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J Am Soc Mass Spectrom. Author manuscript; available in PMC 2010 May 1.

Published in final edited form as:

Published online 2009 January 8. doi: 10.1016/j.jasms.2009.01.001

PMCID: PMC2735887

NIHMSID: NIHMS113998

Department of Chemistry, Indiana University, Bloomington, IN 47405

The publisher's final edited version of this article is available at J Am Soc Mass Spectrom

See other articles in PMC that cite the published article.

The transport of ions through multiple drift regions is modeled in order to develop an equation that is useful for an understanding of the resolving power of an overtone mobility spectrometry (OMS) technique. It is found that resolving power is influenced by a number of experimental variables, including those that define ion mobility spectrometry (IMS) resolving power: drift field (*E*), drift region length (*L*), and buffer gas temperature (*T*). However, unlike IMS, the resolving power of OMS is also influenced by the number of drift regions (*n*), harmonic frequency value (*m*), and the phase number (*ϕ*) of the applied drift field. The OMS resolving power dependence upon the new OMS variables (*n, m*, and *ϕ*) scales differently than the square root dependence of the *E, L*, and *T* variables in IMS. The results provide insight about optimal instrumental design and operation.

When a pulse of ions is injected into a buffer gas, different species separate under the influence of an electric field because of differences in their mobilities through the buffer gas.^{1}^{-}^{4} This phenomenon is the basis for a widely used analytical technique for resolving different components of mixtures called ion mobility spectrometry (IMS). The ability to isolate different species can be understood by considering the resolving power (*t _{d}*/

$$\frac{t}{\mathrm{\Delta}t}={\left(\frac{\mathit{EeL}}{16{k}_{b}T\phantom{\rule{0.2em}{0ex}}\mathrm{In}\phantom{\rule{0.2em}{0ex}}2}\right)}^{1/2}$$

(1)

Of note is the dependence of resolving power on the square root of the various parameters. This relationship imposes limits on the ultimate instrument performance. For example, doubling *L* does not double the resolving power; rather, a two-fold increase in *L* (holding *T* and *E* constant) results in only ~40% increase in resolving power.

In the present paper, we report modeling studies of ion transport through multiple drift regions to which the drift fields are applied at varying frequencies, the experimental setup used in overtone mobility spectrometry (OMS). The understanding that is gained from modeling allows us to develop a simple equation that can be used to estimate the OMS resolving power (*R _{OMS}*). The equation describing the OMS resolving power accounts for a number of geometrical OMS device configurations as well as those parameters used to define

It is important to note that the comparisons to *R _{IMS}* (with respect to

There is currently considerable interest in developing and improving mobility based separation methods. In addition to efforts to improve IMS techniques,^{8}^{,}^{9}^{-}^{13} a new method called field asymmetric (FA)IMS has been developed.^{8}^{,}^{14}^{-}^{22} Additionally, differential mobility analysis (DMA)^{23}^{,}^{24}, which has traditionally been used for sizing of particles,^{23}^{,}^{25} has been extended for analysis of large molecular systems (such as polymers, proteins and protein complexes).^{23}^{,}^{24} The present paper adds to theoretical work associated with understanding these mobility-based techniques.

It is useful to begin by describing the OMS separation process. For convenience, Table I provides a summary of all variable definitions used in this paper. Since IMS techniques, including descriptions of a range of instruments^{26}^{-}^{41} and theoretical treatments^{1}^{,}^{6}^{,}^{42}^{-}^{46} are described elsewhere, further discussion of the various techniques is not provided here.

Each OMS instrument contains a specified number (*n*) of sequential drift regions (*d*) containing both ion transmission (*d _{t}*) and an ion elimination (

The mode of operation of an OMS instrument is determined by the number of phases or drift field application settings. This number is denoted by *ϕ*. Hereafter *ϕ* represents a positive integer indicating the method of applying drift field pulses to transmit ions of specific mobilities through the OMS device. Figure 1 illustrates a portion of an OMS drift region; eight consecutive *d _{t}* regions with accompanying

Illustration of the drift regions of an OMS device. Shown are eight *d* regions (containing both *d*_{t} and *d*_{e} sections). Also shown are the field modulation settings for OMS experiments utilizing phase conditions of *ϕ* = 2, 3, and 4. For the two-phase **...**

To understand the consequences of the field modulation consider the transmission of ions from a continuous ion beam using a two-phase OMS approach. A portion of the continuous ion beam fills the *d _{t}(1)* region (Figure 1). Under the first field settings (A in Figure 1), ions are allowed to pass through the

Having described the field settings for a two-phase system, it is possible to depict the modulation process for a three-phase and a four-phase system. Again the number of distinct field settings is equal to *ϕ*. Additonally, the number of adjacent *d* regions across which the field is applied is also equal to *ϕ* and therefore a complete transmission/elimination cycle is equal to *ϕ* field settings. To understand the impact on ion transmission with increasing *ϕ* it is instructive to consider operation with a greater number of phases as shown in Figure 1. Here, we consider the four-phase system. As the elimination pulse moves from *d _{e}(1)* to

OMS distributions can be obtained by recording the ion signal for a specific ion at given field application frequencies. Figure 2 shows the OMS distributions obtained for the [M + Na]^{+} ion for the oligosaccharide raffinose. The data have been recorded over a range of OMS system phase settings (*ϕ* = 2 to 6). For each setting for *ϕ* a dominant peak corresponding to the fundamental frequency is observed. This peak is observed to broaden with increasing *ϕ*. In addition to the fundamental frequency peak, a number of other peaks are observed for each OMS separation. For example, for the analysis that utilized the two-phase system, peaks are observed at frequencies that are exact multiples of *f _{f}* (including 3·

The origin of overtone peaks can be understood by considering the hypothetical ion beam and transmitted ion distributions that are shown in Figure 3. Here we demonstrate the transmission of ions for the 2^{nd} overtone frequency (*m* = 3 harmonic) for a two-phase OMS separation. During the first field application, the forward most 1/3 of the ions in the *d _{t}(1)* region are transmitted into the

Illustration of the effect of overtone frequency on ion transmission. A continuous ion beam filling the *d*_{t}(1) region followed by transmission of portions of the ion beam is shown. *d*_{t} regions are indicated above each successive region and *d*_{e} regions are **...**

From Figure 3 and the discussion given above, we see that transmission of ions at overtone frequencies will also depend on *ϕ*. For example, for conditions in which the first overtone frequency is applied to a two-phase system, half of the ions in the *d _{t}(1)* region [those nearest the

Because of the relationship between *ϕ* and *m* (via *h*), the focus of the current work is the derivation of an expression for resolving power for those features representing the fundamental and overtone peaks for the different phase systems. These are the major peaks labeled in Figure 2. Here we note that a number of other peaks are observed, most noticeably between the fundamental frequency peak and the *h* = 1 overtone peak for different values of *ϕ* (particularly for greater values). The origin of these peaks is still under investigation and as such they are not treated here in terms of estimation of resolving power. Rather, investigations focus on the dominant peaks (Figure 2) and their relationship to resolving power.

To better understand the separation process of OMS, ion trajectory simulations have been performed. We have used a field array containing 11 *d* regions. The algorithm used to perform the simulations has been written in house and is conceptually similar to those described previously.^{47}^{,}^{48} Briefly, the calculations utilize two-dimensional field arrays similar to those generated in SIMION^{49} to determine time-dependent ion displacements. The displacement calculation of an ion is the sum of the ion motion contributions from the mobility of the ion (*K*) as well as its diffusion. The former contribution is field dependent [i.e., the drift velocity is proportional to the product of the mobility and the electric field (*KE*)] ^{1}, whereas the latter is a randomized contribution in the algorithm (discussed below). Although the algorithm is similar to that described previously,^{48} the field arrays of two dimensions are utilized as opposed to three. This change is made because it reduces the time required to model trajectories of thousands of ions. The two dimensions (instead of three) utilized include the ion axis (z-axis) as well as the height (y-axis). Two-dimensional ion trajectory simulations performed in SIMION^{49} utilize the same field array dimensions.

The modeling algorithm also allows different OMS phase systems to be examined. For simplicity, trajectory simulations for a two-phase system are described below. The modeling requires two field arrays including one in which the *1*+*2j* (*j* = 0, 1, 2, 3, 4, 5, and 6) *d _{e}* regions (see above) contain fields that will transmit ions; the remaining

Displacement of ions due to diffusion is simulated by

$$\sqrt{\overline{{r}^{2}}}={(4D\mathrm{\Delta}t)}^{1/2}$$

(2)

where,
$\sqrt{\overline{{r}^{2}}}$ is the root mean-square displacement of the ion for time increment *Δt* and ion diffusion coefficients can be obtained as mentioned above. For ions used in the simulations, *K* is taken from an experimentally available value (e.g., the mobility of singly-charged bradykinin). To simulate random diffusion in two dimensions,
$\sqrt{\overline{{r}^{2}}}$ is converted into a polar coordinate vector by randomizing a single angle (). The y- and z-axis vector components are determined from
$\sqrt{\overline{{r}^{2}}}\cdot \mathit{sin}\u229d$ and
$\sqrt{\overline{{r}^{2}}}\cdot \mathit{cos}\u229d$, respectively. The diffusion value is then added to the two-dimensional mobility calculation to provide a net ion displacement for each time increment. To perform the trajectory simulation of ions in the two-phase system, a model has been devised where ion origin is referenced with respect to the *d _{t}(1)* region immediately preceding the first ion gate [

The trajectory simulations described above have been performed as a function of frequency to provide the peak profiles shown in Figure 4. Each of 10 to 20 incremental frequencies spanning the fundamental frequency as well as the overtones with harmonic frequencies of *m* = 3 and 5 is used. The modeling peak profiles resemble the experimental results on several fronts. First the relative peak heights which are approximately inversely correlated with *m* are similar to experimentally observed peak heights. Second, the relative resolving powers for the modeled two-phase system [consisting of 11 *d* regions] are similar to values obtained for peptide ions. That is, resolving powers of ~12, ~28, and ~46 are calculated for the *m* = 1, 3, and 5 frequencies, respectively. This increase in resolving power with increasing overtones is similar to that observed experimentally.

OMS peak profiles obtained from ion trajectory simulations. Simulations have been performed for an OMS approach utilizing *ϕ* = 2 and *n* = 11. Peaks have been modeled for the fundamental frequency as well as the *m* = 3 and *m* = 5 harmonic frequencies **...**

It is instructive to consider which ions in the *d _{t}(1)* region are transmitted through the entire mobility device as determined by the ion trajectory simulations. For discussion purposes the

It is also useful to consider the transmission of ions for a single overtone frequency setting. At the 2^{nd} overtone frequency (*m* = 3), ions are transmitted in the left and right third of the *d _{t}(1)* region as shown in Figure 3. Ions in the middle third of the region are eliminated as described above. It is important to note that the smaller transmitted beam portions behave similarly to the larger transmitted portions for the fundamental frequency. That is, it is useful to consider the initial positions of the transmitted ions at the peak base frequencies (frequencies 4 and 6 in Figure 4) as well as the

In order to develop an expression for *R _{OMS}*, consider results from ion trajectory simulations. Of interest is the peak shape produced by the OMS method. By removing the random diffusion component from the ion displacement calculation in ion trajectory simulations (see above), it is possible to construct a peak shape that depends only on mobility filtering. Peaks produced by measuring ion transmission as a function of frequency (in the

An expression for *R _{OMS}* can be obtained by considering the particular ions that limit the overall resolving power. For example, at the base of the

We begin by considering a peak that is associated with ions that are transmitted when the drift field frequency is set to ~1/*t _{d}* (i.e., the resonant frequency,

Now consider conditions where the applied frequency is lower than the ion’s resonant fundamental frequency (frequency 1 in Figure 4). With each pulse all ions arrive at a position that is slightly more displaced within the next *d* region (e.g., for the ion described above, position numbers of 10 and 20 in the *d _{t}(2)* and

With an understanding that the time duration of a single field application setting (*P _{w}*) is equal to

$${P}_{w2}=\frac{{}^{t}\mathit{total}}{n}$$

(3)

$${P}_{w1}=\frac{{}^{t}\mathit{total}}{n-1}$$

(4)

These relationships can be used to relate the frequencies of *f _{1}* and

$${f}_{1}=\frac{(n-1)\phantom{\rule{0.2em}{0ex}}{f}_{2}}{n}$$

(5)

Understanding that OMS peaks are triangular in nature and using Equation 5, it is possible to obtain the simplified expression for ROMS shown in Equation 6 (see supplementary information for equation derivation information details).

$${R}_{\mathit{OMS}}=n$$

(6)

Remarkably, the resolving power of the OMS technique for the fundamental peak depends only on the number of drift region segments. Although this is somewhat intuitive, it stands as an important result because it defines one important difference between OMS, where *R _{OMS}* scales directly with

With an expression for the resolving power of the fundamental frequency, it is useful to include the effects of OMS system phase and harmonic frequency. Consider the case of the four-phase OMS device. Here, four ion gating regions are triggered sequentially to complete one transmission/elimination cycle. As described above, ions occupying the space of three *d* regions are transmitted through the OMS device for every space equivalent to one *d* region that leads to ion elimination (i.e., a duty cycle of 75%). Because the total ion transmission range is three times larger than the comparable two-phase system (i.e., one with d regions of identical length), the difference in number of pulses experienced by ions using field application frequencies *f _{1}* and

The opposite effect is observed with an increase in overtone. Consider the two-phase system operating at the *m* = 3 harmonic frequency (2^{nd} overtone). Because the overall frequency is three times greater than *f _{1}*, the total number of field applications experienced by an ion at the

Until this point, the *d _{e}* regions were treated as infinitely small. It is interesting to consider the consequences of different lengths of

With an understanding of the influence of *n, ϕ, m*, as well as *l _{t}* and

$${R}_{\mathit{OMS}}=\frac{\mathit{mn}}{\phi -1-\frac{{l}_{e}}{{l}_{t}+{l}_{e}}}$$

(7)

This expression reveals several important factors affecting the overall resolving power. First, as observed above, *R _{OMS}* increases linearly with increasing

It is important to note that Equation 7 does not truly represent a worst-case scenario for the greatest mismatch in *f _{1}* and

$${R}_{\mathit{OMS}}=\frac{1}{1-\left[1-\frac{{C}_{2}}{{R}_{\mathit{IMS}}}\right]\phantom{\rule{0.2em}{0ex}}\left[\frac{\mathit{mn}-\left[\phi -1-\frac{{l}_{e}}{{l}_{t}+{l}_{e}}\right]}{\mathit{mn}}\right]}$$

(8)

Here *C _{2}* corresponds to a constant allowing the conversion to

Equation 8 provides insight about limits associated with diffusion. That is, in the limit of high *R _{IMS}* (less diffusion), the denominator approaches a minimum value leading to a maximum

We begin the evaluation of the expression for OMS resolving power by considering several experimental results. Figure 5 shows the experimental peak widths for the [M + Na]^{+} ion of the trisaccharide melezitose for varying numbers (*n*) of *d* regions. The data have been acquired by applying the fundamental frequency for a four-phase OMS device. The results show a substantial decrease in peak width with increasing *n*. At *n* = 11, the measured FWHM is ~1600 Hz, whereas a value of ~345 Hz is obtained at *n* = 43. This large dependence on *n* is intriguing because it suggests that OMS resolving power is not defined by a square root dependence on *L* as is the case for IMS experiments. To evaluate the efficacy of the equation for predicting OMS resolving power as a function of *n*, calculated values have been compared to the experimental results obtained for the four-phase OMS system (Figure 5). The equation predicts a linear increase in *R _{OMS}* from 3.6 to 13.0 at

Experimental OMS distributions for the [M + Na]^{+} ion for the trisaccharide melezitose. The plots show a progression of OMS distributions as a function of *n*. The data span the range of that collected using *n* = 11 *d* regions (bottom trace) to *n* = 43 *d* regions **...**

The ability of the equation to predict resolving power as a function of the applied overtone frequency has also been investigated. The comparison has been evaluated using experimental data for three different systems (*ϕ* = 2, 3, and 4). Additionally, the transmitted overtone peaks (for which *h*=2 to 3) of each system have been compared. The results of the comparison are shown in Figure 6. In general, the agreement is as observed before with the equation providing a resolving power that is ~21% higher on average than experimental results for the fundamental frequency and the overtone frequencies (corresponding to *h*=2 to 3). For all systems, the trend of increasing resolving power with increasing applied frequency is observed. The equation is also able to mimic the observation of increased resolving power with decreasing OMS phase number. In addition to this, the equation, as with the experimental data, shows an increased rate of resolving power improvement (from the fundamental frequency to the overtone frequency corresponding to *h*=2) as a function of decreasing phase number. Finally, it should be noted that for the calculated resolving power, the rate of increase is greatest from the fundamental frequency to the overtone frequency corresponding to *h*=2. At that point, going from the *h* = 2 to 3 overtone frequencies allowing ion transmission, the rate of increase is not as high. This occurs for each OMS phase system. Intuitively, this occurs because of the decrease in size of the transmitted ion beam portion (Figure 3). While this portion of the beam is decreasing with increasing overtone frequency, the overall ion transit time is not. Thus diffusion becomes more significant for this shorter transmitted beam. Because, diffusion is overestimated in our resolving power equation (as the maximum expected value), the theoretical values drop below the experimental values.

The OMS resolving power equation can be used to begin to evaluate factors involving ion diffusion and their influence on the overall attainable resolving power. The resolving power has been evaluated as a function of two parameters from Equation 1 that affect the degree of ion diffusion; these are *E* and *T*. Here we note that from our treatment of *L* [*n*·(*l _{t}* +

Although, diffusion does not appear to affect the resolving power significantly, it is useful to consider a possible explanation for the observation of a larger change at higher overtone numbers. As mentioned above, the transmitted ion beam portions are much smaller for the higher overtones (Figure 3). The contribution of diffusion becomes more significant for these shorter ion beam portions (note the overall transit time is not changed). That is, the distance travelled due to diffusion (counter to the mobility of the ion for the OMS resolving power equation) is a larger percentage of the overall transmission region size.

Another factor that influences the degree of peak broadening due to diffusion is the ratio of the transmit and elimination (*l _{t}* and

The computed values in Table 2 show that the resolving power equation can be used to rapidly test the separation efficiency of specific instrumental configurations. An understanding of the trends for each of the variables suggests the direction in which one should pursue higher resolving power. Another interesting feature of the table is that a relatively high resolving power (~85) can be obtained with a fairly small device.

Until now the discussion has focused on the ability to estimate *R _{OMS}* for a given OMS experimental setup. Briefly the discussion has touched upon the application of OMS as a mobility selection device. As a final thought, an application of ion structure analysis is discussed. As mentioned above, the measurement of

$$\mathrm{\Omega}=\frac{{(18\pi )}^{1/2}}{16}\frac{\mathit{ze}}{{({k}_{b}T)}^{1/2}}{\left[\frac{1}{{m}_{I}}+\frac{1}{{m}_{B}}\right]}^{1/2}\frac{{t}_{D}E}{L}\frac{760}{P}\frac{T}{273.2}\frac{1}{N}$$

(9)

where *P* and *N* correspond to the buffer gas pressure and the neutral number density under STP conditions, respectively. The variable *m _{I}* and

To obtain a collision cross section from an OMS measurement, the distance the ion travels in a specified amount of time must be determined. From an OMS distribution this translates into ion displacement over the time equal to the inverse of the peak apex frequency. For the fundamental frequency, recalling that ion displacement corresponds to one *d* region, the calculation is relatively straightforward. For conditions that employ overtone frequencies (*m*>1), recall that ion displacement is equal to a fraction of the *d* region (or *d*/*m*, see Figure 3). Additionally, because *m* can be expressed in terms of *ϕ* and *h* [i.e., *m*= *ϕ* (*h*-1)+1], it becomes possible to rewrite Equation 25 for OMS measurements as,

$$\mathrm{\Omega}=\frac{{(18\pi )}^{1/2}}{16}\frac{\mathit{ze}}{{({k}_{b}T)}^{1/2}}{\left[\frac{1}{{m}_{I}}+\frac{1}{{m}_{B}}\right]}^{1/2}\frac{E\left[\phi (h-1)+1\right]}{f({l}_{t}+{l}_{e})}\frac{760}{P}\frac{T}{273.2}\frac{1}{N}$$

(10)

One concern about the measurement of cross sections in a device that uses electrodynamic fields is the effect of the voltage slewing rate on the overall measurement. Experiments have shown that ~20 ns are required to achieve stable fields after switching. From a number of experiments, for a variety of ions (different charge states of ubiquitin, peptides, and carbohydrates), the cross sections obtained from OMS measurements differ from those obtained by IMS techniques by 2 to 4% on average. Noting that this difference is larger than the errors obtained for different IMS instruments (typically < 2%), it is possible that a portion may indeed result from field inhomogeneity. It may also result from slight errors in the determination of drift region segment length. The propagation of such an error would be more problematic than a small error in the overall length of a drift tube for an IMS instrument.

The ability to determine accurate collision cross sections may serve a number of applications. One example may include aiding in the identification of mixture components. That is, with significant enhancements to component resolution, cross sections determined for features in OMS distributions can be compared against trial structures.^{43}^{,}^{50}^{-}^{62} Such an approach could be used to select specific structures for further analysis.

Simulations involving the transport of ions through multiple drift regions have been compared with experimental data in order to develop a better understanding of the origin of resolving power in OMS. The results lead to an expression for resolving power that includes experimental variables associated with ion diffusion (*E, L*, and *T*) as well as variables related to OMS instrument operation (*ϕ, n, f, d _{t}*, and

The development of new instrumentation is supported in part by a grants from the National Institutes of Health (AG-024547-01 and P41-RR018942), and the METACyte initiative funded by a grant from the Lilly Endowment. The authors are grateful for numerous stimulating discussions with their colleagues, Liang-shi Li, Caroline C. Jarrold, and Gary M. Hieftje about overtones and instrumentation in general. We also thank John Poehlman and Andrew Alexander for technical support.

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