The force whose sign a frequency chooses

A drop of water with cells suspended in it, a pair of electrodes a few tens of micrometres apart, and an alternating voltage of a few volts. At one frequency of the voltage the cells crowd onto the edges of the electrodes. Turn the frequency down by a factor of ten and the same cells lift off and gather in the weak field above them. Nothing in the liquid has changed, the cells carry no net charge, and the force is not the one a battery exerts on an ion. It is the force on an induced dipole in a field that varies from place to place, and its sign is a property of the frequency.

A dipole in a gradient, and what it is compared with #

The mechanism was set out in the argument about a glass slab drawn into a capacitor: a field polarises a neutral body, the induced dipole lines up with the field, and a dipole in a field that is stronger at one end than the other feels a net force towards the stronger end. The force goes as the gradient of the field’s square, so it does not care about the sign of the field. An alternating field works as well as a steady one, which is what makes the effect usable — a steady field in water would drive currents, electrolyse the electrodes and move every ion in the liquid.

For a sphere of radius $a$ in a liquid, the time-averaged force is

$$\mathbf{F} = 2\pi\varepsilon_m a^3\,\mathrm{Re}\,K(\omega)\,\nabla\left|\mathbf{E}_{\text{rms}}\right|^2, \qquad K = \frac{\varepsilon_p^* - \varepsilon_m^*}{\varepsilon_p^* + 2\varepsilon_m^*},$$

and nearly everything that is interesting is in $K$. Without the stars it is the factor that decides how much field a sphere of dielectric keeps inside itself, and the numerator makes the essential point: the particle is not compared with empty space but with the liquid it displaces. A particle that polarises more than the liquid is pulled up the gradient; one that polarises less is pushed down it, for the same reason a bubble rises in water.

The stars carry the frequency. A material with conductivity $\sigma$ in a field alternating at angular frequency $\omega$ behaves exactly like a dielectric whose permittivity has an imaginary part, $\varepsilon^* = \varepsilon - i\sigma/\omega$ — a conductor is a dielectric that is losing, and which description applies is a question of frequency. At low frequency the conductivity term dominates both starred permittivities and $K$ becomes $(\sigma_p - \sigma_m)/(\sigma_p + 2\sigma_m)$: the comparison is between how well the particle and the liquid conduct. At high frequency the conductivity terms have shrunk away and $K$ becomes $(\varepsilon_p - \varepsilon_m)/(\varepsilon_p + 2\varepsilon_m)$: the comparison is between how well they polarise. Nothing requires the two comparisons to agree.

Charge that cannot keep up #

Why the two regimes are joined by a single step, and where the step falls, is a statement about how quickly charge can move.

In a slowly alternating field, free charge in the particle and in the liquid has time to flow and pile up at the particle’s surface. The accumulated surface charge makes the particle’s dipole whatever the conductivities dictate, and the permittivities are irrelevant. As the frequency rises, a point comes at which the field reverses before the charge has finished accumulating; above it the surface charge cannot form, and the dipole is whatever the bound charge in the two materials can manage. The crossover between the two is a relaxation — the Maxwell–Wagner relaxation, named for the two who worked out the polarisation of interfaces in layered dielectrics more than a century ago — and its characteristic frequency is the inverse of the charge-relaxation time of the combination, $(\sigma_p + 2\sigma_m)/2\pi(\varepsilon_p + 2\varepsilon_m)$.

The two latex beads in the second figure show where the conductivity of an insulator comes from, and it is a surprising place. Polystyrene conducts essentially nothing. What conducts is the thin cloud of counter-ions that every charged surface in water gathers around itself, a few nanometres thick, and its effect on the bead is that of a bulk conductivity equal to twice the surface conductance divided by the radius. The same surface on a smaller bead is spread over less volume, so the smaller bead behaves as the better conductor — ten times better for a bead ten times smaller. In water of 0.01 S/m the 200 nm bead out-conducts the liquid and is pulled at low frequency, while the 2 µm bead of the identical material does not and is pushed. A mixture of the two sizes separates in a field at a few hundred kilohertz, which is a way of sorting particles by size with no filter at all, using nothing but the fact that a surface and a volume scale differently.

A relaxation has two faces, and the second one turns out to be a separate force.

The imaginary part of $K$ is the part of the induced dipole that lags the field. In a field that only alternates in strength, the lagging part averages to no force at all. In a field that rotates — made with four electrodes driven a quarter of a cycle apart — the lagging dipole always points a little behind the field, and the field drags it round. The particle spins, at a rate proportional to $\mathrm{Im}\,K$. This is electrorotation, and it is used to measure the same properties the translational force depends on, by counting revolutions under a microscope instead of watching drift.

The two parts are not independent measurements. They are the real and imaginary parts of one function of frequency that cannot respond before it is driven, and a function with that property has its two halves tied together: the step in one and the peak in the other are the same relaxation seen from two sides, and either can be computed from the other.

Why the bounds are +1 and −½ #

The factor $K$ has limits that no choice of materials can break, and they are worth seeing because they are lopsided. If the particle is enormously more polarisable or conductive than the liquid, $K \to +1$. If it is enormously less, $K \to -1/2$. The asymmetry comes from the $2\varepsilon_m$ in the denominator, which in turn comes from the geometry of the field round a sphere: the field that leaks around an insulating sphere and the field that is sucked into a conducting one are not mirror images.

The practical consequence is that the most a particle can be pulled is twice the most it can be pushed. A liquid can be made to repel almost anything by making it conduct far more than the particles — but not strongly, and never more strongly than a well-chosen particle can be attracted.

A cell is a conductor wrapped in an insulator #

A living cell is not a homogeneous bead. Its interior is a salty solution that conducts nearly as well as seawater, and it is enclosed by a membrane of lipid about five nanometres thick that conducts almost not at all. A membrane that thin has an enormous capacitance per unit area, about a microfarad per square centimetre, and that capacitance is the whole story of the living cell’s curve in the first figure.

At low frequency the membrane blocks current, so the field cannot reach the conducting interior and the cell behaves as an insulating sphere. In a liquid that conducts at all, an insulating sphere has $K$ near $-1/2$, and the cell is pushed out of strong field. As the frequency rises, the membrane’s capacitance lets current through — a capacitor’s impedance falls as the frequency rises — and above a certain frequency the membrane is effectively short-circuited. The cell now presents its conducting interior to the field, and if the interior conducts better than the liquid, $K$ swings positive and the cell is pulled in. At still higher frequency the conductivities stop mattering altogether, the permittivities take over, and since the cytoplasm polarises a little less than water, $K$ ends slightly negative again. Two crossovers, and three regimes.

The first crossover is set by the membrane’s capacitance and the liquid’s conductivity together. For a cell of radius $R$ whose membrane has capacitance $C_m$ per unit area, in a liquid of conductivity $\sigma_m$, it falls close to

$$f_1 \approx \frac{\sqrt2\,\sigma_m}{2\pi R\,C_m},$$

which for a membrane five nanometres thick with a relative permittivity of six — a microfarad per square centimetre — a radius of five micrometres and a liquid of 0.01 S/m comes to about 42 kHz, where the exact curve crosses. The formula says two things the curve alone does not. The crossover moves in proportion to the liquid’s conductivity, so raising it tenfold raises the crossover tenfold; and it moves inversely with the cell’s size, so large cells cross over sooner than small ones in the same liquid.

A cell that has died has a membrane that leaks. Its conductivity rises by orders of magnitude, the ions inside escape towards the concentration outside, and the capacitive barrier that produced the living cell’s low-frequency push is gone. The dead cell in the first figure is pulled at low frequency, where the living one is pushed, and has only one crossover. Anywhere in the shaded bands, one field applied through one pair of electrodes sends the two kinds in opposite directions — which is how living and dead yeast, healthy and infected blood cells, and cancer cells from the blood cells around them have all been sorted, without labels and without touching them.

The crossover frequency is itself a measurement. Since it moves with the membrane’s capacitance per unit area, and that capacitance depends on how much the membrane is folded, a cell with a rougher, more folded surface crosses over at a lower frequency. Measuring the frequency at which a cell stops moving measures the area of membrane it carries per unit of apparent surface.

Pulled particles land on metal; pushed ones can float #

The direction of the force has a consequence that is not obvious from the formula, and it connects the whole subject to a theorem about static fields.

A particle pulled up the gradient goes where the field is strongest, and the field is strongest at the surface of the electrodes, at their edges. It cannot be held anywhere in between, and the reason is exact rather than practical. Each component of a static field in charge-free space satisfies Laplace’s equation, so the square of the field’s strength can have no maximum in the interior of the space: it is largest somewhere on the boundary. That is the same fact that makes it impossible for any arrangement of static charges to hold a charge still, transplanted from the potential to the field strength. A positively polarising particle always ends on an electrode.

A negatively polarising one is in the opposite situation. The square of the field’s strength can have a minimum in free space — it can be zero at a point surrounded by stronger field on every side — and a particle pushed down the gradient sits in that minimum stably. Eight electrodes at the corners of a small cube, driven in the right pattern, make a field with a zero at the centre, and a cell pushed out of strong field floats there in a cage with no walls. The alternating field and the time average are what make the dipole always align, but the reason the cage is possible at all is that the forbidden extremum is the maximum and not the minimum. It is the same asymmetry that lets a rapidly alternating force hold something up where no steady one could.

The grip goes as the volume #

The force is proportional to $a^3$, which means dielectrophoresis is a tool for things of a particular size and becomes useless below it.

What any trap has to beat is the jiggling of thermal motion, which gives every particle an energy of order $kT$ whatever its size. A cell ten micrometres across held by a field of $10^5$ V/m — a few volts across a few tens of micrometres — sits in a trap millions of $kT$ deep and does not move at all except as the field directs. A virus fifty nanometres across needs fields ten times stronger to be held firmly. A protein a few nanometres across needs $10^7$ volts per metre, and at that field the current flowing through any real buffer heats the liquid, and the heating drives flows that push the protein about far harder than the dielectrophoretic force does. The size below which the method stops being worth using is set by that competition rather than by any limit on the field.

The same $a^3$ explains the optical version of the force. At the frequency of light the factor $K$ is set by the refractive indices, the gradient is the focus of a laser beam, and a microscope objective can make a gradient on a scale of a wavelength — which is the tweezer that holds a bead in a beam, dielectrophoresis at $10^{14}$ Hz.

Where the single-shell cell stops being a cell #

The field is taken to be uniform across the particle. The formula treats the particle as a point dipole in the local gradient, which requires the field to change little across it. Near the edge of an electrode, where the field changes on a scale of micrometres, a cell ten micrometres across violates that, and higher multipoles of its induced charge contribute forces the dipole term does not.

The cell is one shell and one interior. Real cells have a nucleus with its own envelope, organelles and a cell wall in plants and yeast, and each adds a relaxation to the factor. The single-shell model captures the first crossover well and the high-frequency behaviour only roughly.

Other forces are ignored. The field heats the liquid in proportion to its conductivity, and the heating makes flows; the field also acts on the charge in the thin layer at the electrode surfaces, making flows of its own at low frequency. Both can move particles faster than dielectrophoresis does, and much of the craft of using the effect lies in choosing the frequency and conductivity at which they are weak.

And the particles do not interact. A dense suspension of polarised particles attracts itself: neighbouring dipoles along the field attract, and particles form chains along the field lines — pearl chains — whose behaviour belongs to the physics of the suspension rather than of any one particle.

A spread of cells, not one crossover #

Every curve here is a property of a single particle computed from bulk numbers — conductivities, permittivities and a membrane thickness — each of which varies from cell to cell by tens of per cent. A real population produces a spread of crossover frequencies, and the shaded bands of the first figure are narrower in practice than they look, because the edges of two distributions overlap. Whether a sorting succeeds is a question about those distributions, and a single curve cannot answer it.

Nor can the field picture show time. A particle in the right-hand panel does not arrive at the field’s minimum instantly; it drifts there against viscous drag at a speed proportional to $a^2$, so small particles are both weakly held and slow to arrive. The directions in the figure say where things go, not how long they take to get there.

Still open: whether the membrane’s electrical signature can diagnose a cell #

That a cell’s crossover frequency shifts when it dies, when it becomes cancerous, when it is infected by a parasite, or when it differentiates into another type is well established; each changes the membrane’s capacitance per unit area or the interior’s conductivity. What is not established is whether those shifts are specific enough to be used as a diagnosis rather than as a preliminary sort. The electrical properties of a cell are a handful of numbers, and a great many different changes to a cell move the same handful. Studies that separate one cancer cell line from blood in a laboratory succeed; whether a crossover measurement on a patient’s cells can distinguish a disease from the ordinary variability between people is still being argued, and it depends on how much the few numbers the membrane can report really say about the cell behind it.

The next question the subject raises is what happens when the particles are no longer far apart. A suspension of polarisable particles in an alternating field stiffens from a liquid into a paste that will hold a load within milliseconds of the field being switched on, which is the basis of electrorheological fluids and the reason the physics of one polarised particle is only the beginning of the physics of many. The habit worth carrying from here is to ask of any polarisation what it is being compared with. The force on a neutral particle measures the difference between the particle and what it displaces, at the frequency being asked — change the liquid or change the frequency, and the same particle is pulled or pushed.