You have a passive process (diffusion) compared to an active process (electrophoresis), and moreover, the active process ALSO includes diffusion, because you cannot prevent diffusion.

So: electrophoresis is faster. Even very very gentle electrophoresis will be faster than diffusion alone.

You don't need equations to determine that X+Y is faster than X alone.

This is assuming your material doesn't exhibit any sort of voltage dependent effects (piezoelectric properties etc: tightening up in an electric field or whatever).

If you ions are bound to the ball then the ball will also undergo electrophoresis. If your ball is charged anyway then you will get a distribution of counterions around the ball which will counter the thermal diffusion.

You need to account for the strength of the electric field in order to estimate whether random diffusion or linear motion will dominate overall. Halide ions have electrophoretic mobilities of the order of 1e-8 to 1e-7 m2s-1V-1. The diffusion coefficient of chloride in water is of the order 1e-12 m2-1. You may not like equations but they are important (and simple in this case). Over a given time, the average distance travelled by the ions will be:

diffusion: the square root of (diffusion coefficient x time)

electrophoresis: the electrophoretic mobility x the voltage / distance between your electrodes x time.

In this instance, diffusion can very easily dominate. This is true for nanoparticles, too, which are much larger than simple ions and therefore diffuse much more slowly. It is a big problem in measuring electrophoretic mobilities of nanoparticles in salt water. That's why I invented a technique that averages the diffusion to zero and keeps the electrophoretic motion.

Unless your ball has the same density as the liquid, you will get sedimentation, too, which could be faster than diffusion.

You have to use some equations to predict the likely behavior or just do the experiment. Measure the bulk conductivity as a function of time.

You can describe your system as a porous particle immersed in an electrolyte solution. From an electrokinetic perspective, this is complicated.

Hermans, J. J. Sedimentation and Electrophoresis of Porous Spheres. J. Polym. Sci. 1955, 18 (90), 527–534. https://doi.org/10.1002/pol.1955.120189008.

Duval, J. F. L.; Ohshima, H. Electrophoresis of Diffuse Soft Particles. Langmuir 2006, 22 (8), 3533–3546. https://doi.org/10.1021/la0528293.

Hsu, H.-P.; Lee, E. Electrophoresis of a Single Charged Porous Sphere in an Infinite Medium of Electrolyte Solution. Journal of Colloid and Interface Science 2013, 390 (1), 85–95. https://doi.org/10.1016/j.jcis.2012.09.036.

John Hildyard - Thank you for the added confidence you provided in this matter.

John Francis Miller - From your explanation I think of doing or designing some sort of experiment. But I understand, I need more strong theoretical understanding before I do anything. Thank you for the references to start off...