Growing drops

Several mercury electrodes combine the features of the DME and HMDE. In particular, one employs a narrow-bore capillary that produces DMEs with drop lives of 50-70 s (14). Another involves a controlled-growth mercury drop (15). For this purpose, a fast-response valve offers a wide range of drop sizes and a slowly (step-by-step) growing drop. 

The problem of convective diffusion toward the growing drop was solved in 1934 by Dionyz Ilkovic under certain simplifying assumptions. For reversible reactions (in the absence of activation polarization), the averaged cnrrent at the DME can be represented as... 

A growing-drop method has been reported [53] for measuring interfacial liquid-liquid reactions, in which mass transport to the growing drop was considered to be well-defined and calculable. This approach was applied to study the kinetics of the solvent extraction of cupric ions by complexing ligands. 

To ensure that the detector electrode used in MEMED is a noninvasive probe of the concentration boundary layer that develops adjacent to the droplet, it is usually necessary to employ a small-sized UME (less than 2 /rm diameter). This is essential for amperometric detection protocols, although larger electrodes, up to 50/rm across, can be employed in potentiometric detection mode [73]. A key strength of the technique is that the electrode measures directly the concentration profile of a target species involved in the reaction at the interface, i.e., the spatial distribution of a product or reactant, on the receptor phase side. The shape of this concentration profile is sensitive to the mass transport characteristics for the growing drop, and to the interfacial reaction kinetics. A schematic of the apparatus for MEMED is shown in Fig. 14. 

A charging current (non-faradaic) due to the formation of an electric double layer on the surface of the growing drop most polarographs permit a... 

In LSV usually a single-sweep procedure, the so-called impulse method, is applied with the result illustrated in Fig. 3.30. In the multi-sweep procedure, formerly called Kipp method, Fig. 3.31 is obtained, which shows the saw-tooth character of the sweep and a series of peak curves of increasing height caused by the growing drop surface. Exceptionally, use is made of a triangular sweep in the impulse method this variant of cyclovoltammetry is depicted in Fig. 3.32... 

For a given vapor pressure, there is a critical drop size. Every drop bigger than this size will grow. Drops at a smaller size will evaporate. If a vapor is cooled to reach over-saturation, it cannot condense (because every drop would instantly evaporate again), unless nucleation sites are present. In that way it is possible to explain the existence of over-saturated vapors and also the undeniable existence of fog. 

Growing drop methods constitute a more recent group of techniques, not discussed so far. They have in common that a drop is formed at the tip of a narrow capillary, inside which the pressure is measured by a very sensitive transducer. A variety of sophisticated designs to carry out the measurement can be found in the recent literature. ... 

In polarography, not enough time is available for the diffusion layer to reach its stationary thickness. Instead, the current per unit electrode area decreases with the square root of time, the signature time-dependence for diffusion. On the other hand, the area of the growing drop expands, proportional to the two-thirds power of drop age r (i.e., time elapsed since the previous mercury drop fell off). These two counteracting effects, diffusion currents per area proportional to r 1/2, and area growth as t2/3, combine to yield polarographic current-time curves with a time dependence of t-1/2 X t2/3 = t1/6, as expressed in the Ilkovid equation. 

FIG. 17 (a) Typical current (/)-time (t) characteristics for bromine transfer from an aqueous phase to an expanding DCE drop measured at a 1 qm diameter Pt UME positioned beneath the growing drop. The images (l)-(4) correspond to the points indicated on the transient. The final drop size in (4) is 1.00 mm. Data are analyzed to produce (normalized) concentration vs. distance profiles, such as that in (b), showing experiment (O) and theory for a transport-limited process (—). 

Even in mass-transfer-limiled processes, excursions in selectivity can be observed at finite contact times. This is predicted by rate models as simple as Eq. (8.4-1) for two metals with different equilibrium constant valoes, The phenomenon involves initially fast coextraction followed by crowding out of the less preferred metal during coijipetidon for extractant. This has been observed during simultaneous extraction of copper and zinc chlorides by TIOA in a growing-drop experiment"1 21 and in extraction or uranyl nitrate and nitric acid by TBP in a Lewis cell.2 as shown in Fig. 8.4-5. 

Tensiometry" as the surface tension passes a maximum due to the superposition of area expansion and adsorption rate of surfactants. Similar behaviour is observed at the surface of growing drops, discussed in the next paragraph. 

CONSIDERA TION OF iNTERFACIAL AREA CHANGES AND RADIAL FLOW FOR GROWING DROPS... 

The kinetics of the adsorption process taking place at the surface of a growing drop or bubble is important for the interpretation of data from drop volume or maximum bubble pressure experiments. The same problem has to be solved in any other experiment based on growing drops or bubbles, such as bubble and drop pressure measurements with continuous, harmonic or transient area changes (for example Passerone et al. 1991, Liggieri et al. 1991, Horozov et al. 1993, Miller at al. 1993, MacLeod Radke 1993, Ravera et al. 1993, Nagarajan Wasan 1993). 

Ilkovic (1934, 1938) was the first who discussed the adsorption at a growing drop surface. A complete diffusion-controlled adsorption model, considering the radial flow inside a growing drop, was derived much later by Pierson Whittaker (1976). The diffusion equation for describing the transport inside or outside a spherical drop or bubble has the following form ... 

Fig. 4.9 Change of the diffiisional layer thickness adjacent to the surface of a growing drop...

Eq. (4.45) can be used to describe the adsorption process at the surface of growing drops. The analysis of this rather complex equation showed that the rate of adsorption at the surface of a growing drop with linear volume increase, as it is arranged in a usual drop volume experiment, is about 1/3 of that at a surface with constant area. This result is supported by experimental findings (Davies et al. 1957, Kloubek 1972, Miller Schano 1986, 1990) and also by an approximate solution first discussed by Delahay Trachtenberg (1957) and Delahay Pike (1958) ... 

A more advanced theory of the adsorption process at growing drop surfaces was made by MacLeod Radke (1994). In contrast to the theory discussed above they do not assume a point source at the beginning of the process but a finite drop size. On the basis of an arbitrary dependence R(t) a theory of diffusion- as well as kinetically-controlled adsorption was then derived. In addition to the diffusion equation (4.43) the following boundary condition is proposed ... 

Although this is a very complex equation, it allows to take into consideration any function of R(t), and consequently A(t), resulting from experiments with growing drops or bubbles. In combination with an adsorption isotherm (diffusion-controlled case) or a transfer mechanism (mixed diffusion-kinetic-controlled model) it describes the adsorption process at a growing or even receding drop. Eq. (4.48) can be applied in its present form only via numerical calculations and an algorithm is given by MacLeod Radke (1994). 

The time for bubble growth to a hemisphere t, is the difference between the total bubble time Tb and the dead time x, p and L are pressure and gas flow rate and p and are the respective values at a critical point. From considerations discussed above about adsorption processes at the surface of growing drops it is concluded that the situation with the growing bubble is comparable, at least until the state of the hemisphere. Then, the process runs without specific control and leads to an almost bare residual bubble after detachment due to the very fast bubble growth. 

This is already the more general case with a time dependent interfacial area, again without taking into account any resulting flow. For A = const, the final term on the right hand side diminishes. MacLeod Radke (1994) discussed the adsorption for growing drops in a second liquid. To do so, they used diffusion equations of the type (4.43) in both liquid phases as well as the boundary condition (4.65). The result has the structure of Eq. (4.46) but consists of two... 

For radially growing drops MacLeod Radke (1994) derived a long time approximation for a drop growing into air. 

Growing drops and bubbles good good no commercial set-up... 

The difference in the design of the other growing drop set-ups consists most of all in the use of a direct pressure transducer instead of a differential one. In all cases the data acquisition is made by an on-line coupled computer. In the instrument of Nagarajan Wasan (1993) the syringe is also controlled by the computer allowing different types of volume, and consequently drop surface area changes to be measured. The instruments of MacLeod Radke... 

Fig. 5.18 Principle of a growing drop experiment according to Passerone et al. (1991) S - motor driven syringe, C - capillary, DPT - differential pressure transducer, LI and L2 - the two liquids...

As discussed by MacLeod Radke (1993) the growing drop instrument in the design shown in Fig. S.19 provides three experimental techniques a maximum drop pressure, a continuously... 

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