Dilute solution theories

The general theory of polymer solutions, in which the nonuniformity at high dilutions was disregarded, yielded Eq. (31"), which is of the same form as Eq. (72). The coefficients of the first terms in the two expansions are identical, of course. In view of Eqs. (45) and (46), the second coefficient as given by Eq. (75 ) differs from that in Eq. (31") by the factor Thus the dilute solution theory predicts a slope... 

A number of other attempts have been made to account for the properties of concentrated aqueous solutions of ionic compounds by procedures that represent further improvements on the simple Debye-Huckel approach. However, they lie outside the scope of the present chapter. The important point to emphasize is that the concentrated aqueous solutions that are generally employed in the preparation of AB cements tend to exhibit significant ion-ion interactions such interactions lead to significant deviations from ideality which may be accounted for by substantial extension of the ideas of simple dilute solution theory. 

Barkey, Tobias and Muller formulated the stability analysis for deposition from well-supported solution in the Tafel regime at constant current [48], They used dilute-solution theory to solve the transport equations in a Nernst diffusion layer of thickness S. The concentration and electrostatic potential are given in this approximation... 

Dilute Solution Theory. Equation 28 is the result of using dilute solution theory.Such an analysis yields the Nernst—Planck equation... 

Dilute solution theory considers only the interactions between each dissolved species and the solvent. The motion of each charged species is described by its transport properties, namely, the mobility and the diffusion coefficient. These transport properties can... 

It appears that one can develop a qualitative understanding of the simple flow properties at moderate concentration without going beyond concepts which are already inherent either in the dilute solution theory of polymers or in the properties of particulate suspensions. The dependence of viscosity on c[i ] is believed to reflect a particle-like or equivalent sphere (127) hydrodynamics in solutions of low to moderate concentration. In particular, the experimental facts do not force the consideration of effects which might arise from the permanent connectedness of the polymer backbones. Situations conducive to the entangling of molecules may be present, e.g., overlap of the coils, but either entanglement contributions are small, or else they are overwhelmed by the c[f ] interactions. 

In this respect, another insufficiency of Lodge s treatment is more serious, viz. the lack of specification of the relaxation times, which occur in his equations. In this connection, it is hoped that the present paper can contribute to a proper valuation of the ideas of Bueche (13), Ferry (14), and Peticolas (13). These authors adapted the dilute solution theory of Rouse (16) by introducing effective parameters, viz. an effective friction factor or an effective friction coefficient. The advantage of such a treatment is evident The set of relaxation times, explicitly given for the normal modes of motion of separate molecules in dilute solution, is also used for concentrated systems after the application of some modification. Experimental evidence for the validity of this procedure can, in principle, be obtained by comparing dynamic measurements, as obtained on dilute and concentrated systems. In the present report, flow birefringence measurements are used for the same purpose. 

Since the dilute solution theory is considered as the basis for the indicated treatment, it will receive considerable attention. Influences of several parameters as molecular weight, molecular weight distribution, thermodynamic and kinetic chain stiffness, intramolecular hydrodynamic inter-action, optical properties of the chain and solvent power will be considered. 

From dilute solution theory the following definitions are well-known, viz ... 

The most evident reason is that dilute solution measurements can preferably be compared directly with the unmodified dilute solution theory as reviewed in Chapter 3. As has already been pointed out in Section 2.6.1, the form birefringence in dilute solution can effectively be suppressed by the choice of a solvent of practically the same refractive index as the polymer. In such a "matching solvent the contrast between the coil of the macromolecule and its surrounding practically disappears. This means that, at the same time, the influence of the shape (form) of the coil disappears. Also the comparison with measurements on con-... 

When doing calculations with colligative properties, it is usually advantageous to make measurements at a number of concentrations and extrapolate the results to zero concentration, where the ideally dilute solution theory is clearly applicable. 

The water distribution within a polymer electrolyte fuel cell (PEFC) has been modeled at various levels of sophistication by several groups. Verbrugge and coworkers [83-85] have carried out extensive modeling of transport properties in immersed perfluorosulfonate ionomers based on dilute-solution theory. Fales et al. [109] reported an isothermal water map based on hydraulic permeability and electro-osmotic drag data. Though the model was relatively simple, some broad conclusions concerning membrane humidification conditions were reached. Fuller and Newman [104] applied concentrated-solution theory and employed limited earlier literature data on transport properties to produce a general description of water transport in fuel cell membranes. The last contribution emphasizes water distribution within the membrane. Boundary values were set rather arbitrarily. 

Analogous equations are written for fluxes along the other faces. Equation (8) is assumed to be linear. Commonly, in electrochemical systems, the constitutive relationship may be nonlinear. For example, assuming dilute-solution theory, the flux of species j is given by... 

Simulations assumed dilute-solution theory, so that the flux of species j is given by Eq. (9). If the origin is attached to the center of the electrode on the moving plate and if UB 1, the velocity parallel to the electrode can be approximated by... 

Dilute solution theory with no interactions is assumed. Conductivity is estimated from ... 

The parameter g is less than unity for polymers with LCB, and sometimes defined as the ratio of the root-mean-square values, Sg,br/5 g,im- The calculation of g for a given branch structure is straightforward, but radii of gyration are hard to measure experimentally. Intrinsic viscosities are readily measurable, but relating g to g, and thus to branch structure is difficult. Calculations of g involve dilute solution theory, as discussed following Eq. (3.10), with the... 

Rider and O Brien (44) have extended the dilute-solution theory to incorporate the order d> correction which allows one to describe the ESA behavior up to particle concentrations of order 10% by volume. For higher con-... 

The efficacy of O Brien s analysis is demonstrated by the data in Table 2 which shows a comparison of the estimated values of zeta potential and of size using the dilute-solution theory and the more elaborate theory of Eqs. (9) to... 

In this section we will then use the simphfied version of Eq. (14.2) as has been customary in dilute solution theories [1,2,3,4] and wnte for a solution with a single solute a ... 

I.C. Sanchez and E.A. DiMarzio, Dilute solution theory of polymer crystal growth a kinetic theory of chain folding, J. Ghem. Phys. 55, 893-908 (1971). 

Improvements in data handling have been suggested. Chee has generalized existing dilute solution theories and has formulated simple equations to estimate unperturbed dimensions and thermodynamic parameters. He has also modified the Schulz-Blaschke equation to give = [>/l (1 -l- / where n varies between... 

Huang and Eichinger have suggested that none of the dilute solution theories are capable of assigning a temperature dependence to At which is sufficiently large to match that of experimental data obtained by them. 

This model has been successfully used by Chin and Sabde [25] for crevice cathodic protection using numerical analysis based on the dilute solution theory and reduction reaction of dissolved oxygen and Aa+, Cl, and OH ions at the crevice surface. Hence, the Nemst-Plank equation, eq. (4.2), can be generalized as a differentiable and continues scalar diffusion molar flux function... 

The friction coefficient fo now represents the frictional force per monomer unit per unit velocity encountered by a submolecule junction as it moves through a medium consisting of other polymer molecules it is ordinarily considerably higher than the fo of dilute solution theory, which is approximately proportional to solvent viscosity. 

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