Donnan Phase Models

FIGURE 13.6 A scheme of a virtual Donnan phase a portion of the solution around the humic particle, effectively containing the diffuse layer (grayed area) is considered as a different phase separated from the bulk solution, in electroehemical equilibrium. 

Inside the Donnan volume, Equation 13.24 no longer holds because of the presence of the negatively charged macromolecule instead, the electroneutrality condition is now (assuming that the Donnan volume is large enough as to neglect border effects) 

Clearly, the concentrations inside cannot be equal to the bulk solution values because of the macromolecule charge, the cation concentration should be higher than the bulk, whereas the anion concentration should be lower. The ions able to cross the membrane must be in electrochemical (osmotic) equilibrium between the two solutions that is, in Equations 3.1 through 3.3, the electrochemical potentials must be equal. 

If the activity coefficients are assumed equal at both sides of the membrane, the Donnan potential for a 1 1 electrolyte of concentration c results to be 

In principle, the Donnan potential and the concentrations inside the Donnan volume can be found if z is known and the activity coefficients can be obtained. However, in several models, as discussed later. Yd is not explicitly calculated, and only Equations 13.25 and 13.27 are used to close the system. 

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One of the main assumptions of the Donnan partition model is that two well-defined phases (polymer and solution) exist and the electrostatic potential presents a sharp transition between them. This approximation is fulfilled when the typical decay length of the electrostatic potential (Debye length) is much shorter than the film thickness. The other limiting situation is that where all the redox sites are located in a plane and thus the Debye length is larger than the film thickness. This situation can be described by the surface potential model ... 

The following relationship is derived for the distributions of ions M " " and Na+ between the polyion domain (Donnan phase) and the bulk solution phase with the Gibbs-Donnan model. 

The simple Donnan model has limitations that are comparable to those of the GC model. The assumption that d,D = 0 just outside the hypothetical membrane is reasonable as long as the particle size is much larger than For particle sizes that are of the order of the hypothetical membrane can be thought at about a distance from the particle surface to incorporate the diffuse layer ions in the Donnan phase [25]. Ultimately this leads to a value of Vd that is larger than the void volume inside the particles and an average Donnan potential that is somewhat lower than the potential without this correction. The magnitude of the differences will depend on... 

A notable limitation of the simple Donnan model is that the potential may not be uniformly distributed through the Donnan phase close to the charged sites the potential will be the highest and the fixed charges may be grouped together in small island like domains [24]. Potentials calculated with Eq. (31) therefore tend to be the average potentials inside... 

A discussion of the Donnan model, including its use in mixed electrolytes, has been given by Bolt [26]. Bolt applied these models for the description of the ion exchange properties of clay minerals. Ohshima and Kondo [27] give more elaborate expressions for the potential inside and outside the Donnan phase. Their results are based on a uniform density of the fixed sites and they show that ipd,D decreases if the outer bound of the Donnan phase is reached. 

NICA isotherm for competitive binding. Diverse electrostatic models are considered, mainly based on one out of two concepts the impermeable sphere model (similar to a mineral particle) or the Dorman phase model, where a certain volume is assumed to be in Donnan equilibrium with the bulk. Considering these contributions, several models are proposed in the end showing similar fitting quality to titration curves because these curves have poor sensitivity to the many adjustable parameters additional, independent data are needed to improve modeling of cation binding to HSs. 

Figure 9.13 Measured dpK/PP/da values for poly(NIPAM-AA) (B0.6-A2.5), (B0.6-A5), (B0.6-A10), (B0.6-A15) and (B0.6-A20) ( ). Predicted dp f PP/da values using a two-phase model based on the Donnan concept ( ).

Hard-sphere or cylinder models (Avena et al., 1999 Benedetti et al., 1996 Carballeira et al., 1999 De Wit et al., 1993), permeable Donnan gel phases (Ephraim et al., 1986 Marinsky and Ephraim, 1986), and branched (Klein Wolterink et al., 1999) or linear (Gosh and Schnitzer, 1980) polyelectrolyte models were proposed for NOM. Here the various models must be differentiated in detail—that is, impermeable hard spheres, semipermeable spherical colloids (Marinsky and Ephraim, 1986 Kinniburgh et al., 1996), or fully permeable electrolytes. The latest new model applied to NOM (Duval et al., 2005) incorporates an electrokinetic component that allows a soft particle to include a hard (impermeable) core and a permeable diffuse polyelectrolyte layer. This model is the most appropriate for humic substances. 

Use of Eq. (18) permits identification of the log(y +/Yp,+ ) term as the one contributing most importantly to differences in affinity of pairs of univalent metal ions, M and N, for a cation-exchange resin. For example, for the equilibrium distribution of Li and Na ions between dilute solutions of Li and Na" " chloride (myQ + m jci = 0.010 m) and the much more highly concentrated Dowex-50 (8% cross-linked by weight with divinylbenzene) phase (niy, + iUnj, = 4.5 m) the p(Vy+ - Vjjj,+ )/2.3 RT and the 2 log(yyQ/y j(-i) term yield a small sum (< 0.04) while the value of log (Yu /Ynj,+ ) approaches 0.35 once correction for interaction between the two metal ions is made. Correlation between experiment and computation of the Gibbs-Donnan-based terms is strongly supportive of the model. 

Another necessary modification of the fibre phase data is related to the fact that the amount of water inside the fibres cannot be allowed to vary fieely during calculations. The simplest solution is to keep its amount constant. This is not precisely tme as the swelling of fibres is affected both by pH and the ionic strength of the solution. However, the calculated ionic distribution is not particularly sensitive to moderate changes in fibre phase volume and the same assumption of a fixed amount of fibre phase water has been commonly made with other Donnan theory-based models as well. 

Adamson (51) proposed a model for W/0 microemulsion formation in terms of a balance between Laplace pressure associated with the interfacial tension at the oil/water interface and the Donnan Osmotic pressure due to the total higher ionic concentration in the interior of aqueous droplets in oil phase. The microemulsion phase can exist in equilibrium with an essentially non-colloidal aqueous second phase provided there is an added electrolyte distributed between droplet s aqueous interior and the external aqueous medium. Both aqueous media contain some alcohol and the total ionic concentration inside the aqueous droplet exceeds that in the external aqueous phase. This model was further modified (52) for W/0 microemulsions to allow for the diffuse double layer in the interior of aqueous droplets. Levine and Robinson (52) proposed a relation governing the equilibrium of the droplet for 1-1 electrolyte, which was based on a balance between the surface tension of the film at the boundary in its charged state and the Maxwell electrostatic stress associated with the electric field in the internal diffuse double layer. 

As was shown in the BAHLM model for transport kinetics, the values of the overaU mass-transfer coefficients govern the location (i ax) and the maximum quantity (Ofemax) of the metal species in the LM phase (see Eqs (27) and (28)). At Ormax Qf, the BAHLM is working mainly as a Donnan dialysis system, in which the loaded carrier solution is a treated feed. In this... 

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