The dilute limit

Tracer Diffusivity Tracer diffusivity, denoted by D g is related to both mutual and self-diffusivity. It is evaluated in the presence of a second component B, again using a tagged isotope of the first component. In the dilute range, tagging A merely provides a convenient method for indirect composition analysis. As concentration varies, tracer diffusivities approach mutual diffusivities at the dilute limit, and they approach selr-diffusivities at the pure component limit. That is, at the limit of dilute A in B, D g D°g and... 

One can easily extend the above analysis to dilute and semi-dilute solutions of EP [65,66] if one recalls [67] from ordinary polymers that the correlation length for a chain of length / in the dilute limit is given by the size R of the chain oc When chains become so long that they start to overlap at I I (X the correlation length of the chain decreases and reflects... 

Before turning to dynamics, we should hke to point out that, because no solvent is explicitly included, the Rouse model [37,38] (rather than the Zimm model [39]) results in the dilute limit, as there is no hydrodynamic interaction. The rate of reorientation of monomers per unit time is W, and the relaxation time of a chain scales as [26,38]... 

The system of H chemisorbed on Cu/Ni is examined both with and without surface segregation. In the case of cs = Cb (i.e., no surface segregation), the curve of AE vs q, is shown in Fig. 6.3(a), and is seen to have a monotonic behaviour, which is almost linear for intermediate values of c, . In the dilute limits (cfe close to 0 or 1), AE is closer to the value for the corresponding pure system than a purely linear relationship would produce, which suggests that the effect of any minority atoms, even near the surface, is cancelled by the averaging process used in the CPA. 

These experiments suggest that as the long time self-diffusion coefficient approaches zero the relaxation time becomes infinite, suggesting an elastic structure. In an important study of the diffusion coefficients for a wide range of concentrations, Ottewill and Williams14 showed that it does indeed reduce toward zero as the hard sphere transition is approached. This is shown in Figure 5.6, where the ratio of the long time diffusion coefficient to the diffusion coefficient in the dilute limit is plotted as a function of concentration. 

Inspection of Equation (5.68) indicates that in the dilute limit if we divide by the solvent viscosity we obtain the intrinsic viscosity ... 

Kerosene can be utilized effectively to reduce the pour point of most distillate fuels. The dilution limits are often based upon whether kerosene dilution will negatively impact fuel properties such as the viscosity, distillation parameters, sulfur limit, or cetane number. 

Moreover, the influence of the motions of the particles on each other (i.e., when the motion of a particle affects those of the others because of communication of stress through the suspending fluid) can also influence the measured diffusion coefficients. Such effects are called hydrodynamic interactions and must be accounted for in dispersions deviating from the dilute limit. Corrections need to be applied to the above expressions for D and Dm when particles interact hydrodynamically. These are beyond the scope of this book, but are discussed in Pecora (1985), Schmitz (1990), and Brown (1993). 

To obtain a useful approximate solution of the PB equation (S8.6-4), we consider the dilute limit in which the electric potential O is weak compared with the ambient thermal energy kT. In this limit, the Boltzmann exponential can be linearized by retaining only the leading term in the power series expansion... 

Interdiffusion of AX-BX2 in the dilute limit (AX is the solvent) has been discussed by [A. B. Lidiard (1957) R. E. Howard, A. B. Lidiard (1964)]. Lidiard proposed to equate the fluxes jBn and j] (, where ) = [B, V) is the (neutral) associate between BA and VA. The argument goes that BA+ is only mobile in form of the pair, when BA and VA are nearest neighbors and can exchange sites. Formally, this means... 

T want to finish this introduction by a short sketch of the history of the subject. The physics of dilute polymer solutions by now has been ail active field for about TO years. Much of the early work is connected to the name of Flory and summarized in his classic books [Flo53, Fit>69], Up to about. 1970 much theoretical or experimental work concentrated on the behavior in the dilute limit, where via virial expansions the problem can be reduced to considering only a few interacting chains. The development led to the so-called two parameter theories , which essentially expand quantities like Rg or A2 in powers of z, In 1971 these developments were most carefully reviewed in a book by Yamakawa [Yam.71]. 

As a final example we consider the density autocorrelation function, where we first discuss the dilute limit... 

We have formulated the method starting from the cluster expansion valid in the dilute limit. The loop expansion is renormalized by exactly the same steps. The only new feature is the occurrence of the parameter W = W(N) (cf. Eq. (5.1)), which under these steps transforms as... 

To summarize, strict e-expansion a priori seems to yield unambiguous results. Closer inspection, however, reveals that in low order calculations considerable ambiguity is hidden in the definition of the physical observables used as variables or chosen to calculate. What is worse, the e-expansion does not incorporate relevant physical ideas predicting the behavior outside the small momentum range or beyond the dilute limit. In particular, it does not give a reasonable form for crossover scaling functions. On the other hand, it can be used to calculate well-defined critical ratios, which are a function of dimensionality only, Even then, however, the precise definition of the ratio matters,... 

The choice of no governs r for small momenta in the dilute limit where Eq. (13.21) reduces to Nr = no- We thus should consider some universal ratio, defined for c —> 0, q -+ 0 which must be most sensitive to Nr. The interpenetration ratio ip is a good candidate, since it depends on Nr even in zero-loop approximation. From Sect. 15.4, Eq, (15.54) we take the one-loop order result... 

In Fig. 13.3 we also plotted lines / = 0.9,/ = 0.1, which bound the excluded volume or the 0-region, respectively. In the dilute limit w —> 1 these lines reduce to z = const. In the semidilute limit they behave as z s, corresponding to... 

If compared to Eq. (9 17) the second equation shows that N/Nr is to be identified with the number of segments n per concentration blob, whereas Nr — Nfn is the number of blobs per chain. The first equation shows that we find a smooth crossover from the dilute limit w — 1 to the semidilute limit w — 0. In the latter limit Eqs. (14 13), (14.14) yield the expected power law... 

Fig. 14,1. (a) logL0 Rg/B as function of log10 s for z = 103. Also included are the dilute limit. s/mri dashes) and the semidilute excluded volume limit (long dashes) (Eq. (14.15)). (b) Rg/RQ as function of s, showing the gradual crossover from dilute to semidilute behavior. The semidilute power Law (Eq. (14.15) is represented by the dashed line... 

In the dilute limit of course equals the radius of gyration. We thus consider the semidilute limit, where is of the order of the screening length. (As will be discussed in Chap. 19, it is not strictly identical to the screening length.) Equation (14.35) reduces to... 

Impurity ions in wurtzite lattices are described by the same expressions for P2, and P3c, with a numerically insignificant difference in P3o. These expressions are only quantitatively accurate in the dilute limit, but many of the doped nanocrystals discussed in this chapter fall in this limit. The reader is referred to Ref. 42 for a generalized treatment of the problem. Figure 2(b) plots the probabilities calculated from Eq. 4a-d as a function of impurity concentration. The fraction of dopants having at least one nearest-neighbor dopant is quite high even at moderate impurity concentrations (<5%). Needless to say, whereas purification to ensure size uniformity is possible (size-selective precipitation), no purification method has yet been developed for ensuring uniform dopant concentrations in an ensemble of nanocrystals. 

The dilute limit emerges when a /z3 dielectric response of dense suspension follows the Lorentz-Lorenz or Clausius-Mossotti relation6 e = [(1 + 2Na/3)/(l - Na/3)] (This is the next approximate form when the number density N is too high to allow the linear relation e = 1 + Na.) Below what density N will this e be effectively linear in polarizability Expand... 

As the segment density p increases from the dilute limit (or as coils collapse for v < 0), higher-order interactions (three-body, four-body,...) become important. The fraction of space excluded by chains may be expressed by the... 

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