Mass transport

Mass transport of the analyte under investigation is governed by the Nemst-Planck equation defined by  

If we consider an electrochemical experiment which is conducted in a solution that has supporting electrolyte and in stagnant solutions (non-hydrodynamic conditions, see later) such that migration and convection can be neglected from Eq. (2.25), this is thus reduced to consider the only relevant mode of mass transport to the electrode surface on the experimental time scale, which is difliision. 

The difhision of species i, from bulk solution to the electrode is described by Pick s first and second laws of diffusion  

Let us consider a simple redox process involving the transfer of one-electron between the electrode and species A in solution to form the product B in solution, as shown below  

Departing from the bulk solution towards the electrode surface, natural convection dies away due to the rigidity of the electrode surface and frictional forces, this is the diffusion layer, and since only concentration changes occur in this zone, dififu-sional transport is in operation. Note that in reality there is no real defined zones and these merge into one another, but it is a useful concept. Under experimental conditions, the diffusion layer is in the order of tens to hundreds of micrometers in size. 

The mass transport through geomembrane liners is described with the help of mathematical models. However, modelling will only supply reasonable results if all the relevant aspects of the basic physical processes are covered by the model, if mass transport parameters relevant for the model calculation and properties of the liner materials relevant for the mass transport modelling are sufficiently well known and if the boundary eonditions above and beneath the liner, under which the pollutants exert their effect, can be specified. Difficulties arise specifically with the last point, sinee it is usual for a mixture of several pollutants in a complex and temporally changing contamination situation to be present. On the other hand, the importance of the effect of the interaction of different substances on the mass transport is sometimes overestimated, and as a rule it is sufficient to look at the behaviour of a few guide substances . 

In addition, often too little attention is paid to the correct representation of the relevant physical processes in the mathematical model. Two examples should be mentioned here  

Holes in the geomembrane can have a considerable influence on mass transport depending on the properties of the subgrade. However, different views exist as to how water flow through a hole should be correctly modelled. 

Diffusion is a process in which the transport of matter through a substance occurs. Selfdiffusion refers to atoms diffusing among others of the same kind (e.g. in pure metals). 

The governing phenomenological equation for ionic conduction, as in electronic conduction, is Ohm s law (Eq. 6.21). Concentration gradient induced processes, on the other hand, follow Pick s laws of diffusion, derived by Adolf Eugen Pick (1829-1901) in 1855 (Pick, 1855). 

The mechanisms of mass transport can be divided into convective and molecular-flow processes. Convective flow is either forced flow by pumps and compressors, for example, in pipes and packed beds, or natural convection driven by density gradients that are induced by temperature gradients in a fluid. For molecular flow we have to distinguish whether we have diffusion in a free fluid phase or in porous solids. These processes are examined below. 

All adsorption and desorption processes depend on transport of solute to and from the interface. There are basically four major transport mechanisms (Fig. 3)  

Turbulent or stirred solutions may incorporate all of the processes noted. 

Integrating gives the total number of molecules, n, adsorbed at the elapsed time (t)  

All fluid interfaces contain an undisturbed layer of solution adjacent to the interface. Mass transport in this boundary layer occurs only by diffusion. The thickness of the boundary layer depends on temperature, stirring, and the interface itself. It is up to 0.1 cm thick in unstirred systems and approaches 10 3 cm in vigorously stirred systems3,31,32). 

Once the interface is partially saturated with adsorbed solute molecules, then the rate of adsorption falls below the rate of diffusion, suggesting an energy barrier to adsorption. 

In general, it is necessary to consider three modes of mass transport in electrochemical systems (1) diffusion (2) migration and (3) convection. 

The treatment of mass transport, more than any other aspect of the subject, highlights the diflercnces between laboratory experiments and industrial-scale electrolyses. In the former, there is great concern to ensure that the mass transport conditions may be described precisely by mathematical equations (which, moreover, are soluble) since this is essential to obtain reliable mechanistic and quantitative kinetic information. The need in an industrial cell is only to promote the desired effect within technical and economic restraints and this permits the use of a much wider range of mass transport conditions. In particular, a diverse range of electrode-electrolyte geometry and relative movement are possible. 

The investigation of the mechanism and kinetics of electrode processes is normally undertaken with solutions containing a large excess of base electrolyte (i.e, so that the migration of electroactive species is unimportant), but two types of experiment are common  

Using unstirred solutions and a short timescale so that natural convection does not interfere. 

Using a form of forced convection which may be described exactly by far the most important system is the rotating-disc electrode and it frequently provides a good but simple analogue of the mass transport conditions in industrial cells. 

Diffusion. Diffusion is the movement of a species down a concentration gradient and it occurs whenever there is a chemical change at a surface. An electrode reaction converts starting material to product (e.g. O R) and hence close to the electrode surface there is a boundary layer (up to 10 cm thick) in which the concentration of O is lower at the surface than in the bulk while the opposite is the case for R and, hence, O will diffuse towards and R away from the electrode. 

Convection is the movement of a species due to mechanical forces. It can be eliminated, at least on a short timescale (it is difficult to eliminate natural convection arising from density differences on a longer timescale, i.e. longer than 10 s) by carrying out the electrolysis in a thermostat in the absence of stirring or vibration. 

In industrial practice it is much more common to stir or agitate the electrolyte or to flow the electrolyte through the cell. These are all forms of forced convection and, when present, it is always the predominant form of mass transfer. 

The rate-limiting step is generally determined by either the surface reaction kinetics or by mass transport. 

In the case of control by surface reaction kinetics, the rate is dependent on the amount of reactant gases available. As an example, one can visualize a CVD system where the temperature and the pressure are low. This means that the reaction occurs slowly because of the low temperature and there is a surplus of reactants at the surface since, because of the low pressure, the boundary layer is thin, the diffusion coefficients are large, and the reactants reach the deposition surface with ease as shown in Fig. 2.8a. 

The main act of the electrochemical process, charge transfer, is localized in a very thin double electric layer. This process can take place continuously only when electrically active particles, that is, particles that participate in the charge transfer step are transferred toward the electrode, and the products formed move in the opposite direction - from the surface of the metal phase of the electrode to the solution volume. When electrochemical deposition of metal takes place in simple (noncomplex) salt solutions, there may be no transport of the product, because the metal atoms formed do not participate in the diffusion process, and they form a new solid phase - a crystal lattice. 

Diffusion, migration, and convection are the three possible mass transport processes accompanying an electrode reaction. In the first case, the particles move due to the formed concentration gradient. The flux in the case of planar (onedimensional or linear) diffusion can be described by Pick s first law  

When diffusion is coupled with migration, the appropriate form of the Nernst - Planck equation has to be solved. Por instance, when the ion with charge ze is distributed at concentration c and the potential cp, a one-dimensional flux of 

Electrochemistry of Metal Complex Applications from Electroplating to Oxide Layer Formation, First Edition. Arvydas Survila. 

The intensity of migration mass transport can be significantly reduced by adding an excess of indifferent electrolyte whose ions are electrically inactive within a wide range of the potentials, that is, they do not participate in the charge transfer process. These ions create the electric field of the opposite direction, which compensates for the aforementioned potential gradient to a great extent. This sim-phfies the theoretical description of the current. 

Here vx and vy are the components of the velocity vector in the coordinate directions. Velocity v in an arbitrary direction is 

The value of groundwater velocity within a rock or sediment, then, invariably exceeds that of specific discharge. 

Chemical mass is redistributed within a groundwater flow regime as a result of three principal transport processes advection, hydrodynamic dispersion, and molecular diffusion (e.g., Bear, 1972 Freeze and Cherry, 1979). Collectively, they are referred to as mass transport. The nature of these processes and how each can be accommodated within a transport model for a multicomponent chemical system are described in the following sections. 

Advective transport, or simply advection, refers to movement of chemical mass within a flowing fluid or gas. For our purposes, it is most commonly migration of aqueous species along with groundwater. In constructing a transport model, we prefer to consider how much of the thermodynamic components - the total masses of the basis entries Aw, A(, A, and Am - move, rather than track migration of the free masses of each individual aqueous species. 

Taking Cw, Ci, Cp, and Cm as the volumetric concentrations (mol cm-3) of basis entries mobile in the groundwater, the advective fluxes (mol cm-2 s-1) of 

As seen previously (Section 2.2), the rate of an electron transfer is also conditioned by the rate with which the electrode is supplied with reagent and cleared of the electrogenerated product. 

3 Diffusion-limited current planar and spherical electrodes 

9 An example of a convective-diffusion system the rotating disc electrode 

In the last chapter it became clear that in the expression for the rate of an electrode reaction 

Reactive iiquid fiim containing both reactants 

According to the above discussion, the metal ions produced under applied potential may dissociate from the anode surface and get into the electrolyte solution due to the electrostatic attraction from polar water molecules and anions in an electrolyte. Driven by the electric field between anode and cathode, all cations move toward the cathode and all the anions move toward the anode. The ion motion driven by electric field is called migration, as shown with white arrows in Fig. 10.4. 

The third form of mass transport is convection driven by pressure. When forced circulation exists in electrolyte, convection may be the dominant form of mass transport. Thus, in general, a flux Jj (mol/s cm) of species j may occur due to the above three types of mass transport mechanisms. The flux can be described by the Nernst-Planck equation [5] 

In a real electrochemical system, convection is usually introduced by such means as rotating electrode, stirring, or other forced circulation. In any case, the electrolyte moves relative to electrode surfaces. Due to the mechanical friction between electrolyte solution and electrode surface, a velocity v(x) variation exists. The velocity of solution flow is generally a constant (vqo) in bulk solution (far from the electrode surface and the wall of solution container) and decreases while approaching the solid surfaces [6]. The solution flow velocity v(x) = 0 at solid surface (x = 0). A hydrodynamic (or Prandtl) boundary layer is defined as [6] 

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The process of the clinker mass transport is simulated as follows ... 

As also noted in the preceding chapter, it is customary to divide adsorption into two broad classes, namely, physical adsorption and chemisorption. Physical adsorption equilibrium is very rapid in attainment (except when limited by mass transport rates in the gas phase or within a porous adsorbent) and is reversible, the adsorbate being removable without change by lowering the pressure (there may be hysteresis in the case of a porous solid). It is supposed that this type of adsorption occurs as a result of the same type of relatively nonspecific intermolecular forces that are responsible for the condensation of a vapor to a liquid, and in physical adsorption the heat of adsorption should be in the range of heats of condensation. Physical adsorption is usually important only for gases below their critical temperature, that is, for vapors. 

If a fluid is placed between two concentric cylinders, and the inner cylinder rotated, a complex fluid dynamical motion known as Taylor-Couette flow is established. Mass transport is then by exchange between eddy vortices which can, under some conditions, be imagmed as a substantially enlranced diflfiisivity (typically with effective diflfiision coefficients several orders of magnitude above molecular difhision coefficients) that can be altered by varying the rotation rate, and with all species having the same diffusivity. Studies of the BZ and CIMA/CDIMA systems in such a Couette reactor [45] have revealed bifiircation tlirough a complex sequence of front patterns, see figure A3.14.16. 

NMR is an important teclnhque for the study of flow and diflfiision, since the measurement may be made highly sensitive to motion without in any way influencing the motion under study. In analogy to many non-NMR-methods, mass transport can be visualized by imaging the distribution of magnetic tracers as a fiinction of time. Tracers may include paramagnetic contrast agents which, in particular, reduce the transverse... 

Diflfiision, convection and migration are the fonns of mass transport that contribute to the essential supply and removal of material to and from the electrode surface [1, 2, 3 and 4],... 

Double potential steps are usefiil to investigate the kinetics of homogeneous chemical reactions following electron transfer. In this case, after the first step—raising to a potential where the reduction of O to occurs under diffrision control—the potential is stepped back after a period i, to a value where tlie reduction of O is mass-transport controlled. The two transients can then be compared and tlie kinetic infomiation obtained by lookmg at the ratio of... 

Improved sensitivities can be attained by the use of longer collection times, more efficient mass transport or pulsed wavefomis to eliminate charging currents from the small faradic currents. Major problems with these methods are the toxicity of mercury, which makes the analysis less attractive from an eiivironmental point of view, and surface fouling, which coimnonly occurs during the analysis of a complex solution matrix. Several methods have been reported for the improvement of the pre-concentration step [17,18]. The latter is, in fact. 

The expression for the mass-transport-limiting current density may be employed together with the Nemst equation to deduce the complete current-potential response in a solution containing only oxidized or reduced species... 

The great advantage of the RDE over other teclmiques, such as cyclic voltannnetry or potential-step, is the possibility of varying the rate of mass transport to the electrode surface over a large range and in a controlled way, without the need for rapid changes in electrode potential, which lead to double-layer charging current contributions. 

A number of different types of experiment can be designed, in which disc and ring can either be swept to investigate the potential region at which the electron transfer reactions occur, or held at constant potential (under mass-transport control), depending on the infomiation sought. 

The solution flow is nomially maintained under laminar conditions and the velocity profile across the chaimel is therefore parabolic with a maximum velocity occurring at the chaimel centre. Thanks to the well defined hydrodynamic flow regime and to the accurately detemiinable dimensions of the cell, the system lends itself well to theoretical modelling. The convective-diffiision equation for mass transport within the rectangular duct may be described by... 

A microelectrode is an electrode with at least one dimension small enough that its properties are a fimction of size, typically with at least one dimension smaller than 50 pm [28, 29, 30, 31, 32 and 33]. If compared with electrodes employed in industrial-scale electrosynthesis or in laboratory-scale synthesis, where the characteristic dimensions can be of the order of metres and centimetres, respectively, or electrodes for voltannnetry with millimetre dimension, it is clear that the size of the electrodes can vary dramatically. This enonnous difference in size gives microelectrodes their unique properties of increased rate of mass transport, faster response and decreased reliance on the presence of a conducting medium. Over the past 15 years, microelectrodes have made a tremendous impact in electrochemistry. They have, for example, been used to improve the sensitivity of ASV in enviroiunental analysis, to investigate rapid... 

Of course, in order to vary the mass transport of the reactant to the electrode surface, the radius of the electrode must be varied, and this unplies the need for microelectrodes of different sizes. Spherical electrodes are difficult to constnict, and therefore other geometries are ohen employed. Microdiscs are conunonly used in the laboratory, as diey are easily constnicted by sealing very fine wires into glass epoxy resins, cutting... 

The effects of ultrasound-enlianced mass transport have been investigated by several authors [73, 74, 75 and 76]. Empirically, it was found that, in the presence of ultrasound, the limiting current for a simple reversible electrode reaction exhibits quasi-steady-state characteristics with intensities considerably higher in magnitude compared to the peak current of the response obtained under silent conditions. The current density can be... 

Compton R G, Ekiund J C, Page S D, Mason T J and Walton D J 1996 Voltammetry in the presence of ultrasound mass transport effects J. Appl. Electrochem. 26 775... 

Birkin P R and SilvaMartinez S 1995 The effect of ultrasound on mass-transport to a microelectrode J. Chem. See., Chem. Commun. 17 1807... 

A solution containing botli reactants and a catalyst may be mixed mechanically to bring tire constituents into efficient contact—otlierwise, tire rate of tire catalytic reaction would be affected by mass transport (e.g., diffusion)... 

Alternatively, the mass transport properties in tire solution can become rate detennining—tire reaction is tlien said to be diffrrsion controlled. 

Under diffusion-controlled dissolution conditions (in the anodic direction) the crystal orientation has no influence on the reaction rate as only the mass transport conditions in the solution detennine the process. In other words, the material is removed unifonnly and electropolishing of the surface takes place. 

Schuck P 1996 Kinetics of iigand binding to receptors immobiiized in a poiymer matrix, as detected with an evanescent wave biosensor, i. A computer simuiation of the influence of mass transport Biophys. J. 70 1230-49... 

The layer of solution adjacent to the electrode in which diffusion is the only means of mass transport. 

Concentration gradients for the analyte in the absence of convection, showing the time-dependent change in diffusion as a method of mass transport. 

The flux of material to and from the electrode surface is a complex function of all three modes of mass transport. In the limit in which diffusion is the only significant means for the mass transport of the reactants and products, the current in a voltammetric cell is given by... 

Influence of the Kinetics of Electron Transfer on the Faradaic Current The rate of mass transport is one factor influencing the current in a voltammetric experiment. The ease with which electrons are transferred between the electrode and the reactants and products in solution also affects the current. When electron transfer kinetics are fast, the redox reaction is at equilibrium, and the concentrations of reactants and products at the electrode are those specified by the Nernst equation. Such systems are considered electrochemically reversible. In other systems, when electron transfer kinetics are sufficiently slow, the concentration of reactants and products at the electrode surface, and thus the current, differ from that predicted by the Nernst equation. In this case the system is electrochemically irreversible. 

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