Computational study of electrostatic focusing of aerosol nanoparticles using an einzel lens

Highlights

• Trajectory simulations are used to study the focusing of aerosol nanoparticles in a 3-electrode einzel lens.

• The focusing in vacuum is greatly influenced by a ratio of electrostatic potential energy to kinetic energy ${\chi }_e$.

• The focal length is seen to vary inversely with ${\chi }_e$.

• Focusing performance deteriorates with increasing gas pressure.

• A maximum operating pressure for efficient particle focusing and a minimum pressure for vacuum-like behavior is identified.

• Considerations for successfully selecting operating parameters (${\chi }_e$ and gas pressure) are discussed.

Abstract

This study computationally explores the possibility of focusing charged aerosol nanoparticles using electrostatics, similar to focusing of electrons and ions. A non-dimensional electrostatic focusing parameter ${\chi }_e$, defined as the ratio of electrostatic potential energy to the kinetic energy of an aerosol nanoparticle, significantly determines focusing performance. The focusing device considered here is a 3-electrode electrostatic (“einzel”) lens. The average focal length of the lens is seen to have an inverse power relationship with ${\chi }_e$. For low values of ${\chi }_e$ ∼3 in this study, the particles are seen to cross the lens axis once, while at higher ${\chi }_e$ multiple axis cross-over points appear. Similar to electron and ion optics, nanoparticle focusing is also limited by spherical aberration and beam divergence due to finite spread of particles in the inlet cross section of the lens and spatial non-uniformity of the focusing electric field. Other factors that influence focusing performance such as the electrostatic lens geometry, and the distribution of velocity and kinetic energy of the particles at the inlet of the lensing region are recognized, but not considered here for simplicity. In vacuum, good focusing performance (i.e.) a narrow beam of nanoparticles with minimum spherical aberration and small divergence angle is theoretically possible if ${\chi }_e$<1 and if spread of particles in the inlet is confined to 20% of radius of the cylindrical lens. The effect of gas pressure is also probed to understand the degradation of focusing performance due to particle-gas interactions. It is seen that, for particles of specified size and density, a certain maximum pressure exists beyond which the device can no longer be efficiently used to focus nanoparticles. Likewise, below a certain pressure, the focusing performance is nearly independent of gas pressure, thereby enabling the selection of an operating pressure for such devices.

Introduction

Focusing of aerosol (gas-phase) nanoparticles into narrow beams is motivated by applications in aerosol mass spectrometry (Deng et al. (2008); Huffman et al. (2005); Schreiner, Schild, Voigt, and Mauersberger (1999)), particle jet printing applications (Lin, Cole, & Jacobs, 2010; Tse & Barton, 2015), micro-patterning (Di Fonzo et al., 2000; Dong, Bapat, Hilchie, Kortshagen, & Campbell, 2004; Qi, McMurry, Norris, & Girshick, 2010), and the fabrication of three-dimensional microstructures (Akedo, Ichiki, Kikuchi, & Maeda, 1998). Murphy and Sears (1964) pioneered the generation of aerosol particle beams by flowing particles through a series of capillaries, later adopted by others (Allen & Gould, 1981; Hall & Beeman, 1976; Kievit, Marijnissen, Verheiljen, & Scarlett, 1992; Seapan, Selman, Seale, Siebers, & Wissler, 1982; Sinha & Friedlander, 1986). Although experimentally demonstrated, this method was not supported by analysis of particle motion to enable the systematic design of such focusing devices. Alternative to vacuum focusing is the use of sheath gas flow to confine particle beams to narrow cross sections by limiting their transverse diffusional broadening. While the sheath flow reduces the beam diameter effectively by a factor of ∼10 (Dahneke & Cheng, 1979; Dahneke & Flachsbart, 1972), it also dilutes the particle concentration leading to decreased particle detection sensitivity for mass spectrometry or low throughput for patterning applications.

To overcome the difficulties associated with the sheath gas and to obtain higher aerosol transport efficacy than capillaries, Liu, Ziemann, Kittelson, and McMurry (1995a) designed the aerodynamic lens that consists of a series of contractions and expansions of flow cross section achieved by the use of orifice plates. For a particle-laden flow, the aerodynamic lens provides the same focusing effect as sheath air without additional gas handling. The aerodynamic focusing of particles is based on their propensity to move towards the centerline of an axisymmetric flow when moving through successive contractions and expansions (Robinson, 1956), provided their inertia is less than the critical inertia to avoid collision with the walls of the flow tube (Hinds, 2012). Prior to Liu et al., De La Mora and Riesco-Chueca (1988) showed that particle inertia (described by a Stokes number that compares particle relaxation time to the fluid advection time scale) leads to focusing of particles onto a single spot and a crossover point on the axis of a flow. Their conclusions were drawn from calculated trajectories of particles in an incompressible flow through a nozzle, with Brownian motion neglected. The computational investigation described in this paper draws inspiration from Fernandez de la Mora's approach of quantifying focusing outcomes as well as the calculation of trajectories with one-way coupling to an advection field (Fernandez de la Mora, 2006; De La Mora and Riesco-Chueca (1988)) – in that work, incompressible flow field was employed, while we investigate the effect of electrostatic field in vacuum and at finite pressures (without a systematic fluid flow field). The minimum beam width achieved using the inertial focusing method of Liu et al. (1995a) approaches ∼0.4 mm, that increases with decreasing particle size as demonstrated using spherical dioctyl sebacate particles in the range of ∼50–250 nm (Liu, Ziemann, Kittelson, & McMurry, 1995b). Several designs of aerodynamic lenses have been used to effectively collimate nanoparticles in the range of 100–900 nm (Schreiner et al., 1999), 340–4000 nm (Schreiner, Voigt, Mauersberger, McMurry, & Ziemann, 1998), 60–600 nm (Zhang et al., 2004), 3–30 nm (Wang, Kruis, & McMurry, 2005), 30–300 nm (Lee, Cho, & Lee, 2008), 5–50 nm (Lee, Kim, & Lee, 2009) and 30 nm–10 μm (Lee, Hwang, Kim, Kim, & Lee, 2013). The beam width produced by this method is limited by Brownian motion and lift forces on the particles during expansion through the orifices and the exit nozzle of the lens. Overcoming the Brownian limit of beam broadening is theoretically impossible without the application of radial forces by external means (such as electric fields for example). Thus, reduction of beam width beyond those achieved by the aerodynamic lens has been challenging and has not been accomplished so far.

Alternate to the inertial particle focusing mechanism of the aerodynamic lens, several attempts have been made to use electrostatic and electrodynamic forces or a combination of both fluid and electric forces to focus particles. Electron and ion focusing devices using applied electric fields have been harnessed for many applications such as electron microscopes, cathode ray tubes, ion beam milling apparatus and drift tube mobility spectrometry (Cumeras, Figueras, Davis, Baumbach, & Gràcia, 2015; Fernández-Maestre, 2012; Oberreit & Hogan, 2015). The ion/electron trajectories in these devices are manipulated using a series of ring/planar electrodes with an applied voltage gradient to confine them to a narrow region around the axis. The analogous use of electric fields to focus aerosol nanoparticles could potentially mitigate beam broadening by Brownian motion and be instrumental in producing narrow beams than is currently possible using inertial focusing alone. The charge and electrical mobility (which is dependent on the gas pressure) of particles determine their response to an applied electric field. Electric fields have been used numerously to manipulate the trajectories of aerosol particles for measurement and patterning. Knutson and Whitby (1975) developed the differential mobility analyzer that spatially separates particles based on their electrical mobility or size (for spheres). The experimental verification Liu et al. (1995b)'s design of aerodynamic lens (Liu et al., 1995a) used electrostatic fields to deflect charged particles to measure their nominal velocity in a focused beam. Kane, Oktem, and Johnston (2001) used an electrostatic lens to concentrate nanoparticles before introducing into the time-of-flight detector of a mass spectrometer for improved sensitivity. They have observed that electrostatic focusing increases the hit rate (sensitivity) by increasing the overlap of the laser beam with the particle beam. The deposition of charged nanoparticles (<5 nm) of diverse materials using photoresists (for selective area deposition) and external biasing of voltages has enable the creation of nano-patterns and are successful demonstrations of the utility of electric fields to control particle motion advantageously (Choi et al., 2015; Kim et al., 2006; Krinke, Deppert, Magnusson, Schmidt, & Fissan, 2002; Lin et al., 2010; Park, Jeong, Kim, & Hwang, 2013; You & Choi, 2007; You et al., 2010).

Masuda, Fujibayashi, Ishida, and Inaba (1972) used a set of parallel cylindrical electrodes, separated by insulating spacers and connected to an alternating voltage source that produced a spatially periodic electric field in the focusing region. Charged aerosol particles were shown to have periodic motion along the curved lines of force and were repulsed from the electrode due to the action of centrifugal force and electric force. Based on the different electrode configurations, the particles can either levitate or levitate and accelerate simultaneously along the lens axis. Based on the same methodology, Hutchins, Holm, and Addison (1991) designed a cone frustum shaped screen having an entrance and exit diameter of 7.0 cm and 2.5 cm respectively with a length of 17.0 cm for electrodynamic focusing of charged particles and achieved minimum beam width ∼ 1 mm. They have observed that 5.2 μm particles could be focused to ∼2–4 mm beam widths for electric elementary charges of 2000–6000, positive or negative charges on the particles. As aerosol particles are much heavier and have lower velocities than electrons and ions, it is conceivable that they require considerably higher number of electric charges to respond to the applied field ($\overrightarrow{F}=q\overrightarrow{E})$.

Heise and Rang (1949) have used a simple 3-electrode einzel lens to focus electron beams experimentally, analogous to light. An einzel lens is made of three ring electrodes (separated by insulating spacers), with the first and third electrodes held at the same voltage (and of the same length) while the second electrode is held at a different voltage to create a voltage gradient for focusing. The numerical calculations of electron focusing using einzel lenses that relate the focal length and the operating parameters (voltage and geometry) developed by Adams and Read (1972) have been used numerously to design charged particle focusing devices (Chang et al., 1996; Odenthal, 1991). Computational studies have been used to understand electrostatic particle deposition and inspires our use of trajectory simulations to parameterize focusing using electrostatic fields (Rusinque, Fedianina, Weber, & Brenner, 2019). A systematic exploration of the motion of charged nanoparticles particles to understand electrostatic focusing using a cylindrical einzel lens with a simple 3-electrode geometry is carried out in this study. Motivated by ion and electron focusing using einzel lenses, it is desirable to deduce the operating parameters (particle velocity and charge, strength of electric fields and gas pressure) for successful focusing of nano- and micro-particles beyond the Brownian diffusion limit. This study, using trajectory simulations, computationally explores the electrostatic focusing of aerosol nanoparticles to understand the effect of particle parameters (material, kinetic energy/velocity, size, number of charges), lens geometry, operating voltage/applied electric field and gas pressure on focusing performance (quantified by the focal length, spherical aberration and divergence angle of particle beams). The comparison between the electric potential energy of the particle to kinetic energy determines the ease with which they are deflected towards the lens axis by the applied electric field. The thermal energy of the particles as well as the drag exerted by the gas medium on their motion are also important in determining focusing outcomes. We also identify conditions in which the spherical aberration and divergence angle of the focused beam can be minimized and deduce the upper limit of gas pressure at which an einzel lens acts as a focusing device without significant distortion by collisions between particles and background gas molecules. Lastly, we elucidate qualitative relationships between focal length, spherical aberration and the divergence angle with the ratio of the electric potential energy to the kinetic energy of the particles, the particle Knudsen number as well as particle diameter and material density.