At finite concentrations

At finite concentration, tire settling rate is influenced by hydrodynamic interactions between tire particles. For purely repulsive particle interactions, settling is hindered. Attractive interactions encourage particles to settle as a group, which increases tire settling rate. For hard spheres, tire first-order correction to tire Stokes settling rate is given by [33]... 

Freed, KF Muthukumar, M, On the Stokes Problem for a Suspension of Spheres at Finite Concentrations, Journal of Chemical Physics 68, 2088, 1978. 

The frictional coefficient varies with concentration, but at infinite dilution it reduces to the coefficient (/o) for an isolated polymer molecule moving through the surrounding fluid unperturbed by movements of other polymer molecules (see Chap. XIV). At finite concentrations, however, the motion of the solvent in the vicinity of a given polymer molecule is affected by others nearby binary encounters (as well as ones of higher order) between polymer molecules contribute also to the observed frictional effects. The influence of these interactions will persist to very low concentrations owing to the relatively large effective volume of a polymer molecule, to which attention has been directed repeatedly in this chapter. Since the sedimentation con-stant depends inversely on the frictional coefficient, s must also depend bn concentration. 

We have recently performed systematical measurements of the intrinsic viscosity of acrylamide-acrylic acid copolymers for large ranges of r and a, in the presence of CaCl2(26). Our results show that the empirical relation (14) can be extended to the case of divalent cations by using the value of 7 given in relation (15). It should then possible to predict the variation of intrinsic viscosity at infinite dilution, but at finite concentration the formation of aggregates makes difficult the determination of the Huggins constant. 

Figure 18 Scaling law of sedimentation coefficient s0 for NaPSS. Measurements of s at finite concentrations (left) and plot of s0(M) (right). Reproduced with permission from Machtle and Borger [78].

At finite concentration, the conditional probabilities will be modified to include a factor, u>, which takes account of the presence of other chains in the system. According to the mean field approximation,... 

A similar multiphase complication that should be kept in mind when discussing solutions at finite concentrations is possible micelle formation. It is well known that for many organic solutes in water, when the concentration exceeds a certain solute-dependent value, called the critical micelle concentration (cmc), the solute molecules are not distributed in a random uncorrelated way but rather aggregate into units (micelles) in which their distances of separation and orientations with respect to each other and to solvent molecules have strong correlations. Micelle formation, if it occurs, will clearly have a major effect on the apparent activity coefficient but the observation of the phenomenon requires more sophisticated analytical techniques than observation of, say, liquid-liquid phase separation. 

At infinite dilution, g(R) — exp (be KRjR). Linearizing the inner exponential and neglecting the second term in the denominator of the last equation we recover the Bjerrum result (Eq. (185)). However, at finite concentrations even if we retain terms to the same order in log y1 and g(R), Eqs. (183) and (186) will not in general give the same value of p. The use of a mass action formalism as a means both of calculating activity coefficients and of studying the pair distribution function via the degree of association p at finite concentrations is not done in a self-consistent manner in the Bjerrum type of treatment. 

As the laws of dilute solution are limiting laws, they may not provide an adequate approximation at finite concentrations. For a more satisfactory treatment of solutions of finite concentrations, for which deviations from the limiting laws become appreciable, the use of new functions, the activity function and excess thermodynamic functions, is described in the following chapters. 

Fig. 6.19 Relaxation rates from single exponential fits to the NSE data from PAM AM den-drimers of generation g=5-8 (5%) in d-methanol. The solid lines are derived from NSE data from the FRJ2-NSE (Jiilich) and MESS (Saclay) spectrometers and show the prediction for simple Stokes-Einstein diffusion of hard spheres at finite concentration. (Reprinted with permission from [306]. Copyright 2002 American Institute of Physics)...

In this section a brief survey over the results achieved so far by small-angle scattering studies is given. The discussion is done in two parts. First data obtained at vanishing concentration are presented that allow one to assess the density distribution inside the dissolved dendrimers. Then data measured at finite concentration are shown which give highly interesting information on the mutual interaction of dendrimers in solution. 

It is clear from equation (7) that the addition of a second surfactant results in further decrease in y the essential requirements being a not too small adsorption of the second surfactant. Whether it replaces the first surfactant or is adsorbed in addition to it is immaterial, just as it is not essential for the two surfactants to form a complex. If the two surfactants are of the same type e.g. both water soluble anionic surfactants, they will form mixed micelles and this will lower the activity of the second surfactant added and decrease both its Fand dp. However, if the two surfactants are different in nature, e.g. one predominantly water soluble and the other oil soluble, they will only slightly affect each other s activity and their combined effect on the interfactial tension may be large enough to bring y to zero at finite concentrations. 

The diffusion coefficients at infinite dilution (D]0, D 0, and Dr0) for the fuzzy cylinder reduce to those for the wormlike cylinder, which can be calculated as explained in Appendix B. On the other hand, these diffusion coefficients, D, Dx, and Dr, for the fuzzy cylinder at finite concentrations can be formulated by use of the mean-field Green function method and the hole theory, as detailed below. 

In Sect. 6.3, we have neglected the intermolecular hydrodynamic interaction in formulating the diffusion coefficients of stiff-chain polymers. Here we use the same approximation by neglecting the concentration dependence of qoV), and apply Eq. (73) even at finite concentrations. Then, the total zero-shear viscosity t 0 is represented by [19]... 

Variations in JeR at finite concentration reflect the influence of intermoleeular interaction on the relative spacings of the long relaxation times. Through JeR these effects may be examined separately from effects on the magnitude of the viscosity (or the average magnitude of the long relaxation times). The ratio Exf/(X T )2 is called S2/Sj by Ferry and co-workers. The above expression for JeR... 

Grand Canonical Description of Solutions at Finite Concentration... 

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