Electrokinetic Instability Research Articles

Mechanism of sinuous and varicose modes in electrokinetic instability.

The flow of miscible electrolyte streams with mismatched electrical conductivities in the presence of a parallel applied electric field is known to exhibit electrokinetic instability (EKI). This paper deals with EKI in a configuration where the base state is established by electro-osmotic flow (EOF) of three coflowing streams, with the center stream having different conductivity than the sheath streams. All reported experiments of this EKI have shown that the instability exhibits either sinuous or varicose modes depending upon whether the center stream has higher or lower conductivity than the sheath streams, respectively. In this paper we elucidate the physical mechanism underlying the selection of these unstable modes in EKI using linear stability analysis. The stability analysis shows that the EOF simply convects the unstable modes besides establishing the base state. The instability occurs due to stationary convection cells, in the reference frame moving with the EOF, resulting from the coupling of the applied electric field with free charge in the regions with conductivity gradients. Importantly, we show that the unstable and stable disturbances for the configuration with a higher conductivity center stream have opposite stability characteristics when the center stream has lower conductivity than the sheath streams. Our analysis correctly explains the numerous experimental observations showing the consistent appearance of sinuous modes for higher conductivity and varicose modes for lower conductivity center streams.

Investigation of Operational Parameters for Nanoelectrokinetic Purification and Preconcentration.

This work reports on experimental investigations into the operational parameters of nanoelectrokinetic purification and preconcentration, especially utilizing on ion concentration polarization (ICP). ICP as a nanoscale electrokinetic phenomenon has demonstrated promising advances in various fields utilizing an ion depletion zone (IDZ) with a steep electric field gradient inside the ICP layer. However, the inevitable electrokinetic instability occurring within the IDZ has posed a challenge in operating the ICP system stably. To address the need for a stable and efficient ICP operation in various devices and applications, we propose an operational strategy along with conducted research to determine optimal operating ranges. In order to investigate the operational parameters, a unit voltage (VTH) is introduced as the threshold for initiating ICP. We examined the applicability of VTH across various operating ranges to ensure its effectiveness and versatility. In ICP purification, we categorize three modes (steady, burst, and unsteady) based on IDZ expansion and stability under varying VTH and flow rate conditions, presenting optimal operational conditions that minimize the voltage margin. In ICP preconcentration, a systematic investigation is conducted to observe the influence of background electrolyte concentration and voltage conditions on preconcentration efficiency, offering insights into the correlation between preconcentration factor, electrical conditions, and preconcentration time. Therefore, this research would contribute to the practical understanding of nanoelectrokinetics, providing insight into experimental designs. These findings are expected to offer valuable guidance to researchers aiming to utilize ICP's potential across a spectrum of applications, from purification to preconcentration, in the realm of micro/nanofluidic systems.

Electrokinetic instability of a highly charged and weakly diffusing analyte in a buffer electrolyte near an ion-selective surface

Electrokinetic instability of a highly charged and weakly diffusing analyte in a buffer electrolyte near an ion-selective surface

Fast electrokinetic mixing in microflows with different electrical conductivities

This work provides a novel three-dimensional numerical analysis of electrokinetic mixing in microflows with different electrical conductivities. Our study examines ferrofluid and water co-flow mixing quantitatively, which has been overlooked in previous micromixers based on electrokinetic instabilities. This has become possible by considering the actual diffusivity of ferrofluid. The findings reveal that thinner channels exhibit stronger suppression of instability due to the pronounced effect of walls on the flow, emphasizing the importance of 3D simulation for shallow channels. Additionally, increasing the electric field intensity was found to have dual opposing effects on instabilities and electrolyte mixing. It enhances the body force and instabilities, improving mixing, but also accelerates flow velocity, reducing fluid residence time and leading to weaker mixing of the flows. A notable observation was a decrease in the mixing index with the augmentation of electric field intensity; for a channel with H = 200 µm, the mixing index reduced by approximately 13% when the electric field strength was increased from 179 to 210 V/cm.

Electrokinetic and Electroconvective Effects in Ternary Electrolyte Near Ion-Selective Microsphere.

The paper presents theoretical and experimental investigations of the behavior of an electrolyte solution with three types of ions near an ion-selective microparticle with electrokinetically and pressure-driven flow. A special experimental cell has been developed for the investigations. An anion-selective spherical particle composed of ion-exchange resin is fixed in the center of the cell. An enriched region with a high salt concentration appears at the anode side of the particle when an electric field is turned on, according to the nonequilibrium electrosmosis behavior. A similar region exists near a flat anion-selective membrane. However, the enriched region near the particle produces a concentration jet that spreads downstream akin to a wake behind an axisymmetrical body. The fluorescent cations of Rhodamine-6G dye are chosen as the third species in the experiments. The ions of Rhodamine-6G have a 10-fold lower diffusion coefficient than the ions of potassium while bearing the same valency. This paper shows that the concentration jet behavior is described accurately enough with the mathematical model of a far axisymmetric wake behind a body in a fluid flow. The third species also forms an enriched jet, but its distribution turns out to be more complex. The concentration of the third species increases in the jet with an increase in pressure gradient. The pressure-driven flow stabilizes the jet, yet electroconvection has been observed near the microparticle for sufficiently strong electric fields. The electrokinetic instability and the electroconvection partially destroy the concentration jet of salt and the third species. The conducted experiments show good qualitative agreement with the numerical simulations. The presented results could be used in future for implementing microdevices based on membrane technology for solving problems of detection and preconcentration, and thus simplifying chemical and medical analyses utilizing the superconcentration phenomenon. Such devices are called membrane sensors, and are actively being studied.

Rapid prototyping of polydimethylsiloxane (PDMS) microchips using electrohydrodynamic jet printing: Application to electrokinetic assays.

Polydimethylsiloxane (PDMS) based microfluidic devices have found increasing utility for electrophoretic and electrokinetic assays because of their ease of fabrication using replica molding. However, the fabrication of high-resolution molds for replica molding still requires the resource-intensive and time-consuming photolithography process, which precludes quick design iterations and device optimization. We here demonstrate a low-cost, rapid microfabrication process, based on electrohydrodynamic jet printing (EJP), for fabricating non-sacrificial master molds for replica molding of PDMS microfluidic devices. The method is based on the precise deposition of an electrically stretched polymeric solution of polycaprolactone in acetic acid on a silicon wafer placed on a computer-controlled motion stage. This process offers the high-resolution (order10 m) capability of photolithography and rapid prototyping capability of inkjet printing to print high-resolution templates for elastomeric microfluidic devices within a few minutes. Through proper selection of the operating parameters such as solution flow rate, applied electric field, and stage speed, we demonstrate microfabrication of intricate master molds and corresponding PDMS microfluidic devices for electrokinetic applications. We demonstrate the utility of the fabricated PDMS microchips for nonlinear electrokinetic processes such as electrokinetic instability and controlled sample splitting in ITP. The ability to rapid prototype customized reusable master molds with order 10 m resolution within a few minutes can help in designing and optimizing microfluidic devices for various electrokineticapplications.

Strong effect of fluid rheology on electrokinetic instability and subsequent mixing phenomena in a microfluidic T-junction

When two fluids of different electrical conductivities are transported under the influence of an electric field, the electrokinetic instability (EKI) phenomenon often triggers in a microfluidic device once the electric field strength and conductivity gradient exceed some critical values. This study presents a detailed numerical investigation of how the rheological behavior of a fluid obeyed by the non-Newtonian power-law constitutive relation could influence this EKI phenomenon in a microfluidic T-junction. We find that as the fluid rheological behavior changes from shear-thickening (n >1) to shear-thinning (n <1), the EKI phenomenon is significantly influenced under the same conditions. In particular, the intensity of this EKI phenomenon is found to be significantly higher in shear-thinning fluids than in Newtonian and shear-thickening fluids. Also, the critical value of the applied electric field strength for the inception of this EKI phenomenon gradually increases as the fluid rheological behavior progressively moves from shear-thinning to shear-thickening. The corresponding mixing phenomenon, often achieved using this EKI phenomenon, is also notably higher in shear-thinning fluids compared to Newtonian and shear-thickening fluids. A detailed analysis of both the flow dynamics and mixing phenomena inside the microdevice is presented and discussed in this study. To perform so, we also employ the data-driven dynamic mode decomposition technique, considered one of the widely used reduced-order models to analyze a dynamical system. This analysis facilitates a better understanding of the EKI-induced chaotic convection and mixing phenomena inside the microdevice. We observe that the spatial expanse and intensity of the coherent flow structures differ significantly as the power-law index changes, thereby providing valuable insight into certain aspects of the underlying flow dynamics that, otherwise, are not apparent from other analyses.

Electrokinetic Instability in Viscoelastic Fluids in Microgravity Conditions

Electrokinetic Instability in Viscoelastic Fluids in Microgravity Conditions

Fluid viscoelasticity suppresses chaotic convection and mixing due to electrokinetic instability

When two fluids of different electrical conductivities are transported side by side in a microfluidic device under the influence of an electric field, an electrokinetic instability (EKI) is often generated after some critical values of the applied electric field strength and conductivity ratio. Many prior experimental and numerical studies show that this phenomenon results in a chaotic flow field inside a microdevice, thereby facilitating the mixing of two fluids if they are Newtonian in behavior. However, the present numerical study shows that this chaotic convection arising due to the electrokinetic instability can be suppressed if the fluids are viscoelastic instead of Newtonian ones. In particular, we observe that as the Weissenberg number (ratio of the elastic to that of the viscous forces) gradually increases and the polymer viscosity ratio (ratio of the solvent viscosity to that of the zero-shear rate viscosity of the polymeric solution) gradually decreases, the chaotic fluctuation inside a T microfluidic junction decreases within the present range of conditions encompassed in this study. We demonstrate that this suppression of the chaotic motion occurs due to the formation of a strand of high elastic stresses at the interface of the two fluids. We further show that this suppression of the chaotic fluctuation (particularly, the span-wise one) inhibits the mixing of two viscoelastic fluids. Therefore, one needs to be cautious when the EKI phenomenon is planned to use for mixing such viscoelastic fluids. Our observations are in line with that seen in limited experimental studies conducted for these kinds of viscoelastic fluids.

Computational framework for resolving boundary layers in electrochemical systems using weak imposition of Dirichlet boundary conditions

We present a finite element based computational framework to model electrochemical systems. The electrochemical system is represented by the coupled Poisson–Nernst–Planck (PNP) and Navier–Stokes (NS) equations. The key quantity of interest in such simulations is the current (flux) at the system boundaries. Accurately computing the current flux is challenging due to the small critical dimension of the boundary layers (small Debye layer) that require fine mesh resolution at the boundaries. We present a numerical framework which resolves this challenge by utilizing a weak imposition of Dirichlet boundary conditions for Poisson–Nernst–Planck equations. In this numerical framework we utilize a block iterative strategy to solve NS and PNP equations. This allows us to efficiently and easily implement the weak imposition of Dirichlet boundary conditions. The results from our numerical framework shows excellent agreement when compared to strong imposition of boundary conditions (strong imposition requires a much finer mesh). Furthermore, we show that the weak imposition of the boundary conditions allows us to resolve the fluxes in the boundary layers with much coarser meshes compared to strong imposition. We also show that the method converges as we refine the mesh near the boundaries at a much faster rate compared to strong imposition of the boundary layer. We present multiple test cases with varying boundary layer thickness to illustrate the utility of the numerical framework. We illustrate the approach on canonical 3D problems that otherwise would have been computationally intractable to solve accurately. Lastly, we simulate electrokinetic instabilities near a perm-selective membrane with weakly imposed boundary conditions on the membrane. This approach substantially reduces the computational cost of modeling thin boundary layers in electrochemical systems.

Novel electroosmotic micromixer configuration based on ion‐selective microsphere

In this paper, a micromixer of a new configuration is presented, consisting of a spherical chamber in the center of which an ion-selective microsphere is placed. Stratified liquid is introduced through the chamber via inlet and outlet holes under an external pressure gradient and an external electric field is directed in such a way that the resulting electroosmotic flow is directed against the pressure-driven flow, resulting in mixing. The investigation is carried out by direct numerical simulation on a super-computer. Optimal values of the applied electric field are determined to yield strong mixing. Above this optimal mixing regime, a number of instabilities and bifurcations are realized, which qualitatively coincide with those occurring during electrophoresis of an ion-selective microgranule. As shown by our calculation, these instabilities do not lead to an enhanced mixing. The resulting electroconvective vortices remain confined near the surface of the microgranule, and do not sufficiently perturb the stratified fluid flow further from the granule. On the other hand, another type of instability caused by the salt concentration gradient can generate sufficiently strong oscillations to enhance mixing. However, this only occurs when the external electric field is sufficiently high that the electroosmotic flow is comparable to the pressure-driven flow. This ultimately leads to creation of reverse flows of the liquid and cessation of the device operation. Thus, it was shown that the best mixing occurs in the absence of electrokinetic instability. Based on the data obtained, it is possible to select the necessary geometric characteristics of the micromixer to achieve the optimal mixing mode for a given set of liquids, which may be ten times more effective than passive mixers at the same flow rates. A comparison with the experimental data of the other authors confirms the effectiveness of this device and its other capabilities. Furthermore, the basic device design can be operated in other modes, for example, an electrohydrodynamic pump, a streaming current generator, or even a micro-reactor, depending on the system parameters and choice of an ion-selective granule.

Electrokinetic instability due to streamwise conductivity gradients in microchip electrophoresis

We present an experimental and numerical investigation of electrokinetic instability (EKI) in microchannel flow with streamwise conductivity gradients, such as those observed during sample stacking in capillary electrophoresis. A plug of a low-conductivity electrolyte solution is initially sandwiched between two high-conductivity zones in a microchannel. This spatial conductivity gradient is subjected to an external electric field applied along the microchannel axis, and for sufficiently strong electric fields an instability sets in. We have explored the physics of this EKI through experiments and numerical simulations, and supplemented the results using scaling analysis. We performed EKI experiments at different electric field values and visualised the flow using a passive fluorescent tracer. The experimental data were analysed using the proper orthogonal decomposition technique to obtain a quantitative measure of the threshold electric field for the onset of instability, along with the corresponding coherent structures. To elucidate the physical mechanism underlying the instability, we performed high-resolution numerical simulations of ion transport coupled with fluid flow driven by the electric body force. Simulations reveal that the non-uniform electroosmotic flow due to axially varying conductivity field causes a recirculating flow within the low-conductivity region, and creates a new configuration wherein the local conductivity gradients are orthogonal to the applied electric field. This configuration leads to EKI above a threshold electric field. The spatial features of the instability predicted by the simulations and the threshold electric field are in good agreement with the experimental observations and provide useful insight into the underlying mechanism of instability.

Spatiotemporal Organization in a Reaction-Diffusion Model for Alloy Electrodeposition

Alloy electrodeposition processes have been experimentally shown to exhibit electrokinetic instability that can lead to compositional heterogeneity in the electrodeposit bulk. Inspired by these experimental evidences we consider the DIB model, a morphochemical reaction-diffusion system introduced in [Bozzini et al., J. Solid State Electr.17, 467–479 (2013)] as a mean of understanding the formation of morphological patterns found in the electrodeposition processes. We discuss its rich spatiotemporal dynamics from both the analytical and numerical points of view, revealing interesting classes of morphogenetic scenarios in good qualitative accordance with experiments. The DIB model can in fact undergo to Turing and Hopf instabilities and classical Turing patterns as well as oscillating Turing patterns can be detected. In the Hopf region, we prove the existence of spiral wave behaviour and observe transition to spatiotemporal chaos via spirals break up. The role of cross-diffusion in pattern formation and selection is also discussed as well as some recent extensions of the model to spherical domains.

Joule heating effects on electrokinetic flows with conductivity gradients.

Instability occurs in the electrokinetic flow of fluids with conductivity and/or permittivity gradients if the applied electric field is beyond a critical value. Understanding such an electrokinetic instability is significant for both improved transport (via the suppressed instability) and enhanced mixing (via the promoted instability) of liquid samples in microfluidic applications. This work presents the first study of Joule heating effects on electrokinetic microchannel flows with conductivity gradients using a combined experimental and numerical method. The experimentally observed flow patterns and measured critical electric fields under Joule heating effects to different extents are reasonably predicted by a depth-averaged numerical model. It is found that Joule heating increases the critical electric field for the onset of electrokinetic instability because the induced fluid temperature rise and in turn the fluid property change (primarily the decreased permittivity) lead to a smaller electric Rayleigh number.

Thermodiffusion in Electrolyte Between Electric Membranes Under External Electric Field

Thermodiffusion of a binary electrolyte situated in the microgap between two ion-selective surfaces under external electric field is theoretically studied. It is assumed that temperature gradient occurs only by Joule heating of the electrolyte; and the concentration gradient may trigger an inner gravitational instability. There are three types of instability in such setup: electrokinetic, thermoelectrokinetic and internal gravitational instability. The direct numerical simulation of the problem shows that for the typical salts solutions such as NaCl or KCl the termodiffusion effect is negligible, while for some salts with bigger values of the reduced Soret coefficients of ions (for example, for tetra-n-butylammonium fluoride) the thermodiffusion may significantly affect the voltage-current characteristic. It is also obtained, that the negative thermodiffusion leads to the destabilization of the one-dimensional steady state regime and shifts the critical values of governing parameters for each aforementioned type of instability. The thermodiffusion mostly influences the classical electrokinetic instability and less — the internal gravitational instability. Under effect of both types of thermodiffusion (normal and negative), the thermoelectrokinetic instability manifests at smaller values of Rayleigh numbers.

Transitions and Instabilities in Imperfect Ion-Selective Membranes.

Numerical investigation of the underlimiting, limiting, and overlimiting current modes and their transitions in imperfect ion-selective membranes with fluid flow through permitted through the membrane is presented. The system is treated as a three layer composite system of electrolyte-porous membrane-electrolyte where the Nernst–Planck–Poisson–Stokes system of equations is used in the electrolyte, and the Darcy–Brinkman approach is employed in the nanoporous membrane. In order to resolve thin Debye and Darcy layers, quasi-spectral methods are applied using Chebyshev polynomials for their accumulation of zeros and, hence, best resolution in the layers. The boundary between underlimiting and overlimiting current regimes is subject of linear stability analysis, where the transition to overlimiting current is assumed due to the electrokinetic instability of the one-dimensional quiescent state. However, the well-developed overlimiting current is inherently a problem of nonlinear stability and is subject of the direct numerical simulation of the full system of equations. Both high and low fixed charge density membranes (low- and high concentration electrolyte solutions), acting respectively as (nearly) perfect or imperfect membranes, are considered. The perfect membrane is adequately described by a one-layer model while the imperfect membrane has a more sophisticated response. In particular, the direct transition from underlimiting to overlimiting currents, bypassing the limiting currents, is found to be possible for imperfect membranes (high-concentration electrolyte). The transition to the overlimiting currents for the low-concentration electrolyte solutions is monotonic, while for the high-concentration solutions it is oscillatory. Despite the fact that velocities in the porous membrane are much smaller than in the electrolyte region, it is further demonstrated that they can dramatically influence the nature and transition to the overlimiting regimes. A map of the bifurcations, transitions, and regimes is constructed in coordinates of the fixed membrane charge and the Darcy number.

Instabilities, bifurcations, and transition to chaos in electrophoresis of charge-selective microparticle

Electro-hydrodynamic instabilities near a cation-exchange microgranule in an electrolyte solution under an external electric field are studied numerically. Despite the smallness of the particle and practically zero Reynolds numbers, in the vicinity of the particle, several sophisticated flow regimes can be realized, including chaotic ones. The obtained results are analyzed from the viewpoint of hydrodynamic stability and bifurcation theory. It is shown that a steady-state uniform solution is a non-unique one; an extra solution with a characteristic microvortex, caused by non-linear coupling of the hydrodynamics and electrostatics, in the region of incoming ions is found. Implementation of one of these solutions is subject to the initial conditions. For sufficiently strong fields, the steady-state solutions lose their stability via the Hopf bifurcation and limit cycles are born: a system of waves grows and propagates from the left pole, θ = 180°, toward the angle θ = θ0 ≈ 60°. Further bifurcations for these solutions are different. With the increase in the amplitude of the external field, the first cycle undergoes multiple period doubling bifurcation, which leads to the chaotic behavior. The second cycle transforms into a homoclinic orbit with the eventual chaotic mode via Shilnikov’s bifurcation. Santiago’s instability [Chen et al., “Convective and absolute electrokinetic instability with conductivity gradients,” J. Fluid Mech. 524, 263 (2005)], the third kind of instability, was then highlighted: an electroneutral extended jet of high salt concentration is formed at the right pole (region of outgoing ions, θ = 0°). For a large enough electric field, this jet becomes unstable; the perturbations are regular for a small supercriticality, and they acquire a chaotic character for a large supercriticality. The loss of stability of the concentration jet significantly affects the hydrodynamics in this area. In particular, the Dukhin–Mishchuk vortex, anchored to the microgranule at θ ≈ 60°, under the influence of the jet oscillations loses its stationarity and separates from the microgranule, forming a chain of vortices moving off the granule. This phenomenon strongly reminds the Kármán vortices behind a sphere but has another physical mechanism to implement. Besides the fundamental importance of the results, the instabilities found in the present work can be a key factor limiting the robust performance of complex electrokinetic bio-analytical systems. On the other hand, these instabilities can be exploited for rapid mixing and flow control of nanoscale and microscale devices.

Electrokinetic instability in microchannel viscoelastic fluid flows with conductivity gradients

Electrokinetic instability (EKI) is a flow instability that occurs in electric field-mediated microfluidic applications. It can be harnessed to enhance sample mixing or particle trapping but has to be avoided in particle separation. Current studies on EKI have been focused primarily on the flow of Newtonian fluids. However, many of the chemical and biological solutions exhibit non-Newtonian characteristics. This work presents the first experimental study of the EKI in viscoelastic fluid flows with conductivity gradients through a T-shaped microchannel. We find that the addition of polyethylene oxide (PEO) polymer into Newtonian buffer solutions alters the threshold electric field for the onset of EKI. Moreover, the speed and temporal frequency of the instability waves are significantly different from those in the pure buffer solutions. We develop a three-dimensional preliminary numerical model in COMSOL, which considers the increased viscosity and conductivity as well as the suppressed electroosmotic flow of the buffer-based PEO solutions. The numerically predicted threshold electric field and wave parameters compare favorably with the experimental data except at the highest PEO concentration. We attribute this deviation to the neglect of fluid elasticity effect in the current model that increases with the PEO concentration.

Numerical Investigation of the Effect of Two-Dimensional Surface Waviness on the Current Density of Ion-Selective Membranes for Electrodialysis

Ion-selective membranes are an important component of electrodialysis stacks for desalination. Manufacturing imperfections or slight inhomogeneity of the material can lead to minute membrane surface imperfections. Two-dimensional solutions of the coupled Poisson–Nernst–Planck and Navier–Stokes equations were sought for a perfectly smooth membrane and for membranes with well-defined small-amplitude harmonic surface roughness. The simulations were carried out with the validated rheoEFoam solver by Pimenta and Alves. In the overlimiting regime, the electric field is strong enough for an electrokinetic instability to occur. The instability leads to disturbance growth and the formation of electro-convection cells, which strongly increase the current density. The present simulations show that with an increasing ion concentration and applied voltage, the instability becomes stronger and the overlimiting regime is reached earlier. The limiting current density shows a noticeable dependence on the wavelength of the surface roughness. When the wavelength of the surface roughness is incommensurate with the wavelength of the naturally occurring instability, the limiting current density is increased. Since production membranes will always have some degree of surface roughness, this suggests that membrane surface treatments which favor certain wavelengths may have an effect on the overall membrane performance.

Electro-Kinetic Instability in a Laminar Boundary Layer Next to an Ion Exchange Membrane.

The electro-kinetic instability in a pressure driven shear flow near an ion exchange membrane is considered. The electrochemical system, through which an electrical potential drop is applied, consists in a polarization layer in contact with the membrane and a bulk. The numerical investigation contained two aspects: analysis of the instability modes and description of the Lagrangian transport of fluid and ions. Regarding the first aspect, the modes were analyzed as a function of the potential drop. The analysis revealed how the spatial distribution of forces controls the dynamics of vortex association and dissociation. In particular, the birth of a counter-clockwise vortex between two clockwise vortices, and the initiation of clusters constituting one or two envelopes wrapping a vortex group, were examined. In regards to the second aspect, the trajectories were computed with the fourth order Runge Kutta scheme for the time integration and with the biquadratric upstream scheme for the spatial and time interpolation of the fluid velocity and the ion flux. The results for the periodic mode showed two kinds of trajectories: the trochoidal motion and the longitudinal one coupled with a periodic transverse motion. For the aperiodic modes, other mechanisms appeared, such as ejection from the mixing layer, trapping by a growing vortex or merging vortices. The analysis of the local velocity field, the vortices’ shape, the spatial distribution of the forces and the ion flux components explained these trajectories.