Numerical Simulation of Ion Transport in a Nano-Electrospray Ion Source at Atmospheric Pressure
Understanding ion transport properties from the ion source to the mass spectrometer (MS) is essential for optimizing device performance. Numerical simulation helps in understanding of ion transport properties and, furthermore, facilitates instrument design. In contrast to previously reported numerical studies, ion transport simulations in a continuous injection mode whilst considering realistic space-charge effects have been carried out. The flow field was solved using Reynolds-averaged Navier-Stokes (RANS) equations, and a particle-in-cell (PIC) method was applied to solve a time-dependent electric field with local charge density. A series of ion transport simulations were carried out at different cone gas flow rates, ion source currents, and capillary voltages. A force evaluation analysis reveals that the electric force, the drag force, and the Brownian force are the three dominant forces acting on the ions. Both the experimental and simulation results indicate that cone gas flow rates of ≤250 slph (standard liter per hour) are important for high ion transmission efficiency, as higher cone gas flow rates reduce the ion signal significantly. The simulation results also show that the ion transmission efficiency reduces exponentially with an increased ion source current. Additionally, the ion loss due to space-charge effects has been found to be predominant at a higher ion source current, a lower capillary voltage, and a stronger cone gas counterflow. The interaction of the ion driving force, ion opposing force, and ion dispersion is discussed to illustrate ion transport mechanism in the ion source at atmospheric pressure.
KeywordsNano-ESI-MS Ion transport Space-charge effect Numerical gas dynamic simulation
Electrospray ionization mass spectrometry (ESI-MS) is a powerful analytical technique that provides both qualitative and quantitative information in chemical and biological applications [1, 2, 3, 4]. MS with nano-electrospray ionization (nano-ESI) is an important branch of ESI sources [5, 6, 7, 8]. A nano-ESI emitter has a micro-scale orifice that promotes low analyte consumption, which benefits the mass spectrometric analysis of analyte of limited availability. In a nano-ESI-MS, the analyte ionization is induced by applying a high DC voltage to the emitter at a variable distance from a counter electrode. After ionization, the charged droplets/ions are transported into the MS through the MS inlet. In addition to the electric driving force, drag force from the buffer gas at atmospheric pressure and the Coulombic repelling force lead to ion dispersion and reduction of the ion transmission efficiency. These competing processes complicate the characteristics of the ion transmission. From an experimental perspective, it is difficult to measure the relative importance of the underlying mechanisms in ion transmission since the MS only records the “end product” of transmitted ions. Therefore, it is essential to implement numerical simulation tools in order to understand the transport properties of the ion source and, furthermore, to improve the efficiency of transfer of ions into the analyzer.
The detection limit of a mass spectrometer is highly dependent on the analyte ionization efficiency in the ion source and the ion transmission efficiency through the MS inlet. The analyte ionization efficiency is determined by several factors, including analyte type, analyte concentration, solvent composition, emitter design, etc. . Owing to a number of unknowns in the ionization mechanism, it is difficult to quantify these influencing factors.
The ion transmission efficiency is the ratio of the ions that reach the MS detector to the total ions generated in the ionization process. The loss of ions during the ion transfer process to the MS detector can be large, and the majority are lost in the ion source , especially for atmospheric pressure ion sources. Factors influencing the ion transmission efficiency in the ion source include the applied voltage on the emitter, the source current, the gaseous flow field, the ESI-MS interface design, etc. This study will focus on the ion transport properties of the ion source at atmospheric pressure and will not consider the ion generation mechanism in the nano-ESI plume.
Studies of the ion transmission efficiency in ESI sources at atmospheric pressure have advanced in the past decade [11, 12, 13]. Experimental studies with different electrospray (ES) conditions have found that for low flow rate electrosprays, the decreased ES current reduces space-charge effects and increases the transmission efficiency [9, 14]. However, the generally divergent nature of the charged spray leads to a discrepancy between the plume size and the MS inlet size (limited by practical considerations), which ultimately limits the ion transmission efficiency. As the ion transmission efficiency in the ESI source is highly dependent on the geometric design of the ESI-MS interface, novel designs of the interface [15, 16, 17] were experimentally carried out to manipulate the flow field and to further increase the ion transmission. These experimental works reveal the dependence of the ion transmission efficiency on several factors, but details about characteristics of ion transport from an atmospheric pressure ion source still need further study.
In addition to experimentation, numerical simulation is an alternative and potentially powerful tool due to its relatively low cost, short turnaround time, flexibility to change geometric and operating conditions, and reliable results. Numerical simulation of ion trajectories overcomes limited measurements in experiments to provide a full picture of ion transport properties and, furthermore, facilitates instrument design. However, numerical simulation is still challenging owing to the contradictory relationship between accuracy and computational cost in solving the full governing equations of particle motion under combined gas dynamics and electric forces. Transport of ions in a combined electric field and buffer gas field can be described by treating the ions as either a continuum (e.g., a scalar field of the ion concentration) or discrete particles. From a macroscopic perspective, an electro-kinetic transport equation, the Poisson-Nernst-Planck (PNP) equation, is applied to describe dynamics of the ion flux in a buffer gas and a weak electric field [18, 19]. From a microscopic perspective, motions of ions can be tracked individually in a discrete model, which is the particle tracking method. Based on different particle representations and treatments of buffer gas field, particle tracking methods can be combined with a continuous buffer flow field in the Euler-Lagrange approach. If both the tracking particles and the buffer gas molecules are treated as individual particles, the direct simulation Monte-Carlo (DSMC) method can be adopted to simulate the ion-gas collisions directly [20, 21, 22]. As the PNP equation for a continuous ion concentration field is not as stable and numerically reliable as particle tracking methods [19, 23], it is rarely applied to the field of mass spectrometry. The DSMC method suffers from prohibitively expensive computational cost at an elevated pressure of the buffer gas. When it is used in an ESI-MS system, it is usually combined with a computational fluid dynamics (CFD) method for the ion source at atmospheric pressure . For the Euler-Lagrange approach with discrete particles and a continuous buffer gas field, the interaction between the ion and the buffer gas can also be calculated by several different methods. One is to use the viscous damping method (the Stokes’ drag law) to describe the net effect of buffer gas on the particles , for example, the discrete phase model (DPM)  in ANSYS Fluent. The Stokes’ drag law, however, does not accurately represent the forces on particles of submicron-scale at high particle Knudsen numbers. A corrected drag law has been considered in this work and will be discussed in detail in the following section on methods. In order to consider the diffusional effects for ion transport, a statistical diffusion simulation (SDS) was introduced [27, 28] to account for both the ion mobility-based drift and random diffusion. The SDS method employs user-defined ion mobility or a predetermined field-dependent ion mobility function to achieve an average drift motion of ion and uses collision statistics to simulate hard-sphere collision-based diffusion of ions. With this method, the combined effect of mobility and diffusion on ion trajectories at a wide pressure range can be predicted with affordable computational resources. It has some applications in the MS system [17, 29]. Another method in the Euler-Lagrange frame is a hard-sphere collision-based Monte Carlo approach  to simulate ion-gas collision in a continuous buffer gas field, which is different from the DSMC method with collisions between discrete ion particles and buffer gas molecules. This Monte-Carlo method is widely used in ESI-MS systems [31, 32]. However, under atmospheric pressure conditions, the Monte-Carlo method is computationally prohibitive for a continuous injection of ions due to the extremely high ion-gas collision frequency at elevated pressures.
There are several commercial software packages for ion tracking in an MS system. The ion optics simulation software, SIMION [33, 34, 35], is a common tool used in the MS industry to solve gas-phase ion transport. Nevertheless, SIMION cannot solve the gaseous flow field but relies on the flow variables to be imported from an external CFD solver. COMSOL Multiphysics is another simulation tool commonly used in MS [24, 32, 36, 37]. It is based on a finite element method, which has limited numerical techniques to accurately solve very complex flows. For an ion source at atmospheric pressure involving complex flow and electric fields, both impacting the motions of ions, CFD solvers such as ANSYS Fluent with customized functionality is an excellent choice for ion transport simulations. ANSYS Fluent simulations of MS at atmospheric pressure have been reported previously [31, 38]. A major advantage of these approaches is that it is capable of computing ion trajectories within any arbitrary 2D or 3D geometry under the combined influence of complex gaseous flow and electric field using a single software package.
In order to take account of the space-charge effects, SIMION introduces a charge repulsion method based on empirical estimations for an ion beam or ion cloud. This method is also adopted by Jurcicek et al.  in their ANSYS Fluent simulation. Another method is to apply an analytical expression for the Coulomb force for a simple geometry , which cannot, however, be extended for arbitrary geometries as in the case of an ion source chamber. Therefore, most of the previously published numerical works simulated a fixed initial ion number and distribution (single pulsed injection), where only the ion transport is captured and the real space-charge effects are most likely to be inaccurate due to the lack of a realistic ion distribution that would result from a continuous injection of ions. In this numerical study, ANSYS Fluent was customized to simulate ion trajectories, taking account of realistic space-charge effects in a continuous injection mode of nano-ESI in a cone-gas influenced ion source flow field. Using the developed methods, electric field distortion by space-charge effects and the ion transmission efficiency at different cone gas flow rates, capillary voltages, and the source currents were areas of focus for the study. Experiments were also carried out to complement and validate the simulations, wherever possible.
The Governing Equations
ANSYS Fluent is a commercial CFD software package based on the finite volume method to discretize partial differential equations for complex arbitrary geometries. It allows users to customize functionalities to enhance its capabilities with the aid of user-defined functions (UDF), user-defined material properties, user-defined particle injection files, etc. The governing equations for the flow field are the conservation equations of mass, momentum, and energy . In the current study, a compressible turbulent flow is modeled using a Reynolds-Averaged Navier-Stokes (RANS) method. The density-based solver is chosen because of a rapid change of pressure and temperature that occurs near to the MS inlet and the k-ε turbulence model with an enhanced wall treatment is applied to capture turbulence statistics. The second order upwind scheme is used to discretize the convection terms in the governing equations. The library “Metis”  is applied for partitioning the computational domain for parallel computation.
The motion of ions in a mass spectrometer ion source at atmospheric pressure is governed mainly by forces from the gas flow field, the electric field, and the space-charge effects. The forces on ions from the gas flow field can be modeled either using gas-ion collision models or suitable drag force models. The kinetic hard-sphere collision models typically used for computing ion-molecule collisions are suited to only low pressure (vacuum) conditions. At atmospheric pressure, the collision frequency is extremely high and the time step involved is extremely small, so that the associated computational cost for ion transport modeling in realistic geometries is prohibitively high, even using the fastest computers. In this study, focusing on the ion source chamber at atmospheric pressure, the drag forces and other forces from the gas flow field are calculated explicitly. The Lagrangian discrete phase model in ANSYS Fluent is applied to model ion transport in continuous injection mode under an unsteady electric field and a steady flow field.
Validation of the Corrected Drag Model for Gas-Phase Ions
The drag model up to a Reynolds number of 106 can be found in Table 5.2 of Clift et al. .
Reduced Ion Mobility of Ions in a Drift Tube. The Simulation Data has been Computed Using Electric Fields of 100, 200, 300, 400, and 500 V/cm and is Shown in the Format of Mean and Variance
Ion mass (Da)
K0 [cm2/(V sec)]
Estimated by the Mason-Schamp equation
Simulated by the hard sphere collision model
Simulated by the drag model
Results and Discussion
Computational Model and Configuration
Because of a faster desolvation rate in nano-ESI compared with conventional ESI, this study makes an initial assumption as in previous studies  that fully desolvated gas-phase ions are instantaneously produced at the emitter tip. The gas-phase ion injection frequency is 10 MHz for a continuous injection mode. This injection interval is about one-thousandth of a typical transmission time for ions in a nano-ESI. After a study of different injection time intervals, the current one is validated to generate an effectively continuous injection. Based on the shape of a nano-ESI spray plume observed in experiments , the injected verapamil ions (m/z = 455.6, charge state = +1) were initially randomly seeded in a circular plane with a radius of 0.5 mm and located 0.5 mm in front of the emitter tip. The velocity of particles is initialized to follow the Maxwell-Boltzmann particle velocity distribution.
All simulations in this work were performed in parallel mode using Open MPI  on an IBM iDataPlex system called the Blue Wonder. Blue Wonder is located at the Hartree Center in the UK, and each node in this system consists of 2×12 core Intel Xeon processors (Ivy Bridge E5-2697v2 2.7GHz) and 64GB RAM. Each parallel simulation case in this study used 32 cores. The computational time mostly depends on the number of tracking ions existing in the source chamber. For example, for the case with an ion source current of 30 nA, a cone gas flow rate of 50 slph, and a capillary voltage of 2 kV, the transient physical time to reach the equilibrium state is about 0.2 ms and the equilibrium tracking particle number is about 10,000. For this case, it took around 6.5 h of wall-clock time with 32 cores to reach the equilibrium state. Upon reaching equilibrium, it took about 0.5 h of wall-clock time to advance 0.01 ms of physical time in the unsteady particle tracking.
The flow field is determined by a combination of the low pressure at the MS inlet and the counter-current flow of the cone gas. The exact pressure at the MS inlet is unknown beforehand, as it is related to the pressure in the first vacuum region of the MS. MS inlet pressures from 0.1 to 1 atm have been studied, and show that if the pressure is lower than approximately 0.45 atm in the current configuration, the velocity profiles in the dominant region of the source chamber show no discernible differences due to choked flow at the MS inlet. A gas flow study of a different type of nanospray ESI source (a heated capillary connecting the atmosphere and a fore-vacuum volume)  found that MS inlet pressures lower than around 0.4 atm had little influence on the mass flow rate into the MS. In order to rule out the influences of the MS inlet pressure on flow field and to focus on the flow field in the source chamber only, a pressure of 0.4 atm at the MS inlet in this study was chosen.
The electric potential distribution depends on the applied voltage on the emitter, the ion source chamber geometry, and the instantaneous charge density distribution. When the geometry remains unchanged, the real-time electric field is determined by the voltage of the capillary, the ion source current, and the ion distribution.
Comparison of Forces on Ions
In order to improve the accuracy of MS source simulations, it is necessary to consider the relative importance of all the dominant forces on the ions. In this section, a typical nano-ESI spray case with an ion source current of 30 nA, a cone gas flow rate of 50 slph and a capillary voltage of 2 kV is considered.
The corrected drag force and electric forces have been described in the previous section. Additional forces on the ion particle include: (a) a thermophoretic force that is linearly dependent on the normalized temperature difference ΔT/T; (b) a Brownian force that is corrected with a Cunningham factor; (c) a Saffman lift force that is a shear lift originating from the inertia effects in the viscous flow around the particle. The turbulence dispersion of ions is predicted by the stochastic tracking method. When particle rotation in the fluid is not considered, the Magnus lift force and the rotation lift force can be neglected. As the particle density of the ion is much higher than the buffer gas (nitrogen) density, the “virtual mass” force to accelerate the surrounding fluid and the pressure gradient forces are negligible. The definition and physical meanings of these forces can be found in [26, 54, 55].
The influence of the cone gas flow rate on the ion destinations is plotted in Figure 6b. When the cone gas flow rate is less than 250 slph, changes to the buffer gas flow only redistribute the ion loss between deposition on the cone wall and on the other walls, and have little influence on the total ion loss, ion transmission, and the ion concentration in the chamber. When the cone gas flow rate is greater than 250 slph, the buffer gas becomes counter-flowing (an opposing force for the ion transport), where the stronger cone gas slows ion transport towards the MS inlet and increases the number of surviving ions in the source chamber, corresponding to enhanced space-charge effects and increased ion dispersion. The ion transmission efficiency into the MS decreases from 16% to 4% when the cone gas flow rate increases from 250 to 450 slph. Since the total source current is fixed in all these cases, the trend of the current into the MS is equivalent to the ion transmission efficiency. The ion current into the MS with a flow rate less than 250 slph is approximately 5 nA, whereas it reduces to around 1 nA with a flow rate of 450 slph.
The influence of the capillary voltage on the ion destination is shown in Figure 6c. For a cone gas flow rate of 50 slph and a source current of 30 nA, the ion transmission efficiency almost doubles for an increase in capillary voltage from 1 to 2 kV, but the enhancement is less marked from 2 to 3 kV. This trend is accompanied by a dramatic reduction in the number of surviving ions in the source from 1 to 2 kV and only a slight decrease from 2 to 3 kV. Less sensitivity to a higher voltage was also observed in . The reason is mainly that a lower capillary voltage generates a smaller electric driving force for ions, leading to a higher ion concentration under continuous ion injection and, in turn, to greater space-charge ion losses. However, the benefit of an increased driving force (capillary voltage) is not boundless, since the shrinking plume and increasing ion concentration is counterbalanced by increased space-charge that ultimately leads to a dynamic equilibrium. This accounts for the observed flattening of the response for ions transported in the MS at increasing capillary voltages (shown in Figure 6c).
The influence of the ion source current on the transmission efficiency at a fixed cone gas flow rate of 50 slph has also been studied, and the computed results are shown in Figure 6d and e. An increase in the source current enhances the space-charge effects and encourages ion dispersion. At the same time, the increased ion concentration in the chamber promotes an enhanced electric driving force for ions in the main transport path (Figure 4a), which reduces the time duration of ion dispersion. As a result of these competing effects, the simulations show that the ion transmission efficiency decreases exponentially with increasing source current, but the total ion current into the MS increases slowly from 3 nA at a source current of 10 nA to 6 nA at a source current of 110 nA (Figure 6e). This trend of an exponential decrease of the ion transmission was also observed in other types of MS .
Summarizing the above analysis, the basic mechanism of ion transport depends on the interaction between the assisting forces, the opposing forces, and ion dispersion. Stronger driving/assisting forces (e.g., the buffer gas with a cone gas flow rate of less than 250 slph and the electric force) speed the transport of ions and reduce ion concentration in the source chamber, corresponding to smaller space-charge effects on ion dispersion and ion loss. Opposing forces (e.g., cone gas flow rates greater than 250 slph) have an opposite effect. The ion dispersion mainly comes from the space-charge effects, which depends on the ion concentration (or charge density). However, factors that increase the number of surviving ions in the source (e.g., a higher ion source current, a weaker driving force, or stronger opposing force) do not promote a linear increase of the ion concentration due to the expansion of the ion plume under the influence of space-charge. Thus, the ion dispersion and ion loss in the ion source chamber has a feature of self-confinement to some extent, which explains the asymptotic characteristics of the ion transmission efficiency under reduced cone gas flow rates, increased capillary voltage, and increased ion source current. In all, for this study, it is observed that a reasonable balance between high transmission efficiency and useful MS signal is obtained at a cone gas flow rate not exceeding 250 slph, a capillary voltage of about 2 kV, and a source current of between 30 and 40 nA. These observations agree with empirical optimization of typical experimental settings.
Figure 8e shows the evolution of the ion current through the MS inlet from an initial injection at t = 0 s for the continuous injection of ions at different cone gas flow rates. The ion signal is recorded every 0.002 ms and a centered moving average running at a time interval of 0.02 ms is applied to smooth signals. The time taken to reach an equilibrium current is longer for higher cone gas flow rates. Having established an equilibrium current, it is found that the ion transmission time is around 0.2 ms for cone gas flow rates of less than 250 slph. A further increase of the cone gas flow rate leads to a longer transmission time, for example, 0.4 ms for a cone gas flow rate of 450 slph. The dynamic process of reaching an equilibrium state also implies that a continuous injection of ions is necessary to obtain realistic ion transport characteristics.
In this study, ANSYS Fluent was customized to simulate flow fields, electric fields, and ion transport in a nanospray ion source under different operating conditions. The particle-in-cell method was implemented to take into account the local real-time charge density. The real-time space-charge effects on electric field distortion and ion transport characteristics in a continuous ion injection mode have been numerically studied. A drag model with the Cunningham slip correction factor has been validated to accurately predict ion transport properties at atmospheric pressure conditions based on a comparison with an ion-gas hard sphere collision model. After comparing the five main types of forces on ions, it was found that the drag force, the electric force, and the Brownian force are the dominating influences, whereas the thermophoretic force and the Saffman lift force are negligibly small for most of the ion transport path. Ion transmission efficiency and ion destinations have been studied at different cone gas flow rates, different source currents, and different capillary voltages, and the computed results reveal valuable insights. Both the experimental and simulation results show that cone gas flow rates of ≤250 slph are necessary for high ion transmission efficiency, as higher cone gas flow rates reduce the ion signal significantly. The computed results also show that the ion transmission efficiency reduces exponentially with an increased ion source current. In this study, only gas-phase ions (and droplets with fixed diameters) are simulated. The developed code in this study also provides a framework to simulate droplets with a given initial diameter and charge distribution for investigating droplet desolvation and transport in a standard ESI source, which will be carried out in a future work.
The work is part of a Knowledge Transfer Partnership (KTP) project supported by Innovate UK (project ID: KTP010047). The authors thank all members of the local management committee of this KTP project. D.R.E. and B.J. thank the Engineering and Physical Sciences Research Council (EPSRC) under program grant EP/N016602/1 and EPSRC grants EP/K038664/1, EP/K038621/1, and EP/K038427/1. The authors also express their gratitude to David Langridge and Gordon Jones from Waters Corporation for helpful discussions.
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