Simulations of Ion Motion at Atmospheric Pressure
The simulation of gaseous ion trajectories has been used extensively in the development of ion optics for mass spectrometry (MS), ion mobility spectrometry (IMS), electron microscopes (EM), and focused ion beam (FIB) systems. In the case of systems operating in high-vacuum (EM and FIB) the simulation environment is often simplified and assumed to be collision-free and ion motion is influenced purely by electric and magnetic fields; however, IMS and many MS systems operate in a pressure regime in which collisions must be taken into account to accurately predict ion motion . The primary simulation tool used in this work, SIMION 8.0 includes two collision models: a hard sphere collision model (HS1) and a statistical diffusion simulation model (SDS). Either can be incorporated into a simulation environment for the treatment of collisions between ions and a background gas. HS1 employs hard-sphere collision kinetics to compute the resulting ion trajectory change for ion-molecule collisions individually . This approach is not computationally feasible at atmospheric pressures as the mean free path in air at 25 °C is approximately 67 nm . Rather than treating individual collisions, the SDS algorithm uses a combined approach of diffusion and ion mobility to simulate ion motion in electric fields.
The motion of ions at atmospheric pressure is heavily influenced by the diffusion of ions in the medium, as well as by external forces exerted on the ions (electric fields, bulk gas flow, etc.). Diffusion can be expressed as:
where J, D, and Vn are the number of ions passing through an area normal to the gas flow, a proportionality constant, and the concentration gradient, respectively . In the SIMION-SDS algorithm, diffusion is simulated by imposing a jump onto the ion trajectory. The direction of the jump is randomized and it’s magnitude is determined by an interpolation between collision statistics tables (selected based on the mass ratio of the ion to a background gas molecule) and scaled appropriately based on an expected number of collisions in the simulation time step .
When subjected to an electric field (E), the velocity of an ion in a gas with no bulk flow is determined by its mobility (K) in the buffer gas :
K is determined experimentally and is directly proportional to D and the charge (e) on the ion and inversely proportional to temperature (T) multiplied by the Boltzmann constant (k) :
Equation 1.3 is known as the Nernst-Townsend relation and holds for the cases in which ions are thermalized. The mobility can further be expressed as :
where N is the density of the neutral molecules, p is the reduced mass of the collision pair, and QD is the collisional cross section (CCS). Due to the range of working conditions used in IMS instruments, the mobility of an ion is often reported as the reduced mobility (K0) which is corrected for 273 K and a pressure (P) of 760 Torr:
At each time step within a SIMION-SDS simulation, the velocity of an ion is subjected to the effects of gas flow and the applied electric field, in the form of ion mobility (Eq. 1.2). A simulated diffusion in the form of a random jump is superimposed on this motion to determine the location of the ion during the start of the next time-step. A detailed description of the SDS algorithm is given by Appelhans et al.  and in the SIMION 8.0 documentation. The SDS algorithm is capable of either using a defined mobility for each ion, or in the cases in which this data is not available, known information (particle diameters, masses, etc.) is used to estimate a value for ion mobility. Spatial variations in gas flow, pressure, and temperature may also be incorporated into the SDS algorithm to more accurately model conditions in which these parameters are known. Effects due to space charge can also be included in the modeling; however, ions must be flown as a group when incorporating space charge effects into an SDS simulation.
The implementation of the SDS algorithm into the SIMION workspace has been shown to be an accurate predictor for ion motion at or near atmospheric pressure and has been validated experimentally in several cases including traditional drift cell ion mobility spectrometry (IMS) and high field asymmetric waveform ion mobility spectrometry (FAIMS) [49-51].