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As I know, in electroplating method, the type of voltage used is DC voltage. My question is, does replacing DC current supply by AC one affect the functioning of the cell? And why?

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The difference is quite huge. Firstly, the deposition will occur on both electrodes (if possible). Secondly, if the frequency is high enough, there will be no deposition at all as only polarization and depolarization effect take place. The frequency effects mass, transport and electrical double layer. Depending on the electrolyte composition, the dissolution (if possible) will reduce the deposition. Usually pulse electroplating is used. The problematic is huge, please see some articles for more info:

An Overview of Pulse Plating - Norman M. Osero

and

Pulse and pulse reverse plating—Conceptual, advantages and applications, M.S. Chandrasekar, Malathy Pushpavanam

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