(Invited) Short Introduction of Brenner’s Paradigm on the Steady State Diffusion Layer Model of Alloy Electrodeposition Composition Control, and Its Possible Extensions
Authors: Sun Yunkai , Zangari Giovanni 
Published: December 2020

Understanding the composition control of alloy electrodeposition is the prerequisite of pursuing fundamental understandings of phase and morphology evolution in the deposit. It is well known that a constant electrolyte concentration boundary, called diffusion layer, will form when the system is under forced convection or natural convection.[1] Given enough time, the thickness of diffusion layer, which in the beginning follows the Cottrell equation, will reach a steady state value, and thus a steady state deposition flux rate and deposition current could be estimated using the ionic diffusivity and diffusion layer length. For alloy electrodeposition, using the steady state deposition flux rate, ideally, the composition of the deposit could be predicted by the concentration ratio in the electrolyte and the mass-transfer coefficient ratio of the depositing ions.
Despite being proposed by Essin in the 1930s and systematically documented in Brenner’s famous alloy electrodeposition book in the 1960s,[2, 3] nowadays, this behavior seems to be set aside when investigating the alloy electrodeposition properties. We decided to summarize the recent evidence for the composition control at the limiting case of diffusion layer control (limiting current condition) and discuss the confusions we are facing when applying this paradigm of alloy electrodeposition composition control at steady state.
In this presentation, we will tentatively present the following topics:

Levich in the 1960s has estimated the diffusion layer thickness of the electrodeposition on a vertical plate without any forced convection: when depositing in low viscosity electrolytes without forced convections, the natural convection will be induced by the tiny variation of local electrolyte density of the electrolyte and lead to the formation of diffusion layer.[4, 5] We will briefly present his model and derivation. From this model, the diffusion layer thickness due to 0.01mM Ni(II) at the height of 1 cm on the vertical electrodecould be as much as 0.21 cm. The time for reaching the steady state diffusion layer thickness and the initial layer thickness of the deposit (assuming two elements start to be deposited simultaneously) is estimated using the Cottrell equation.

Expansion of mass-transfer coefficient to higher orders with respect to concentration, under the circumstances of multi-component, multi-element alloy electrodeposition. The matrix for multicomponent system could be fitted from the experimental data with a designated order of approximation on the mass-transfer coefficient. We will give the example for fitting the simplest case: binary system with linear behavior, based on data mentioned in our AgFe DMH-citrate work, where both ions are complexed as anions and deposited under limiting current condition. Meanwhile, a tentative construction of multi-element electrodeposition is given from the tabulated ionic diffusivities and a fixed diffusion layer thickness.

Non-linear behaviors of deposit composition ratio - electrolyte concentration ratio sometimes happen in the alloy electrodeposition comparing with the prediction from the diffusion layer model. We will conceptually discuss the several possible causes for this non-linear behavior: the existence of homogeneous reaction inside the diffusion layer due to the pH and ligand concentration variation,[6] the transition from laminar flow to turbulent flow of the electrolyte near the electrode due to hydrogen bubbles and high deposition rate,[1] the formation of extended charge region at the cathode under overlimiting current,[7] and the situations where charge transfer kinetics governs the overall deposition rate.

References:

Denpo, K., et al., Turbulent natural convection along a vertical electrode. Metallurgical Transactions B, 1983. 14(4): p. 633-643.

Essin, O., Überspannungstheorie und gemeinsame Entladung der Ionen. Zeitschrift für Physikalische Chemie, 1934. 171(1): p. 341-347.

Brenner, A., 9-3 Literature Survey of Theory of Alloy Deposition, in Electrodeposition of Alloys, A. Brenner, Editor. 1963, Academic Press. p. 268-273.

Levich, V.G., 23 Diffusional Flow in Natural Convection. The Case of a Vertical Plate, in Physicochemical hydrodynamics. 1962, Prentice-Hall: Englewood Cliffs, Nj. p. 127-136.

Agar, J., Diffusion and convection at electrodes. Discussions of the Faraday Society, 1947. 1: p. 26-37.

Critelli, R.A., M. Bertotti, and R.M. Torresi, Probe effects on concentration profiles in the diffusion layer: Computational modeling and near-surface pH measurements using microelectrodes. Electrochimica Acta, 2018. 292: p. 511-521.

Rosso, M., J.-N. Chazalviel, and E. Chassaing, Calculation of the space charge in electrodeposition from a binary electrolyte. Journal of Electroanalytical Chemistry, 2006. 587(2): p. 323-328.