Finite concentration

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]... 

A normal diffusion process, however, runs at a finite concentration of particles different from zero. In this situation it was found [101] that a fractal character (73) of the resulting structure is restricted to an interval a < R < if), where d is the diffusion length (67). Larger clusters have a constant density on a length scale larger than They are no longer fractal there. These observations have various consequences for crystal growth, and will be discussed in the next section. 

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. 

Monte Carlo simulation shows [8] that at a given instance the interface is rough on a molecular scale (see Fig. 2) this agrees well with results from molecular-dynamics studies performed with more realistic models [2,3]. When the particle densities are averaged parallel to the interface, i.e., over n and m, and over time, one obtains one-dimensional particle profiles/](/) and/2(l) = 1 — /](/) for the two solvents Si and S2, which are conveniently normalized to unity for a lattice that is completely filled with one species. Figure 3 shows two examples for such profiles. Note that the two solvents are to some extent soluble in each other, so that there is always a finite concentration of solvent 1 in the phase... 

A comparison of the equivalent conductance at some finite concentration (Ac) with that at infinite dilution (AJ gives a measure of the fraction of electrolyte dissociation at the higher concentration. One introduces a, the degree of dissociation or ionization, and writes... 

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. 

Finite concentrations are treated in terms of a virial expansion... 

Finite concentrations are accounted for by a virial expansion of both D and s-1... 

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,... 

This equation has the expected behavior that AG< becomes more positive with decreasing solubility of the solute. However, free energies of solvation for different solutes cannot be related to their relative solubilities unless the vapor pressures of the different solutes are similar or one takes account of this via Equation 76. Furthermore, if the solubility is high enough that Henry s law does not hold, then one must consider finite-concentration activity coefficients, not just the infinite-dilution limit. 

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. 

In the biomedical literature (e.g. solute = enzyme, drug, etc.), values of kf and kr are often estimated from kinetic experiments that do not distinguish between diffusive transport in the external medium and chemical reaction effects. In that case, reaction kinetics are generally assumed to be rate-limiting with respect to mass transport. This assumption is typically confirmed by comparing the adsorption transient to maximum rates of diffusive flux to the cell surface. Values of kf and kr are then determined from the start of short-term experiments with either no (determination of kf) or a finite concentration (determination of kT) of initial surface bound solute [189]. If the rate constant for the reaction at the cell surface is near or equal to (cf. equation (16)), then... 

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. 

At a small but finite concentration, the conductivity of an electrolytic solution is no longer given by Eq. (212). Indeed, due... 

The rod is visualised as being constrained to a tube in a similar fashion to entanglements constraining a polymer in reptation theory. So for a finite concentration our diffusion coefficient and rotary Peclet number changes ... 

All measurements, of course, have to be made at a finite concentration. This implies that interparticle interactions cannot be fully neglected. However, in very dilute solutions we can safely assume that more than two particles have only an extremely small chance to meet [72]. Thus only the interaction between two particles has to be considered. There are two types of interaction between particles in solution. One results from thermodynamic interactions (repulsion or attraction), and the other is caused by the distortion of the laminar fiow due to the presence of the macromolecules. If the particles are isolated only the laminar flow field is perturbed, and this determines the intrinsic viscosity but when the particles come closer together the distorted flow fields start to overlap and cause a further increase of the viscosity. The latter is called the hydrodynamic interaction and was calculated by Oseen to various approximations [3,73]. Figure 7 elucidates the effect. 

The preceding discussion was limited to the artificial case of a single ion. When multiple ions are present, in addition to the issues discussed, there is the problem of ion-ion interactions and correlations. The main motivation for such studies is to come close to the realistic situation in which a finite concentration of ions exists near the metal surface that is in equilibrium with ions in the bulk. Another important specific goal is to investigate the applicability of continuum models, such as the Gouy-Chapman theory. " Although this has been the subject of several Monte Carlo... 

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. 

We can observe from Figure 16.4 that for the particular system depicted, a solution of some finite concentration exists for which the activity is 1. Nevertheless, it would be... 

Integrating Equation (17.31) from the infinitely dilute solution to some finite concentration, we obtain, with the assumption of a Raoult s-law standard state for the solvent and a Henry s-law standard state for the solute,... 

One method of overcoming this difficulty is as follows. Instead of setting the lower limit in the integration of Equation (17.33) at infinite dilution, let us use a temporary lower limit at a finite concentration X2. Thus, in place of Equation (17.34), we obtain... 

This equation can be integrated from the infinitely dilute solution ( 2 = 0) to any finite concentration to give... 

However, for more precise calculations, it is necessary to consider that the mobility (hence, the conductance) of ions changes with concentration, even when dissociation is complete, because of interionic forces. Thus, Equation (20.20) is oversimplified in its use of Aq to evaluate a, because at any finite concentration, the equivalent conductances of the and Ac ions, even when dissociation is complete, do not equal Aq. 

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)...

The foregoing considerations have been devoted solely to the discussion of Ig(q) of a single isolated molecule in solution. Experiments require finite concentrations, however, which may be of the order of a few percent. In dilute solutions of dendrimers the influence of concentration can be taken into account in terms of the structure factor S (q) ... 

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