Semi-dilute solutions screening

A comparison with Burchard s first cumulant calculations shows qualitative agreement, in particular with respect to the position of the minimum. Quantitatively, however, important differences are obvious. Both the sharpness as well as the amplitude of the phenomenon are underestimated. These deviations may originate from an overestimation of the hydrodynamic interaction between segments. Since a star of high f internally compromises a semi-dilute solution, the back-flow field of solvent molecules will be partly screened [40,117]. Thus, the effects of hydrodynamic interaction, which in general eases the renormalization effects owing to S(Q) [152], are expected to be weaker than assumed in the cumulant calculations and thus the minimum should be more pronounced than calculated. Furthermore, since for Gaussian chains the relaxation rate decreases... 

It is generally accepted that in semi-dilute solutions under good solvent conditions both the excluded volume interactions and the hydrodynamic interactions are screened owing to the presence of other chains [4,5,103], With respect to the correlation lengths (c) and H(c) there is no consensus as to whether these quantities have to be equal [11] or in general would be different [160],... 

The hydrodynamic interaction is introduced in the Zimm model as a pure intrachain effect. The molecular treatment of its screening owing to presence of other chains requires the solution of a complicated many-body problem [11, 160-164], In some cases, this problem can be overcome by a phenomenological approach [40,117], based on the Zimm model and on the additional assumption that the average hydrodynamic interaction in semi-dilute solutions is still of the same form as in the dilute case. 

It is admitted that asymptotically the coherent contribution reaches a constant limit when the transfer wave number q becomes much larger than the inverse of the correlation length of the system. This correlation length is, in dilute solutions, the size of the chains in semi-dilute solutions, the screening length. In... 

We see that this length er which tells us how the density-density correlations decrease with r is actually proportional to the size of the chains. In a semi-dilute solution, it is larger than the length e defined by (13.2.102). Of course, e is really what we usually call the screening length but the very existence of shows that, for finite chains, there is no complete screening. 

In a semi-dilute solution, the chains screen one another. Then, in order to determine the end-to-end distance of a polymer, we can perform an approximate perturbation calculation, replacing the real interaction by the screened interaction. In the following, we reproduce a calculation first made by Edwards.48 It consists in determining the mean square end-to-end distance by perturbation to first order with respect to the screened interaction b(q) or more precisely its approximate value (13.2.143)... 

The same criterion does not apply in the semi-dilute region because, then, screening effects must be taken into account. Let us thus consider a semi-dilute solution of Brownian chains. The distance 0 between Brownian chains, the so-called overlap Brownian length , is given by (13.2.4) and, for d - 3,... 

Thus, the transition between poor and good solvent in a semi-dilute solution corresponds to a concentration C = C at which the screening effect begins to... 

Region III - Semi dilute solution with moderate screening... 

In a semi-dilute solution, the interactions between polymers are screened but, if in a semi-dilute solution of monodisperse polymers, a very long polymer is introduced, the screening for this polymer cannot be complete. Indeed, for a scale of the same order as the size of the very long polymer, the semi-dilute solution may look practically homogeneous, but as was noticed by Flory51 and de Gennes,52 the very long chain is subjected to an excluded volume interaction our aim here is to calculate this effective interaction. 

Of course, in a semi-dilute solution, because of screening effects, the influence of the interactions on the swelling diminishes. In other words, in the vicinity of the tricritical point, a chain is nearly Brownian with reduced logarithmic corrections. In first approximation, we can deduce an expression of the swelling from (14.6.54) by using for h the value given by (14.6.69). In this way, we obtain... 

When the concentration p increases, the screening length decreases, and this is one of the most revealing facts of the structure of good semi-dilute solutions. 

Figure 5.7. A schematic diagram of a semi-dilute solution, which may be thought of as a transient network characterised by a screening or correlation length...

The repulsive core is what we might call the bare interaction it would be present even in the absence of polymer chains on surrounding sites. The attractive part is due to the chains and tends to reduce or screen out the bate interaction. This notion of screening was first introduced— in a somewhat different context— for semi-dilute solutions by S. F. Edwards. ... 

In entangled solutions screening becomes effective at a characteristic distance, called the screening-length , and denoted s- Figure 3.7 provides an experimental example of the evidence. We see the scattering intensity measured for a semi-dilute solution of polystyrene in CS2, in a plot of versus 5/3 interested in the single chain structure factor in the semi-dilute... 

Another interesting phenomenon in semi-dilute solutions is that the molecular coils decrease in size with increasing polymer concentration. The segments of a given chain becomes screened by the nearby... 

The adsorbed layer is well described by a self-similar picture[4] generated by a geometrical constraint on the screening length of the semi-dilute solution near the wall the only distance in the problem being the the distance z from the wall one must have i = z which leads to a concentration profile... 

Equilibrium Properties of Dilute and Semi-dilute Solutions. Excluded volume causes a polymer to favor expanded states. Large fluctuations, analogous to critical fluctuations, lead to non-ideal power law relations between properties. In the semi-dilute regime excluded volume correlations are screened. Scaling laws relate dilute and semi-dilute exponents. 

Some of the questions to be addressed in Section 1 are i) what are excluded volume and the 0 point ii) why are polymer coils expanded due to excluded volume, and what limits that swelling iii) what is meant by a semi-dilute solution iv) what is screening v) how do screening distance, radius of gyration, and osmotic pressure vary with concentration in the semi-dilute regime vi) what are scaling laws and how are they used and vii) what are blobs ... 

We shall consider first the screening of hydrodynamic interactions, which turns out to be of major importance in a semi-dilute solution. We will then introduce the scaling laws of collective modes in semi-dilute solutions, emphasizing here the role of universal behavior as a function of the concentration variable. The experimental results are then discussed within the framework of this universal behavior. 

The main results for the dynamics of dilute solutions reflect the importance of hydrodynamic interactions each moving monomer in the solvent creates a backflow field which decays very slowly with distance. In a semi-dilute solution, the interference between all these velocity fields induces a screening of the backflow field of a given monomer, which falls off exponentially after a characteristic distance A. This idea was originally proposed by Edwards and Freed we shall briefly summarize their theory for ideal Gaussian chains. 

Another important implication of the screening of the hydrodynamic interactions is that the effective friction of the monomers on the solvent is concentration dependent in a semi-dilute solution. The finite concentration of the surrounding monomers renormalizes the friction constant... 

In a semi-dilute solution the pair correlation function is given by equation (11) and the velocity correlation function is given by the screened Oseen tensor... 

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