Modeling collisional kinetic energy damping, heating, and cooling of ions in mass spectrometers: a tutorial perspective

Abstract

Many powerful methods in mass spectrometry rely on activation of ions by high-energy collisions with gas particles. For example, multiple Collision Induced Dissociation (CID) has been used for many years to determine structural information for ions ranging from small organics to large, native-like protein complexes. More recently, Collision Induced Unfolding (CIU) has proved to be a very powerful method for understanding high-order protein structure and detecting differences between similar proteins. Quantifying the thermochemistry underlying dissociation/unfolding in these experiments can be quite challenging without reliable models of ion heating and cooling. Established physical models of CID are valuable in predicting ion heating but do not explicitly include mechanisms for cooling, which may play a large part in CID/CIU in modern instruments. Ab initio and Molecular Dynamics methods are extremely computationally expensive for modeling CID/CIU of large analytes such as biomolecular ions. In this tutorial perspective, limiting behaviors of ion kinetic energy damping, heating, and cooling set by “extreme” cases are explored, and an Improved Impulsive Collision Theory and associated software (“Ion Simulations of the Physics of Activation”, IonSPA) are introduced that can model all of these for partially inelastic collisions. Finally, examples of modeled collisional activation of native-like protein ions under realistic experimental conditions are discussed, with an outlook toward the use of IonSPA in accessing the thermochemical information hidden in CID breakdown curves and CIU fingerprints.

Graphical Abstract

1. Introduction

Collisional activation of ions in the gas phase is one of the oldest and most commonly used means to study ion fragmentation, often providing rich information as to ion composition and structure.[1–3] This method is known as both “Collision Induced Dissociation”, CID, and “Collisionally Activated Dissociation”, CAD. (We shall use CID here, which seems to be more common nowadays. Ref. 1 by Wells and McLuckey provides a particularly excellent overview of different types of CID instrumentation and their merits.) Collisional activation is also the basis of the emerging Collision Induced Unfolding (CIU) technique, in which native protein and protein complex ions are activated in the gas phase with progressively greater energy to induce them to unfold.[4, 5] Plotting the extent of ion unfolding (in the form of a drift time or collisional cross section distribution) as a function of laboratory-frame collision voltage results in a CIU “fingerprint” that can be used to characterize and compare their native folded structures.[6–8] Generally, collisional activation can be achieved with a wide variety of instrumentation, including simple drift tubes filled with buffer gas and having a constant electric field;[9] traveling wave ion guides;[10–13] linear multipole,[14, 15] Penning[16] or Paul[17] traps; and in the transition between atmosphere and vacuum at the entrance to the mass spectrometer,[18–20] among other options. CID is one of the most important tools in the “omics” fields, and some form of user-adjustable collisional activation is available on the vast majority of commercial mass spectrometry platforms with ion fragmentation capabilities.

Most types of CID instrumentation use buffer gas pressures of a few microbar (a few millitorr), whereas ion mobility spectrometry (IMS) experiments, which also rely on many successive ion-gas collisions, are typically conducted at pressures two or more orders of magnitude higher.[21] These experiments can even be paired together in modern IM-MS instruments to elucidate ion structure as a function of size/shape (via their collision cross section, CCS[22, 23]) or determine the size/shape of fragments, depending on the order in which the experiments are conducted inside the instrument.[21, 24, 25] Despite the extremely high versatility and importance of ion-gas collision instrumentation and experiments in modern mass spectrometry, a unified theory for large biomolecular ion heating, cooling, and motion spanning the range from CID/CIU conditions to those of IMS and covering a broad range of activation potentials has remained elusive. Nevertheless, such a theory would be invaluable in a number of applications, including the following:

1. Determining biopolymer structural information from CID/CIU barrier thermochemistry. Depending on the internal temperature of the ion, the temperature and dynamics of the buffer gas, and the kinetic energy of the ion, inelastic collisions between ion and buffer gas can result in either heating or cooling of the ion. In principle, if the internal energy distribution (or a surrogate for it, such as a microcanonical or “effective” vibrational temperature) of the ion as a function of the experimental parameters (i.e., any electric potentials used to accelerate the ion) is known, kinetic modeling of the observed degree of fragmentation (for CID) or unfolding (CIU) can be used to determine thermochemical values for these processes. Indeed, determining barrier enthalpies and entropies (or Arrhenius activation energies and pre-exponential factors[26–28]) for CID/CIU pathways not only enables one to predict and even control how these processes will evolve under a wide variety of experimental conditions but could also provide insight into the mechanisms of different pathways. For example, a large positive entropy of activation may indicate a sudden, concerted opening of a structure into a much “floppier” one; and a large negative entropy of activation may indicate a “tight”, concerted rearrangement of covalent bonds at the rate-limiting step. Optimistically, one could imagine eventually interpreting unfolding transitions in CIU fingerprints in terms of the protein ion substructures involved, such as helical regions or salt-bridged interfaces, based on both the change in CCS and the barrier enthalpy and entropy of the transition.

2. Direct comparison of experiments on different instruments and between laboratories. Understanding differences in how ions heat and cool in different instruments and with different experimental parameters can be vital for determining the consistency of apparently disparate results obtained across laboratories.[20, 29] For example, the relationship between the collision voltage applied to ion optics to accelerate the ions and the maximum temperature an ion reaches (and how long it stays at that temperature) may be different in drift-tube type CIU experiments versus traveling-wave or in-source activation experiments.[20] This may have important implications for interpreting and differentiating CIU fingerprints across platforms[29] and between labs.[20]

3. Calibrating other types of ion dissociation/unfolding experiments. Having a firm basis for quantitatively understanding CID/CIU may be useful for calibrating[12] Surface Induced Dissociation/Unfolding (SID/SIU) experiments,[30–32] in which ions are accelerated into a metallic surface (sometimes coated with a monolayer of thioalkane or polyfluoroalkane) with an adjustable kinetic energy.[33] SID most often tends to fragment native protein complex ions preferentially along the smallest protein-protein interfaces and yield compact products, whereas CID tends to unfold and eject the smallest protein at the surface of a complex as a high-charge, unfolded product.[30–32] In SID/SIU experiments, the precise efficiency of energy transfer between the surface and ion is very difficult to determine,[12, 34–37] and comparison to CID/CIU may elucidate further differences between the two techniques and provide valuable data for molecular dynamics modeling of the SID/SIU mechanism.

4. Instrument design. Modeling the damping of the ion’s initial (10’s to low 1000’s of eV) kinetic energy, and, in particular, determining the typical time and distance it takes it to slow down as a function of ion, gas, and electric field properties, is valuable for instrument design.[1] For example, in Electron Capture/Transfer Dissociation (ExD) experiments, or experiments in which a laser is used to excite ions, it may be desirable to overlap a kinetically cooled ion beam with the electron donor population[38] or combine collisional activation with laser photon excitation[39] for maximal dissociation efficiency.

Detailed modeling of ion collisional dissociation and experimental determination of ion dissociation barrier thermochemistry, especially for small ions, has a long history.[3, 12, 40–54] Of those theoretical treatments that focus on CID, most focus on the change in kinetic energy and internal energy of an ion in a single collision. In the Impulsive Collision Theory (ICT) of CID introduced by Mahan[43] and developed further by Uggerud and Derrick[41] and Douglas,[47] small organic ions, such as oligopeptides, in kiloelectronvolt CID were coarse-grained as a collection of beads or “pseudo-atoms” to make the collision kinematics tractable with relatively simple calculations. In the ICT, collisions are pictured as locally elastic, head-on, hard-spheres collisions between the buffer gas particle and a single pseudo-atom (which is at rest with respect to the ion’s center of mass), resulting in a change in momentum of the pseudo-atom. This momentum change propagates to both a change in momentum of the entire ion and a change in its internal energy (which is assumed to occur entirely due to the change of kinetic energy of the pseudo-atom, that is, vibrational energy, in the rest frame of the ion). By averaging over all possible impact parameters and collision velocities, the average efficiency of energy transfer from the center-of-mass collision energy of the pseudo-atom and gas particle to internal energy of the ion can be computed. Given the gas mass $\left(m_g\right)$ and an initial guess of pseudo-atom mass $\left(m_a\right)$, the average efficiency $\left({\chi }_{ag}\right)$ for a single collision can be straightforwardly predicted via the following equation:

$${\chi }_{ag}=4\frac{m_a{m}_g}{{\left(m_a+m_g\right)}^2}$$ (1)

This is the fraction of the initial kinetic energy of the gas particle and pseudo-atom (in their center-of-mass frame) that is transferred to new kinetic energy of the pseudo-atom. Notably, the original ICT model starts with the ion having effectively zero vibrational energy, and there is no mechanism within the model whereby the ion can transfer internal energy into kinetic energy of the gas particle to cool off. Within these limitations, Uggerud and Derrick performed molecular dynamics simulations of kiloelectronvolt collisions to relate internal energy change to collision energy and determined average pseudo-atom masses for a 682 Da polystyrene ion of 7, 32, and 49 Da for He, Ar, and Xe buffer gases.[41] These correspond to ${\chi }_{ag}$ of 23%, 25%, and 20%, respectively. However, they also noted that efficiency may be different for different ions and not be monotonic as a function of ion mass. For much lower collision energies, i.e., a few 10’s of eV, Hase determined (using high-level ab initio computations) a collisional energy transfer efficiency of up to ~76% for collisions of Ar with small molecules intended to represent peptide functional groups but concurred with Uggerud and Derrick that efficiency tends to increase with gas particle mass.[55] Uggerud and Derrick also noted that knowledge of ions’ trajectories as they traverse the collision region (and the rest of the instrument), as well as the nature of their internal energy distributions when they enter the collision region, are likely necessary for a complete picture of ion heating.[41] Furthermore, in modern commercial instruments with CID/CIU available for protein-sized ions, hundreds or thousands of collisions between the ion and gas particles can take place with much greater retention of the ion population than in studies of small molecule-ion collisions in earlier high-energy CID instruments, thus modeling of ion behavior of much longer (~millisecond) timescales has become more relevant.

Still other challenges exist for modeling ion heating, cooling, and kinetic energy damping in CID/CIU for large biomolecular ions and complexes. While modern ion trajectory simulation software, such as SIMION^®[56] and COMSOL Multiphysics^®,[57] can model ion motion in electric and magnetic fields (including dynamic ones), SIMION^® currently offers the option to model collisions with gas particles only as hard spheres (i.e., elastic, with no possibility to heat or cool ions internally), and COMSOL Multiphysics^® offers both hard-spheres and inelastic collisions, but the user must specific the kinetic energy loss for the collision. Much more sophisticated RRKM modeling and master equation kinetic modeling have been used to study dissociation barrier energetics for computationally tractable ions, such as small peptides and cluster ions, in Guided Ion Beam Mass Spectrometry,[45] Blackbody Infrared Radiative Dissociation,[44, 58] CID,[49] SID,[3, 59] and other types of activation. However, this level of modeling can be highly demanding of computational resources.

Inspired by these challenges, we present an Improved Impulsive Collision Theory (IICT) that aims to account for heating and cooling of large biomolecular ions via collisions in a way that can be easily mapped onto ion trajectories as they travel through the mass spectrometer. This model includes an explicit mechanism for ion cooling by super-elastic collisions that is absent in existing ICTs but likely plays a large role in the behavior of ions in many instruments available currently. At its heart a Monte Carlo method, in which collision parameters are sampled randomly with each successive collision, the IICT is also amenable to both finite difference and continuous (i.e., analytical) approximations that are much faster to compute. For the prototypical example of an ion with tens to thousands of eV of initial translation energy injected into a tube of thermal gas, a variety of useful quantities are estimated from the analytical form of the model, including kinetic energy damping constants, heating and cooling rate constants, and the number of collision required for the ion to achieve its maximum vibrational energy. The dependence of these quantities on pseudo-atom mass, translational-to-vibrational energy transfer efficiency, gas temperature, ion heat capacity, and other parameters are also examined. To our knowledge, this is the first model that provides reasonable agreement under both CID/CIU and typical drift-tube ion mobility spectrometry conditions without the need for explicit collision modeling by atomistic Molecular Dynamics simulation or calculation of an interaction potential.

2. Theory

The IICT predicts ion heating, cooling, and translational energy damping via an inelastic scattering process, thus it is useful to compare these results to those from simpler, elastic scattering models (specifically, one in which the impact parameter is assumed to be completely free and one in which collisions are restricted to being head-on) and a totally inelastic scattering model. Because real collisions likely fall somewhere in between the elastic, free impact parameter model and the totally inelastic model, these two models should provide reasonable bound estimates for realistic collisional activation behavior. Note thatinternal energy is not explicitly partitioned here into vibrational and rotational energy, as rotational energy is unlikely to play a large role in CID/CIU of large ions (which can have many hundreds or thousands of vibrational degrees of freedom and only three rotational degrees of freedom),[55] and rotational energy can also exchange in and out of hindered internal rotations. It is thus assumed that rotational energy rapidly equilibrates to the kinetic temperature of the buffer gas, and vibrational energy (which is much more important in CID/CIU) evolves much more slowly. “Molecular chaos” is also assumed, that is, collisions are separated enough in space and time that gas particle velocities for each collision are uncorrelated. These are reasonable assumptions under typical CID/CIU conditions in modern experiments, where individual (center-of-mass frame) collision energies do not exceed a few eV, the mean free path is much greater than the ion diameter, and collisions are typically many nanoseconds apart. Cooling effects of photon emission are also ignored, as dissociation/unfolding lifetimes in modern multiple-collision CID/CIU experiments are typically on the upper microsecond timescale and are much faster than BIRD lifetimes (many milliseconds to seconds for protein-sized ions at a few hundred Kelvin above room temperature).

Numerous times throughout the following physics derivations, concepts and results from introductory mechanics and statistical mechanics are invoked. The four most important of these, which the reader may wish to keep in mind, are the following: 1) Many physical quantities are easier to compute in one frame of reference (say, that in which in which the ion is stationary, rather than the laboratory frame), but results in different reference frames can be easily related by adding/subtracting the velocity of one frame relative to the other (a Galilean transformation). 2) in the center of mass frame for two rigid objects colliding, they leave the collision with precisely the opposite velocities (and momenta) with which they entered the collision. 3) If a random particle (e.g., a gas particle) has a velocity sampled from an isotropic distribution (here, often a Maxwell-Boltzmann distribution) in a particular frame of reference, the statistically averaged dot product of its velocity with any other uncorrelated velocity (such as that of an ion moving through a gas-filled collision chamber with a known velocity) is zero in that frame of reference. 4) A basic result from the Equipartition Theorem of statistical mechanics is that an ideal (Maxwell-Boltzmann) gas at temperature $T_g$ has an average kinetic energy of $\frac{1}{2}{m}_g{v}_{g,j}^2=\frac{1}{2}{k}_B{T}_g$ in each Cartesian direction $j$, for a total average kinetic energy of $\frac{3}{2}{k}_B{T}_g$. From this result, its root-mean-square speed is easily determined to be $v_{g,rms}=\sqrt{3k_B{T}_g/m_g}$. Statistical mechanics tells us that this result also holds for atoms in solids and liquids if they are large enough to have a Boltzmann-like distribution of vibrational state energies. For a more detailed derivation of these fundamental results, the reader is referred to Chandler’s Introduction to Modern Statistical Mechanics.[60] Throughout the following derivations, important mathematical results are restated in bold, italicized “key points” to help guide the unfamiliar reader.

2.1. Elastic hard-spheres scattering models.

Likely the simplest explicit model for translational energy damping in CID is that in which the ion and gas are assumed to be hard spheres, with the gas well-described by a Maxwell-Boltzmann velocity distribution at temperature $T_g$. Suppose the ion and gas have mass $m_i$ and $m_g$, respectively, and the ion has been accelerated initially to kinetic energy $K{E}_i^L=zeV$ ($z$ is the charge state of the ion, $e$ is the charge of an electron, and $V$ is the injection potential, and the superscript $L$ indicates the lab frame). Let us further assume the ion is initially moving along the optical axis of the instrument at velocity ${\overrightarrow{v}}_i^L(t=0)$, with $v_i^L\equiv \left|{\overrightarrow{v}}_i^L\right|={\left(2zeV/m_i\right)}^{1/2}$. Let the gas particle with which it initially collides have velocity ${\overrightarrow{v}}_g^L$. Define the relative velocity vector ${\overrightarrow{v}}_{rel}^L\equiv {\overrightarrow{v}}_i^L-{\overrightarrow{v}}_g^L$, and let $\hat{n}$ be a unit vector pointing along the line from the center of the ion to the center of the gas particle. A straightforward result from classical mechanics, based on conservation of linear momentum and total kinetic energy, is that

$${\overrightarrow{v}}_i^{L^{\prime }}={\overrightarrow{v}}_i^L-2\frac{m_g}{m_g+m_i}\left({\overrightarrow{v}}_{rel}^L\cdot \hat{n}\right)\hat{n}={\overrightarrow{v}}_i^L-2\frac{m_g}{m_g+m_i}{v}_{rel}\left(\text{cos}\theta \right)\hat{n}$$ (2)

where $\theta$ is the angle between ${\overrightarrow{v}}_{rel}^L$ and $\hat{n}$, which must be between 0 and $\pi /2$ for a collision to occur. Key point: The above equation means the collision is “specular”, or mirror-like. in the center-of-mass frame, the ion and gas leave the collision with velocities in which their components along $\widehat{\mathit{n}}$ are exactly reversed and components perpendicular to $\widehat{\mathit{n}}$ are unaltered. (We use the prime symbol, ′, to indicate post-collision values throughout this work.) Figure 1a illustrates the geometry of the collision in the center-of-mass frame of the ion and gas particle.

Figure 1.

Schematic illustration of collision geometries and temperature changes in the five models described in the text. Color range from blue to red indicates relatively cold to warm temperatures, respectively. Collisions resulting in ion heating are shown for the totally inelastic and original ICT models, whereas an example of a collision resulting in ion cooling is shown for the IICT model.

In this single collision, we see that the change in kinetic energy of the ion in the lab frame is

$$\begin{gathered} \Delta K{E}_i^L=\frac{1}{2}{m}_i{\left(v_i^{L^{\prime }}\right)}^2-\frac{1}{2}{m}_i{\left(v_i^L\right)}^2 \\ =\frac{1}{2}{m}_i{\left(2\frac{m_g}{m_g+m_i}{v}_{rel}\left(\text{cos}\theta \right)\hat{n}\right)}^2-2\frac{m_g{m}_i}{m_g+m_i}{\overrightarrow{v}}_i^L\cdot \left(v_{rel}\left(\text{cos}\theta \right)\hat{n}\right) \\ =\frac{1}{2}{m}_g\left(\frac{4m_i{m}_g}{{\left(m_g+m_i\right)}^2}\right)v_{rel}^2{\text{cos}}^2\theta -2\frac{m_g{m}_i}{m_g+m_i}{\overrightarrow{v}}_i^L\cdot \left(v_{rel}\left(\text{cos}\theta \right)\hat{n}\right) \end{gathered}$$ (3)

This expression is exact thus far. Let us see its value when averaged (indicated by angle brackets) over the Maxwell-Boltzmann distribution of ${\overrightarrow{v}}_g^L$ and an isotropic distribution of $\hat{n}$ (over a half-sphere). When the lab-frame velocity of the ion is still relatively high, ${\overrightarrow{v}}_{rel}^L\approx {\overrightarrow{v}}_i^L$, and the second term on the right hand side of the above equation is approximately

$$-2\frac{m_g{m}_i}{m_g+m_i}{\overrightarrow{v}}_i^L\cdot \left(v_i^L\left(\text{cos}\theta \right)\hat{n}\right)=-2\frac{m_g{m}_i}{m_g+m_i}{\left(v_i^L\right)}^2{\text{cos}}^2\theta$$ (4)

Whence

$$\begin{gathered} \Delta K{E}_i^L\approx \frac{1}{2}{m}_g\left(\frac{4m_i{m}_g}{{\left(m_g+m_i\right)}^2}\right)v_{rel}^2{\text{cos}}^2\theta -2\frac{m_g{m}_i}{m_g+m_i}{\left(v_i^L\right)}^2{\text{cos}}^2\theta \\ =\frac{1}{2}{m}_g\left(\frac{4m_i{m}_g}{{\left(m_g+m_i\right)}^2}\right)v_{rel}^2{\text{cos}}^2\theta -\frac{1}{2}\left(\frac{4m_i{m}_g}{{\left(m_g+m_i\right)}^2}\right)\left(m_g+m_i\right){\left(v_i^L\right)}^2{\text{cos}}^2\theta \end{gathered}$$ (5)

Noting that the average value of ${\text{cos}}^2\theta$ over all possible values of $\theta$ on a half-circle (two-body collisions are confined to a plane) is just $\frac{1}{3}$, let us also define the unitless quantity ${\chi }_{ig}=4m_i{m}_g{\left(m_g+m_i\right)}^{-2}$. (${\chi }_{ig}$ is maximal, with a value of 1, when $m_i=m_g$, and it is very small when $m_i\gg m_g$.) Then we have

$$\begin{gathered} ⟨\Delta K{E}_i^L⟩\approx \frac{1}{3}*\frac{1}{2}{m}_g\left(\frac{4m_i{m}_g}{{\left(m_g+m_i\right)}^2}\right)⟨v_{rel}^2⟩-\frac{1}{3}*\frac{1}{2}\left(\frac{4m_i{m}_g}{{\left(m_g+m_i\right)}^2}\right)\left(m_g+m_i\right)⟨{\left(v_i^L\right)}^2⟩ \\ =\frac{1}{6}{m}_g{\chi }_{ig}\left(⟨{\left(v_i^L\right)}^2⟩+⟨{\left(v_g^L\right)}^2⟩-2⟨{\overrightarrow{v}}_i^L\cdot {\overrightarrow{v}}_g^L⟩\right)-\frac{1}{6}\left(m_g+m_i\right){\chi }_{ig}⟨{\left(v_i^L\right)}^2⟩ \\ =-\frac{1}{6}{m}_i{\chi }_{ig}\left(⟨{\left(v_i^L\right)}^2⟩\right)+\frac{1}{6}{m}_g{\chi }_{ig}{\left(v_g^L\right)}^2 \end{gathered}$$ (6)

because ${\overrightarrow{v}}_g^L$ is isotropically distributed in the lab frame. Invoking the Maxwell-Boltzmann distribution of the gas kinetic energy to compute the average of ${\left(v_g^L\right)}^2$, we finally have:

$$⟨\Delta K{E}_i^L⟩\approx -\frac{1}{6}{m}_i{\chi }_{ig}\left(\frac{2K{E}_i^L}{m_i}\right)+\frac{1}{6}{m}_g{\chi }_{ig}\left(\frac{3k_B{T}_g}{m_g}\right)=-\frac{1}{3}{\chi }_{ig}K{E}_i^L+\frac{1}{2}{\chi }_{ig}{k}_B{T}_g$$ (7)

Key point: on average, the ion tends to lose kinetic energy in a collision proportional to the energy with which it entered the collision, but it also tends to gain a small amount back in the form of thermal kinetic energy from the gas. Intuitively, the first term on the right-hand side is ~1/3 of the ion’s kinetic energy in the lab frame, scaled down by ${\chi }_{ig}$. By treating this as a differential equation with respect to collision number, $Z$, that is,

$$\frac{dK{E}_i^L\left(Z\right)}{dZ}=-\frac{1}{3}{\chi }_{ig}K{E}^L\left(Z\right)+\frac{1}{2}{\chi }_{ig}{k}_B{T}_g$$ (8)

we can see that, at least initially (when $v_i^L\gg v_g^L$), the solution is approximately

$$K{E}_i^L\left(Z\right)=\left(K{E}_i^L\left(0\right)-\frac{3}{2}{k}_B{T}_g\right)e^{-\frac{{\chi }_{ig}Z}{3}}+\frac{3}{2}{k}_B{T}_g$$ (9)

Which has a “lifetime” (with respect to $Z$) of ${\zeta }_{\mathit{\text{elastic}}}^{\mathit{\text{free}}\mathit{\text{impact}}\mathit{\text{parameter}}}=\frac{3}{{\chi }_{ig}}\approx \frac{3m_i}{4m_g}$. It should be emphasized that this result becomes less accurate as the kinetic energy of the ion decreases to the point where $v_i^L$ becomes comparable to $v_g^L$, so it is most useful in understanding how ions behave immediately after injection into the collision cell during CID/CIU.

If we instead imagine that all collisions are constrained to be head-on (i.e., the impact parameter is always zero; see Figure 1b), then $\text{cos}\theta =1$, and

$$\begin{gathered} \Delta K{E}_i^L=\frac{1}{2}{m}_i{\left(v_i^{L^{\prime }}\right)}^2-\frac{1}{2}{m}_i{\left(v_i^L\right)}^2=\frac{1}{2}{m}_i{\left(2\frac{m_g}{m_g+m_i}{v}_{rel}\right)}^2-2\frac{m_g{m}_i}{m_g+m_i}{\overrightarrow{v}}_i^L\cdot {\overrightarrow{v}}_{rel} \\ =\frac{1}{2}{m}_i{\left(2\frac{m_g}{m_g+m_i}\left({\overrightarrow{v}}_i^L-{\overrightarrow{v}}_g^L\right)\right)}^2-2\frac{m_g{m}_i}{m_g+m_i}{\overrightarrow{v}}_i^L\cdot \left({\overrightarrow{v}}_i^L-{\overrightarrow{v}}_g^L\right) \end{gathered}$$ (10)

Invoking the Maxwell-Boltzmann distribution of ${\overrightarrow{v}}_g^L$, this averages to

$$\begin{gathered} \frac{1}{2}{m}_i{\left(2\frac{m_g}{m_g+m_i}\right)}^2\left({\left(v_i^L\right)}^2+{\left(v_g^L\right)}^2\right)-2\frac{m_g{m}_i}{m_g+m_i}{\left(v_i^L\right)}^2 \\ =\frac{1}{2}{m}_g{\chi }_{ig}\left(2\frac{K{E}_i^L}{m_i}+3\frac{k_B{T}_g}{m_g}\right)-\frac{1}{2}{\chi }_{ig}\left(m_g+m_i\right)\left(2\frac{K{E}_i^L}{m_i}\right)=-{\chi }_{ig}K{E}_i^L+\frac{3}{2}{\chi }_{ig}{k}_B{T}_g \end{gathered}$$ (11)

Note that, unlike Eq. 9, this result is still valid when the ion is moving much slower than the gas particle in the lab frame. Treating this result as a differential equation with respect to collision number, $Z$, we have

$$\frac{dK{E}_i^L\left(Z\right)}{dZ}=-{\chi }_{ig}K{E}_i^L\left(Z\right)+\frac{3}{2}{\chi }_{ig}{k}_B{T}_g$$ (12)

for which the solution is readily obtained as:

$$K{E}_i^L\left(Z\right)=\left(K{E}_i^L\left(0\right)-\frac{3}{2}{k}_B{T}_g\right)e^{-{\chi }_{ig}Z}+\frac{3}{2}{k}_B{T}_g$$ (13)

This head-on elastic hard-spheres solution clearly has “lifetime” (with respect to collision number) ${\zeta }_{\mathit{\text{elastic}}}^{\mathit{\text{head-on}}}=\frac{1}{{\chi }_{ig}}\approx \frac{m_i}{4m_g}$ and decays to an asymptotic kinetic energy of $\frac{3}{2}{k}_B{T}_g$, the thermal kinetic energy of the gas, which is reasonable. Key point: comparing this result with Eq. 9, we see that the collisional lifetime of kinetic energy in the head-on case is exactly one third of that for the case with an isotropic impact parameter. In both cases, the larger the ion is compared to the gas particle, the more collisions will be required to slow it down by the same fractional amount.

2.2. “Thermalizing” inelastic collisions.

For the elastic models described above, there is, of course no change in the internal energy or vibrational temperature of the ion. It is interesting to consider the opposite extreme, in which the ion and gas particle form a long-lived collision complex in every collision, with the gas and ion being re-emitted isotropically in the center-of-mass frame after the collision and possessing thermal velocities (see Figure 1c). (The physics of complex formation by this mechanism was explored previously by Durup for collisions of gas particles with very small, e.g., diatomic, ions.[61]) It is reasonable to impose the condition that the relevant “temperature” is that of the collision complex after the center-of-mass frame collision energy has been thoroughly redistributed among all vibrational modes. Call this temperature $T_{ig}^{*}$. Since the average center-of-mass frame collision energy at this temperature, according to the kinetic theory of gases, is

$$K{E}_{tot}^{c{m}^{\prime }}=\frac{1}{2}{\mu }_{ig}{v}_{rel}^2=\frac{3}{2}{k}_B{T}_{ig}^{*}$$ (14)

we can invoke conservation of linear momentum to determine the final speeds of the gas and ion in the center-of-mass frame. That is,

$$m_i{v}_i^{c{m}^{\prime }}=m_g{v}_g^{c{m}^{\prime }}$$ (15)

requires that

$$\begin{gathered} K{E}_{tot}^{c{m}^{\prime }}=\frac{1}{2}\left(m_i{\left(v_i^{c{m}^{\prime }}\right)}^2\right.\left.+m_g{\left(v_g^{c{m}^{\prime }}\right)}^2\right)=\frac{1}{2}\left(m_i{\left(v_i^{c{m}^{\prime }}\right)}^2+m_g{\left(\frac{m_i}{m_g}{v}_i^{c{m}^{\prime }}\right)}^2\right) \\ =\frac{1}{2}{m}_i\left(1+\frac{m_i}{m_g}\right){\left(v_i^{c{m}^{\prime }}\right)}^2 \end{gathered}$$ (16)

From this, it is clear that

$$K{E}_i^{c{m}^{\prime }}={\left(1+\frac{m_i}{m_g}\right)}^{-1}K{E}_{tot}^{c{m}^{\prime }}=\frac{m_g}{m_g+m_i}K{E}_{tot}^{c{m}^{\prime }}=\frac{m_g}{m_g+m_i}\frac{3}{2}{k}_B{T}_{ig}^{*}$$ (17)

To convert this back to the lab frame, we first note that

$${\overrightarrow{v}}_i^{L^{\prime }}={\overrightarrow{v}}_i^{c{m}^{\prime }}+{\overrightarrow{v}}_{cm}^L$$ (18)

so that

$$K{E}_i^{L^{\prime }}=\frac{1}{2}{m}_i{\left(v_i^{L^{\prime }}\right)}^2=\frac{1}{2}{m}_i\left({\left(v_i^{c{m}^{\prime }}\right)}^2+2{\overrightarrow{v}}_i^{c{m}^{\prime }}\cdot {\overrightarrow{v}}_{cm}^L+{\left(v_{cm}^L\right)}^2\right)$$ (19)

However, the dot product vanishes by hypothesis (since ${\overrightarrow{v}}_i^{c{m}^{\prime }}$ is assumed isotropic and is uncorrelated with ${\overrightarrow{v}}_{cm}^L$). Thus, we have

$$⟨K{E}_i^{L^{\prime }}⟩=\frac{1}{2}{m}_i\left(⟨{\left(v_i^{c{m}^{\prime }}\right)}^2⟩+⟨{\left(v_{cm}^L\right)}^2⟩\right)=⟨K{E}_i^{c{m}^{\prime }}⟩+\frac{1}{2}{m}_i⟨{\left(v_{cm}^L\right)}^2⟩$$ (20)

We also note that

$${\overrightarrow{v}}_{cm}^L=\frac{m_i{\overrightarrow{v}}_i^L+m_g{\overrightarrow{v}}_g^L}{m_g+m_i}$$ (21)

so that, invoking the Maxwell-Boltzmann distribution of ${\overrightarrow{v}}_g^L$, we get

$$⟨{\left(v_{cm}^L\right)}^2⟩=2{\left(\frac{m_i}{m_i+m_g}\right)}^2\frac{K{E}_i^L}{m_i}+3{\left(\frac{m_g}{m_i+m_g}\right)}^2\frac{k_B{T}_g}{m_g}=\frac{2m_{i}K{E}_i^L}{{\left(m_i+m_g\right)}^2}+\frac{3m_g{k}_B{T}_g}{{\left(m_i+m_g\right)}^2}$$ (22)

We can put the above terms together to get

$$⟨K{E}_i^{L^{\prime }}⟩=\frac{m_g}{m_g+m_i}\frac{3}{2}{k}_B{T}_{ig}^{*}+\frac{m_i^{2}K{E}_i^L}{{\left(m_i+m_g\right)}^2}+\frac{3}{2}\frac{m_i{m}_g{k}_B{T}_g}{{\left(m_i+m_g\right)}^2}$$ (23)

To proceed further, we need to define $T_{ig}^{*}$. In principle, the canonical temperature of a single ion is not well-defined, as this value is defined by the distribution of internal energies across a large ensemble of ions. Let us hypothesize that vibrational energy within the ion-gas complex redistributes completely before re-emission, that the ion contains many more vibrational degrees of freedom than the number of new vibrational degrees of freedom (i.e., 3) added by formation of the complex, and that the energy flow within the ion-gas complex is ergodic. This is plausible, as internal vibrational energy redistribution within proteins is expected to occur on the upper femtosecond to low picosecond timescale,[62] much faster than the nanosecond-scale intervals between collisions in typical CID/CIU experiments, and the density of states for protein-sized ions is relatively high. Then it is reasonable to conclude that these new degrees of freedom possess a distribution of energy approximately equivalent to a Boltzmann distribution at temperature $T_{ig}^{*}$ such that the entire complex has mean internal energy $u_{ig}^{*}=u_i+\frac{1}{2}{\mu }_{ig}{v}_{rel}^2$. Key point: viewed another way, for a relatively small region of the large ion like the pseudo-atom, the rest of the ion acts as a heat bath at temperature ${\mathit{T}}_{\mathit{i}\mathit{g}}^{\mathbf{*}}$, the temperature at which the ion would have average internal energy ${\mathit{u}}_{\mathit{i}\mathit{g}}^{*}$. In statistical mechanical terms, this results in a roughly canonical distribution of vibrational energy within the small regions when averaged over many vibrations.[60] (The small binding energy of the gas to the ion can be included, but it will be used up in dissociating the complex, anyway, so we ignore it here.) Explicitly,

$$u_{ig}^{*}=u_i+\frac{1}{2}\frac{m_i{m}_g}{m_g+m_i}{\left({\overrightarrow{v}}_i^L-{\overrightarrow{v}}_g^L\right)}^2$$ (24)

Averaging this quantity over the Maxwell-Boltzmann distribution for ${\overrightarrow{v}}_g^L$, we get:

$$⟨u_{ig}^{*}⟩=u_i+\frac{1}{2}\frac{m_i{m}_g}{m_g+m_i}\left(2\frac{K{E}_i^L}{m_i}+3\frac{k_B{T}_g}{m_g}\right)$$ (25)

Let us also assume, for simplicity, that the heat capacity, $\frac{d{u}_{ig}}{d{T}_{ig}^{*}}\equiv c\left(T_{ig}^{*}\right)$, of the complex is negligibly different from that of the ion, i.e., $c\left(T_{ig}^{*}\right)\approx c\left(T_i\right)$, which is reasonable for large biomolecular ions. Then we have

$$T_{ig}^{*}=T_i+\frac{1}{2c\left(T_i\right)}\frac{m_i{m}_g}{m_g+m_i}\left(2\frac{K{E}_i^L}{m_i}+3\frac{k_B{T}_g}{m_g}\right)$$ (26)

Key point: this result indicates that, the higher the heat capacity of the ion, the less a collision at a given lab-frame kinetic energy will raise the vibrational temperature of the collision complex.

Thus,

$$\begin{gathered} ⟨K{E}_i^{L^{\prime }}⟩=\frac{m_g}{m_g+m_i}\frac{3}{2}{k}_B\left(T_i+\frac{1}{2c\left(T_i\right)}\frac{m_i{m}_g}{m_g+m_i}\left(2\frac{K{E}_i^L}{m_i}+3\frac{k_B{T}_g}{m_g}\right)\right)+\frac{m_i^{2}K{E}_i^L}{{\left(m_i+m_g\right)}^2} \\ +\frac{3}{2}\frac{m_i{m}_g{k}_B{T}_g}{{\left(m_i+m_g\right)}^2} \end{gathered}$$ (27)

And

$$\begin{gathered} ⟨\Delta K{E}_i^L⟩=⟨K{E}_i^{L^{\prime }}⟩-K{E}_i^L \\ =\frac{m_g}{m_g+m_i}\frac{3}{2}{k}_B\left(T_i+\frac{1}{2c\left(T_i\right)}\frac{m_i{m}_g}{m_g+m_i}\left(2\frac{K{E}_i^L}{m_i}+3\frac{k_B{T}_g}{m_g}\right)\right)+\frac{m_i^{2}K{E}_i^L}{{\left(m_i+m_g\right)}^2} \\ +\frac{3}{2}\frac{m_i{m}_g{k}_B{T}_g}{{\left(m_i+m_g\right)}^2}-K{E}_i^L \\ =\left(-1+\frac{m_i^2}{{\left(m_i+m_g\right)}^2}+\frac{3k_B}{2c\left(T_i\right)}\frac{m_g^2}{{\left(m_g+m_i\right)}^2}\right)K{E}_i^L \\ +\left(-\frac{3k_B}{2c\left(T_i\right)}\frac{m_i{m}_g}{{\left(m_g+m_i\right)}^2}+\frac{m_i{m}_g}{{\left(m_i+m_g\right)}^2}\right)\frac{3}{2}{k}_B{T}_g+\frac{m_g}{m_g+m_i}\frac{3}{2}{k}_B{T}_i \\ =\left(-\frac{2m_i{m}_g+m_g^2}{{\left(m_i+m_g\right)}^2}+\frac{3k_B}{2c\left(T_i\right)}\frac{m_g^2}{{\left(m_g+m_i\right)}^2}\right)K{E}_i^L+\left(\frac{3k_B}{8c\left(T_i\right)}{\chi }_{ig}+\frac{1}{4}{\chi }_{ig}\right)\frac{3}{2}{k}_B{T}_g \\ +\frac{m_g}{m_g+m_i}\frac{3}{2}{k}_B{T}_i \\ =\left(-\frac{1}{2}{\chi }_{ig}+\left(\frac{3k_B}{2c\left(T_i\right)}-1\right)\frac{m_g^2}{{\left(m_i+m_g\right)}^2}\right)K{E}_i^L+\left(\frac{3k_B}{8c\left(T_i\right)}{\chi }_{ig}+\frac{1}{4}{\chi }_{ig}\right)\frac{3}{2}{k}_B{T}_g \\ +\frac{m_g}{m_g+m_i}\frac{3}{2}{k}_B{T}_i \end{gathered}$$ (28)

This result is exact with the above assumptions in place. Because the ion typically contains thousands of vibrational degrees of freedom, $c\left(T_i\right)\gg k_B$ at all but extremely low temperatures, and $m_g\ll m_i$, so we can make a further level of approximation to yield (valid at all $K{E}_i^L$ and realistic $T_g$ and $T_i$):

$$⟨\Delta K{E}_i^L⟩\approx -\frac{1}{2}{\chi }_{ig}K{E}_i^L+\frac{3}{8}{\chi }_{ig}{k}_B{T}_g+\frac{3}{8}{\chi }_{ig}{k}_B{T}_i$$ (29)

Key point: in the thermalizing, totally inelastic model, the ion tends to lose kinetic energy in a collision proportional to its initial kinetic energy, but now it also gains some kinetic energy from both the thermal energy of the gas and the ion’s internal vibrational energy. From this result we can conclude that the steady-state kinetic energy of the ion, after it has slowed down and cooled down, is

$$K{E}_i^L(Z\to \infty )=\frac{3}{4}{k}_B\left(T_g+T_i\right)$$ (30)

In the absence of an external electric field, we anticipate $T_i\to T_g$ approximately, so that $K{E}_i^L(Z\to \infty )\approx \frac{3}{2}{k}_B{T}_g$, in agreement with the Equipartition Theorem.

We can now also calculate the expected change in the ion’s internal energy as a result of a collision for this model:

$$u_i^{\prime }=u_{ig}^{*}-\frac{3}{2}{k}_B{T}_{ig}^{*}$$ (31)

Using results from above, we can find the expected value of these quantities:

$$\begin{gathered} ⟨u_i^{\prime }⟩=u_i+\frac{1}{2}\frac{m_i{m}_g}{m_g+m_i}\left(2\frac{K{E}_i^L}{m_i}+3\frac{k_B{T}_g}{m_g}\right) \\ -\frac{3}{2}{k}_B\left(T_i+\frac{1}{2c\left(T_i\right)}\frac{m_i{m}_g}{m_g+m_i}\left(2\frac{K{E}_i^L}{m_i}+3\frac{k_B{T}_g}{m_g}\right)\right) \end{gathered}$$ (32)

Thus,

$$\begin{gathered} ⟨\Delta u_i⟩=\frac{1}{2}\frac{m_i{m}_g}{m_g+m_i}\left(2\frac{K{E}_i^L}{m_i}+3\frac{k_B{T}_g}{m_g}\right)-\frac{3}{2}{k}_B\left(T_i+\frac{1}{2c\left(T_i\right)}\frac{m_i{m}_g}{m_g+m_i}\left(2\frac{K{E}_i^L}{m_i}+3\frac{k_B{T}_g}{m_g}\right)\right) \\ =\frac{m_g}{m_g+m_i}\left(1-\frac{3k_B}{2c\left(T_i\right)}\right)K{E}_i^L-\frac{3}{2}{k}_B{T}_i+\frac{3}{2}\frac{m_i}{m_g+m_i}\left(1-\frac{3k_B}{2c\left(T_i\right)}\right)k_B{T}_g \end{gathered}$$ (33)

Again noting that $c\left(T_i\right)\gg k_B$ and $\frac{m_g}{m_g+m_i}\approx \frac{{\chi }_{ig}}{4}$, we can approximate

$$⟨\Delta u_i⟩\approx \frac{{\chi }_{ig}}{4}K{E}_i^L+\frac{3}{2}{k}_B\left(T_g-T_i\right)$$ (34)

Key point: this result indicates that the ion will tend to gain vibrational energy when it has relatively high kinetic energy in the lab frame, but when it slows down, it will lose vibrational energy and cool down due to the difference between its vibrational temperature and the kinetic temperature of the gas. The second term also keeps the ion from cooling down further once it equilibrates to the temperature of the gas translationally and vibrationally, as should be expected in real experiments.

The last component of this model relates to the expected change in vibrational temperature of the ion after a collision, which is simply

$$⟨\Delta T_i⟩=\frac{⟨\Delta u_i⟩}{c\left(T_i\right)}$$ (35)

It would be highly desirable to decouple the above two differential for $K{E}_i^L\left(Z\right)$ and $u_i\left(Z\right)$ to obtain analytical results for the collision number at which the maximum ion internal temperature is reached as well as what that temperature is. However, the heat capacities of dry bulk protein powders (which should be close to those of protein ions except for small phonon contributions to the powders) are known to be far from the thermodynamic limit (i.e., $k_B$ per vibrational degree of freedom) and to increase relatively rapidly over tens of Kelvin near room temperature.[63] Our group has used ab initio methods and statistical mechanics to compute heat capacity curves per vibrational degree of freedom for several classes of protonated/deprotonated biomolecular ions (proteins/peptides, nucleic acids, sugars, lipids, and small drug-like molecules). Qualitatively, because the coupled differential equations for $⟨u_i⟩$ and $⟨K{E}_i^L⟩$ are linear and first-order, the solution is approximately (due to the temperature-varying heat capacity) biexponential, with kinetic energy falling monotonically and internal energy quickly rising, then more slowly falling. It is somewhat tedious to show, but the corresponding fast and slow rate constants (with respect to collision number) for both processes are the opposite of the eigenvalues of the corresponding rate constant matrix:

$$k_{\mathit{\text{fast}}/\mathit{\text{slow}}}\approx \frac{{\chi }_{ig}}{4}+\frac{3k_B}{4c\left(T_i\right)}\pm \frac{{\chi }_{ig}}{8}\sqrt{4+6\frac{k_B}{c\left(T_i\right)}-24\frac{k_B}{c\left(T_i\right){\chi }_{ig}}+36\frac{k_B^2}{{\left(c\left(T_i\right){\chi }_{ig}\right)}^2}}$$ (36)

where $k_{\mathit{\text{fast}}}$ (respectively, $k_{\mathit{\text{slow}}}$) refers to the variant of this expression using the plus (respectively, minus) sign. These rate constants are independent of gas pressure and very nearly independent of initial ion kinetic energy (hence, injection potential). For N₂, Ar, and heavier gases, $k_{\mathit{\text{fast}}}\approx \frac{{\chi }_{ig}}{2}$, and $k_{\mathit{\text{slow}}}\approx \frac{3k_B}{4c\left(T_i\right)}$ is about one order of magnitude slower near room temperature. (For He, which is atypically light among common CID/CIU collision gases, $k_{\mathit{\text{fast}}}$ is slightly faster than $\frac{{\chi }_{ig}}{2}$, and $k_{\mathit{\text{fast}}}$ and $k_{\mathit{\text{slow}}}$ are much more similar to one another.) Intriguingly, the fast rate constant, which dominates kinetic energy loss and heating of the ion, is between those of the free-impact-parameter and head-on elastic models described above. An estimate of the collision number $Z_{max}$ at which the internal energy of the ion reaches its maximum is the “logarithmic average” of these two rate constants:

$$Z_{max}\approx \frac{\text{ln}\left(\frac{k_{\mathit{\text{fast}}}}{k_{\mathit{\text{slow}}}}\right)}{k_{\mathit{\text{fast}}}-k_{\mathit{\text{slow}}}}\approx \frac{2}{{\chi }_{\mathit{\text{ig}}}}\text{ln}\left(\frac{2{\chi }_{\mathit{\text{ig}}}c\left(T_i\right)}{3k_B}\right)$$ (37)

Because ${\chi }_{ig}$ varies inversely with ion mass, and $c\left(T_i\right)$ is proportional to mass, this number increases approximately linearly with ion mass. Key point: by taking into account the mean time and distance between collisions (see Section 3, below), ${\mathit{Z}}_{\mathit{\text{max}}}$ can be used to estimate when and where inside the collision cell the ion reaches it maximum vibrational temperature, i.e., when and where ion dissociation/unfolding should be fastest.

2.3. Partially inelastic collisions: the original ICT.

Real collisions in collisional activation experiments are unlikely to be either perfectly elastic (even with a free impact parameter) or completely inelastic (as in the “thermalizing” model described above). Thus, we should expect accurate models to lie somewhere between the extremes described above. In particular, kinetic energy damping in field-free CID/CIU should decrease exponentially with collision number, with a damping “lifetime” $\zeta$ somewhere between ${\zeta }_{\mathit{\text{elastic}}}^{\mathit{\text{head-on}}}=\frac{1}{{\chi }_{\mathit{\text{ig}}}}$ and ${\zeta }_{\mathit{\text{elastic}}}^{\mathit{\text{free}}\mathit{\text{impact}}\mathit{\text{parameter}}}\approx \frac{3}{{\chi }_{ig}}$, and it should not be possible to transfer more kinetic energy into vibrational energy of the ion than in the thermalizing inelastic model. (We exclude the possibility of clustering of numerous gas particles onto the ion, as this is unrealistic in typical CID/CIU experiments.) However, the thermalizing inelastic model contains no “tunable” parameters that can be used to adjust the efficiency of energy transfer.

The ICT, by contrast, is an inelastic collision model with a single tunable parameter that otherwise shares many similarities with the kinematics of the above-described models. In the original ICT developed by Mahan,[43] Uggerud and Derrick,[41] and Douglas,[47] the ion is coarse-grained into imaginary pseudo-atoms (“beads”), and collisions take place between a gas particle and a single pseudo-atom. These collisions are perfectly elastic from the perspective of the gas particle and pseudo-atom (with mass $m_a$), but they transfer momentum into the pseudo-atom that changes the pseudo-atom’s vibrational energy from the perspective of the ion (see Figure 1c). In their treatment, these authors start every collision with the pseudo-atom moving at the velocity of the ion (i.e., it has no additional vibrational energy), and they assume an isotropic distribution of impact parameters (as in the first elastic collision model described here). Instead of ${\chi }_{ig}$, the relevant transfer efficiency parameter turns out to be ${\chi }_{ag}=4\frac{m_a{m}_g}{{\left(m_a+m_g\right)}^2}$, which is entirely dependent on the pseudo-atom mass, an adjustable parameter.[41] As this value determines what fraction of the center-of-mass collision energy of the pseudo-atom and gas particle $\left(\frac{1}{2}{\mu }_{ag}{v}_{rel}^2\right)$ is deposited as internal energy into the ion, it provides a natural means to adjust the inelasticity of the ion-gas collisions that is absent from the three models described above. The typical pseudo-atom mass needed to account for the inelasticity observed in Molecular Dynamics simulations of Uggerud and Derrick are several Da (for He gas) up to several tens of Da (for N₂ or Ar buffer gas).[41] They rationalize this by considering that the “size” of the reactive region of the ion in the high-energy CID experiments they performed (with collision energies of up to 1000’s of eV) is likely close to the size of the gas particle and should constitute a few atoms. We would add that, from room temperature up to ~2000 K, $k_{B}T$ is a few hundred to just over 1000 cm⁻¹, i.e., thermally populated vibrational normal modes over this temperature range will typically include at least three or four atoms.

2.4. Partially inelastic collisions: the Improved ICT (IICT).

Collisions in the original ICT (originally developed to study high-energy, few-collision experiments) act only to heat the ion and slow it down in the lab frame. Thus, there is no way for ions to cool off, as should be expected in the simple free-flight CID/CIU experiment at much lower (up to a few eV) collision energies under discussion, which is intended to model modern CID/CIU in commercially available instruments. In the Improved ICT (IICT), it is hypothesized that ions cool through super-elastic collisions, in which vibrational energy of the pseudo-atom is transferred into kinetic energy of the outgoing gas particle (a vibrationally “hot” pseudo-atom “punches” the incoming gas particle; see Figure 1e). It is also assumed that there is no need to average over impact parameters in these collisions, as the pseudo-atom is not a well-defined object within the ion, but can be thought of as merely whatever region of the ion reacts immediately to the collisions. The mass of the pseudo-atom can simply be defined to be the mass needed to explain the average inelasticity of collisions via perfectly head-on collisions between the pseudo-atom and gas particle in the ICT. This greatly reduces the computational complexity of the problem, as there is no need to pick a random impact parameter for each collision of the pseudo-atom and gas particle. Finally, it should be emphasized that this model is not really coarse-graining in the sense most commonly used in Molecular Dynamics, in which beads explicitly represent groups of atoms, polymer subunits, amino acids, etc., with well-defined locations within the ion. Rather, the pseudo-atom in a given collision is located wherever at the ion’s surface is impinged by the gas particle, with the rest of ion acting as a heat bath after the collision. (Note that multiple scattering is ignored here, as to model it accurately requires knowledge of the protein’s structure, and we seek utility in cases where the protein’s structure may not be accurately known but is at least roughly globular and spherical.)

Let us use the same notational conventions as above, with subscript “a” representing the pseudo-atom. Let us also define three frames of reference: that of the ion itself (“I”), that of the center of mass of the pseudo-atom and gas particle alone (“C”), and the laboratory frame (“L”). We begin our analysis in frame $C$, where the atom enters the collision with velocity ${\overrightarrow{v}}_a^C$ and the gas particle with velocity ${\overrightarrow{v}}_g^C$. Conservation of linear momentum requires that $m_a{\overrightarrow{v}}_a^C=-m_g{\overrightarrow{v}}_g^C$, and the impulsive collision hypothesis of the (I)ICT requires that, after the collision, ${\overrightarrow{v}}_a^{C\prime }=-{\overrightarrow{v}}_a^C$ and ${\overrightarrow{v}}_g^{C^{\prime }}=-{\overrightarrow{v}}_g^C$.

We now move back to the rest frame of the ion. We must have that ${\overrightarrow{v}}_a^I={\overrightarrow{v}}_a^C+{\overrightarrow{v}}_{\mathit{\text{agcom}}}^I$, with the last term being the velocity of the center of mass of the pseudo-atom and gas particle in frame $I$. Explicitly, ${\overrightarrow{v}}_{\mathit{\text{agcom}}}^I=\frac{m_a{\overrightarrow{v}}_a^I+m_g{\overrightarrow{v}}_g^I}{m_a+m_g}$. After the collision, we have ${\overrightarrow{v}}_a^{I\prime }=2{\overrightarrow{v}}_{\mathit{\text{agcom}}}^I-{\overrightarrow{v}}_a^I$. Substituting in the expression for ${\overrightarrow{v}}_{\mathit{\text{agcom}}}^I$, we have:

$${\overrightarrow{v}}_a^{I^{\prime }}=\frac{m_a{\overrightarrow{v}}_a^I+2m_g{\overrightarrow{v}}_g^I-m_g{\overrightarrow{v}}_a^I}{m_a+m_g}$$ (38)

In the lab frame, this is:

$${\overrightarrow{v}}_a^{L^{\prime }}=\frac{m_a{\overrightarrow{v}}_a^I+2m_g{\overrightarrow{v}}_g^I-m_g{\overrightarrow{v}}_a^I}{m_a+m_g}+{\overrightarrow{v}}_i^L=\frac{m_a{\overrightarrow{v}}_a^L+2m_g{\overrightarrow{v}}_g^L-m_g{\overrightarrow{v}}_a^L}{m_a+m_g}$$ (39)

Adding together the final momentum of the pseudo-atom to the pre-collision momentum of the rest of the ion, we obtain the final velocity of the ion in the lab frame:

$${\overrightarrow{v}}_i^{L^{\prime }}=\frac{\left(m_i-m_a\right){\overrightarrow{v}}_i^L+m_a{\overrightarrow{v}}_a^{L^{\prime }}}{m_i}=\frac{\left(m_i-m_a\right){\overrightarrow{v}}_i^L}{m_i}+\frac{m_a}{m_i}\frac{\left(m_a-m_g\right){\overrightarrow{v}}_a^L+2m_g{\overrightarrow{v}}_g^L}{m_a+m_g}$$ (40)

And the change in the ion’s velocity due to the collision is:

$$\begin{gathered} \Delta {\overrightarrow{v}}_i^L={\overrightarrow{v}}_i^{L^{\prime }}-{\overrightarrow{v}}_i^L=\frac{-m_a{\overrightarrow{v}}_i^L+m_a{\overrightarrow{v}}_a^{L\prime }}{m_i}=-\frac{m_a{\overrightarrow{v}}_i^L}{m_i}+\frac{m_a}{m_i}\left(\frac{m_a{\overrightarrow{v}}_a^I+2m_g{\overrightarrow{v}}_g^I-m_g{\overrightarrow{v}}_a^I}{m_a+m_g}+{\overrightarrow{v}}_i^L\right) \\ =\frac{m_a}{m_i}\left(\frac{m_a{\overrightarrow{v}}_a^I+2m_g{\overrightarrow{v}}_g^I-m_g{\overrightarrow{v}}_a^I}{m_a+m_g}\right) \end{gathered}$$ (41)

Statistical mechanics tells us that, for vibrational modes with frequencies well below $\frac{k_{B}T}{h}$ (207 cm⁻¹ at room temperature and ~1400 cm⁻¹ at 2000 K), the speed of the reduced mass of the mode is approximately Maxwell-Boltzmann distributed in the reference frame of the ion.

From the above, we can estimate the expected change in kinetic energy, internal energy, and vibrational temperature of the ion in a collision. Explicitly,

$$\begin{gathered} \Delta K{E}_i^L=\langle \frac{1}{2}{m}_i\left({\left(v_i^{\text{'}L}\right)}^2-{\left(v_i^L\right)}^2\right)\rangle \\ =\frac{1}{2}{m}_i\langle {\left({\overrightarrow{v}}_i^L-\frac{m_a{\overrightarrow{v}}_i^L}{m_i}+\frac{m_a}{m_i}\frac{\left(m_a-m_g\right){\overrightarrow{v}}_a^L+2m_g{\overrightarrow{v}}_g^L}{m_a+m_g}\right)}^2-{\left(v_i^L\right)}^2\rangle \\ =\frac{1}{2}{m}_i\langle {\left({\overrightarrow{v}}_i^L+\frac{m_a}{m_i}\frac{\left(m_a-m_g\right){\overrightarrow{v}}_a^I+2m_g{\overrightarrow{v}}_g^I}{m_a+m_g}\right)}^2-{\left(v_i^L\right)}^2\rangle \\ =\frac{1}{2}{m}_i\langle {\left(v_i^L\right)}^2+2{\overrightarrow{v}}_i^L\cdot \left(\frac{m_a}{m_i}\frac{\left(m_a-m_g\right){\overrightarrow{v}}_a^I+2m_g{\overrightarrow{v}}_g^I}{m_a+m_g}\right)+{\left(\frac{m_a}{m_i}\frac{\left(m_a-m_g\right){\overrightarrow{v}}_a^I+2m_g{\overrightarrow{v}}_g^I}{m_a+m_g}\right)}^2-{\left(v_i^L\right)}^2\rangle \\ =\frac{1}{2}{m}_i\langle -2{\overrightarrow{v}}_i^L\cdot \left(\frac{m_a}{m_i}\frac{2m_g{\overrightarrow{v}}_i^L}{m_a+m_g}\right)+{\left(\frac{m_a}{m_i}\frac{\left(m_a-m_g\right){\overrightarrow{v}}_a^I+2m_g{\overrightarrow{v}}_g^I}{m_a+m_g}\right)}^2\rangle \\ =\frac{1}{2}{m}_i\langle -4\frac{m_a}{m_i}\frac{m_g}{m_a+m_g}{\left(v_i^L\right)}^2+\frac{m_a^2}{m_i^2}\frac{1}{{\left(m_a+m_g\right)}^2}{\left(\left(m_a-m_g\right){\overrightarrow{v}}_a^I+2m_g{\overrightarrow{v}}_g^I\right)}^2\rangle -4\frac{m_a}{m_i}\frac{m_g}{m_a+m_g}\frac{2K{E}_i^L}{m_i} \\ =\frac{1}{2}{m}_i\langle +\frac{m_a^2}{m_i^2}\frac{1}{{\left(m_a+m_g\right)}^2}\left({\left(m_a-m_g\right)}^2{\left(v_a^I\right)}^2+4m_g^2{\left(v_g^L\right)}^2+4m_g^2{\left(v_I^L\right)}^2\right)\rangle -4\frac{m_a}{m_i}\frac{m_g}{m_a+m_g}\frac{2K{E}_i^L}{m_i} \\ =\frac{1}{2}{m}_i\langle \frac{m_a^2}{m_i^2}\frac{1}{{\left(m_a+m_g\right)}^2}\left({\left(m_a-m_g\right)}^2\frac{3k_B{T}_i}{m_a}+4m_g^2\frac{3k_B{T}_g}{m_g}+4m_g^2\frac{2K{E}_i^L}{m_i}\right)\rangle \\ =-4\frac{m_a}{m_i}\frac{m_g}{m_a+m_g}K{E}_i^L+\frac{3}{2}{k}_B{T}_i\frac{m_a}{m_i}\frac{{\left(m_a-m_g\right)}^2}{{\left(m_a+m_g\right)}^2}+\frac{3}{2}{k}_B{T}_g\frac{m_a}{m_i}\frac{4m_a{m}_g}{{\left(m_a+m_g\right)}^2}+\frac{m_a^2}{m_i^2}\frac{4m_g^2}{{\left(m_a+m_g\right)}^2}K{E}_i^L \\ =-\frac{{\chi }_{ag}}{m_i}\left(m_a+m_g-\frac{m_g{m}_a}{m_i}\right)K{E}_i^L+\frac{m_a}{m_i}\left(1-{\chi }_{ag}\right)\frac{3}{2}{k}_B{T}_i+\frac{m_a}{m_i}{\chi }_{ag}\frac{3}{2}{k}_B{T}_g \end{gathered}$$ (42)

Key point: this rather complicated expression tells us that, in the IICT, the ion tends to lose kinetic energy in a collision proportional to its initial kinetic energy but gains a small amount back due to both its own vibrational energy and the kinetic energy of the gas. The relative size of these contributions all depend on the mass of the pseudo-atom.

As for the internal energy of the ion, in the (I)ICT, the change in internal energy is by hypothesis exactly equal to the change in kinetic energy of the pseudo-atom in the reference frame of the ion. Thus,

$$\begin{gathered} \Delta u_i=u_i^{\prime }-u_i=K{E}_a^{\prime I}-K{E}_a^I \\ =\frac{1}{2}{m}_a\left({\left(v_a^{\prime I}\right)}^2-{\left(v_a^I\right)}^2\right)=\frac{1}{2}{m}_a\left({\left({\overrightarrow{v}}_a^I-\frac{2m_{g}c}{m_a+m_g}\left({\overrightarrow{v}}_a^I-\left({\overrightarrow{v}}_g^L-{\overrightarrow{v}}_i^L\right)\right)\right)}^2-{\left(v_a^I\right)}^2\right) \\ =\frac{1}{2}{m}_a\left(\frac{4m_g^2{c}^2}{{\left(m_a+m_g\right)}^2}\left({\left(v_a^I\right)}^2+{\left(v_g^L\right)}^2+{\left(v_i^L\right)}^2\right)-\frac{4m_{g}c}{m_a+m_g}\left({\overrightarrow{v}}_a^I\cdot {\overrightarrow{v}}_a^I-{\overrightarrow{v}}_a^I\cdot {\overrightarrow{v}}_g^L+{\overrightarrow{v}}_a^I\cdot {\overrightarrow{v}}_i^L\right)-\frac{4m_g^2{c}^2}{{\left(m_a+m_g\right)}^2}\left({\overrightarrow{v}}_a^I\cdot {\overrightarrow{v}}_g^L-{\overrightarrow{v}}_a^I\cdot {\overrightarrow{v}}_i^L+{\overrightarrow{v}}_g^L\cdot {\overrightarrow{v}}_i^L\right)\right) \end{gathered}$$ (43)

In computing the expected value of this quantity, we again note that all the dot products in the above expression between unlike quantities vanish by assumptions about symmetry, and we are left with:

$$⟨\Delta u_i⟩=\frac{1}{2}{m}_a⟨\left(\frac{4m_g^2}{{\left(m_a+m_g\right)}^2}\left({\left(v_a^I\right)}^2+{\left(v_g^L\right)}^2+{\left(v_i^L\right)}^2\right)-\frac{4m_g}{m_a+m_g}{\left(v_a^I\right)}^2\right)⟩$$ (44)

Taking thermal averages for the pseudo-atom and gas average squared velocities, we obtain:

$$⟨\Delta u_i⟩=\frac{1}{2}{m}_a⟨\left(\frac{4m_g^2}{{\left(m_a+m_g\right)}^2}\left(\frac{3k_B{T}_i}{m_a}+\frac{3k_B{T}_g}{m_g}+\frac{2K{E}_i^L}{m_i}\right)-\frac{4m_g}{m_a+m_g}\frac{3k_B{T}_i}{m_a}\right)⟩$$ (45)

We note that

$$\frac{m_g^2}{{\left(m_a+m_g\right)}^2}-\frac{m_g}{m_a+m_g}=-\frac{m_a{m}_g}{{\left(m_a+m_g\right)}^2}$$ (46)

Substituting our definition for the “efficiency”, ${\chi }_{ag}$, we at last obtain:

$$⟨\Delta u_i⟩=\frac{{\chi }_{ag}}{2}3k_B\left(T_g-T_i\right)+{\chi }_{ag}\frac{m_g}{m_i}K{E}_i^L$$ (47)

If we again let $m_a=m_g$ (so that ${\chi }_{ag}=1$), this solution becomes

$$⟨\Delta u_i⟩=\frac{3}{2}{k}_B\left(T_g-T_i\right)+\frac{m_g}{m_i}K{E}_i^L\approx \frac{3}{2}{k}_B\left(T_g-T_i\right)+\frac{{\chi }_{ig}}{4}K{E}_i^L$$ (48)

Importantly, the IICT is essentially identical to the totally inelastic model (Eqs. 29 and 34) when the region of the ion that effectively undergoes an elastic collision with the gas particle (i.e., the pseudo-atom) has the same mass as the gas particle. We can therefore recover any desired result from the totally inelastic model by simply setting $m_a=m_g$. Key point: any other value of ${\mathit{m}}_{\mathit{a}}$ will yield ${\mathit{\chi }}_{\mathit{a}\mathit{g}}<1$ and decrease the inelasticity of the model, providing a tunable parameter with exactly the desirable properties we laid out at the beginning of this section. Additionally, the IICT provides a mechanism for vibrational energy transfer out of the pseudo-atom into kinetic energy of the outgoing gas particle (i.e., ion cooling) that is exactly the reverse of the heating mechanism and therefore satisfies microscopic reversibility.

We finally note that, although a treatment of ion-gas collisions within electric fields is beyond the scope of this manuscript, both the IICT and totally inelastic models described here agree with the Wannier temperature from ion mobility theory when the ion is assumed to have achieved a steady-state drift velocity.[64] At steady-state, we have

$$⟨K{E}_i^L⟩=⟨K{E}_{i,\mathit{\text{drift}}}^L⟩=\frac{1}{2}{m}_i{v}_{i,\mathit{\text{drift}}}^2$$ (49)

and

$$\begin{gathered} ⟨\Delta u_i⟩=0=\frac{{\chi }_{ag}}{2}3k_B\left(T_g-T_i\right)+{\chi }_{ag}\frac{m_g}{m_i}K{E}_{i,\mathit{\text{drift}}}^L=\frac{{\chi }_{ag}}{2}3k_B\left(T_g-T_i\right) \\ +{\chi }_{ag}\frac{m_g}{m_i}\frac{1}{2}{m}_i{v}_{i,\mathit{\text{drift}}}^2 \end{gathered}$$ (50)

We can rearrange this to:

$$\frac{3}{2}{k}_B{T}_i=\frac{3}{2}{k}_B{T}_g+\frac{1}{2}{m}_g{v}_{i,\mathit{\text{drift}}}^2$$ (51)

which implies

$$T_i=T_g+\frac{1}{3}{m}_g{v}_{i,\mathit{\text{drift}}}^2$$ (52)

Key point: under steady-state drift conditions, the ion is slightly hotter than the gas due to its field-induced motion. Eq. 52 is exactly the result obtained in two-temperature ion mobility theory.[64–66]

2.5. Treatment of external electrostatic fields, distance, and time.

Both constant and spatially varying electric fields can be straightforwardly added to the IICT by simply adding the appropriate change in kinetic energy of the ion due to acceleration in the field between collisions. That is,

$$\begin{gathered} ⟨\Delta K{E}_i^L⟩=-\frac{{\chi }_{ag}}{m_i}\left(m_a+m_g-\frac{m_g{m}_a}{m_i}\right)K{E}_i^L+\frac{m_a}{m_i}\left(1-{\chi }_{ag}\right)\frac{3}{2}{k}_B{T}_i+\frac{m_a}{m_i}{\chi }_{ag}\frac{3}{2}{k}_B{T}_g \\ +{\int }_{{\overrightarrow{x}}_i}^{{\overrightarrow{x}}_i^{\prime }}q\overrightarrow{E}\left(\overrightarrow{x}\right)\cdot d\overrightarrow{x} \end{gathered}$$ (53)

where ${\overrightarrow{x}}_i$ is the position of the ion, $q=ze$ is the ion’s charge, and $\overrightarrow{E}\left(\overrightarrow{x}\right)$ is the electric field. (The integral term simply represents the kinetic energy gained by acceleration of the ion in the external electric field between one collision and the next.) It is straightforward to keep track of the ion velocity as a vector in three-dimensional space and use the kinetic theory of gases to determine when and where the next collision will occur either by using the mean collision time or by sampling the appropriate distribution of collision times. Specifically, the mean free path until the next collision, $\bar{\lambda }\left(Z\right)$, can be computed as a function of collision number from its lab-frame velocity $⟨v_i^L\left(Z\right)⟩\equiv \sqrt{2⟨K{E}^L\left(Z\right)⟩/m_i}$, the gas number density $N_g$, and the collision cross section of the ion and gas ${\sigma }_{ig}$:[67]

$$\lambda \left(Z\right)=\frac{⟨v_i^L\left(Z\right)⟩}{{\sigma }_{ig}{N}_g⟨v_{rel}^L\left(Z\right)⟩}$$ (54)

Where

$$⟨v_{rel}^L\left(Z\right)⟩\approx \sqrt{\frac{8k_B{T}_g}{\pi m_g}}\left(\left(s+\frac{1}{2s}\right)\sqrt{\frac{\pi }{2}}\text{erf}\left(s\right)+\frac{1}{2}{e}^{-s^2}\right)$$ (55)

and

$$s\equiv \frac{⟨v_i^L\left(Z\right)⟩}{\sqrt{\frac{2k_B{T}_g}{m_g}}}$$ (56)

Key point: this is different from the simpler expression often encountered in introductory physical chemistry textbooks, in which the speed of the gas is ignored because it is assumed to be very small compared to the ion; in modern CID/CIU experiments, the ion and gas often have similar speeds, so Eqs. 54 and 55 should be used instead. In the Monte Carlo version of IonSPA, the distance to the next collision is selected from an exponential distribution with this mean value. In Monte Carlo IonSPA, spatially-varying characteristics of the buffer gas can also be easily implemented by treating $N_g$ and $T_g$ as spatially varying via a grid (“look-up table”) computed with COMSOL^® or similar gas dynamics software, or as an explicit function of ion position. If there is a finite gas flow velocity, the Maxwell-Boltzmann velocity distribution of the gas and the velocity of the ion are simply added to this flow velocity at each step of the computation in Monte Carlo IonSPA.

When the ion’s kinetic energy is still very high compared to that of the gas, diffusion away from travel along the instrument optical $\left(Z-\right)$ axis should be small enough that we can approximate the ion’s progression along this axis after $Z$ collisions as

$$z\left(Z\right)\approx \sum _{j=0}^Z\lambda \left(j\right)$$ (57)

The time between each successive collision can be likewise estimated as

$$\tau \left(Z\right)\approx \frac{\lambda \left(Z\right)}{⟨v_i^L\left(Z\right)⟩}=\frac{1}{{\sigma }_{ig}{N}_g⟨v_{rel}^L\left(Z\right)⟩}$$ (58)

The total time until the Z^th collision is thus approximately

$$t\left(Z\right)\approx \sum _{j=0}^{Z-1}\tau \left(j\right)$$ (59)

under these conditions. These approximations are used in IonSPAavg, which does not use Monte Carlo sampling, to generate axes representing lab-frame distance traveled and ion flight time in kinetic energy, ion vibrational energy, and ion temperature plots. Computations of ion trajectories and vibrational energies in IonSPAavg are much faster than with “full” Monte Carlo IonSPA. However, Monte Carlo IonSPA can be used to compute and analyze the distribution of trajectory properties and vibrational energies across an ensemble of ions arising from randomness of collisions, whereas IonSPAavg treats only their averages.

2.6. Treatment of time-varying electric fields.

For instrumentation with time-varying electric fields, such as the Waters Synapt series, which uses “Traveling Wave Ion Guides” (TWIGs),[10] temporal variation of the electric field can be easily incorporated into Monte Carlo IonSPA. The electric field itself can be represented by a time-varying grid or by an analytical expression as a function of ion position and time.[68] Integration of trajectories is performed with time steps much smaller than both the mean free time between collisions and the temporal cycle of the Traveling Wave.

3. Simulation Results and Discussion

To give a few visually instructive examples of the above theoretical results, we performed simulations on native-like cytochrome c⁷⁺ (12365 Da) and tetradecameric GroEL⁷⁰⁺) (~801000 Da). Collision cross sections of 12.8 and 15.9 nm², respectively, were used for cytochrome c in He and N₂ gas, and the cross section in Ar was assumed to be the same as in N₂. A N₂ collision cross section of 218 nm² was used for GroEL⁷⁰⁺. Instrument conditions were set to be similar to those in the Collision Cell of an Agilent 6545XT quadrupole-time-of-flight mass spectrometer, with a length of 18 cm, a pressure of 21 μbar, and constant elution field of 39 V/m, except for high-pressure simulations, in which a constant field-to-concentration (E/N) value of 1 Td was used for each gas studied (He, N₂, and Ar) at a pressure of 2.1 mbar. IonSPAavg, which uses discrete collisions and expected kinetic energy and internal energy changes as described in the Theory section, was used for all simulations. Pseudo-atom masses for each gas type were derived from the average inelasticity of collisions between native-like ions and the gas particle estimated using GROMACS with the AMBER97 force field (approximately 31, 68, and 75 Da for He, united-atom N₂, and Ar for an ion vibrational temperature near 298 K, respectively).

Figure 2 compares kinetic energy damping for cytochrome c⁷⁺ for all of the models (other than the original ICT) using an injection potential of 20 V. As expected from the results derived in the Theory section, the totally inelastic and IICT models fall in between the elastic models with free impact parameter and with head-on collisions. In fact, the IICT model falls in between the totally inelastic and free-impact-parameter hard spheres models at the one extreme and the head-on elastic model at the other.

Figure 2.

Kinetic energy damping for native-like cytochrome c⁷⁺ with 20 V injection potential into N₂ buffer gas (21 μbar) predicted with IonSPA for collision models described in the section 2. See section 3 for simulation details.

In Figure 3, vibrational energy vs. collision number curves predicted using IonSPAavg are plotted for cytochrome c⁷⁺ and GroEL⁷⁰⁺ in N₂ gas, each at two different injection potentials (20 and 50 V), using the totally inelastic and IICT models. As expected, the IICT predicts less heating than does the totally inelastic model for the pseudo-atom masses used. (Curves predicted by the IICT are identical to the totally inelastic model when the pseudo-atom mass is set equal to the mass of the gas particle.) Note that different total numbers of collisions are encountered for each curve before the ion elutes from the 18 cm Collision Cell due to the interplay between ion vibrational energy and kinetic energy. Also note that $Z_{max}$ depends only weakly on injection potential in these simulations. This slight dependence is entirely attributed to the small elution field, which contributes a small additional amount of ion heating to ions that slow down closer to the entrance of the Collision Cell; the effect disappears in the absence of a field. Note that, whereas cytochrome c⁷⁺ achieves its maximum vibrational energy well inside the Collision Cell for both models, GroEL⁷⁰⁺ requires many more collisions to heat and cool and barely, if at all, reaches its maximum vibrational energy within the Collision Cell under the experimentally realistic conditions studied.

Figure 3.

Ion heating and cooling predicted with IonSPA for the totally inelastic in IICT models for native-like (a) cytochrome c⁷⁺ and (b) GroEL⁷⁰⁺ at injection potentials of 20 and 50 V. For GroEL⁷⁰⁺, solid lines indicate trajectories through the 18-cm Collision Cell, and dotted lines indicate continuation of these trajectories for another 18 cm under the same constant elution field and gas pressure.

Using the transformations from collision number to time and distance described above in the Theory section, collision heating and cooling of native-like cytochrome c⁷⁺ were predicted using IonSPAavg IICT model with He, N2, and Ar buffer gas (see Figure 4). Due to its high corresponding pseudo-atom mass, He is very inefficient at slowing down the ion, which elutes from the Collision Cell in just over 120 μs before achieving a (relatively low) maximum vibrational energy. By contrast, N₂ and Ar are much more efficient at slowing down the ion and also at heating and cooling it. For these gases, the ion reaches its maximum vibrational temperature near the entrance to the Collision Cell and spends the majority of its time in the Collision Cell cooling down. Thus, modeling the ion temperature in this or similar instrumentation as constant during the dissociation/unfolding process could result in significant inaccuracies.

Figure 4.

Heating and cooling of native-like cytochrome c⁷⁺ predicted by IonSPAavg for the IICT model in three different buffer gases. (a) illustrates vibrational energy vs. distance through the Collision Cell, and (b) illustrates vibrational energy vs. time.

IonSPAavg can also be used to predict the behavior of large biomolecular ions at much higher pressures (and lower E/N). In Figure 5, collision heating and cooling modeled with the IICT in IonSPAavg is shown for native-like cytochrome c⁷⁺ at 2.1 mbar and a constant E/N of 1 Td, roughly representing a conventional drift-tube type ion mobility spectrometer. An injection potential of 50 V was used to illustrate how quickly the gases heat and cool the ion. Although the ion’s vibrational energy increases by a factor of ~2–5 (in fact, by the same amount as in Figure 4), it cools back down in less than 30 μs, and the ion spends only 1–2 μs within 10% of its maximum vibrational energy for each gas.

Figure 5.

Heating and cooling of native-like cytochrome c⁷⁺ at an E/N of 1 Td and a gas pressure of 2.1 mbar in He, N₂, and Ar buffer gas with an injection potential of 50 V. Inset shows zoomed-in region where heating and cooling are most rapid.

4. Conclusions

In this article, the importance and potentially untapped utility of ion internal and kinetic energy modeling in modern CID/CIU experiments have been discussed. Various extreme scenarios for field-free collisional activation have also been explored: perfectly elastic, totally inelastic, and partially inelastic collisions (via the original ICT and Improved ICT). By including a mechanism for ion cooling (super-elastic collisions with gas particles), the Improved Impulsive Collision Theory (incorporated into the software suite IonSPA) that addresses many of the shortcomings of the original ICT. The IICT exhibits behavior intermediate between the totally inelastic and head-on elastic models and includes a single tunable parameter (pseudo-atom mass) that controls where the model lands between these extremes. Despite not having been developed with ion mobility spectrometry in mind, the model also agrees surprisingly well with results from two-temperature ion mobility theory, especially at field-to-concentration ratios greater than a few Td. The IICT in IonSPA can be used to predict the trajectories of ions within many types of mass spectrometry instrumentation by adjusting the gas and external electric field properties, both of which can be either constant or vary in space and time. Crucially, IonSPA provides a link between instrumental parameters (buffer gas identity and pressure, collisional activation potentials, etc.) and the internal temperatures (and internal energy distributions) experienced by ions as a function of flight time, i.e., $T_i\left(t\right)$. Because ions may heat and cool significantly between initial injection of the ion until quenching or detection, this information will be crucial for accurate kinetic modeling, for example, using a time-dependent Eyring rate expression:

$$\frac{d\left[i\right]\left(t\right)}{dt}=-k\left(T_i\left(t\right)\right)\left[i\right]\left(t\right)$$ (60)

with

$$k\left(T_i\left(t\right)\right)=\frac{k_B{T}_i\left(t\right)}{h}{e}^{\frac{\Delta S^{‡}}{k_B}}{e}^{-\frac{\Delta H^{‡}}{k_B{T}_i\left(t\right)}}$$ (61)

For scenarios in which non-Boltzmann internal energy distributions are likely to play a large role in kinetics, RRKM modeling can also be performed using time-dependent internal energy distributions from Monte Carlo IonSPA. Future work will explore the use of time- and instrument parameter-dependent ion energies and temperatures predicted with IonSPA to interpret CID breakdown curves and CIU fingerprints.

Highlights.

• Existing theories for collisional activation of gas-phase ions are briefly reviewed

• Improved Impulsive Collision Theory predicts ion motion, heating, and cooling inside mass spectrometers

• Results from this model are compared to elastic and totally inelastic theories

• IonSPA software enables modeling by the IICT with Monte Carlo and ensemble-average modes

Biography

James S. Prell received his bachelor’s degree summa cum laude from Washington University in St. Louis in 2005 in German, Mathematics, and Chemistry, with minors in Religious Studies and Music. He completed his Ph.D. in 2011 in the Department of Chemistry at the University of California, Berkeley, studying gas-phase ion spectroscopy and thermochemistry under the supervision of Prof. Evan R. Williams. He then worked as a postdoctoral scholar in the group of Prof. Stephen R. Leone at the University of California, Berkeley, studying solid-state attosecond spectroscopy. He joined the faculty of the Department of Chemistry and Biochemistry at the University of Oregon in 2014, where his research group investigates the structure and thermochemistry of biomolecular ion and their complexes as well as develops methods and software to interpret highly congested mass spectra. He is the recipient of an American Society for Mass Spectrometry Research Award as well as a National Science Foundation CAREER Award, and he is an editorial advisory board member for the Journal of the American Society for Mass Spectrometry. He is co-founder and faculty advisor for the University of Oregon’s chapter of the Society for the Advancement of Chicanos/Hispanics and Native Americans in Science. In his spare time, he enjoys cooking, biking, and playing classical piano.