The Dynamic Diffusion Layer in Branched Growth of a Conductive-Polymer Aggregate in a 2-D Electrolysis Cell D. P. Barkey* and P. D. LaPorte Department ofChemicalEngineering, University ofNew Hampshire, Durham, New Hampshire03824-3591 INTRODUCTION Electrodeposition (ECD) in two- dimensional radial cells has been widely used as a model pattern forming system. Parallels between ECD aggregates and patterns of viscous fingering, annealing and solidification, among other systems, have led to generalizations about growth dynamics far from equilibrium (1-3). The emergent length scales in pattern- forming processes have their origins in the fields and interfacial dynamics that drive growth. In ECD there are 2 fields, a concen- tration field and an electric field. Dif- fusion-controlled structure is observed on a scale of a few tens of microns, correspond- ing to the mass-transfer boundary-layer thickness (4,5). The electric field deter- mines structure that scales with cell size~ while the surface free energy imposes a low- er limit on texture. The large-scale branch- ing observed in 2-D ECD is determined by the interaction of the electric field with the interracial boundary conditions (4). In this report, we show that the dynam- ically-determined diffusion layer in 2-D ECD cells can be modeled as a nearly smooth front that forms an envelope around the branches. We find that the electrolyte may not be depleted at the interface even at current densities above the limiting current to a stationary electrode. While total concentration overpotential calculated from the derived concentration profile may be small, its gradient ~s large at high growth velocities. We also present experimental evidence, from electrodeposition of a conducting polymer, that the diffusion layer forms a smooth envelope around the branches. MASS TRANSPORT MODEL A typical ECD aggregate formed by elec- trodeposition far from equilibrium consists of a dense array of branches advancing within a well-defined front. The electrolyte is partially depleted at the front, and ions migrate and diffuse to the *Electrochemical Society Active Member. growing interface. If all of the growth takes place at the tips, and if the branches are closely spaced, the concentration of ion inside the aggregate is uniform and equal to its value at the growth front. A steady-state solution to the diffusion equa- tion exists when the average concentration within the aggregate, including deposited material as well as material in solution, is equal to the bulk solution value. -~Z C = C b - (Cb-Ci)e D [i] v is the growth velocity, D the diffusivity of the material being deposited, and z the distance from the front. C is the concen- tration, C i the concentration at the inter- face and C b that of the bulk solution. The sum of the migration and diffusion fluxes must equal the cell current. I(1-t) = -DnF = -vnF(Cb-C i) [2] i t is the transference number of the material being deposited. For Ci = O, I ~ = -~FCb/(1-t) [3] I* is a dynamic limiting current propor- tional to growth velocity. An effective diffusion-iayer:thickness is given by D/v. Consistent with previous investigations we find that the growth velocity depends strongly on cell potential but weakly on concentration (4). For ob_mic ceils in which velocity is proportional to cell potential, the ratio I/I is a constant, and the cell current and growth velocity are not limited by diffusion. Velocities as high as tens of microns per second have been observed (4). If a resistive aggregate is modeled as a disk of uniform composition, adhering to a central electrode in a 2-D cell with a con- centric-ring counter electrode, the ohmic cell current is given by ! = [4] I 1 [(!- K!)inr +! inr o- K!'inri] + ~ 2~hV ~' K V 1655