Electromagnetic Forces and Torques: From Dielectrophoresis to Optical Tweezers

Abstract

Electromagnetic forces and torques enable many key technologies, including optical tweezers or dielectrophoresis. Interestingly, both techniques rely on the same physical process: the interaction of an oscillating electric field with a particle of matter. This work provides a unified framework to understand this interaction both when considering fields oscillating at low frequencies─dielectrophoresis─and high frequencies─optical tweezers. We draw useful parallels between these two techniques, discuss the different and often unstated assumptions they are based upon, and illustrate key applications in the fields of physical and analytical chemistry, biosensing, and colloidal science.

Figure 1

Examples of some of the objects that can be manipulated using electromagnetic forces and torques. From the smallest to the largest: atoms, biomolecules (Reprinted with permission from ref (26). Copyright 2001 American Association for the Advancement of Science), nanomaterials, and cells (Reprinted with permission from ref (27). Copyright 2009 The Company of Biologists).

Figure 2

Comparison between low- and high-frequency fields.

Figure 3

Comparison between the equivalent multipole method and the Maxwell’s stress tensor approach. The different multipoles are indexed by their order n in such a way that the first order multipole is the dipole, the second order multipole is the quadrupole, followed by the octupole, and so on. Similarly, the first three field derivatives (calculated in r₀ = 0) are ∇E_t, ∇²E_t = ∇∇E_t, and ∇³E_t = ∇∇∇E_t. Only the first three electrical multipoles and electric field derivatives are shown, but the comparison is easily extended to higher orders and to magnetic fields.

Figure 4

Intensity gradient force in different field distributions. (a) Gaussian field intensity profile producing a potential well. In this case, the particle will fall into the well and find a stable position where the force is equal to 0. (b) Linear field intensity profile. Here, the particle will experience a constant force along the positive direction. p has been set to 2 in both cases, and arbitrary units are employed on the x and y axis.

Figure 5

Generation of different intensity and phase gradient forces depending on the position of the polarization phasor in the complex plane. (a) Positive intensity and phase gradient forces for and > 0. (b) Negative intensity and phase gradient forces for and < 0.

Figure 6

Real and imaginary parts of the polarizability (top) and the corresponding Argand plots (bottom) for a homogeneous sphere with a 100 nm diameter immersed in different media for the cases (a) ϵ_p^′ = 10ϵ₀, ϵ_m^′ = 2.5ϵ₀ and σ_p^′ = 10^–8 S/m, σ_m^′ = 4 × 10^–8 S/m and (b) ϵ_p^′ = ϵ₀, ϵ_m^′ = 10ϵ₀ and σ_p^′ = 10^–7 S/m, σ_m^′ = 10^–8 S/m. In the Argand plots, the red dot represents the center of the locus of α, and the black line is its magnitude.

Figure 7

Real and imaginary parts of the polarizability (top) and the relative Argand plots (bottom) for a sphere with a 100 nm diameter made of (a) silver and (b) gold immersed in air. The phasor moves anticlockwise for increasing frequencies and, as clearly seen in (a), crosses the imaginary axis at the plasmon resonance condition. ϵ_p^′ and ϵ_p^″ have been derived from ref (76), and the radiative corrections of eq 8 have been taken into account.

Figure 8

Intensity gradient generation with low- (blue panels) and high- (pink panels) frequency fields. (a) Traditional DEP setup employing metallic electrodes (Reprinted with permission from ref (25). Copyright 2011 Wiley). (b) iDEP employing insulating posts (Reprinted with permission from ref (25). Copyright 2011 Wiley). (c) LIDEP exploiting the photoconductive properties of a-Si:H (Reprinted with permission from ref (124). Copyright 2008 Springer Nature). Note that planar LIDEP geometries have also been proposed. (d) Near-field hotspot generated around a metallic nanostructure (Reprinted with permission from ref (129). Copyright 2010 American Chemical Society). (e) Optical tweezer generated at the focal point of an objective (Reprinted with permission from ref (130). Copyright 2003 Springer Nature). (f) Optical standing wave generated by the interference between an incident laser beam on a mirror and the reflected beam (Adapted with permission from ref (131). Copyright 2011 Optical Society of America).

Figure 9

Phase gradient generation with low- (blue panels) and high- (pink panels) frequency fields. (a) A typical TWDEP setup, where multiple electrodes are excited with a polyphase AC voltage generating both a vertical (repulsive) and a horizontal (translating) force (Reprinted with permission from ref (148). Copyright 2003 IEEE). (b) TWDEP devices employed to trap analytes (Reprinted with permission from ref (55). Copyright 2009 Springer Nature). (c) Evanescent wave illumination for the near-field excitation of a particle at the interface between two different media (Reprinted with permission from ref (74). Copyright 2002 Optical Society of America). For dielectric media, illumination at an angle greater than the total reflection angle is needed, while for metallic substrates, momentum-matching conditions need to be satisfied to excite plasmonic evanescent modes at the interface. (d) A magneto-optical trap for atoms and molecule cooling and manipulation (Reprinted with permission from ref (170). Copyright 2014 Springer Nature).

Figure 10

Spin and orbital torque. (a) A particle spinning around its center of mass. (b) The same particle can be set into orbit around an external axis. The directions of the intensity and phase gradients for a typical vortex beam are also shown. Here, the torque is a vector coming out of the page toward the reader.

Figure 11

Condition for spin torque generation. On the left, the polarization vector in an isotropic and lossless material points in the same direction as the incoming electric field, thereby preventing the generation of a torque. On the right, the polarization in an anisotropic and/or lossy material can be noncollinear with the electric field, which gives rise to a torque.

Figure 12

Spin torque generation. (a) Phasor (left) and real space (right) diagrams for a right circularly polarized field interacting with a lossy (top) or gain (bottom) particle. (b) Material (left) or shape (right) anisotropies induce different polarizations along two different directions, which generates a torque.

Figure 13

Generation of circularly polarized fields. (a) A typical ROT setup to produce left circularly polarized RF fields (Reprinted with permission from ref (64). Copyright 2017 Wiley), where each electrode is dephased by 90° with respect to the adjacent ones. (b) A combination of a linear polarizer and a quarter-wave plate converts an unpolarized laser beam to a circularly polarized electromagnetic wave (Adapted with permission from ref (180). Copyright 2013 American Chemical Society).

Figure 14

Generations of optical fields carrying orbital angular momentum. In particular, the intensity (top) and phase (bottom) profiles of a vortex beam having (a) l = 1 (Reprinted with permission from ref (205). Copyright 2020 Springer Nature) and (b) l = 3 (Adapted with permission from ref (204). Copyright 2013 Optical Society of America) are shown.

Figure 15

Analyte trapping with the use of intensity gradients. Panel (a) shows, in clockwise order from the top left figure, the use of DEP forces for mitochondrion removal (Reprinted with permission from ref (15). Copyright 2019 Springer Nature), particle trapping (Reprinted with permission from ref (11). Copyright 2014 American Chemical Society), and the use of optical forces for molecular trapping and sensing (Reprinted with permission from ref (226). Copyright 2016 American Chemical Society) and particle manipulation (Adapted with permission from ref (227). Copyright 2006 Optical Society of America). Panel (b) shows, on the top, the use of DEP force spectroscopy to detect Ag ions (Reprinted with permission from ref (228). Copyright 2016 American Chemical Society) and, on the bottom, the use of optical tweezers for molecular force spectroscopy (Adapted with permission from ref (229). Copyright 2013 Biophysical Society).

Figure 16

Analyte trapping with the use of intensity gradients. Panel (a) shows the use of trapping forces to study myosin (Reprinted with permission from ref (23). Copyright 2009 Royal Society of Chemistry) and kinesin molecular motors (Adapted with permission from ref (248). Copyright 2011 Springer Nature). Panel (b) shows, in blue, the fabrication of AFM probes using DEP forces (Reprinted with permission from ref (257). Copyright 2005 American Chemical Society) and, in pink, the fabrication and actuation of a Jack-in-the-box using optical tweezers (Reprinted with permission from ref (258). Copyright 2019 Royal Society of Chemistry).

Figure 17

Electromagnetic binding. (a,b) Binding forces acting on a particle dimer excited with a plane wave propagating along k = (k, 0, 0) and polarized along E^I = (0, 0, E^I) (Reprinted with permission from ref (302). Copyright 2010 American Physical Society). The near field case is shown on the left, while the far-field interaction is shown on the right. Panel (a) depicts the case where R = [0, R sin(φ), R cos(φ)] is perpendicular to k, and the green lines follow the trajectories of particle B toward its equilibrium position, as denoted by dotted edges. Panel (b) shows the case where R = (R, 0, 0) is parallel to k. (c) On the left are 3D metallic electrodes fabricated by assembling and welding multiple Galinstan droplets through dielectrophoresis (Adapted with permission from ref (304). Copyright 2015 Wiley). On the right is a 2D crystal of gold nanoparticles assembled through optical binding (Reprinted with permission from ref (305). Copyright 2015 American Chemical Society).

Figure 18

Analyte transport with the use of intensity gradients. Panel (a) shows particle manipulation, in blue, by utilizing a DEP probe (Reprinted with permission from ref (163). Copyright 2013 Wiley) and, in red, by moving a plasmonic hotspot in space (Reprinted with permission from ref (154). Copyright 2017 American Chemical Society) or by exploiting polarization effects on a metasurface (Reprinted with permission from ref (334). Copyright 2014 American Chemical Society). The top figure in Panel (b) shows the working principle of a Brownian ratchet (Reprinted with permission from ref (337). Copyright 2016 American Chemical Society), while the bottom one shows its application to the transport of DNA molecules (Reprinted with permission from ref (338). Copyright 1999 The National Academy of Sciences).

Figure 19

Strategies for analyte transport and sorting by exploiting phase gradients. (a) Exploitation of propagating plasmonic modes for particle transport (Reprinted with permission from ref (346). Copyright 2013 American Chemical Society). (b) TWDEP electrodes for cells and particles actuation (Reprinted with permission from ref (289). Copyright 2001 Wiley). (c) From top to bottom, the panel shows the use of SLMs for particle sorting (Adapted with permission from ref (347). Copyright 2021 AIP Publishing) and manipulation (Adapted with permission from ref (14). Copyright 2020 American Chemical Society).