Electro-osmotic coefficient

This hydrodynamic contribution to n is determined by the dielectric constant (e) and the viscosity of water (u), the surface charge density of the pore (Z), the pore radius (rp), and the proton conductivity of the pore (cTpore)- The hydrodynamic electro-osmotic coefficient for a typical pore with Tp = 1 nm is found in the range of [i.e., n ydr -1-10]. 

The total electro-osmotic coefficient = Whydr + mo includes a contribution of hydrodynamic coupling (Whydr) and a molecular contribution related to the diffusion of mobile protonated complexes—namely, H3O. The relative importance, n ydr and depends on the prevailing mode of proton transport in pores. If structural diffusion of protons prevails (see Section 6.7.1), is expected to be small and Whydr- If/ ori the other hand, proton mobility is mainly due to the diffusion of protonated water clusters via the so-called "vehicle mechanism," a significant molecular contribution to n can be expected. The value of is thus closely tied to the relative contributions to proton mobility of structural diffusion and vehicle mechanism. ... 

In Table III the specific conductance and electro-osmotic coefficient (3) for the SPS membrane are shown together with the data for a conventional ion-exchange membrane, AMF C103 (16,17) (polyethylene-styrene graft copolymer containing sulphonic acid groups). It appears that there is a close similarity in properties of both membranes. ... 

In addition, the simple phenomenological relation (6.1.4), with a constant electro-osmotic coefficient lc, was replaced by a more elaborate one, accounting for the w dependence on the flow rate and the concentrations Ci, C2 via a stationary electro-osmotic calculation. This approach was further adopted by Meares and Page [7] [9] who undertook an accurate experimental study of the electro-osmotic oscillations at a Nuclepore filter with a well-defined pore structure. They compared their experimental findings with the numerically found predictions of a theoretical model essentially identical to that of [5], [6]. It was observed that the actual numerical magnitude of the inertial terms practically did not affect the observable features of the system concerned. 

Below we shall need some order of magnitude estimates for the hydraulic permeability i> and the electro-osmotic coefficient u>. Such estimates are provided by the expressions... 

Equation (6.3.3b) results from assuming a Poiseille flow in a filter s pore of typical radius r, i is the dynamic viscosity of the fluid. Equation (6.3.3c) is a common expression for the electro-osmotic coefficient [13], with d and , respectively, the dielectric constant of the fluid and the -potential of the pore wall. For the time being, we shall assume il> constant (independent of C(x,t)). 

In accordance with (6.3.7a) define the dimensionless electro-osmotic coefficient ui as... 

Figure 3. Water electro-osmotic coefficient vs. anolyte concentration for Nafion 295 membrane, 80° C, 2 kA/m2. Key Q, measurements with new cell design, identical anolyte/catholytes , chlorate present in anolyte A, measurements with old cell design, identical catholyte/anolytes.

For the transport properties, the conductivity is explained based on the pereolation eoneept where the percolation threshold occurs around X = 2 and is correlated with enough clusters being hydrated and connected by hydrated sulfonie acid sites to form a complete conductive pathway across the membrane [34,41]. With the addition of more water into the system, more elusters and channels form and the pathways become less tortuous the eonductivity increases. The electro-osmotic coefficient depends on the type... 

The physical model can be used to describe trends seen in experimental data. For example, the interconnectivity of the cluster network is predicted to have a profound effect on a membrane s transport properties. The percolation threshold for conductivity should increase when the clusters become smaller, which could be due to a stiflfer and/or more crystalline polymer matrix. These smaller clusters would also mean that the membrane would exhibit lower electro-osmotic coefficients, larger liquid water uptakes, and a greater dependence of the various properties on water content than in Nafion . In fact, these predictions are what is seen in such systems as sulfonated polyetherketones [19, 72] and Dow membranes [73, 74] or when the equivalent weight [22] or drying temperature [4, 6] of Nafion is increased. 

The reported electro-osmotic coefficient of water typically lies between 1 and 2 (but can be as high as ten at high water content, see, e.g., ref [1]), i.e., protons moving from the anode to the cathode drag , on overage, one or two water molecules with them [2]. This leads to the obvious question how this can happen if the proton moves, as originally assumed, in a purely Grotthuss style relay fashion (in which protons hop from one water molecule to the next one [3, 4]) that does not require the motion of protonic aqua complexes as a... 

The nature of the bottlenecks for proton conductance in the dry membrane state or on the way to it is, however, still the subject of debates. This wiU only be resolved after more detailed experimental studies (of macroscopic transport parameters such as proton conductance and electro-osmotic coefficients as a function of water content, or gas and liquid permeability before and after operation, and of microscopic structural probes such as small-angle neutron and X-ray scattering) will have discriminated between competing models. By and large, the direction of effects that go with dehydration is obvious enough to be introduced into phenomenological models of overall cell performance. 

Section VI ends with a general relation in the limit of large double layer thickness that, to the best of our knowledge, is novel. This propert) yields a single reduced representation of the electro-osmotic coefficients for all the studied configurations as functions of a dimensionless double layer thickness. 

The dimensionless conductivity 0/0 and coupling coefficients depend linearly upon C The conductivity and electro-osmotic coefficients 6- = (cr/a - l)/c ) and are plotted in Figs 4a and 4b. respectively, versus the solid volume concentrations ( ) for the three reduced surface potentials = -1,0, +1. For = 0, one can see that 6- tends to -3/2 as (j) 0. Moreover, for uncharged particles, the... 

For any given C, the electro-osmotic coefficient p is a decreasing function of the solid volume fraction, as may be seen in Fig, 4b. In the dilute limit, the fluid in the bulk of the pore space is driven by the slip velocity at short distances of the particles. For lower porosities, the double layers overlap, and the free streaming zone shrinks and ultimately disappears. 

The ratio of the velocity of the EOF to the applied electric field, which expresses the velocity per unit field, is defined as electro-osmotic coefficient or, more properly, electro-osmotic mobility ([Xeo). 

The total electro-osmotic coefficient nd = nhydr + nmoi includes terms due to hydrodynamic coupling nhydr, and a molecular coupling nmoi, that is related to the structural diffusion of protonic defects. The relative contributions of nhydr and nmol depend on the mechanism of proton transport in pores. 

The problems involved in formulating a consistent theory of binding are illustrated when one measures the thermodynamic and dynamic properties of a limited number of polyelectrolytes with different counter-ions and then attempts to formulate a simple theory. For example, Gregor measured the selective uptake, self diffusion coefficients, electrical conductivity and electro-osmotic coefficients of a number of different ions in ion-exchange membrane and resin systems. The measurement of selective uptake is unequivocal and its correlation with binding is straightforward. Data on self-diffusion coefficients are complicated to interpret because the narrow pores of these insolu-bilized polyelectrolytes place steric and hydrodynamic restrictions upon the diffusive process. These can be overcome, at least in a semi-quantitative manner, by the use of appropriate correction terms [7]. The measurement of the electro-osmotic coefficient is simple and its interpretation is similarly straightforward. Data on electrical conductivity require interpretation because of the steric and hydrodynamic restraints of the pore nature of the system there is an electro-osmotic correction to the electrical conductivity. Table I tabulates normalized values for different counter-ions with... 

In the case of the chloride and iodide Table I shows that Stern layer binding is probably responsible. It is interesting to observe that the electro-osmotic coefficients correlate well with the (hydrated) hydrodynamic size of all ions. It is evident further that in spite of the extremely high molality of these systems (their internal molality is of the order of 5-7 M) there is apparently not a substantial dehydration of the lithium ion. The equal electro-osmotic coefficients of the chloride and iodide ions show that by whatever mechanisms they are transported through the membrane, the volumes transported are very much the same as for the free ions themselves. ... 

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