Diffusion in concentrated solutions

The correlations discussed earlier pertain to diffusion in dilute solutions. With increased concentration, some things are different and the considerations will thus be different. Diffusion coefficients vary with the volume fraction of the solute, often in a complex manner with extrema. Diffusion coefficients are no longer a proportionality constant, but vary with concentration and become concentration 

The correction for diffusion coefficient given in Equation (9.56) may be attributed to a cluster of molecules in the solution. 

Self-diffusion becomes much slower in high-molecular-weight concentrated solutions. A theory for this process has been developed by deGennes. A chain of N monomers in a semidilute or concentrated solution can be represented as a concatenation of blobs of rms length. The chain is constrained in a tube of other chains of length L = (N/g%, where g is the number of monomers per blob. The chain itself has a mean-squared end-to-end length (r = N / g if. 

The chain diffuses along its tube in a Brownian motion determined by the friction of the chain in the tube. The local friction per blob is given by  

The time necessary to diffuse a mean-squared distance along the tube of V- is called the reptation time  

The self-diffusion coefficient can then be expressed as a scaling relation  

The number of monomers per blob is proportional to And for a good solvent, the screening leng is proportional to c This means that the self-diffusion coefficient scales as  

Diffusion causes convection. To be sure, convective flow can have many causes. For example, it can occur because of pressure gradients or temperature differences. However, even in isothermal and isobaric systems, diffusion will always produce convection. This was clearly stated by Maxwell in 1860 Mass transfer is due partly to the motion of translation and partly to that of agitation. In more modem terms, we would say that any mass flux may include both convection and diffusion. 

This combination of convection and diffusion can complicate our analysis. The easier analyses occur in dilute solutions, in which the convection caused by diffusion is vanishingly small. The dilute limit provides the framework within which most people analyze diffusion. This is the framework presented in Chapter 2. 

In some cases, however, our dilute-solution analyses do not successfully correlate our experimental observations. Consequently, we must use more elaborate equations. This elaboration is best initiated with the physically based examples given in Section 3.1. This is followed by a catalogue of flux equations in Section 3.2. These flux equations form the basis for the simple analyses of diffusion and convection in Section 3.3 that parallel those in the previous chapter. 

The material in this chapter is more complicated than that in Chapter 2 and is unnecessary for many who are not trying to pass exams in advanced courses. Nonetheless, this material has fascinating aspects, as well as some tedious ones. Those studying these aspects often tend to substitute mathematical manipulation for thought. Make sure that the intellectual framework in Chapter 2 is secure before starting this more advanced material. 

---

The beauty of the reptation model is that it is able to make predictions about molecular flow both in solution and at fracture by assuming that the molecules undergo the same kind of motions in each case. For both self-diffusion in concentrated solutions and at fracture, the force to overcome in pulling the polymer molecule through the tube is assumed to be frictional. 

Equ0on 10.96 does not apply to either electrolytes or to concentrated solutions. Reid, Praushitz and Sherwood discuss dilfusion in electi olytes. Little information is available on diffusivities in concentrated solutions although it appears that, for ideal mixtures, the preduct is a linear function of the molar concentration. 

In an excellent review article, Tirrell [2] summarized and discussed most theoretical and experimental contributions made up to 1984 to polymer self-diffusion in concentrated solutions and melts. Although his conclusion seemed to lean toward the reptation theory, the data then available were apparently not sufficient to support it with sheer certainty. Over the past few years further data on self-diffusion and tracer diffusion coefficients (see Section 1.3 for the latter) have become available and various ideas for interpreting them have been set out. Nonetheless, there is yet no established agreement as to the long timescale Brownian motion of polymer chains in concentrated systems. Some prefer reptation and others advocate essentially isotropic motion. Unfortunately, we are unable to see the chain motion directly. In what follows, we review current challenges to this controversial problem by referring to the experimental data which the author believes are of basic importance. 

The diffusivity in concentrated solutions differs from that in dilute solutions because of changes in viscosity with concentration and also because of changes in the degree of nonideality of the solution [16]... 

G. Fleischer. Self-diffusion in concentrated solutions of polystyrene in toluene No evidence for large-scale heterogeneities. Macromolecules, 32 (1999), 2382-2383. 

Diffusion in concentrated solutions is complicated by the convection caused by the diffusion process. This convection must be handled with a more complete form of Pick s law, often including a reference velocity. The best reference velocity is the volume average, for it is most frequently zero. The results in this chapter are valid for both concentrated and dilute solutions so they are more complete than the limits of dilute solutions given in Chapter 2. 

---