(163h) Dielectrophoresis of Nano-Colloids in Strong Electrolytes

Dielectrophoresis (DEP) has become a popular microfludic technique for identifying, concentrating and sorting bacteria, blood cells and even large biological molecules. DEP manipulation of immuno-colloids with functionalized antibodies and genetic beads with DNA probes is an active area of research. Fabricating nano-circuits by DEP assembly of nanowires and colloid-crystal assembly with DEP traps are also heavily pursued by a number of labs around the world.

However, fundamental understanding is still lacking for how a colloid dipole is induced by a high frequency (~1 MHz) AC field to produce a time-averaged DEP force on the colloid in a non-uniform field. The classical leaky dielectric (Maxwell-Wagner theory) has been found to be woefully inadequate at high ionic strength when the Debye screening length is is comparable or smaller than the particle size a (Greeen and Morgan, 2003). Correction to the classical theories by O'Konski, Hunter, O'Brien and Dukhin to include Stern layer and diffuse layer conduction and convection are also inadequate. In particular, the DEP cross-over frequency of low-permittivity nanocolloids, corresponding to the frequency with zero induced dipole moment, is observed to rise by as much as two order of magnitude to 10 MHz before dropping by another four orders of magnitude when the medium conductivity is increased. None of the existing theories can capture and explain these large variations in the dipole relaxation time of the nano-colloids in strong electrolytes.

We present a new theory here that extends the classical theory to all electrolyte strengths and is in quantitative agreement with literature and our own data for latex and silica nano-colloids. We find that when the Debye length is comparable to the particle size, both tangential and conduction currents are important in the diffuse layer. As a result, the diffuse layer effect cannot be lumped into an effective particle conductivity and/or a tangential surface current, as in earlier theories based on the leaky dielectric charge accumulation model. Instead, we explictly resolve the temporal and spatial charge distribution within the diffuse layer by constructing the complex spherical harmonics of the Poisson equation in the frequency domain. The relevant linear eletro-static and charge transport equations are derived with a linear Huckel theory for symmetric electrolytes. As in ACEO flow, the diffuse layer tangential flux is related a length scale comparable to the particle size while the normal charging current has a length scale comparable to the screening length. (For these thick diffuse layer, however, Stern layer adsorption needs to be included to allow charge accumulation.) The cross-over frequency hence corresponds to the RC time which is the geometic mean of the the tangential and normal diffusive times. The induced space charges in the diffuse layer do not produce appreciable net body force of electro-osmotic flow. They, however, screen and enhance the external field at different sides of the cross-over to reduce and amplify dielectric and conductive polarization, respectively. We find that the precipitous rise in the cross-over frequency at a specific medium conductivity (when it is equal to the particle conductivity) occurs for nano-colloids whose dimension is of the order of the ion diffusivity multiplied to the medium permittivity and divided by the surface conductance of the Stern layer, or about 10 nm for latex and silica colloids.

The precipitous drop in the cross-over occurs when the medium conductivity exceeds the particle conductivity due to Stern layer conductance and when the screening length is much smaller than the particle size. In this limit, the external field is completely screened by the surface field and the only charging current into the diffuse layer occurs at one pole of the particle where the surface field is aligned with the external field. By resolving the charging flux at this polar inner region, whose dimension is comparable to the screening length, we show that the diffuse layer polarization by tangential electro-migration is determined by this polar flux that drives the migrating front. The diffuse layer polarization hence never reaches Poisson-Boltzmann equilibrium in both the normal and tangential directions for this open system and the cross-over frequency is not determined by the usual diffusive relaxation towards equilibrium. Instead, the polarization (tangential migration) time increases with decreasing polar flux when the conductivity increases and the dimension of the polar charging region diminishes. This flux-driven induced dipole relaxation time explains the four-orders of magnitude drop in cross-over frequency at very high medium conductivity. We also offer fluorescent video evidence of this tangential charging front around larger particles. The front consists of charged fluorescent dye that can be driven into the diffuse layer at the entry pole.

Both analytical theories offer convenient scalings that allow us to collapse all latex and silica colloid cross-over data for strong electrolytes, where the classical leaky dielectric theories fail completely.