Convection as a Source of Self-Organization in Electrochemical Systems
The fluid convection is described as the phenomenon self-organizing into spatial patterns of similar morphology: cooperating counter-rotating rolls, fingerprint-like patterns, and hexagonal patterns—for different driving forces. The classical Bénard–Rayle
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Convection as a Source of Self-Organization in Electrochemical Systems
5.1 5.1.1
Convection as a Self-Organized Phenomenon The Navier-Stokes Equation
Convection is the macroscopic collective flow of particles of the fluids and gases, which can be caused by various driving forces. It is a nonlinear phenomenon which is mathematically described by the Navier–Stokes equation for the motion of viscous liquid: @q 1 þ ðq rÞq ¼ rp þ r2 q þ F @t r
(5.1)
where q denotes the vector of fluid velocity; r, the fluid density; , its dynamic viscosity; p, pressure evolving in the fluid; and F is the volume driving force (N m3) for convection. For the cases described in this work, the pressures developing in the fluid are moderate and then the incompressibility of the fluid can be assumed: divq ¼ 0
(5.2)
as the condition significantly simplifying numerical integration of Eq. (5.1), yielding the vector field of velocity q. The following driving forces F can operate as a source of convection: 1. Buoyancy forces, caused by density gradients of the fluid in the gravitational field which gradients, in turn, can be induced by the thermal gradients or by the concentration gradients (also under isothermal conditions); the thermally induced convection is usually called the Be´nard–Rayleigh instability. 2. Surface (interfacial) tension gradients, which may be induced by thermal or concentration gradients, associated also with (electro)chemical processes at the interface; this type of convection is usually termed the Be´nard–Marangoni instability. M. Orlik, Self-Organization in Electrochemical Systems II, Monographs in Electrochemistry, DOI 10.1007/978-3-642-27627-9_5, # Springer-Verlag Berlin Heidelberg 2012
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5 Convection as a Source of Self-Organization in Electrochemical Systems
3. Volume electric forces, arising from the violation of the electroneutrality principle that may occur under specific conditions. The fluid motion of that type is called the electrohydrodynamic (EHD) convection. In physical systems, in order to induce the EHD convection, the high voltage (of the order of tenths kVs) is usually required for the nonconducting, nonelectroactive media, and then even the unipolar charge injection may occur. For electrochemical systems, the sufficiently high spatial charge in the solution may develop even for a few volt voltage if the thin layer of the electrolyzed solution contains only trace amount of the conducting species. Irrespective of the particular nature of driving force, convection of all kinds may exhibit, under appropriate conditions, the self-organized nature: the motion of the fluid attains then the shape of cooperating rolls, forming also fingerprint-like shapes or hexagonal cells. Patterns of this type become examples of the spatial convective structures which are typical of one-component systems with thermal gradient. Often the terms: Be´nard-Rayleigh and Be´nard-Marangoni instabilities are referred to the formation of such self-organized spatial patterns. More complicated, multicomponent (in
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