Modeling of ion motion and experimental investigations of ion excitation in a linear quadrupole trap with a 4% added octopole field are described. The results are compared with those obtained with a conventional round rod set. Motion in the effective potential of the rod set can explain many of the observed phenomena. The frequencies of ion oscillation in the x and y directions shift with amplitude in opposite directions as the amplitudes of oscillation increase. Excitation profiles for ion fragmentation become asymmetric and in some cases show bistable behavior where the amplitude of oscillation suddenly jumps between high and low values with very small changes in excitation frequency. Experiments show these effects. Ions are injected into a linear trap, stored, isolated, excited for MS/MS, and then mass analyzed in a time-of-flight mass analyzer. Frequency shifts between the x and y motions are observed, and in some cases asymmetric excitation profiles and bistable behavior are observed. Higher MS/MS efficiencies are expected when an octopole field is added. MS/MS efficiencies (N 2 collision gas) have been measured for a conventional quadrupole rod set and a linear ion trap with a 4% added octopole field. Efficiencies are chemical compound dependent, but when an octopole field is added, efficiencies can be substantially higher than with a conventional rod set, particularly at pressures of 1.4 ϫ 10 Ϫ4 torr or less. . The most widely discussed distortion is the "stretched" ion trap [2], which has the end cap electrodes moved out so that the distance to the end cap, z 0 , is increased over that of an ideal field, z 0 ϭ r 0 ⁄ ͙ 2, where r 0 is the distance from the center to the ring electrode. It has been argued that the addition of higher order multipole fields of the correct sign to 3-D traps improves MS/MS efficiency [1c, 1f, 2], and allows faster ejection at the stability boundary [2,3], to give higher scan speeds and improved mass resolution.There is increasing interest in using linear quadrupoles as ion traps, both as stand alone mass analyzers with radial [4] or axial [5] ejection, or in combination with other mass analyzers (for a recent review see [6]).There is also interest in trapping and exciting ions for MS/MS at the relatively low pressures typical for operation of the last mass analyzing quadrupole in triple quadrupole systems, ca. 3 ϫ 10 Ϫ5 torr [7]. Addition of higher multipoles to a linear ion trap might be expected to provide benefits similar to those seen with 3-D traps. Douglas and coworkers [8] have shown that an octopole field can be added to a linear quadrupole by using rod sets with rods equally spaced from the central axis but with one pair of rods different in diameter than the other pair, as shown in Figure 1. The electric potential within this rod set is given to a good approximation bywhere x is the distance from the center towards a smaller rod, y is the distance from the center towards a larger rod, r 0 is the distance from the center to any rod, and U and V rf are the amplitudes...

Molecular dynamics simulations of the ion clouds stored in a linear quadrupole rf ion trap are performed at low temperatures ͑0.1 mK-15 K͒. The introduction of periodic boundary conditions allowed us to study the translationally uniform ion cloud geometry. The time evolution of the kinetic energy of the ion cloud is monitored during the simulation and the increase in the kinetic energy of the cloud ͑rf heating͒ is determined; its dependence on temperature, trapping voltage, and the number of ions is studied. The dependence of rf heating rate on temperature shows that rf heating is undetectable until the temperature reaches approximately 0.5 K; it then increases rapidly until it reaches a maximum at around 2 K. It starts to decrease slowly at higher temperatures. The rf heating rate is shown to increase very sharply with the amplitude of the trapping voltage. The dependence of the rf heating rate on the number of ions shows the influence of the ion crystal shell structure at low temperatures, and has a simple linear dependence at higher temperatures.

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We have constructed, and tested as mass filters, linear quadrupoles with added hexapole fields of 4%, 8%, and 12%, with and without added octopole fields. A hexapole field can be added to the field of a linear quadrupole by rotating the two y rods toward an x rod. This also adds an octopole field which can be removed by making the x rods greater in diameter than the y rods. In comparison to conventional quadrupole mass filters these rod sets have severely distorted quadrupole fields, with a mix of both even and odd higher spatial harmonics. They allow evaluating the performance of rod sets with strong geometric and field distortions as mass filters. Conventional mass analysis at the tip of the stability diagram has been compared to mass analysis using islands of stability. The stability islands are produced by applying an auxiliary quadrupole excitation field to the quadrupole. We show that with normal mass analysis at the tip of the stability diagram, the transmission, resolution, and peak shapes are relatively poor in comparison to a conventional rod set. However, the use of islands of stability dramatically improves the resolution and peak shape, and in some cases ion transmission, suggesting that mass analysis with islands of stability may provide a method to overcome a wide range of field imperfections in linear quadrupole mass filters.

This paper presents an efficient sharp interface model to study the morphological transformations of misfit particles in phase separated alloys. Both the elastic anisotropy and interfacial energy are considered. The geometry of the material interface is implicitly described by the level set method so that the complex morphological transformation of microstructures can be accurately captured. A smoothed extended finite element method is adopted to evaluate the elastic field without requiring remeshing. The equilibrium morphologies of particles are shown to depend on the elastic anisotropy, interfacial energy as well as the particle size. Various morphological transformations, such as shape changes from spheres to cuboids, directional aligned platelets and particle splitting, are observed. The simulated results are in good agreement with experimental observations. The proposed model provides a useful tool in understanding the morphological transformation of precipitates, which will facilitate the analysis and design of metallic alloys.

Mass analysis with linear quadrupole mass filters is possible by forming "islands" in the stability diagram with auxiliary quadrupole excitation. In this work, computer simulations are used to calculate stability boundaries, island positions, and peak shapes and ion transmission for mass analysis with linear quadrupole mass filters that have added octopole fields of about 2 to 4%. Rod sets with exact geometries that have quadrupole and octopole fields only in the potential, and round rod sets, with multipoles up to N ϭ 10 (the twenty pole term) included in the calculations, show the same stability boundaries, island positions, and peak shapes. With the DC voltage applied to the rods so that the Mathieu parameter a Ͻ 0, conventional mass analysis is possible without the use of an island. With the DC polarity reversed so that a Ͼ 0, the resolution and transmission are poor preventing conventional mass analysis. In principle, mass analysis in an island is possible with operation at either of two tips. Provided the correct island tip is chosen for mass analysis, peak shapes comparable to those with a Ͼ 0 and no excitation are possible, both with a Ͼ 0 and with a Ͻ 0. In the latter case, the use of an island of stability allows mass analysis when the added octopole otherwise prevents conventional mass analysis. As with three-dimensional Paul traps [8], the addition of field distortions to a linear quadrupole ion trap can improve MS/MS efficiency [9] or give faster ejection of ions at a stability boundary [10]. The field distortions are described mathematically by the addition of higher spatial harmonics or multipole fields to the quadrupole field. Methods to add octopole [9c, 11] or hexapole [12] fields to linear quadrupoles have been described.In some applications [3], it is desirable that a linear trap used for MS/MS is capable of mass analysis. Various methods of mass analysis with linear quadrupoles have been described. Conventionally, DC and RF potentials can be applied between the rod pairs [13] to place ions to be mass analyzed at a tip of a stability region to produce a mass filter. The addition of higher multipoles to the field has, in the past, been expected to degrade the performance of a linear quadrupole operated as a mass filter in this way [14]. Nevertheless, it has been found that linear quadrupoles with added octopole [15] and hexapole fields [12a] can in fact be operated as mass filters, provided the DC voltage is applied to the electrodes with the correct magnitude and polarity. Ions can also be mass analyzed in a linear quadrupole by radial ejection through slots in the rods [2]. Alternatively, axial ejection can be used for mass analysis with a linear quadrupole, either with or without trapping of ions. Ions within the quadrupole are excited by dipole or quadrupole excitation, gain sufficient kinetic energy to overcome a potential barrier at the quadrupole exit, and are ejected [3, 16]. Preliminary experiments show that this method can be used for mass analysis with a linear quadrupole that h...

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An efficient parallel Stokes’ solver has been developed for complete description of hydrodynamic interactions between Brownian particles in bulk and confined geometries. A Langevin description of the particle dynamics is adopted, where the long-range interactions are included using a Green’s function formalism. A scalable parallel computational approach is presented, where the general geometry Stokeslet is calculated following a matrix-free algorithm using the general geometry Ewald-like method. Our approach employs a highly efficient iterative finite-element Stokes’ solver for the accurate treatment of long-range hydrodynamic interactions in arbitrary confined geometries. A combination of mid-point time integration of the Brownian stochastic differential equation, the parallel Stokes’ solver, and a Chebyshev polynomial approximation for the fluctuation-dissipation theorem leads to an O(N) parallel algorithm. We illustrate the new algorithm in the context of the dynamics of confined polymer solutions under equilibrium and non-equilibrium conditions. The method is then extended to treat suspended finite size particles of arbitrary shape in any geometry using an immersed boundary approach.