You can read in the polyelectrolyte project page of this site or in our review paper about how the counterion sheath surrounding a polyelectrolyte causes hydrodynamic interactions to be screened, which makes polyelectrolytes “free-draining”. In brief, this occurs because the electric field acts on the polymer, which moves through the surrounding fluid shearing it. However, the total force on the counterion sheath is equal and opposite to the force on the polymer and the counterions also shear the fluid. The shear virtually cancels out on length scales longer than the thickness of the counterion sheath (the Debye length).

But imagine what happens if both an electric field and also a mechanical force act on the polyelectrolyte chain. The counterions do not form a connected object and so the mechanical force doesn’t act on them – only the electric field acts on the counterions. Therefore, the shear doesn’t cancel out and there is a net flow of counterions relative to the polyelectrolyte’s reference frame.

When the polymer is subject to a mechanical force (such as a tethering tension) there is (EOF) at the surface of the chain generated by the electrophoresis of the counterion sheath relative to the stationary chain. This is all very well described by the principle of Electro-Hydrodynamic Equivalence Principle.
The Equivalence Principle states that when polyelectrolytes with a thin counterion sheath are acted on by an electric and mechanical force simultaneously, one can replace the electrostatic and hydrodynamic equations of motion with an effective local flow equal to the translational velocity that the polyelectrolyte would have during in free-solution electrophoresis. It can not be overstated how significant this is for electrophoresis of polyelectrolytes. According to the Equivalence Principle, researchers who are able to design devices that apply any mechanical force in concert with an electric field may achieve length-dependent size separation.

It is well established that the Equivalence Principle can be used to replace the complicated electrohydrodynamics with an equivalent incident flow with respect to chain conformation. But what is more difficult to demonstrate is how approximate the equivalence is for the generated electro-osmotic flow (EOF) of the internal and surrounding fluid.

MPCD (with our mean field MPCD-MD Debye-Hückel algorithm) is the ideal computational method for testing the accuracy of the Equivalence Principle from the perspective of the fluid.