The Physics of Dielectrophoresis
Dielectrophoresis (often abbreviated DEP) was first observed early in the 20th century, but described fully (and named) by Herbert Pohl in 1950. It is an electrostatic phenomenon, distantly related to electrophoresis, and describes the motion of suspended particles resulting from polarisation forces produced by a nonuniform electric field. It can be used to uncover the electrophysiological workings of cells, and as a unique label-free approach to cell sorting that promises to revolutionise personalised medicine. This page is designed to familiarise those new to the field with the necessary basis to get started using DEP.
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Lets start with the familiar electrostatics situation. Particles can be charged (+/-) or possess neutral charge. Within a uniform electric field, charged particles move to the electrode of opposite charge, see (A) right.
If the particle is neutrally charged it will not move. If the particle is polarisable the electric field will move charges within the particle to form a dipole across the particle (B). In a uniform electric field this will still not induce movement as the dipole forces on each side will oppose each other.
In a non-uniform electric field the field strength varies between the electrodes leading to the electric force on one side of the particles dipole being slightly stronger than the other. The particle will now move in the direction of the resultant dielectrophoretic force, $ F_{DEP} $.
For this simple introduction we won’t go into details but it is important to be aware that the dipole charges at each side of the particle are dependant on the differences in polarizability between both the particle and its surrounding liquid. Both polarizations are affected by the frequency of the applied electric field. This gives arise to a graph of polarisability for a given particle in a given medium at a range of frequencies, called the ‘dielectrophoretic signature’ or ‘Clausius-Mossotti factor’ polarisation curve.
In order to understand DEP, we begin with its more famous relative, Electrophoresis (EP). This is the physical manifestation of Coulomb’s law (F=QE). Where F is the electrophoretic force, Q is charge and E is the electric field at the charge. This only works for charged particles, neutral particles will not move (A).
As its name suggests, dielectrophoresis is related to electrophoresis, but the “di-” suffix (Greek for “two”) comes from the fact that this is not the force arising from a single charge (or “monopole”), but from a dipole – something with both positive an negative charges that do not share the same position.
According to Coulomb’s law, a dipole in an electric field will experience forces applied to both dipole charges acting in opposite directions. If the electric field is equal everywhere, then the forces on the dipole are equal and opposite, and the particle will not move (Above figure, B).
However, if the electric field is non-uniform, then the force on the dipole at the higher electric field will be greater than the force on the dipole in the lower electric field.
The particle will then move in the direction dictated by the charge of the dipole in this highest field. The force can be in the direction of increasing or decreasing electric field strength – these are termed “positive dielectrophoresis” (pDEP) and “negative dielectrophoresis” (nDEP) respectively.
as, $|μ_1| = |μ_2|$ and, $E_1 > E_2$
therefore, $|F_1| > |F_2|$
What governs whether the particle experiences positive or negative DEP?
In reality, all materials have capacitance and resistance, which can be treated as a resistor and capacitor in parallel. Any charged capacitor will accumulate charges at its surfaces. Where two materials – say, a cell and water – meet, different amounts of charge accumulate on either side of the interface, depending on the polarisability of each material. As the ratios of charge either side of the interface changes, the cumulative effect is that the dipole can change in both magnitude and direction. This is called the Maxwell-Wagner interfacial polarisation (see figure, left).
If the polarisability of the medium is greater than the particle then the accumulated charges look like (a) and produces negative DEP. If the inverse domination then the charges look like (b) and produces positive DEP.
It is important to remember that the dipole charges at each side of the particle are dependant on the differences in polarizability between both the particle and its surrounding liquid. Both polarizations are affected by the frequency of the applied electric field. This gives arise to a graph of polarisability for a given particle in a given medium at a range of frequencies, called the ‘dielectrophoretic signature’ or ‘Clausius-Mossotti factor’ polarisation curve.
Symbols
- ε, Permittivity
- σ, Conductivity
- ε*, Polarisability / Complex Permittivity, ε* = ε + iσ/ω
- ω, Angular frequency
- K, Dielectric constant / Relative Permittivity, K = ε / ε₀
- F, (Dielectrophoretic) Force
- E, Electric Field Strength
- μ, Dipole strength
DEP for Cell Sorting
Different particles exhibit different signatures; for cells, this typically results from differences in membrane morphologies, cellular volumes and cytoplasmic composition (amongst other variables), whilst inorganic particles are affected by molecular/atomic bonding, particle shape and polar chemistry.
If a mixture contains two or more homogenous subpopulations, DEP can be used to exploit differences in their dielectrophoretic signatures.
Take the above example: between frequencies of 120kHz to 220kHz, the orange particles experience positive dielectrophoresis, whilst the blue particles experience negative dielectrophoresis. By applying an electrical field of this frequency, DEP can be used to separate these populations.
DEP for Cell Analysis
At any given frequency there is an equation that links the dielectrophoretic force to the electrophysiology of a cell. The dielectrophoretic equation is given below where $ r $ is particle radius, $ K_m $ the relative permittivity of the medium, $ e_0 $ the (absolute) permittivity of free space, $ E $ the electric field strength.
If we have a way of measuring the dielectrophoretic force (conveniently we do! – the 3DEP) then we can calculate electrophysiology using the Clausius-Mossotti factor (CM). If the particle being analysed is a homogenous particle of single-material then this expression can be used to find that materials permittivity, ε, and conductivity, σ.
Cells however rarely satisfy this, at a minimum they require treating as a distinct membrane region surrounding a cytoplasmic region. To do this we apply the ‘single-shelled’ model, adjacent.
$$ CM=((ε_p^*-ε_m^*)/(ε_p^*+2ε_m^* )) $$
Extracting membrane and cytoplasmic properties using this expression can be performed manually, using general trends as simplified in the adjacent figure, or semi-automatically using curve fitting algorithms, as included in the 3DEPs accompanying software.
Advanced Electrophysiology
As studies continue into the cellular “Electrome” it is becoming possible to extract additional properties associated with surface chemistry such as membrane potential and Zeta potential.
🔗 On the low frequency dispersion observed in dielectrophoresis spectra – (Hughes, 2024)
Further Shelled Models
If the nucleus is large relative to the cell, then you may wish to apply a double shelled model, shown above, but this increases mathematical complexity.