _{n=1}^{\infty }{a}_n{(E_D/N)}^{2n}\langle F_{2n+1}\rangle }$$ (10)

For any a_n set, d depends on the 〈F_2n+1〉 values for specific F(t).

Earlier F(t) optimizations^33,39,40,45 have represented a(E/N) by the leading (n = 1) term of eq (1) that commonly dominates the separation in “full-size” FAIMS systems⁵ operated at E/N <∼100 Td. However, terms with n ≥ 2 are often important even here and grow quickly at higher E/N, becoming dominant at >120 Td employed in latest miniaturized^3,7,48 and reduced-pressure⁴⁹ FAIMS devices. Also, the E(t) profiles were optimized for fixed peak (E_D) or peak-to-peak (E_P-P) amplitude, implying the maximum possible amplitude to be best. Here we show that lowering E_D or E_P-P may improve separation, hence both the waveform profile and amplitude must be optimized. This is done here for realistic a(E/N) functions.

Global Waveform Optimization

The rectangular F(t) by eq (7) is called “ideal” as it maximizes |d| and thus FAIMS resolution. This happens because E in E₊(t) and E₋(t) is fixed, while other forms comprise a range of E in either or both and hence are less asymmetric. Then:

$$\langle F_{2n+1}\rangle =(1-f^{-2n})/(f+1)$$ (11)

All 〈F_2n+1〉 by eq (11) and thus d by eq (10) are trivially null for f = 1 when F(t) is symmetric f⇒∞ and when F = 0. Hence |d| reaches maximum (d_max) at an intermediate f, with the optimum (f_opt) depending on E_D/N and relative a_n values. For the leading term of eq (10):

$$\langle F_3\rangle =(f-1)/f^2,$$ (12)

that reaches the maximum absolute 〈F₃〉, or 〈F₃〉_max, of 1/4 at³⁶ f_opt = 2. The maximum is not abrupt, particularly on the high-f side: e.g., the 〈F₃〉 value is below 〈F₃〉_max by ≈11% at f = 1.5 or f = 3 and 25% at f = 4 (Fig. 3 a). This allows other effects to greatly shift f_opt, as discussed below.

Fig. 3.

Form-factors of rectangular E(t) constrained by E_D (a, c) and E_p-p (b, d) for hypothetical ions with various a_n values. In (a, b), a_n = 1 for n = 1 – 3 as labeled and a_n = 0 for other n. In (c, d), a_n = 0 for n >2 and a_R values are labeled, curves are for a_R = 0 (solid line), a_R >0 (long dash), and a_R <0 (short dash). The maxima are marked by arrows up, minima by arrows down. The dotted line in (c) is for 〈F〉 = 0 (no separation).

This optimization assumed constant E_D, which is often limited by the electrical breakdown threshold. The optimum for rectangular E(t) with fixed peak-to-peak amplitude (E_P-P), that often results from engineering limitations, differs because shifting f above 2 increases E_D (Fig. 2), and |d| initially rises despite decreasing for constant E_D and f_opt > 2. Indeed:

$$E_D=f{E}_{\text{P-P}}/(f+1)$$ (13)

and eq (10) converts to:

$$d=\frac{K_0(0)f{t}_c}{{(f+1)}^2}\frac{E_{P-P}}{N}{\displaystyle \sum _n{a}_n}{\left(\frac{E_{P-P}}{N}\right)}^{2n}\frac{f^{2n}-1}{{(f+1)}^{2n}}$$ (14)

The leading term of eq (14) is:

$$d=K_0(0)a_1{t}_c\frac{f(f-1)}{{(f+1)}^3}{\left(\frac{E_{P-P}}{N}\right)}^3$$ (15)

and |d| has an (also gradual) maximum⁴⁰ at $f=2+\sqrt{3}\cong 3.73$ Fig. 3 b) when

$$d_{\max }=K_0(0)\left|a_1\right|t_c\sqrt{3}(E_{\text{P-P}}/N{)}^3/18$$ (16)

The trends of eq (12) and eq (15) were verified by measurements,^40,50 producing f_opt ∼2 with constant E_D and ∼3.7 with constant E_PP. Constraints on both E_P-P and E_D lead to 2< f_opt <3.73.

However, accepting f = 2 or 3.73 as the optima^{33,39,40,45,51} for rectangular F(t) is inaccurate because the 〈F_2n+1〉 quantities for n >1 are not null and maximize at different f (Fig. 3 a, b; Table 1). With E_D constraint, f_opt decreases for higher n because the 2n power over E_D magnifies the dissimilarity between F₊(t) and F₋(t), and the same ion motion disbalance requires a smaller difference in E. With the E_P-P constraint, f_opt increases for higher n. So |d| always maximizes at f ≠ 2 or 3.73, unless at very low E_D/N (or for ions with unusually small a_n for n >1) where terms with n >1 are negligible. As good FAIMS separations require substantial E_D/N, the terms with n = 2 are usually important and those with n = 3 and even 4 may also be significant.^44,52 The present discussion is limited to n ≤ 2, which often suffices⁵² at moderate E_D/N (< ∼80 – 100 Td).

Table 1.

Optimum f and maximum m 〈F_2n+1〉 F values up to n = 5 for rectangular F(t).

Separation order	Rectangular (ED fixed)		Rectangular (EP-P fixed)
	fopt	〈Fn〉	fopt	〈Fn〉
n = 1	2	0.250	3.73	0.0962
n = 2	1.65	0.326	5.04	0.0669
n = 3	1.49	0.365	7.00	0.0491
n = 4	1.40	0.388	9.00	0.0387
n = 5	1.34	0.404	11.0	0.0320

The differences between f_opt values at n = 1 – 4, especially 1 and 2, are modest compared to the breadth of maxima of 〈F_2n+1〉(f) curves (Fig. 3 a, b). Hence 〈F_2n+1〉 values for one n are close to their maxima at f_opt for other n. For example, in Fig. 3 a, the value of 〈F₅〉 at f = 2 is ∼96% of 〈F₅〉_max found at f = 1.65. However, the terms with n >1 matter for optimum F(t) because |d| may maximize outside of the range between f_opt for specific n when the signs of at least two a_n differ. With only two n (e.g., 1 and 2), this happens when a_n have opposite signs. In such cases, f_opt may greatly differ from that for n = 1 when the ratio of n = 2 and n = 1 terms in eq (1),

$$a_R=a_2(E_D/(N{)}^2/a_1$$ (17)

is not far from −1. For instance, at a_R = −0.8 (with E_D constraint), d_max is located at f ∼1.24 while at f = 2 we find d = 0, i.e., no separation occurs (Fig. 3 c)! This extreme example clearly shows that n >1 terms are crucial for waveform optimization when a₁ and a₂ have opposite signs, a common situation as discussed below. For further analysis, we parse eq (10) and eq (14) as

$$d=K_0(0)a_1(E_D/N{)}^3{t}_c\langle F\rangle =K_0(0)a_1(E_{P-P}/N{)}^3{t}_c{f}^3\langle F\rangle /(f+1{)}^3$$ (18)

where 〈F〉 is the “effective form-factor”:

$$\langle F\rangle =\langle F_3\rangle +a_R\langle F_5\rangle$$ (19)

As the separation power depends on |E_C| and |d|, what matters is absolute 〈F〉.

First, we optimize f for constant E_D. When a₁ and a₂ have same signs, f_opt shifts from 2 for n = 1 to ≅1.65 for n = 2 as a_R increases (Fig. 3 c); as shown above, f = 2 is only slightly suboptimum even at highest a_R. With opposite a₁ and a₂ signs, f_opt rapidly rises with decreasing a_R (Fig. 3 c) and keeping f = 2 can drastically decrease absolute 〈F〉. For a_R <−0.5, a region of 〈F〉 <0 appears at f near 1.0. As a_R decreases, the minimum moves to higher f and deepens while the maximum lowers, and for a_R ≅ −0.75 the value of |〈F〉| in the minimum (at f ≅1.21) reaches that in the maximum (at f ≅3.59). Then the maximum shifts to still higher f and disappears at a_R = −1 and f ⇒ ∞ while the minimum further deepens and also shifts to higher f, approaching ≅1.65 for a_R ⇒ –∞ (Fig. 3 c). Fixing E_P-P instead of E_D produces similar behavior (Fig. 3 d), with |〈F〉| in the minimum and maximum equalizing for a_R ≅ −0.85 when f ≅1.46 and ≅5.97, respectively.

Similarly to the case of f = 2, the optimum |〈F〉| minimizes close to a_R = −0.8 (Fig. 4 a). Unlike at f = 2, the minimum is not null and thus permits some separation, but its height is only ∼15% of 〈F₃〉_max at a_R = 0 and the resolution at a_R close to −0.8 would be poor. An analogous picture for fixed E_P-P follows from Fig. 3 d. Then reducing |a_R by use of below-maximum E_D/N or E_P-P/N may be profitable, despite lower (E/N)³ factors in eq (18). To optimize E_D and F(t) simultaneously, we may combine eq (17) and eq (18) with either E_D or E_P-P constraint into

$$d=K_0(0)({a_1}^5{a_2}^{-3}{a_R}^3{)}^{1/2}{t}_c\langle F\rangle$$ (20)

At any given a_R, the value of |d| is greatest at the maxima of |〈F〉|. For f = 2, that value grows with decreasing a_R up to a_R ≅−0.48, then drops to 0 at a_R = −0.8, and rises again (Fig. 4 b). In the result, the values of |d| are lower for −0.92< a_R <−0.48 than for a_R = −0.48. So |d| can be increased by decreasing E_D/N until a_R = −0.48, which means reducing E_D by up to 28%. For optimum f, the minimum of |d| becomes shallower and the sub-optimum region (S) shrinks to −0.80< a_R <−0.52 (Fig. 4 b), but maximizing |d| may still require decreasing E_D by up to 19%.

Fig. 4.

Properties of separations using rectangular E(t) with E_D constraint: (a) absolute form-factor (derived from Fig. 3 c) and (b, c - left axis) absolute ion displacements per FAIMS cycle [in the units of a₁^5/2a₂^−3/2K₀(0)t_c] for fixed (b) and variable (c) E_D. Thin solid lines (a) are for maximum or minimum 〈F〉 as labeled, thick solid lines are for maximum |〈F〉|, dashed lines are for fixed f as marked, and circles (c, region L) are for f = 1.35. Vertical lines show the region boundaries: dash-dot-dot (b) for fixed f and dash-dot (b, c) for variable f. In (c), the arrow points to the greatest difference between |d| with optimum f and f = 2 (that is best at a_R = 0) and the dotted line shows optimum f (right axis).

Hence, to maximize |E_C|, one should (Fig. 4 c): (i) for a₂ and a₁ with same signs, raise E_D/N to the allowed maximum while decreasing f from 2 to f ≅1.65 – 2.0, depending on E_D/N; (ii) for a₂ and a₁ with opposite signs, raise E_D/N until a_R reaches −0.52 while increasing f from 2 to ≅2.6, then (if limitations on E_D/N permit) jump to a_R = −0.8 and f = 1.24 and raise E_D/N to the maximum while increasing f to ≅1.24 – 1.65, again depending on E_D/N. To enable all those capabilities, the value of f must be adjustable from 1.24 to 2.6. However, the fixed f = 2 provides |E_C| within 7% of the maximum for a_R >−0.52 and other f have real worth only on the low-a_R side of region S (in the following, region L), especially a_R∼ −(0.9 – 1.2) where f_opt ≅1.3 grossly differs from 2 and |E_C| at f_opt can reach 2.2× that at f = 2 (Fig. 4 c). Adopting f = 2 on the high-a_R side (region H) and f = 1.35 in region L provides |E_C| within 9% of the maxima at any a_R (Fig. 4 c) while reducing the needed waveform flexibility to switching between two f values.

The present optimization may be extended to a(E/N) including terms with n >2 and/or waveforms constrained by E_P-P. With either constraint, the evolution of 〈F_2n+1〉(f) dependences for n >2 continues the trend from n = 1 to 2 (Fig. 3, Table 1). Hence the effect of adding a term with any n >2 to the n = 1 term is akin to that of adding the n = 2 term considered here, but (for equal a_n/a₁ ratio) greater because the difference between 〈F_2n+1〉 and 〈F₃〉 increases at higher n for any f value (Fig. 3 a, b). The addition of term(s) with n >2 to the presently studied superposition of n = 1 and 2 terms may produce more complex dependences, which may be important at highest E/N values where the terms with n >2 become substantial.

Relevance to Actual FAIMS Measurements

As the best waveforms of any class are determined by a_R, one may wonder what values are realistic. Of particular interest are the cases of a_R ∼−(0.5 – 1.5) for which the optimum forms are most sensitive to a_R and notably differ from those for a_R = 0. The a_R for any ion/gas pair scales as (E/N)2 by eq (17), hence in theory one may reach any |a_R| at strong enough fields and the notion of a “typical” a_R makes sense only for specific E_D/N magnitude. The original “full-size” FAIMS design largely adopted in Thermo Fisher systems features gap widths (g) of ∼1.5 – 2.5 mm and operates at ambient pressure, normally employing E_D ∼15 – 25 kV/cm or E_D/N ∼60 – 100 Td: at weaker fields the drift nonlinearity rarely suffices for good separation while the electrical breakdown threshold precludes much stronger fields (in N₂ or air).⁵³ That threshold increases for narrower gaps according to the Paschen’s law,⁵³ and micromachined FAIMS devices (e.g., SDP-1 by Sionex with g = 0.5 mm)⁷ allow E/N up to 140 Td. Same may be achieved by reducing the gas pressure, e.g., E/N = 180 Td was established at ∼390 Torr.⁴⁹ The recent development of FAIMS “chips” with g ∼10 µm by Owlstone has allowed raising E/N to ∼400 Td.⁴⁸

The value of a_R also depends on the ion(s) and gas through the a₂/a₁ ratio, and we shall now estimate those for global and targeted separations.

Global separations

In global analyses, one seeks to maximize the overall separation space, which in FAIMS means typical |E_C| values as reviewed in the Introduction. For species with a₁ >0, typically a₂ <0 and a increases up to a maximum at certain E/N and decreases at greater E/N. This behavior (called “type B”)⁵ is ubiquitous for both atomic and polyatomic cations and anions with m < ∼400 Da in N₂ or air at room temperature, including 13 of 17 protonated and 15 of 17 deprotonated amino acids,⁵⁴ protonated benzene and all 7 amines studied,⁵⁵ 8 protonated ketones up to decanone and 5 of their proton-bound dimers,⁵⁶ all 10 protonated organophosphorus compounds investigated and 7 of their dimers,⁵⁷ and I⁻ and anions of 5 common explosives and their degradants: 1,3-dinitrobenzene, 1,3,5-trinitrobenzene, p-mononitrotoluene, 2,4-dinitrotoluene, and 2,4,6-trinitrotoluene (TNT).²⁸ The magnitude of a₂/a₁ for those 72 species spans >3 orders of magnitude from <10⁻⁶ to >10⁻³ Td⁻² (Fig. 5), but most values are on the order of 10⁻⁵ – 10⁻⁴ Td⁻² regardless of the ion mass and the median a₂/a₁ is −5.5 × 10⁻⁵ Td⁻², for which a_R = −0.5 at E/N ∼95 Td that is typical for either micromachined or “full-size” FAIMS. Exemplary species close to this median are (Glu – H)⁻, (TNT – H)⁻ (Table 2), and anions of other 4 explosive traces with a₂/a₁ = −(5.2 – 6.0) × 10⁻⁵ Td⁻². Half of the ions have higher |a₂/a₁| values and a_R = −0.5 is reached at lower E/N; for some, e.g., H⁺(Decanone) (Table 2), that occurs already at the lower end of practical FAIMS range (∼60 – 70 Td). For most other ions, |a₂/a₁| >10⁻⁵ and a_R reaches −0.5 at E/N <220 Td, i.e., well within the range of Owlstone devices. Rarely, the a₂/a₁ values are so miniscule that a_R remains insignificant at E/N used in current FAIMS systems (Fig. 5). For example, for (Ala – H)⁻ (Table 2), a_R would reach −0.5 only at E/N ∼10³ Td. As present a₁ and a₂ values were fit to FAIMS measurements at E_D/N ∼70 – 120 Td, they cannot be used to accurately extrapolate a(E/N) to much stronger fields where terms with higher n become important. Hence we compute a_R values at higher E_D/N not to maximize |E_C| for specific ions, but to illustrate the E_D/N magnitude at which the optimum waveforms in typical scenarios materially deviate from those derived for a_R = 0.

Fig. 5.

Measured values of a₂/a₁ for representative type B ions: protonated (Δ) and deprotonated (▲) amino acids,⁵⁴ protonated benzene and amines (▼), protonated ketone monomers (●) and dimers (○),⁵⁶ protonated monomers (■) and dimers (□) of organophosporus compounds,⁵⁷ and deprotonated or radical anions of explosives²⁸ (◊).

Table 2.

Values of a₁ and a₂ for some ions in air or N₂ (at T = 300 K) measured using FAIMS.^a

Species	Cl−	Ala	Pro	Ser	Leu	Ile	ProOH	Glu	H+(Dec.)	TNT
Mass, Da	35	88	114	104	130	130	130	146	157	226
a1, 10−6 Td−2	18.7	12.0	7.82	12.4	5.43	5.15	5.55	7.12	4.6	4.4
a2, 10−10 Td−4	64	−0.075	0.50	-1.77	−1.85	−0.58	−0.08	−4.0	−5.2	−2.7
a2/a1, 10−5 Td−2	34	−0.063	0.64	−1.4	−3.4	−1.1	−0.14	−5.6	−11.3	−6.1

^aData are from [47, 52] for Cl⁻, [54] for deprotonated amino acids (alanine, proline, serine, leucine, isoleucine, hydroxyproline, and glutamic acid), [56] for H⁺(Decanone), and [28] for the deprotonated TNT.

The specific a_n and thus a_R at certain E_D for any ion depend on the gas composition, and |a₂/a₁| values in some exceed those in N₂. For example, the humidity in ambient air (often used in field analyses) modifies a(E). At any water vapor pressure tried⁵⁸ (P_w = 120 – 6000 ppm), ions of all 4 explosives and their degradants measured retain a₁ >0 and a₂ <0, but |a₂/a₁| increases at higher P_w up to a maximum of 8.3 × 10⁻⁵ Td⁻² that leads to a_R <−0.5 already at E_D = 80 Td.

For “type A” ions,⁵ the a(E/N) curves measured by FAIMS are fit by a₁ >0 and a₂ >0. In N₂ or air, this applies primarily to the smallest ions (e.g., Cl⁻), but also some medium-size ones such as (Pro – H)⁻ (Table 2). Though a₂/a₁ values can be quite high and produce substantial a_R even at low E_D/N (for Cl⁻, a_R = 1.7 already at 70 Td), positive a_R hardly warrant waveform re-optimization, as discussed above. However, a(E/N) functions cannot increase indefinitely: at E/N ⇒ ∞, the ion/molecule potential always approaches the hard-shell limit where K drops³² at higher E/N. Thus, when a₁ >0, the value of a maximizes at finite E/N (exhibiting type B behavior) and observation of type A ions is a mere artifact of limited E/N range sampled in FAIMS. (Most type A ions are small because the maxima of K shift to greater E/N for deeper ion/molecule potentials that are more common to smaller and particularly atomic ions where the gas molecule can come close to the charged site.) That type A ions inevitably convert to type B at higher E/N implies that a_n <0 for some n >1. Though that n may equal 3 or greater, the effect on optimum waveform at E/N near or above the maximum K will overall resemble that explored here for type B behavior due to a₂ <0.

Targeted separations

As a filtering technique, FAIMS (like quadrupole MS) is mainly useful for targeted analyses, where removal of other species does no harm. In quadrupole MS, the conditions for maximum resolution of targeted analyses (in the selected ion monitoring mode) and global analyses (in the scanning mode) are identical. That is not quite true in FAIMS.

Targeted separations depend on the spread between d and thus E_C values of two or more species and not E_C of a single ion, as indicated by eq (9). To optimize E(t) for resolution of analytes X and Y, we should replace the coefficients a_n for one ion by (a_n,X – a_n,Y). Then the dependences of optimum waveforms on a_R found above continue to apply, with a_R still given by eq (17) but a₂/a₁ defined as:

$$a_2/a_1=(a_{2,X}-a_{2,Y})/(a_{1,X}-a_{1,Y})$$ (21)

A prototypical isomeric separation in biological analyses is that of leucine and isoleucine amino acids. Those were resolved by FAIMS as deprotonated anions⁵⁹ in N₂ and protonated cations⁴ in 1:1 He/N₂, in both cases just barely. For anions, the experimental E_D/N was 67 Td where a_R equals −0.15 for (Leu – H)⁻ and −0.05 for (Ile – H)⁻ (Table 2). Both values suggest that the best F(t) for this separation is essentially identical to that for a₂ = 0. However, the difference between a(E/N) of Ile and Leu anions has {a₁ = 0.28 × 10⁻⁶ Td⁻²; a₂ = −1.27 × 10⁻¹⁰ Td⁻⁴}, leading to very high |a₂/a₁| = 45 × 10⁻⁵ Td⁻² and a_R = −2.0 at same 67 Td. Hence this separation can likely be improved using the waveforms optimized for region L.

This situation is not limited to isomers. For the deprotonated hydroxyproline (ProOH) (Table 2) that is isobaric to (Leu – H)⁻, the value of a_R at 67 Td equals −0.01, and the optimum E(t) is determined solely by the n = 1 term. However, the differential a(E/N) of (ProOH – H)⁻ and (Leu – H)⁻ has {a₁ = 0.12 × 10⁻⁶ Td⁻²; a₂ = 1.77 × 10⁻¹⁰ Td⁻⁴}, and a_2/a₁ is an extreme ∼150 × 10⁻⁵ Td⁻² leading to a_R = 6.6. So, even at this low E_D/N, the optimum F(t) is determined almost only by the n >1 terms. That is of little consequence here because a_R >0, but isobars with similarly large negative a_2/a₁ certainly exist.

Much greater magnitude of a₂/a₁ by eq (21) compared to a₂/a₁ for either X or Y in above cases reflects⁴⁴ a lower correlation of a₂ values for different ions compared to that of a₁. Opposite examples exist: the differential a(E/N) of (Ser – H)⁻ and (Leu – H)⁻ has {a₁ = 6.97 × 10⁻⁶ Td⁻²; a₂ = 0.08 × 10⁻¹⁰ Td⁻⁴}, and a₂/a₁ = 0.11 × 10⁻⁵ Td⁻² is much lower than the values for either species (Table 2). However, the median values of |a₂/a₁| (in 10⁻⁵ Td⁻²) for 17 amino acids studied⁵⁴ and their 136 possible pairs are, respectively, 1.9 vs. 4.7 for cations and 1.7 vs. 5.6 for anions. The results for subsets of ions and pairs with a₂/a₁ <0 are similar: the medians are 1.9 vs. 4.6 for cations and 2.1 vs. 7.9 for anions. That is, statistically the mean effective |a_R| values for pairs of amino acid ions at any E/N are ∼3× those for individual ions and a_R ∼−0.5 for pairs with a₂/a₁ <0 will be reached at E/N lower by a factor of ∼$\sqrt{3}$: on average ∼90 Td typical in standard FAIMS systems vs. ∼160 Td used at reduced pressure or in miniature chips. Thus, the distinction between optimum waveforms at a_R = 0 and ∼−1 is likely more important in targeted analyses.

Conclusions

The asymmetric waveforms E(t) that maximize resolving power (R) of FAIMS materially depend on the a(E/N) profile(s) for ion(s) of interest. The optimum E(t) is defined by E/N and ratios of coefficients a_n with the terms of a(E/N) expansion in a power series: truncating to two terms, the key quantity is a_R = a₂(E/N)²/a₁. For positive a_R (when the terms add), the effect is small: E(t) optimized without considering the a(E/N) profile (i.e., for a_R = 0) provide R within ∼7% of the maximum. In this case, one should always maximize the E(t) amplitude, E_D, within the power supply or electrical breakdown constraints. With negative a_R, the 2^nd term is subtracted from the 1^st and, at a_R ∼−1, the difference that underlies FAIMS separation is not close to either. This produces “sub-optimum” (S) regions at a_R within the −(0.5 – 0.9) range, where separation is improved via reducing E_D by up to ∼20 – 30% from the maximum until |a_R| decreases to the region boundary. The optimum E(t) remain close to those at a_R = 0 in regions H on the high-a_R side of S, but substantially differ in regions L on the low-a_R side, where using the E(t) optimized at a_R = 0 reduces R for all three classes by up to 2.2 times. In L, the optimum E(t) also depends on a_R, but >90% of maximum R can always be achieved using fixed forms intermediate between those optimum on the L/S boundary and for a_R ⇒ ∞. Thus re-optimization of E(t) for each a_R can in practice be emulated by selecting one of the two forms (Fig. 6). In H, we can use E(t) optimized for a_R = 0 and employed in present FAIMS systems, while the new E(t) found here can produce significant gains in L. Though we included only the first two terms of the a(E/N) expansion, the optimum waveforms are primarily dictated by (constructive or destructive) interference of terms, and not their specific powers. Hence, addition of further terms (which are often quite significant in advanced FAIMS designs using E/N >100 Td) will produce similar effects that can be treated using the present framework.

Fig. 6.

Near-optimum waveforms proposed for use in H and L regions, values of f are marked.

Ions in FAIMS have been grouped into type A where the a(E/N) function increases, B where it has a maximum, and C where it decreases.⁵ Analyses of type A and C ions fall into the H region, and new waveforms proposed here for the L region would be used for type B species that include most ions of explosives in air or N₂ over a broad range of humidity. However, all type A ions convert to type B at higher E/N values, and thus new waveforms become relevant for species deemed type A as new miniaturized or reduced-pressure FAIMS systems using higher E/N are introduced. The key applications of FAIMS are to targeted analyses that are based not on individual a(E/N) functions but on their spread that can behave as ”type B” even when neither ion does. Hence new waveforms intended for L region may improve resolution of specific ions despite R for each maximized by existing E(t).

Though presently optimized waveforms maximize FAIMS specificity for an ion or ion pair, different E(t) may be desired for other reasons. In particular, a F(t) that sort ions by a_n values for n >1 without regard to a₁ may enable higher-order differential (HOD) IMS⁴⁴ analyses that should be substantially orthogonal to FAIMS based on whole a(E/N). As optimum E(t) for separation of ion pairs depend on their differential a_n values, the best forms for resolution of three or more ions will generally deviate from those for each pairwise combination; their optimization remains to be explored. In contrast, absolute ion mobilities that underlie conventional IMS depend on E weakly or not at all within the measurement accuracy, and ions with equal K at some E are unlikely to be resolved at any practical E. The possibility to tailor analyses by modifying F(t) is one manifestation of the unique flexibility of differential IMS.