Frequency‑Modulated Dielectrophoresis of Vesicles and Cells: Predictable U‑Turn Trajectories at the Crossover Frequency

Background

Cellular polarizability is dominated by the membrane and cytoplasmic electrical properties, which vary with the applied field frequency. This frequency‑dependent dielectric spectrum enables label‑free discrimination of cell phenotypes via impedance spectroscopy, AC DEP, traveling‑wave DEP (twDEP), and electrorotation [1–16]. Compared with antibody‑based or adhesive labeling methods, these electrical techniques avoid cell modification and potential activation, making them attractive for noninvasive cell sorting.

DEP arises in a nonuniform electric field, generating a force proportional to the gradient of the field amplitude and the real part of the Clausius–Mossotti (CM) factor. The phase gradient gives rise to twDEP or electrorotation, which provide complementary dielectric information. Our work extends the conventional AC DEP by introducing FM electric fields, allowing continuous traversal of the crossover frequency f_X and enabling rapid dielectric characterization.

In typical AC DEP setups, the field is applied between fixed microfabricated electrodes, limiting spatial flexibility. Electronic tweezers—where micromanipulated electrode needles deliver the field—offer a versatile, contact‑free alternative. However, dielectric measurements require short exposure times (<30 s) to reduce solvent flow effects, which is challenging with multi‑electrode systems. FM‑DEP addresses this by modulating the frequency while keeping the electrode geometry constant.

Our plug‑in system (Fig. 1) employs two tungsten needle electrodes, each 0.5 µm tip diameter, positioned 100 µm apart. The needles are driven by an arbitrary waveform generator and current amplifier, and the entire assembly is mounted on a patch‑clamp micromanipulator beneath an inverted microscope. This configuration permits precise, noncontact manipulation of individual vesicles or cells above the substrate.

Experimental setup. A schematic of the dielectrophoretic manipulation system illustrating the AC or FM electric field applied to a target particle via a pair of electrode needles controlled by patch‑clamp micromanipulators.

The FM wave is defined by

𝐸(𝐫,t)=𝐴(𝐫)cosθ(t),

with the phase related to the instantaneous frequency f(t) by θ̇=2πf(t). We use a wide‑band FM modulation:

f(t)=fc+Δf cos(2πfmt),

where the modulation frequency f_m is much smaller than the carrier frequency range, satisfying the wide‑band limit (WBL) condition Δf/f_m≫1.

Methods

Materials

MLVs were prepared from 1,2‑dioleoyl‑sn‑glycero‑3‑phosphatidylcholine (DOPC; Avanti Polar Lipids). A 1 mL DOPC solution (20 mM) in chloroform/methanol (2:1 v/v) was dried under nitrogen and vacuum‑sealed for >12 h. The thin film was rehydrated with deionized water and incubated at 25 °C for several hours.

Human T‑cell leukemia (JKT‑β‑del) and B‑cell leukemia (CCRF‑SB) cell lines were cultured in RPMI 1640 supplemented with 10 % fetal bovine serum and 100 mM sodium pyruvate at 37 °C, 5 % CO₂. After 1 week, cells were pelleted (370 g, 3 min, twice) and resuspended in 1 mL RPMI 1640. Suspensions were then diluted in isotonic 200 mM sucrose to achieve desired conductivities.

Human RB cells were isolated from whole blood of healthy volunteers (early twenties). Cells were suspended in a 3.1 % hematocrit RPMI 1640 mix and diluted with 200 mM sucrose. All RB experiments were performed within 10 min of blood draw.

Experimental Setup

Conductivity of suspensions was measured with a SevenMulti meter (Mettler‑Toledo). The FM or AC field was generated by an Agilent 33220A waveform generator and amplified by an F30PV current amplifier. Two 0.5 µm tungsten needles, separated by 100 µm, delivered the field. The needle pair was positioned 50 µL of suspension on a heated stage (25 °C). Imaging was performed with a Nikon TE2000‑U microscope and a Retiga Exi CCD camera (25 fps). The FM waveform’s frequency resolution was verified to be within experimental error.

Additional Movie S1–S3 demonstrate AC DEP manipulation of diatom cells in deionized water: (S1) rotation at 30 kHz, (S2) pushing toward a glass wall at 100 kHz, and (S3) retrieval at 20 MHz.

Results and Discussion

Experimental Observation of a Leukemia Cell Experiencing FM‑DEP

The plug‑in needles allow field application above the substrate, facilitating precise cell selection. Movie S4 shows a triangular diatom cell responding to a 100–500 kHz frequency jump at 0.5 s intervals. Movies S5 and S6 present typical U‑turn trajectories of TL cells under FM‑DEP (0.25 Hz modulation, 200 kHz ≤ f(t) ≤ 3 MHz). Fig. 2 displays a 3‑D trajectory (x, y, t) where the origin corresponds to the needle tip. The x‑axis aligns with the tangent to the electrode surface; the y‑axis reflects the U‑turn amplitude.

3‑D trajectory of a targeted TL cell. Periodic U‑turns due to the frequency modulation are demonstrated for the TL cell undergoing the FM‑DEP.

U‑turns comprise three phases: leaving the needle, approaching after a U‑turn, and staying on the electrode surface. Solvent flow often prevents exact return to the same spot, but the timing of leaving and U‑turn initiation is clear. The observed U‑turn period (≈4 s) matches the modulation period of 0.25 Hz.

Theoretical Study on the FM‑DEP

For a spherical particle with interior permittivity ε_in and conductivity σ_in and exterior ε_out, σ_out, the DEP force is

𝐹DEP(𝐫,t)=𝐩(𝐫,t)·∇𝐸(𝐫,t),

with the dipole moment

𝐩(𝐫,t)=4πR3εoutKH{𝐸(𝐫,t)+ (τ/Δτ) Ẽ(𝐫,t)},

where K_H=(ε_in–ε_out)/(ε_in+2ε_out), Δτ^–1=τ₀^–1–τ^–1, and τ₀ and τ are the characteristic times defined in the text. Averaging over an AC cycle yields the mean DEP force

⟨𝐹DEP⟩=χ(fAC)∇𝐴RMS2,

where χ(f) is proportional to the real part of the CM factor.

Replacing the AC field with an FM wave and invoking the WBL, the averaged DEP force becomes

⟨𝐹DEP(𝐫,t)⟩=χ{f(t)}∇𝐴RMS2,

showing that the force direction changes whenever f(t) crosses f_X. This explains the periodic U‑turns observed experimentally.

Solving for the instant when f(t)=f_X gives

fX=fc–Δf cos(πfmΔtn),

where Δt_n is the measured time from leaving the electrode to the U‑turn. Thus, the FM‑DEP method provides a rapid determination of f_X from the observed trajectory.

Comparing Crossover Frequencies of a Single MLV Determined from the FM‑ and AC‑DEPs

Using a salt‑free MLV suspension, we compared f_X obtained from FM‑DEP (10–50 kHz FM range, f_m=0.1 Hz, 10 s period) and AC‑DEP (30–100 kHz sinusoid). From the FM trajectory, the mean leaving time was 5.8 ± 0.2 s, yielding f_X=35 ± 1 kHz. AC‑DEP force measurements via a moving needle yielded a force–frequency curve that fitted the model

FDEP(f)= [L+(2πfτ)2H]/[1+(2πfτ)2],

with fitted parameters L=–21.02 pN, H=19.03 pN, τ=4.9 µs, giving f_X=34.15 kHz, in excellent agreement with the FM result.

Conductivity Dependencies of the Crossover Frequencies for Biological Cells

We applied FM‑DEP to TL, BL, and RB cells in conductivity ranges 60–160 mS m^–1 (Δf≈700 kHz, f_m=0.25 Hz). For each cell type, we measured the leaving times of 10 cells per drop, averaged over two drops, and derived f_X using Eq. (10). Figure 6 plots the resulting f_X versus σ_out. The data were fitted to the extended single‑shell model

fX= [σout–(σout2)/(2σcyt)]/(√2 π R Cm) + fX0,

yielding membrane capacitances C_m and cytoplasmic conductivities σ_cyt that agree with literature values (C_m≈0.5–1.5 µF cm^–2, σ_cyt≈0.3–0.8 S m^–1). The consistency across cell types confirms the reliability of FM‑DEP for dielectric characterization.

Conductivity dependence of crossover frequencies. Mean crossover frequency, ⟨f_X⟩, of TL cells (blue triangles), BL cells (green diamonds), and RB cells (red circles) versus solution conductivity σ_out. The best‑fit curves of Eq. (15) are shown as solid lines.

Conclusions

Our WBL‑based theory and experiments confirm that FM‑DEP produces predictable U‑turns at the crossover frequency, enabling rapid, <30‑second determination of f_X. The FM‑DEP measurements of MLVs match direct AC‑DEP force measurements, while the conductivity‑dependent f_X data yield membrane capacitances that align with literature. FM‑DEP therefore offers a robust, noncontact method for dielectric spectroscopy of vesicles and cells, and can be extended to sub‑micron and nanoscopic particles in on‑chip configurations for real‑time electrokinetic spectroscopy.