Improved Momentum-Transfer Theory for Ion Mobility. 1. Derivation of

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Improved Momentum-Transfer Theory for Ion Mobility. 1. Derivation of the Fundamental Equation William F. Siems,*,† Larry A. Viehland,‡ and Herbert H. Hill, Jr.† †

Department of Chemistry, Washington State University, Pullman, Washington 99164-4630, United States Science Department, Chatham University, Pittsburgh, Pennsylvania 15232, United States

‡

S Supporting Information *

ABSTRACT: For the first time the fundamental ion mobility equation is derived by a bottom-up procedure, with 5 real atomic ion−atomic neutral collisions replaced by 5 repetitions of an average collision. Ion drift velocity is identified as the average of all pre- and postcollision velocities in the field direction. To facilitate velocity averaging, collisions are sorted into classes that “cool” and “heat” the ion. Averaging over scattering angles establishes mass-dependent relationships between pre- and postcollision velocities for the cooling and heating classes, and a combined expression for drift velocity is obtained by weighted addition according to relative frequencies of the cooling and heating encounters. At zero field this expression becomes identical to the fundamental low-field ion mobility equation. The bottom-up derivation identifies the low-field drift velocity as 3 /4 of the average precollision ion velocity in the field direction and associates the passage from low-field to high-field conditions with the increasing dominance of “cooling” collisions over “heating” collisions. Most significantly, the analysis provides a direct path for generalization to fields of arbitrary strength.

I

connection between ion drift experiments and the underlying collisions:

ons transported through a neutral gas under the influence of an electric field, E, reach a terminal drift velocity in the field direction, vd. The ratio of drift velocity to field strength is called the mobility, K, traditionally expressed in units of square centimeters per volt per second: v K= d E

(1)

The macroscopically constant vd is the net result of a series of microscopic accelerations, collisions, and decelerations, and K is thus a joint property of the ion and neutral species. In ideal mobility experiments the ion number density, n, is sufficiently low that no observable bulk change in the neutral gas is produced by ion motion and the ions themselves are unaffected by mutual Coulombic repulsion. The energy and momentum a drifting ion acquires from E are unaltered by parallel changes in field and neutral number density, N, that leave E/N unaltered, since the change in acceleration is exactly compensated by the change in proximity of collision partners. The ratio E/N is called the “reduced field”, and it is traditionally given in units of Townsends (1 Td = 1 × 10−17 V·cm2). The term “low-field” denotes situations where the collision frequency is indistinguishable from its value due to thermal motion alone, in contrast to “high-field” cases, in which velocity acquired from the field increases collision frequency well above the thermal value. Terminology such as “low-field” and “high-field” should always be taken as descriptive of E/N, not simply of E. For more than 100 years the following equation, with its precursors and variants,1−8 has served as the key theoretical © 2012 American Chemical Society

1/2 3 ⎛ 2π ⎞ e K= ⎜ ⎟ 16 ⎝ μkT ⎠ N Ω D

(2a)

1/2 3 ⎛ 2π ⎞ eE ΩD = ⎜ ⎟ 16 ⎝ μkT ⎠ vdN

(2b)

The second form emphasizes the way analysts often want to use the equation, as a means to obtain ΩD from experimentally measured temperature, pressure, electric field, and ion drift velocity. The primary concept of ionic size, a property distinct from ionic mass, is embodied in ΩD, which has units of area and is usually called “collision cross-section” by experimentalists. To theoreticians who compute mobilities, the name “collision integral” is preferable, because cross-section is reserved for a microscopic property evaluated at a single collision energy, while ΩD is a macroscopic weighted average of cross sections belonging to the distribution of collision energies that actually occur in the experimental apparatus. In eqs 2a and 2b, e is the charge on the ion, k is Boltzmann’s constant, T is the absolute temperature, and μ is the reduced mass for the ion and neutral (masses m and M, respectively): Received: June 26, 2012 Accepted: October 24, 2012 Published: October 24, 2012 9782

dx.doi.org/10.1021/ac301779s | Anal. Chem. 2012, 84, 9782−9791

Analytical Chemistry

μ=

Mm M+m

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scopic level cannot remain correct after ensemble averaging. The interrelationships include negative correlation between size and relative collision velocity, positive correlation between momentum transfer and relative collision velocity, and the recurring dependence of the average of the ith power of ion velocity on the (i + 1)th power of the velocity. In zero-field cases a 5% error is typical for eq 2a/2b, although errors as high as 13% have been encountered,11 and the errors generally vary as field strength increases.8,10,11 The situation is further blurred by the lack of a sharp demarcation between low- and high-field conditions, as discussed below. The 2−5% relative uncertainty sometimes accepted for mobility measurements can make eq 2a/2b seem sufficient for low-field situations, but applications, ranging from tandem IMS−MS of complex biological samples to stand-alone detection of explosives and chemical warfare agents, would benefit if relative uncertainties could be reduced to 0.2−0.5% or better. It is likely that measurements with this precision will eventually become routine, so the interpretive apparatus should be trustworthy at this same level of confidence. Modern research instruments are designed and operated to maximize IMS resolving power or to maximize ion transmission through IMS−MS interfaces, and to achieve such goals they frequently employ reduced fields intermediate between low and high field. Field strength is sometimes judged to be low if K in eq 1 does not vary noticeably with E/N, but proper classification of field strength should be based on the degree to which collision energetics are altered from the thermal baseline. One natural metric is comparison of vd to the zerofield average relative ion−neutral speed,12 vT, which we refer to simply as the thermal speed:

(3)

In 1905, Langevin1 obtained a relationship equivalent to eq 2a/ 2b for the mobility of ions in weak electric fields, work that went largely unnoticed until its rediscovery in the 1920s by Hassé and Cook.2,3 The equation resurfaced in the 1950s, in an extension to mobility of the Chapman−Enskog iterative techniques for diffusion,9 through theoretical work by Kihara6 and subsequent computations by Mason and Schamp.7 There has been a recent trend to refer to eq 2a/2b as the Mason or Mason−Schamp equation, but because of the equation’s ancient pedigree and because another relationship is named for Langevin, we recommend that eq 2a/2b be referred to simply as the fundamental low-field ion mobility equation. The extension of mobility theory to electric fields of arbitrary strength was first attempted by Hershey,4, and later work by Wannier5 led to Viehland and Mason’s convolution8,10,11 of thermal and field contributions to collision energy to produce an “effective temperature”, Teff, to replace T in a high-field analogue of eq 2a/2b. When eq 2a/2b is modified to be suitable for all field strengths, we drop the adjective and refer to it as the fundamental ion mobility equation. Equation 2a/2b has two surprising properties that warn of limits to its applicability. First, it is symmetrical to interchange of ion and neutral masses, assuming ΩD to be identical for the two cases. However, even at fairly low fields a massive ion in a low-mass drift gas will clearly have its velocity little altered by collisions, while a low-mass ion in a massive drift gas will be strongly scattered.10 We therefore expect differing relationships among drift and pre- and postcollision velocities for the massinterchanged cases, yet eq 2a/2b has no hint of such behavior. While the high-field analogue of eq 2a/2b does introduce mass dependence through Teff, only collision frequency is modified, not the relationships among drift and pre- and postcollision velocities. A second troubling characteristic of eq 2a/2b is that it may be obtained from the first Chapman−Enskog approximation to the diffusion coefficient9 by substituting the zero-field (equilibrium) Nernst−Townsend−Einstein relationship between K and D:10 K=

De kT

⎛ 8kT ⎞1/2 vT = ⎜ ⎟ ⎝ πμ ⎠

(5)

If it is assumed that the measurables of IMS possess three significant digits, then in a general sense as vd grows to within 2 orders of magnitude of vT, we may expect all the parameters related by eq 2a/2b to begin changing simultaneously, since they are all are related to collisions. Relative collision speed, collision frequency, momentum transfer, and ion−neutral size, as well as correlations among these quantities, will all begin to vary with E/N. Due to the interrelationships among the quantities, it is thus possible for eq 2a/2b to become untrustworthy before dependence of K on E/N becomes noticeable. In particular, as will become apparent in the work to follow, for the common case of high-mass ions in a low-mass drift gas, compensating changes in momentum transfer and collision frequency can minimize change in K, even as ΩD becomes sensitive to E/N. A low-field criterion based on experimentally determined quantities13 can be written equivalently in terms of E/N or the ratio of vd to vT:

(4)

The effect of an electric force on ions is thus deduced from a case where an electric field is absent! The Chapman−Enskog result for diffusion assumes equilibrium Maxwellian velocity distributions for both ion and neutral, but for ion drift velocity to be experimentally observable, the velocity distribution must obviously be non-Maxwellian. Thus the fundamental low-field ion mobility equation could be strictly correct only in the limit of zero field. The fundamental ion mobility equation, at any field strength, is an approximate relationship, and misunderstanding of the magnitude, sources, and remedies for its approximations clouds the analytical use of ion mobility spectrometry (IMS). Even in the limit of zero field strength with idealized hard-sphere or Maxwell-model systems, eq 2a/2b is only approximately correct. Approximation is inherent in an expression that supposes simple multiplicative relationships among ensemble average values of drift velocity, collision frequency, momentum transfer, and size. These quantities all arise from interactions at the microscopic level, as reviewed below, but they are so intimately interrelated that formulaic equalities at the micro-

1/2 E 1 ⎛ 3kT ⎞ ⎜ ⎟ ≤ N c′NK ⎝ m + M ⎠

(6a)

1/2 vd (mM ̂ ̂ )1/2 1 ⎡⎛ m ⎞⎛ M ⎞⎤ ⎟⎜ ⎟ ≤ ⎢⎜ ≡ ⎥ vT c ⎣⎝ m + M ⎠⎝ m + M ⎠⎦ c

(6b)

The mass fractions m̂ and M̂ have been introduced, and the factor c = c′(8/3π)1/2 is to be chosen empirically to ensure the quantity on the left-hand side is sufficiently small. The mass dependence of eq 6a/6b makes it clear that the low-field 9783



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low-field ion mobility equation can possibly deliver. Such concerns have motivated us to create a proper momentumtransfer derivation of the fundamental low-field ion mobility equation, in hopes that it will reveal details of the mass dependence of mobility, provide a conceptual framework for understanding changes in collision characteristics as E/N is increased, and establish more solid footing for deducing, from mobility measurements, temperature-dependent collision integrals that will have the same qualitative behavior expected for microscopic cross sections. The derivation has proven more fruitful than expected, for it leads naturally to a modification of eq 2a/2b containing three physically sensible correction factors that we anticipate being able to compute or estimate over a wide range of field strengths for any ion and neutral species.

criterion is more difficult to satisfy when the ion and neutral masses differ greatly from one another, requiring lower E/N and vd than cases where the masses are close in value. Examples can be readily cited from the IMS literature where low-field conditions are assumed and eq 2a/2b is used, although the ionic drift velocity is a significant fraction of the thermal speed according to eq 6a/6b. An ambient temperature study of a 100 000 Da multiprotein complex in Ar drift gas with a Waters Synapt, including a traveling wave IMS with an average field strength of 27 Td, measured vd/vT = 0.116,14 compared to a c = 10 criterion of vd/vT ≤ 0.002. Studies of +1 bradykinin (1016 Da) in He in two ambient temperature IMS devices, one operating at “low pressure” (2.0 Torr, 12.8 V/cm) and one at “high pressure” (180 Torr, 142.2 V/cm), reported vd/vT = 0.095 and vd/vT = 0.012, respectively,15 compared to a c = 10 criterion of vd/vT ≤ 0.0063. Finally, a recently described 2 m high-resolution IMS (ambient T, P = 12−15 Torr He, E/N = 4−7 Td)16 if applied to +1 bradykinin (Ω = 239 Å2)15 would show vd/vT = 0.015−0.033, compared to the same c = 10 criterion of vd/vT ≤ 0.0063. Although we must revisit the contention once our new derivation has been presented, we claim that many, if not most, of the IMS measurements being made today do not come from deep within the low-field region but from the border between low- and high-field regimes, where drift velocity affects collision energy noticeably and where modification of the fundamental low-field ion mobility equation becomes necessary. All rigorous derivations of eq 2a/2b for atomic ions and neutrals have been “top-down” deductions of ion velocity distribution characteristics using the Boltzmann kinetic equation. The term “momentum-transfer derivation” has been used for “bottom-up” justifications of eq 2a/2b based on repetitions of a single average collision and general conservation principles.10,17 However, it is incorrect to call these prior bottom-up treatments “derivations” because two crucial inaccuracies were knowingly permitted for the sake of simplicity. The resulting expressions have expected dependences on experimental variables and physical constants but differ in the numerical coefficient [3−1/2 appears instead of 3(2π)1/2/16], with the correct numerical value grafted on at the end from the top-down derivations. One of the two inaccuracies, use of root-mean-square velocity instead of average speed, is easily remedied. But it is more difficult to correct the second inaccuracy: use of vd as an approximation for the average precollision ion velocity. Obviously ions must be traveling with a drift velocity less than the average precollision velocity, since ions are continually accelerated between collisions and reach their largest velocity just before the next collision! This second approximation is less defective for massive ions in low-mass drift gases, but it is always somewhat wrong and becomes seriously so with ions of low mass in a massive drift gas. In previous momentum-transfer treatments,10,17 eq 2a/2b has been clearly identified as a first approximation, an expedient rough solution that can be corrected later for better agreement with experiment by iterative methods based on the Boltzmann equation. But analytical practitioners of IMS usually employ no follow-up adjustment to their mobility measurements before using eq 2b to calculate cross sections for comparison with structure candidates. One is left with the uncomfortable realization that mobility measurements are being used to generate cross sections that are not microscopic and that are expected to be more precisely known than the fundamental

■

DEFINITIONS AND BACKGROUND We assume all collisions to be elastic. Allowing exchange of energy between translation and internal modes of motion does not alter our basic result but does require careful modifications of the definitions of cross section, collision integral, and velocity distribution. To preserve simplicity, treatment of inelastic collisions is deferred for subsequent work. Likewise, we restrict attention to spherically symmetric ions and neutrals. Generalization to polyatomic species does not alter the basic result but introduces the distracting need to average over multiple geometric variables of encounter. Lowercase letters are used for ionic properties and capital letters for neutral properties; primed quantities (v′, vr′, etc.) refer to postcollision values, and unprimed quantities refer to precollision conditions. Angle brackets and overbars denote averages over the entire ensemble of ions and neutrals in an experiment. The electric field is taken parallel to the laboratory z-axis. Relative Velocity and Center-of-Mass Velocity. While the ion and neutral velocities in the laboratory frame of reference, v and V, respectively, are the natural descriptors for experiments, the relative and center-of-mass velocities, vr and vcm, respectively, are more convenient for treating conservation of energy and momentum and for averaging of collisions: vr ≡ v − V

(7a)

vcm ≡ m̂ v + M̂ V

(7b)

If an ion and neutral collide, vr points toward the neutral before the collision and away from the neutral after the collision. For conservation of energy in an encounter, it is necessary and sufficient that the relative ion−neutral speed is unchanged: |vr′| = |vr| ≡ vr. Equivalently, a collision does not change the length of vr but turns it through a scattering angle π ≥ θ ≥ 0. Because vr is a difference vector, it is unchanged by Galilean transformation of the reference frame, so it may be shifted unchanged between laboratory, center of mass, and other uniformly translating frames of reference. This invariance makes vr the key descriptor of collisions, and it determines both the maximum momentum that can be transferred in a collision (2μvr) and the maximum energy that can be transferred, called the relative energy of collision: ε≡

1 2 μvr 2

(8)

Linear momentum is conserved in a collision if and only if the center-of-mass velocity of the colliding pair is unaltered by the event; that is, vcm′ = vcm. 9784



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Classification of Collisions. Figure 1 is the “archetype” collision to which all others will be related. Although the figure

Figure 2 also presents a third categorization of collisions into “cooling” and “heating” types, based on the direction of energy transfer in the laboratory frame of reference. A cooling collision transfers energy from ion to neutral and has vr·vcm > 0, while a heating collision transfers energy from neutral to ion and has vr·vcm < 0. Other r+ collisions with the same scattering angle can be obtained from the archetype of Figure 1 by giving the encounter a uniform translational z-velocity with respect to the rest frame. Such translations change vcm and the ion−neutral energy transfer of the archetype collision, but the momentum transfer is the same in every case: the ion transfers zmomentum μvr(1 − cos θ) to the neutral. It is also possible to obtain an r− collision from Figure 1 by switching the positions of the ion and the neutral in the figure. This mass reversal gives a collision with the directions of vr = −V and vr′ reversed in relation to the figure, the positions of v′ and V′ switched, and zmomentum μvr(1 − cos θ) transferred from neutral to ion. Other r-reverse collisions with the same scattering angle can be obtained by again giving the mass-reversed encounter a uniform translational z-velocity to create a family of collisions transferring z-momentum μvr(1 − cos θ) from neutral to ion. Average Relative Speed. The average relative ion−neutral speed in the absence of fields is determined entirely by thermal motion, equal to the thermal speed of eq 5:12

Figure 1. Archetype ion−neutral collision in a frame of reference at rest with respect to the precollision neutral. Unprimed and primed velocities denote immediately pre- and postcollision values, respectively. vr = v − V is the relative ion−neutral velocity, θ is the scattering angle of the relative velocity, and ϕ = 1/2(π − θ) is the neutral scattering angle.

depicts the colliding species as hard spheres, the relationships among the vectors are independent of the nature of the ion− neutral interaction potential. By use of conservation of energy and momentum, the postcollision ion and neutral velocity vectors may be deduced from the masses, vr, and the scattering angle of the relative velocity, θ, or the equivalent neutral scattering angle ϕ = (π − θ)/2. In particular, the z-momentum transferred to the neutral by the collision is μvr(1 − cos θ). By creating a preferred direction for ion motion, the electric field also becomes the basis for classification of collisions according the orientation of vr and vcm. A collision is “r+” if vr·E > 0, “r−” if vr·E < 0, “cm+” if vcm·E > 0, and “cm−” if vcm·E < 0. These collision classes are summarized in Figure 2. A [+,+] encounter was chosen as the archetype of Figure 1 because of its presumed resemblance to an overall average collision, with neutral and ion precollision velocity vectors corresponding to ensemble thermal averages, ⟨V⟩ = 0 and ⟨v⟩ = vz, respectively.

vr ≡ vr = vT

(9)

Neutral velocities are always entirely thermal, but ions have a field velocity added to their thermal motion whenever E/N > 0, so eq 9 approximates vr only at very low fields. One possible way of modifying eq 9 to better describe mobility at higher fields is by simple linear addition: vr = vT + βvd

(10)

Setting the unitless parameter β = 1 would be appropriate if all of the additional ion velocity is in the field direction, but in general β > 1 allows for increased ion velocity perpendicular to the field due to scattering from neutrals, an effect that becomes increasingly important when M ≫ m. In general we would expect β to be a function of M and m only, and in view of its origin we call it the “transverse velocity coefficient”.

Figure 2. Collision categories based on parallel (+) or antiparallel (−) alignment of vr and vcm with E. The large orange circles represent ions and the small green circles are neutrals. The four classes correspond to the boxes of the diagram. Each box contains an archetype collision, a velocity vector in the laboratory frame of reference, and the ± designations for (vr, vcm). The boxes may also be read as the quadrants of the continuous twodimensional (vr, vcm) space in the laboratory frame of reference, with the horizontal axis corresponding to (vr, 0) and the vertical axis to (0, vcm). The background color indicates whether the collisions in the quadrant are “cooling” (vr·vcm > 0, blue background) or “heating” (vr·vcm < 0, pink background). 9785



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ab initio potential surface is assumed, QD(vr) can be computed once and for all at the beginning of any calculation of transport properties, since experimental conditions do not affect the potential energy surface. Because collisions in real experiments occur with a distribution of vr values, we need a weighted average of QD over the distribution. At zero field, both ions and neutrals have equilibrium Maxwellian velocity distributions, and the weighted average of QD(vr) may be written in terms of the relative collision energy:

The Discussion will reconsider how to combine thermal and drift contributions to average relative collision speed, but at this stage we merely observe that such a combination is necessary. We will derive an expression that includes vr as a factor, but we leave open for the moment how the factor is chosen. The linear combination of eq 10 is one possibility, but the twotemperature theory8 uses a sum-of-squares convolution of ion drift motion with thermal motion, with a transverse velocity coefficient very nearly given by m̂ −1/2. The two-temperature and other theories start with eq 5 but replace T with an “effective temperature”, Teff.8,10 The use of Teff focuses attention on ion energy, leads to more compact expressions, and is adaptable to exact kinetic theory treatments in which Teff becomes a quantity that can be determined iteratively in a similar way to drift velocity. As discussed above, the relative magnitudes of the thermal and drift factors in eq 10 provide a basis for classifying mobility field strength regimes. For atmospheric pressure drift tubes, 1− 10 Torr drift tubes, and differential mobility devices, vd is generally ∼1%, ∼10%, and ∼100% of vT, respectively, but by the criteria of eq 6a/6b it is also necessary to include the ion and neutral masses in judging effective field strength. Cross Sections. Cross sections abound in physics, each condensing an interaction probability over an unbounded plane of closest encounter into an equivalent target area with an interaction probability of 1. Thus a cross section for momentum transfer is an effective area, not a geometric area, and any collision frequency derived by use of such a cross section is an effective f requency, not a counting frequency. The momentum-transfer cross section can be thought of as a target size for a specific ion and neutral pair and a measure of the ability of the pair to transfer momentum during a single collision. The definition of a microscopic cross section is also a computational recipe. Supposing both neutral and ion to be spherically symmetric, construct a plane perpendicular to vr and position the neutral at the origin of this plane. Launch a series of test ions with speed vr at points uniformly distributed across the plane, and in each case take note of the impact parameter b (radial distance from the origin to the intersection of vr with the plane) and the scattering angle θ. The angle will lie between 0 (no scattering and no momentum transfer) and π (bull’s-eye backscattering and momentum transfer of 2μvr). As mentioned above, the momentum transferred parallel to vr is μvr(1 − cos θ). The momentum transfer (or diffusion) cross section, QD, is the integral of the transfer factor (1 − cos θ) over the whole interaction plane: Q D(vr) = 2π

∫0

ΩD =

∫0

∞

Q De−ε / kT ε 2 dε

(12)

Here e−ε/kTε1/2 is the equilibrium Maxwellian distribution of relative energy, an additional factor of ε1/2(≈ vr) accounts for the probability of an ion−neutral encounter, another factor of ε arises because the plane of encounter must be allowed to move with vcm, and the normalization factor is chosen to give ΩD the value πd2 for hard-sphere ion and neutral. For non-zero field strength, the relative velocity distribution becomes nonMaxwellian, in general described by an unknown function - (ε)ε1/2, and instead of eq 12 we would write the following: ΩD =

1 (kT )−3 2

∫0

∞

Q D -ε 2 d ε

(13)

As mentioned above, we reserve the symbol ΩD and the terms “diffusion collision integral” or “average cross section” for a weighted average of QD over distributions of relative velocity, as in eq 12 or 13, and although analysts often employ the term “cross-section” for ΩD, we reserve this term for the microscopic QD. Approximate Nature of the Fundamental Ion Mobility Equation. The form of the fundamental ion mobility equation and its numerical coefficients have frequently been accepted as exact, but if one adopts this view, any discrepancy between theory and experiment must be absorbed into ΩD. But ΩD has a rigorous definition and its measurement is the core meaning of mobility experiments; it should not be altered in this loose way. Modern ab initio computations for atomic systems can yield cross sections more accurate than experiment,19,20 and we may expect further progress toward computation of polyatomic cross sections.21 Thus ΩD turns out to be the least flexible component of the fundamental ion mobility equation in the discussion of sources of and remedies for approximation. On the other hand, the initial 3/16 numerical coefficient, arising from the average momentum transferred in a collision, and the expression within the square root in eq 2a/2b, arising from the average ion−neutral collision speed, do need to be included in the discussion, as does the central hidden assumption of the momentum-transfer derivation of eq 2a/ 2b: that mobility can be expressed as a product of factors relating to momentum transfer, collision speed, and target area. Because these three factors are correlated with one another, not independent, even such a simple case as a hard-sphere ion and neutral at vanishingly low electric field is only approximately treated by eq 2a/2b. Momentum-transfer derivations or justifications of the fundamental ion mobility equation build on axioms of conservation and steady-state: (i) energy and momentum are conserved in every ion−neutral collision, and (ii) ions do not accumulate momentum or energy but transfer these at steady rates to neutrals. The steady-state axiom reflects the existence of a terminal drift velocity, eq 1. Because ions acquire

∞

[1 − cos θ(vr , b)]b db

1 (kT )−3 2

(11)

The subscript on QD denotes a cross section appropriate for diffusion (and mobility), in distinction to cross sections for viscosity, thermal diffusion, etc. If one or both of the collision partners are not spherical, then two-dimensional integration replaces eq 11 and the cross section must be further averaged over all relative orientations,18 but the result remains a function only of vr with units of area. If the ion and neutral are hard spheres with radii r and R, respectively, then QD = π(r + R)2 = πd2, independent of vr, but for real cases of elastic scattering, QD usually decreases as vr increases. The detailed dependence of the momentum transfer cross section on relative velocity is determined by the ion− neutral potential energy surface. If the existence of an accurate 9786



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Figure 3. Three views of ion velocity in a drift tube. The “statistical collision” view shows vz, the ion velocity parallel to E, decreasing or increasing erratically due to collisions but increasing at a constant rate eE/m between collisions. The “average collision” view replaces the statistical collisions with an equal number of average collisions, in each of which ion velocity drops from the average precollision velocity, ⟨vz⟩, to the average postcollision velocity, ⟨vz′⟩, while still increasing at the rate eE/m between collisions. In the “drift velocity” view, the ion has average macroscopic velocity vd = 1/2(⟨vz + vz′⟩) at all times.

momentum from the field at a rate eE, a “momentum balance” may always be written:

The “correlation coefficient” ζ is the quantity that converts the approximation, eq 2a/2b, into a true equality at low field, so if eq 2b is used to calculate an average cross section from experimental data, it is actually ΩD/ζ that is computed. The correlation coefficient depends in a complex way on the distribution of ion−neutral relative speeds and on the dependences of cross-section and momentum loss on vr, and it is of course a function of E/N. The value of ζ generally lies within about 5% of 1, as noted in the introduction, and its quantitative determination lies in the realm of accurate kinetic theory. We must acknowledge a mathematical maneuver that accompanies the replacement of an average of products with a product of averages in the cases of eqs 14a/14b and 15a/15b. In eq 14a/14b, the momentum transfer factor (1 − cos θ) is needed as part of the definition of the cross section, as in eq 11, and to compute average momentum loss in a collision as discussed below, so the separation requires that the transfer coefficient appear twice. Similarly, in the case of eq 15a/15b, vr is needed as part of the definition of both collision frequency and the collision integral (eq 12), and again the separation requires a quantity to appear twice, this time vr. Only in two extreme cases is it not necessary to double vr: with hard spheres, Ω is independent of vr, and with the Maxwell model of constant collision frequency, vrΩ is constant and no separation into a product of averages is required. The approximations above are well-known in kinetic theory, which also provides ways to construct and implement iterative corrections to calculated values.10 Accurate potential energy surfaces, from ab initio quantum mechanical calculations for example, are employed to compute cross sections as a function of relative collision speed. These cross sections are used to compute velocity moments (average values of v, v2, ..., etc.) by methods that evolved from the Chapman−Enskog treatment of mutual diffusion of neutrals.9 For atomic ion−neutral systems, these top-down methods begin with the Boltzmann equation for f(v, r, t) where f(v, r, t) dv dr is the relative probability of

eE = rate of mom. gain = rate of mom. transfer = (ion mom. loss in a collision)(frequency for this mom. loss) (14a)

Angle brackets indicate an average over collisions. The key maneuver needed to arrive at the fundamental ion mobility equation from a momentum balance is to replace an average of products (eq 14a) by a product of averages (eq 14b): eE ≅ ion mom. loss in a collision collision frequency (14b)

Furthermore, in a standard procedure from elementary kinetic theory,12 collision frequency may be identified as the product of neutral number density, relative ion−neutral speed, vr, and collision cross section. To arrive at the mobility equation, it is again necessary to approximate an average of products (eq 15a) as a product of averages (eq 15b): ⟨collision frequency⟩ = ⟨Nvr(cross section)⟩

(15a)

⟨collision frequency⟩ ≅ N ⟨vr⟩⟨cross section(vr)⟩

(15b)

Thus the fundamental ion mobility equation inherently approximates the correlations that exist between momentum transfer, relative collision speed, and cross section: eE ≅ ⟨momentum transfer⟩⟨vr⟩⟨cross section(vr)⟩ N

(16)

This approximate treatment of correlation is essential to produce a compact expression such as eq 2a/2b, and a correction factor is needed to convert eq 16 and the fundamental ion mobility equation itself into true equalities: eE = ⟨momentum transfer⟩⟨vr⟩⟨cross section(vr)⟩ζ N

(17) 9787



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Figure 4. Weighted contributions to an average collision from the heating and cooling collision classes . The dashed horizontal line corresponds to vd. In the limit of zero field, ⟨vz⟩h = ⟨vz′⟩c, ⟨vz′⟩h = ⟨vz⟩c, and fc = f h = 1/2, while at high field fc ≫ f h.

finding an ion having a velocity within dv of v and a position within dr of r at any time. The general Boltzmann equation is first specialized to the case of steady-state momentum transfer (axiom 2) by neglecting all time derivatives except that of the ion number density. One may then proceed by guessing an initial ion velocity distribution, f(0), selecting an orthogonal set of “basis” functions to serve as weighted corrections to f(0), and solving coupled sets of linear moment equations for the weights. The process is open-ended because equations for lower moments depend on values of higher moments, but if the initial guess and the basis functions are selected judiciously, the lower moments rapidly converge as the size of the basis set is expanded. Functions of velocity, such as the K of eq 2a/2b, emerge from the process as indexed sets of successive approximations, converging in the limit as the basis set is incrementally expanded. The fundamental low-field ion mobility equation or a near relative appears as the first approximation for several natural choices of f(0), and when it appears in the character of a starting point for a convergent series it may be exhibited with a subscript, say [K ]1, indicating “first approximation”. But, while numeric agreement with experiment may ultimately be achieved, the numerical integrations do not yield any closed-form symbolic expressions for the moments or the correlation factor, ζ. What we seek here is a proper derivation of eq 2a/2b as well as a more general symbolic expression that can correlate experiments and computations over a wide range of E/N and T.

speed; that is, vd > vT. In such a case vd is larger than the average velocity change during collision, and there are correspondingly few instances in Figure 3 of collisions having negative vz′ or of collisions with vz′ > vz. In low-field cases such collisions would become much more prevalent, and as E → 0 the proportion of these collisions approaches 50% and the ion path becomes nearly indistinguishable from a random walk. The collisions in Figure 3 may be related to the classes of Figure 2. Cooling collisions move the ion’s velocity toward v = 0 either from v > 0, [+,+], or from v < 0, [−,−]; heating collisions move the ion’s velocity away from v = 0, either from a positive value to a more positive value, [−,+], or from a negative value to a more negative value, [+,−]. At all field strengths, if M ≫m, the velocity excursions between collisions are much larger. If we generalize the velocity changes of Figure 3 to arbitrary fields and masses, it is clear that velocity changes due to collisions depend on m, M, E/N, and T but on modifications of the ion− neutral interactions. Finally, no generality is lost by restricting attention to z-velocity in Figure 3. Momentum is separately conserved in each of the three dimensions, and momentum acquired by ions from the field and transferred to neutrals enters the system in the z-dimension and must always remain in the z-dimension. Because an ion’s z-velocity always increases linearly with time, the average z-velocity between the kth and the (k + 1)th collisions is v = 1/2(v′zk + vz(k+1)), and the average ion velocity, vd, over all 5 collisions must be as follows:

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MOMENTUM-TRANSFER DERIVATION OF THE FUNDAMENTAL ION MOBILITY EQUATION To derive eq 2a/2b from eq 17, it only remains to calculate the average momentum loss in a collision, since we already have eq 10 for average relative velocity and eq 13 for average cross section. An ion moving under the influence of E, represented by the jagged line of Figure 3, does not accumulate momentum because it experiences collisions with neutrals (105−108 collisions for typical mobility experiments). Momentum is steadily acquired and transferred to neutrals at the rate eE. Collisions are instantaneous events in Figure 3, as ion velocity abruptly changes from a precollision value vz to a postcollision value vz′ and then smoothly increases at a rate of eE/m until the next collision. For a strong graphic presentation, Figure 3 depicts the case of a massive ion, m ≫ M, in a high field, where ion drift velocity is large in comparison to the relative thermal

vd =

1 ⎛⎜ 1 2 ⎜⎝ 5

5

∑ [v′zk k=0

⎞ 1 + vz(k + 1)]⎟⎟ = [v′zk + vz(k + 1) ] 2 ⎠ (18)

If we ignore transient effects such as the initial run up to drift velocity when E is switched on, we may imagine that the first and last velocities of the ion’s path do not significantly affect the average in eq 18 and write the following instead: vd =

1 (vz + vz′) 2

(19)

The explanatory power of this simple statement seems to have gone unnoticed. It is the momentum-transfer counterpart to the kinetic-theory invariance of ion velocity distribution to the opposing effects of field acceleration and collision. It tells us that we may think of an average velocity formed from the starting and ending velocities of a single collision, and to suppose 9788



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the mobility experiment equivalent to the result of an average collision repeated 5 times, as sketched in the sawtooth curve of Figure 3. The momentum transferred in a collision depends on both the precollision relative velocity and the scattering angle. Averaging over scattering angles will allow us to express vz′ in terms of vz, while eq 19 will give us a way to eliminate vz in favor of the experimentally determined vd. However, the relationship connecting vz′ and vz with vd is different for the cooling and heating collisions, a fact we acknowledge ahead of time by separating eq 19 into contributions from the two collision types: vd =

1 [⟨vz + vz′⟩c fc + ⟨vz + vz′⟩h fh ] 2

α=

ΩD =

̂ rz h = M̂ (vz h − Vz h ) vz h′ = Mv

(21b)

1 [(1 + m̂ )vzc fc + (1 + M̂ )vz h fh 2 ̂ z h f )] − (mV ̂ zc f + MV c

h

1 3 [1 + mf̂ c + Mf̂ h ]vz = αvz 2 4

(25)

DISCUSSION The preceding arguments are the first time that the fundamental low-field ion mobility equation has been derived by reasoning upward from a single collision rather than downward from the ion velocity distribution. But it is more significant that we have arrived at eq 2b via a general expression, eq 25, that can be tested over a range of field strengths for a variety of approximate or exact ion neutral interaction potentials. The potential value of eq 25 arises from the prevalence of analytical IMS and IMS/MS measurements made on the borderline between the low- and high-field regimes, coupled with the need for accurate average cross sections, comparable across a range of instruments and operating conditions. Such practical implementation of eq 25 requires computation or estimation of the momentum-transfer coefficient α of eqs 23 and 24, the transverse velocity coefficient β and the form of eq 10, and the correlation coefficient ζ of eq 17. These matters will be the subjects of a subsequent paper, now in preparation, but it is possible to make some general observations about methods and expected results here. Momentum-Transfer Coefficient α. The main novelty of our theory is the recognition of the mass and field dependence of momentum transfer in IMS and its explicit incorporation into the fundamental ion mobility equation through the α coefficient. In the absence of an electric field there is no preferred direction for ion motion, heating and cooling collisions are of equal frequency, there is no net flow of either momentum or energy between ion and neutral, and α is equal to 1 for any ion and neutral mass. Even a very weak electric field introduces a preferred direction for ion motion, produces a small predominance of cooling collisions over heating collisions, leads to small net flows of momentum and energy from ion to neutral, and has α slightly greater than 1 if the ion mass is greater than the neutral mass, slightly less than 1 if the ion mass is less than the neutral mass, and equal to 1 if the ion and neutral masses are the same. As field strength increases, the cooling collisions become increasingly predominant, the net momentum and energy flows become stronger, and finally α approaches 2(1 + m̂ )/3 at very high field. If ion and neutral masses are equal, although cooling collisions still grow in dominance as E/N increases, α remains equal to 1 regardless of the field strength. It is only with quite unequal ion and neutral

(22)

By definition, vz = vzc fc + vz h fh , fc + f h = 1, and m̂ + M̂ = 1. Because the asymmetric contributions of the neutral distribution to the cooling and heating collisions have opposite signs, Vzc + Vz h ≅ 0, and we then expect the third term in eq 22 to be small. While it is always true that vzc > vz h , the broadness of the velocity distributions ensures the quantities will be close in value. If we assume equality of these average precollision velocities and neglect the minor contribution of the neutral velocity term, eq 22 becomes much simpler: vd =

3 α eE ζ 4 μvr vdN

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The neutral velocities appear in eq 21a/21b because the + and − high-velocity tails of the neutral distribution contribute differently to the four collision classes. Substitution of eq 21a/ 21b into eq 20 produces vd =

(24)

The vr in eq 25 is given by eq 10, or some other combination of vd and vT with a transverse velocity coefficient β that is yet to be evaluated. As E/N increases, cooling collisions are increasingly dominant, fc approaches 1, and α approaches 2/3[1 + m̂ ], with the high-field limit being 4/3 for a high-mass ion (m̂ → 1) and 2 /3 for a low-mass ion (m̂ → 0). On the other hand, in the zerofield limit fc = f h = 1/2, α approaches 1, vr = vT , and eq 25 becomes eq 2b, the low-field fundamental ion mobility equation.

Here fc and f h are the fractions of collisions in the cooling and heating classes, respectively, and the quantities in angle brackets are separate averages for the cooling and heating collisions. The relationships among the quantities in eq 20 are illustrated in Figure 4. Averaging over scattering angles, a straightforward geometric exercise, is presented in the Supporting Information, where the resulting angularly averaged [+,+] collision is shown in Figure S1. For all four classes of collisions, the average neutral scattering angle is 45° for hard spheres and the momentum transferred from ion to neutral is μvrz, provided we recognize vrz to be a positive number for [+,+] and [+,−] collisions and a negative number for [−,+] and [−,−] collisions. If it is assumed that real ions and neutrals have the same average scattering angle as hard spheres, then the average momentum transferred in a collision may be evaluated with the guidance of Figure 2. For every value of |vrz|, the average collision must lie in the [+,+] quadrant, so we may claim that the average momentum transferred in a collision is μvz . However, the postcollision velocities of the cooling and heating collisions have mass-reversed relationships to the respective precollision velocities: (21a)

2 4 ≤α≤ 3 3

Because α expresses the mass-mediated dependence of momentum transfer on the ion velocity distribution, we will refer to it as the “momentum-transfer coefficient.” Substituting eqs 13 and 23 into eq 17 and rearranging produces an expression for ΩD at any field strength:

(20)

vzc′ = mv̂ rzc = m̂ (vzc − Vzc )

2 [1 + mf̂ c + Mf̂ h ] 3

(23) 9789



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masses that the effect of α becomes dramatic in eq 25, reflecting the fact that high-field velocities of massive ions in a low-mass drift gas change little with collisions, while very low-mass ions in a massive drift gas are nearly brought to a halt with each collision at high field. Our intent is to compute α by explicitly counting the contributions of cooling and heating collisions to the vr distribution, through convoluting a neutral velocity distribution (always Maxwellian at T) with an approximate ion velocity distribution (for example, a Maxwellian at T or Teff displaced by vd). It is certainly possible to do this numerically, but it may finally be possible to write a formula for α versus vd/vT as a function of m̂ for such idealized velocity distributions. From another angle, the apparatus of accurate kinetic theory22−24 will be used with atomic ion−molecule systems over a wide range of E/N, using realistic ab initio potentials with both realistic and artificially selected masses, to assess the degree to which the α function varies with the interaction. Because the ion and neutral velocity distributions become narrower with increasing mass, it is clear from the outset that the threshold of high-field momentum transfer (α > 1 or α < 1) occurs at lower vd when m ≫M or m ≪ M, respectively, in accord with the predictions of eq 6a/6b. This same dependence of the width of the velocity distributions on mass leads us to expect the transition zone between low- and high-field conditions to be narrower when m and M are more disparate. Transverse Velocity Coefficient β and the Functional Form of vr . At very low fields vr is nearly equal to vT, and at very high fields it is nearly equal to vd, but at the intermediate fields of practical importance in IMS both vT and vd contribute significantly to vr . What functional form should be used to combine vT and vd to best represent vr ? Our provisional choice for momentum-transfer theory is the linear combination of eq 10, suggested by the simplest approximation one can imagine for the ion velocity distribution: a Maxwellian shifted by vd. On the other hand, the successful two-temperature theory8,10,11 replaces T in eq 2a/2b by Teff, which is itself given in terms of an ion temperature, Ti, defined by analogy to T from the average ion kinetic energy. The two-temperature theory thus combines vT and vd via a sum of squares, which causes the transition from low field (vT dominant) to high field (vd dominant) to occur over a narrower range of vd/vT than with the linear combination of eq 10. However, there is no α parameter in two-temperature theory. For the most common case of m > M, increases in both α and vr (as given by eq 10) would tend initially to cancel in eq 25. Thus the onset of highfield behavior in momentum-transfer theory could resemble the onset in two-temperature theory arising from the root-meansquare behavior of the average ion−neutral relative velocity, and this circumstance increases our confidence in eq 10 for momentum-transfer theory. We expect the transverse velocity coefficient in eq 10 to range from a minimum β = 1 for m ≫ M to a maximum β = √2 for M ≫ m. A massive ion in a low-mass drift gas is little deflected by collisions, so that the nonthermal transverse velocity will be very small. On the other hand, low-mass ions are strongly scattered by collisions with the massive drift gas, so its nonthermal transverse velocity could become as large as vd. Of course the nonthermal transverse velocity could not become larger than vd because the motion along the field axis is the source of the energy contained in the transverse motion. One avenue to obtaining β for intermediate cases would be to “tilt”

the angularly averaged collision of Figure S1 (Supporting Information) to obtain an arrangement that does not accumulate energy in the transverse motion, as occurs with the present form of the figure. Alternatively, we could approach the ion’s combined parallel and transverse velocities through the Wannier equation:5,10 3 3 1 1 kTi = kT + mvd 2 + Mvd 2 2 2 2 2

(28)

In this approximate relationship, which is the source of the field strength criteria of eq 6a/6b, the penultimate term on the righthand side is the ion energy explicitly contained in the drift motion, and the final term is energy obtained from the field that is contained in random motion, both transverse and parallel to the field. Correlation Coefficient ζ. The correlation coefficient is a quantity that all users of eq 2a/2b ignore and are thus tacitly combining with ΩD. As the strength of E diminishes, the momentum-transfer coefficient becomes unity, vr becomes vT, and the diffusion collision integral becomes indistinguishable from its zero-field value, but the correlation coefficient remains unequal to unity, except possibly by accident. To convert the approximate equation 2a/2b into a true equality, the right-hand side must be multiplied by ζ(E/N = 0). As vr increases, ΩD nearly always decreases, so ⟨vrΩD⟩ < ⟨vr⟩⟨ΩD⟩ and the contribution of this negative correlation tends to reduce ζ below 1. On the other hand the positive correlation between vd and vr tends to increase ζ above 1. The interrelationships among the moments ⟨vn⟩ as expressed in the actual ion velocity distribution also will affect ζ, but in ways we do not know how to predict. To estimate the size and sensitivity of ζ to instrument and analyte properties, we intend to use the fundamental ion mobility equation (eq 25) in conjunction with accurate kinetic theory22−24 applied to atomic ion−molecule systems. Residual differences between accurate computations and eq 25 will be used to assess both the correctness of the α and vr treatments and the behavior of the ζ coefficient.

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CONCLUSIONS There has long been imperfect understanding between practitioners of gas kinetics, who have supplied the theory for IMS, and analytical scientists, who need to use the theory in pioneering new applications of IMS/MS. Partly this is the result of little overlap between domains of current activity, since kinetic theorists can make very few accurate predictions for species of interest to analysts, while analysts are unexcited about the species for which kinetic theory can make predictions in exact accord with experiment. Also, kinetic theorists and analysts approach the tiny domain of overlap from opposite directions: analysts moving from experiment toward structure and kinetic theorists moving from structure toward measurable quantities. This difference in orientation can defeat understanding, even over matters that are of common concern and interest. For example, kinetic theorists who compute mobilities do not know the actual ion velocity distribution function, and in consequence must make use of an indexed set of irreducible collision integrals ΩD(l,s)similar to eqs 11 and 12 but with higher powers of cosl θ and εs+1 in the integrands. On the other hand, in an important sense analysts do not need to worry about the ion velocity distribution, because the ions manifest the distribution in the experimental results they produce. Thus the irreducible collision integrals are of no utility in interpreting 9790



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experiments, but the experimental definition of ΩD in eq 13 is useful to the analyst. Conversely, while a theorist can accept the definition in eq 13, it is of no use in computations. It has the form of ΩD(1,1), but it explicitly involves the unknown ion velocity distribution and so leads nowhere. Part of the value of IMS to chemical analysis comes from its ability to separate complex mixtures, like very high speed chromatography. But IMS also has the potential to provide hard quantitative information about size and shape, and to this end analysts are currently refining methods for extracting structural information from mobility spectra, for example, deducing conformations of gas-phase proteins,25 and building schemes for calibrating mobility spectra and relating results obtained on different instruments.26 Our message is that these important efforts are likely to founder when results are compared between research groups, unless the theoretical underpinnings explicitly include the changes that occur at the onset of high field, changes we are attempting to describe with the improved momentum-transfer theory. We have also approached these same issues with a new top-down method of truncating the coupled moment equations derived from the Boltzmann expression.27 The practical application of our ideas would necessitate few adjustments by analytical scientists, since measurements are frequently performed in sets within narrow ranges of T, P, E, and even of M and m, so that the net corrections of eq 25 compared to eq 2a/2b will be fairly constant within the set. But we remain aware that optimal exploitation of IMS for analysis requires further work on both sides to enlarge the domain of overlap between kinetic theory and chemical analysis.

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(8) Viehland, L. A.; Mason, E. A. Ann. Phys. (N. Y., NY, U. S.) 1975, 91, 499−533. (9) Chapman, S.; Cowling, T. G. The Mathematical Theory of NonUniform Gases, 3rd ed.; Cambridge University Press: London, 1970. (10) Mason, E. A.; McDaniel, E. W. Transport Properties of Ions in Gases; Wiley: New York, 1988. (11) Viehland, L. A.; Mason, E. A. Ann. Phys. (N. Y., NY, U. S.) 1978, 110, 278−328. (12) Present, R. D. Kinetic Theory of Gases; McGraw-Hill: New York, 1958. (13) Yousef, A.; Shrestha, S.; Viehland, L. A.; Lee, E. F. P.; Gray, B. R.; Ayles, V. L.; Wright, T. G.; Breckenridge, W. H. J. Chem. Phys. 2007, 127, No. 154309. (14) Ruotolo, B. T.; Giles, K.; Campuzano, I.; Sandercock, A. M.; Bateman, R. H.; Robinson, C. V. Science 2005, 310, 1658−1661. (15) Counterman, A. E.; Valentine, S. J.; Srebalus, C. A.; Henderson, S. C.; Hoagland, C. S.; Clemmer, D. E. J. Am. Soc. Mass Spectrom. 1998, 743−759. (16) Kemper, P. R.; Dupuis, N. F.; Bowers, M. T. Int. J. Mass Spectrom. 2009, 287, 46−57. (17) Revercomb, H. E.; Mason, E. A. Anal. Chem. 1975, 47, 970− 983. (18) Shvartsburg, A. A. Differential Ion Mobility Spectrometry; CRC Press: Boca Raton, FL, 2009. (19) Gardner, A. M.; Gutsiedl, K. A.; Wright, T. G; Breckenridge, W. H.; Chapman, C. Y. N.; Viehland, L. A. J. Chem. Phys. 2010, 133, No. 164302. (20) White, R. D.; Ness, K. F.; Robson, R. E.; Li, B. Phys. Rev. E 1999, 60, 2231. (21) Viehland, L. A.; Chang, Y. Mol. Phys. 2012, 110, 259−266. (22) Viehland, L. A. Chem. Phys. 1982, 70, 8. (23) Viehland, L. A. Chem. Phys. 1984, 85, 291−305. (24) Viehland, L. A.; Chang, Y.-B. Comput. Phys. Commun. 2010, 181, 1687−1696. (25) Bleiholder, C.; Wyttenbach, T.; Bowers, M. T. Int. J. Mass Spectrom. 2011, 308, 1−10. (26) Bush, M. F.; Campuzano, I. D. G.; Robinson, C. V. Anal. Chem. 2012, 84 (16), 7124−7130. (27) Viehland, L. A.; Siems, W. F. J. Am. Soc. Mass Spectrom. 2012, DOI: 10.1007/s13361-012-0450-7.

ASSOCIATED CONTENT

S Supporting Information *

Additional text and equations and one figure, describing averaging over scattered angles. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Fax: 509-335-8867. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Portions of this work were supported at Washington State University by contract P010058810 from Science Applications International Corporation. Parts of this work were presented at meetings of the International Society for Ion Mobility Spectrometry (ISIMS), at Albuquerque, NM in 2010 and Edinburgh, Scotland in 2011, and at the meeting of the American Society for Mass Spectrometry (ASMS) at Vancouver, BC in 2012. We are grateful for discussions with Richard Knochenmuss, Kevin Giles, and C. Steven Harden.

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REFERENCES

(1) Langevin, M. P. Ann. Chim. Phys. 1905, 8, 245−288. (2) Hasse, H. R. Philos. Mag. 1926, 1, 139−160. (3) Hasse, H. R.; Cook, W. R. Philos. Mag. 1931, 12, 554−566. (4) Hershey, A. V. Phys. Rev. 1939, 56, 916−922. (5) Wannier, G. H. Bell Syst. Tech. J. 1953, 32, 170−254. (6) Kihara, T. Rev. Mod. Phys. 1953, 25, 844−852. (7) Mason, E. A.; H. W. Schamp, J. Ann. Phys. (N. Y., NY, U. S.) 1958, 4, 233−270. 9791


Improved Momentum-Transfer Theory for Ion Mobility. 1. Derivation of

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