We have developed a hybrid MD-MPCD algorithm to simulate the electrophoresis of polyelectrolytes in a way that coarse-grains electrostatic interactions.

# The Problem

A charged monomer in solution attracts oppositely charged ions out of the fluid to form a counterion sheath. This is called the diffuse Debye layer and the charge density of the Debye layer obeys the Poisson-Boltzmann equation for which there is no analytical solution for a spherical monomer. If the potential is rather small then the Debye layer can be described by the Debye-Hückel approximation (the counterion density decays exponential over the characteristic Debye length).

Beyond the Debye length, electrostatic interactions are screened by the oppositely charged cloud of counterions. Perhaps more surprisingly, hydrodynamic interactions due to electrophoretic motion are also screened. To understand why imagine that the monomer moves under the action of an electric force. It shears the fluid, pulling it along with it. But the counterion sheath has an equal but opposite charge. The field forces it in the opposite direction and it also shears the fluid. The shears (almost) cancel out. (I throw in the “almost” because there is a rapidly decaying shorter-ranged field that remains).

If you were reading carefully you noticed that I explicitly said that “hydrodynamic interactions due to electrophoretic motion are screened”. Normal, random motion from thermal noise or drag from pressure driven flows are not screened. When the polymer moves, its counterion cloud moves along with it (not against it as it did for an electric field). This means that the electrophoretic mobility looks like a free-draining (or Roussian) polymer but the diffusion coefficient (hydrodynamic mobility) looks like a hydrodynamically oblique (Zimmian) polymer.

# Electrophoresis: When Hydrodynamics Matter

So you see that hydrodynamic interactions are usually screened in charged polymers. But this isn’t always the case. You can read all about cases when this screening isn’t perfect in our review if you are interested.

In order to study systems in which hydrodynamics aren’t perfectly screened a coarse-grained algorithm that can properly account for the electrohydrodynamics is needed. Usually simulations that account for both hydrodynamics and electrostatics are compuationally expensive (both hydrodynamics and electrostatics are long-range and computationally intense) or are only valid in the infinitely thin Debye layer limit. We developed an algorithm for the electrohydrodynamics of polyelectrolytes that is less computationally intense but is still accurate for finite Debye layers.

We embed a charged MD polymer in an MPCD fluid (as a detail the MD beads are included in the MPCD collision step to couple the polymer to the fluid). At each time step, we use the counterion distribution from the Debye-Hückel approximation to bestow MPCD particles with a temporary charge. When an electric field is applied the MD beads are forced in one direction (electrophoresis) and their not-explicitly-included-counterions are forced in the opposite direction (electro-osmosis). During the MPCD collision step this momentum is mixed – the MD beads shear the fluid and the electro-osmotic flow drags on the MD beads. For long polymers in free-solution this cancels out so that the mobility is length independent. For short chains, the experimentally observed rise is obtained (one point that I didn’t mention is that MPCD particles can get charge from multiple monomers and if their charge becomes to great they are said to “condense” to the chain, which is to say they loose all their charge and the monomers that they got their charge from also loose it. This Manning condensation is very important but if you are worried about these sorts of details you really should go read our paper).

Now we use this algorithm to study situations in which we are either unsure whether hydrodynamic interactions are totally screened or the impact of partial screening. This work was extended to look at electro-osmotic flow generated by tethered polyelectrolytes when I visited Dr. Pai-Yi Hsiao at the National Tsing Hua Univeristy in Taiwan. Currently, Martin Bertrand and I are using it to investigating some surprising effects of confinement on electrophoresis.

This algorithm was developed with Owen Hickey.