Outer solution

The traditional methods for measuring the swelling degree [100] are, as a rule, limited due to the difficulties in quantitative separation of the swollen gel from the outer solution because of extremely low strength of the former. These difficulties can be avoided by measuring the dimensions of a regular shape sample directly in an excess of liquid [19, 101,102], The other example is the modified volumetric method recently developed by us especially for SAH [103],... 

Of particular importance for the application are the effects of the external compression and the ionic composition of the outer solution on the swelling degree. The reason is that hydrogels usually exist in mineralized aqueous solutions (soil solution) and are affected by compression, for example, produced by the surrounding particles of the soil. Even in the absence of any external load the compression develops due to the gel swelling in a constrained volume. 

Fig. 4. Swelling ratio of PAAm hydrogels containing varying amount of AAc units as a function of CuCl2 concentration in the outer solution. Curves are numbered with respect to increasing ionic group content 0 (0), 36.5 (J), 107 (2), 145 (3), and 212 mM (4). From Rieka and Tanaka [101]...

These equations allow either to predict the swelling degree (w = l/(p) as a function of external conditions or to calculate the network parameters from the correlation between the theoretical and experimental dependencies w(q) or w(p) [22, 102], An example of such a correlation is given in Fig. 2 and 5. As can be seen, theoretical predictions are in good agreement with experimental data. However, when the outer solution contains multivalent cations, only a semi-quantitative agreement is attained. 

We may point out parenthetically that it is usually customary to attribute the expansion to electrostatic repulsions between the net (positive) charges on the polymer molecule which are uncompensated due to loss of counter-ions to the outer solution. It may be shown that the osmotic force owing to the excess of mobile ions within the molecule must be equal to the force of electrostatic repulsion when the molecule is in equilibrium with its surroundings. Hence either point of view is equally satisfactory in principle. The two are, of course, mutually related no net charge would develop in the molecule were it not for the mobile counter-ions, and no excess of mobile ions would be retained to exert an osmotic pressure if it were not for the charges on them. 

Complete equilibration of two solutions separated by a membrane is a very slow process. Often quasiequilibrium systems are used, where there is no equilibrium between the outer solutions (their composition is that arbitrarily given at the outset), although each of these solutions is in equilibrium with an adjacent thin membrane surface layer there is no equilibrium within the membrane between these surface layers. 

The mathematical formulations of the diffusion problems for a micropippette and metal microdisk electrodes are quite similar when the CT process is governed by essentially spherical diffusion in the outer solution. The voltammograms in this case follow the well-known equation of the reversible steady-state wave [Eq. (2)]. However, the peakshaped, non-steady-state voltammograms are obtained when the overall CT rate is controlled by linear diffusion inside the pipette (Fig. 4) [3]. 

Recently [8b,30], a new IT feedback mode of SECM was introduced, in which the tip process is a simple or assisted ion transfer. In this mode, a micropipette filled with solvent (e.g., aqueous) immiscible with the outer solution (e.g., organic) serves as an SECM tip. 

Similarly the outer solution r = rg with a sink at infinity n(< >) = 0) is given by... 

Figure 17. The flux balance equations gives rise to three possible solutions While the outer solutions are stable, the middle state is unstable If the actual concentration S(t) is below the nominal value S°, the net flux is negative. The concentration S(l) will decrease even more. Vice versa, if the actual concentration S(l) is above the nominal value, V°, the net flux is positive, leading to a further...

Figure 25.3 The reaction zone configuration used in the present analysis. On the left side solid lines for T, T02, and YcH.4 represent the outer solution, and the dashed lines show profiles resulting from finite reaction rates in the oxygen-consumption layer. The right side corresponds to an expanded view of the regions around in the left sketch, represented by a single fine there, showing the structure of the radical-equilibrium and fuel-consumption layers A — location of fuel and radical layers, B — oxidation layer, C — radical-equilibration layer, and D — fuel-consumption layer...

In both AEA and RRA, there are inert convective-diffusive regions on the fuel and oxidizer sides of the main reaction regions of the diffusion flame. Conservation equations are written for each of the outer inert regions, and their solutions are employed as matching conditions for the solutions in the inner reaction regions. The inner structure for RRA is more complicated than that for AEA because the chemistry is more complex [53]. The inner solutions nevertheless can be developed, and matching can be achieved. The outer solutions will be summarized first, then the reaction region will be discussed. 

The most common analytical applications require one of the two electrodes to be characterized by an unchanging potential, known and independent of the characteristics of the solution being analyzed. Such a device is called the reference electrode. One of the most commonly used is the Ag/AgCl electrode, which consists of a silver wire coated with silver chloride and immersed into a solution saturated by chloride ions a porous plug serves as a cormection bridge with the outer solution. 

First of all, the behavior of the enzymes in the membrane differs markedly from the behavior of the unbound enzymes in solution. It is pertinent to note that the medium in which the enzyme bound to a membrane acts might be determined not only by the composition and structure of the membrane itself, but also by the local concentration distribution of substrate and products. The microenvironment in the membranes is the result of a balance between the flow of matter and enzyme reactions. The substrate and product concentrations in the membrane differ from point to point across the membrane and also from those at the outer solution. By electron microscopy this was experimentally demonstrated beyond doubt with the DAB-peroxidase system by Barbotin and Thomas.16 The effects of these profiles were studied with... 

The outer solutions ul x, e), ur(x,e) have to be smoothly matched with the aid of the appropriate inner solution. Moreover, the matching procedure must specify the location of the front . The inner scale is easily found from (3.3.19), (3.3.22) to satisfy... 

The proper singular perturbation treatment has thus to take care of this initial stage. Probably the simplest way to do this is via a matched asymptotic expansion procedure, with the outer solution of the type (5.2.13), (5.2.14), valid for t = 0(1), matched with an initial layer solution that has an internal layer at x = 0. 

The initial conditions for the b.v.p. (5.2.21), (5.2.22) are to be worked out from matching with the initial layer solution. (It can be easily inferred that the latter yields identical zeros as the initial values to the 0(e) order in the outer solution.) The solution of (5.2.21), (5.2.22) can be easily written out explicitly. Indeed, let us introduce the notation... 

The balance between the exponentials and the AN term in (5.5.13a) yields up to the O( j ) order a piecewise constant outer solution of the form... 

This outer solution, discontinuous at x = 0, has to be smoothed out via an internal layer solution around this point. In this internal layer we distinguish the inner region around x = 0 in which the potential is close to zero and the derivatives term is balanced by N, flanked by two transition layers. In those layers, three terms balance—the derivative, the N term, and one of the two exponents (the positive one for x < 0 and the negative one for x > 0). 

On the other hand, matching with the left outer solution (5.5.15a) implies (5.5.20d) B = -1. 

Equality of osmotic pressures inside the network and in the outer solution which can be written as ... 

Now, we turn to the case when some low-molecular salt is present in the outer solution (ns 4= 0). 

In the polyelectrolyte regime, due to the presence of low-molecular salt, the osmotic pressure of ions becomes less pronounced because the concentration of salt within the network turns out to be less than the concentration of salt in the outer solution n [27]. As the concentration ns grows, the amplitude of the jump of the dependence a(x) decreases and the jump shifts to the region of better solvents (Fig. 2, curve 2). At some critical value of n, the jump on the curve a(x) disappears, i.e. collapse of the network becomes smooth (Fig. 2, curve 3). Under the subsequent increase of n, the curve a(x) becomes closer and closer to the swelling curve of corresponding neutral network (Fig. 2, curves 4). 

In the case of the isoelectric regime of swelling, the concentration of salt in the network is greater than in the outer solution [27] n > ns. The net effect is equivalent to the introduction of effective repulsion between the units of the network chains. The higher concentration of salt, the larger is the size of the network (Fig. 3). 

For an athermal case, the continuous deswelling of the network takes place (Fig. 9, curve 1) which in the result of compressing osmotic pressure created by linear chains in the external solution (the concentration of these chains inside the network is lower than in the outer solution, cf. Ref. [36]). If the quality of the solvent for network chains is poorer (Fig. 9, curves 2-4), this deswelling effect is much more pronounced deswelling to strongly compressed state occurs already at low polymer concentrations in the external solution. Since in this case linear chains are a better solvent than the low-molecular component, with an increase of the concentration of these chains in the outer solution, a decollapse transition takes place (Fig. 9, curves 2-5), which may occur in a jump-like fashion (Fig. 9, curves 3-4). It should be emphasized that for these cases the collapse of the polymer network occurs smoothly, while decollapse is a first order phase transition. 

Also, it is necessary to write down the expression for the free energy Fout of the outer solution (cf. Sect. 2.1.2). We will assume that the concentration of surfactant molecules Co in the outer solution is lower than the critical concentration of micelle formation. Then,... 

In spite of the fact that the concentration of surfactants in the outer solution is assumed to be smaller than the critical micelle concentration, inside the network, micelles are supposed to be formed. The reason for this assumption is, first of all, intensive adsorption of surfactants on the network as a result of the ion exchange reaction. Moreover, in Refs. [38, 39], it was shown that critical concentration of micelles formation c c" within a polyelectrolyte network is much less than that in the solution of surfactant c° . Indeed, when a micelle is formed in solution immobilization of counter ions of surfactant molecules takes place, because these counter ions tend to neutralize the charge of micelles (see Fig. 13), whereas there is no immobilization of counter ions when the micelles are formed in the network the charge of micelles is neutralized by initially immobilized network charges which do not contribute to the translational entropy (Fig. 13). 

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