In the dc discharge case, it is well known that the ions streaming toward the cathode surface and follow the flux conservation from sheath edge to the wall. According to collisional dominated child's lang. law, the current associated with ion flux is related to the sheath thickness and potential on the cathode. If the potential on the cathode surface is changed to higher negative value that leads increase in the current to the cathode and reduces the thickness of sheath. For a given pressure, the change in sheath thickness has impact on the ionization rate and electric field gradient. With increasing the ionization rate, ions density is increased. On other side, higher field gradient may increase the ions velocity at the sheath edge. Since, total flux is multiplication of density and velocity therefore it is difficult to isolate the contribution from density and ions velocity to total increased flux. is there any paper or letter that can give some estimation about density and ion velocity variations with the sheath thickness or cathode potential ?
I think you are over-complicating this somewhat, and need to remember that changes in one aspect ("If the potential on the cathode surface is changed to higher negative value...") tend to propagate through the system and cause changes elsewhere. I would look at DC plasma as bulk plasma (field free region of some plasma density) that loosely couples behavior at anode and cathode surfaces. The main criteria here is anode and cathode currents must be equal, so changing the applied voltage won't necessarily result in the behavior you suggest. In considering total flux to the cathode, generally it should result from +ion flux at the cathode sheath interface so should be around nv/4; bulk +ions meander around until they enter sheath and accelerated toward cathode. Ion density n and ion velocity v here will change relatively slowly when making small changes to voltage, pressure, etc. Turning up the voltage does not always increase ionization rate or ion density. Also consider effects of secondary electrons which can have profound effects. Lastly, look at the physics of hollow cathode discharges.
https://www.researchgate.net/publication/274635279_Fluid_model_of_the_sheath_in_front_of_a_floating_electrode_immersed_in_a_magnetized_plasma_with_oblique_magnetic_field_Some_comments_on_ion_source_terms_and_ion_temperature_effects
Article Fluid model of the sheath in front of a floating electrode i...
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