Overlapping chains

Fig. 8. Schematic representation of the possible conformations of adsorbed (co)polymers prepared using macromonomer technique a brush adsorption of graft copolymer b terminally-attached adsorption c the mushroom-brush transition for strongly overlapping chains proposed by de Gennes [130] and Alexander [155]...

Several improvements to Equation (2D-4) have been suggested. Primarily these modifications involve a more exact treatment of the polymer chain in the lattice such as including the probability of overlapping chains. These improvements are not generally applied in view of the approximations inherent in the lattice model of the fluid and the marginal increase in accuracy resulting from these improvements. 

The major simplifications involved in Equation (2D-4) are that it does not account for the probability of overlapping chains and the volume change upon mixing of the polymer and solvent. The volume change cannot be accounted for in a lattice model when all of the lattice sites are assumed to be filled. The probability that a lattice site is filled, however, can be calculated. Huggins (1941,1942a,b,c) included in his calculations probabilities thata polymer molecule would encounter a filled lattice site. This led to a slightly different form for Equation (2D-4), but Flory (1970) states that the refinement probably is beyond the limits of reliability of the lattice model. 

For situations of overlapping chains, where lateral fluctuations in the segment concentration become rather small, mean-field descriptions become appropriate. The most successful of this type of theoiy is the lattice model of Scheutjens and Fleer (SF-theoiy). In chapter II.5 some aspects of this model were discussed. This theory predicts how the adsorbed amount and the concentration profile 0(z) depend on the interaction parameters and x and on the chain length N. From the statistical-thermodynamic treatment the Helmholtz energy and, hence, the surface pressure ti can also be obtcdned. When n is expressed as a function of the profile 0(z), the result may be written as ... 

In solvents near the -temperature, the thermal blob is larger than the chain gr>N meaning z < 1 or T—9 jTexcluded volume interactions are weak. The interaction energy of two overlapping chains is less than the thermal energy kT, so chains can easily inter-... 

Polymer solutions with volume fractions

g, linear chains interpenetrate each other. At volume fractions above overlap (0 > 0 ) at T—9, polymer solutions are called semidilute 9. This name originates from the fact that at low volume frac-tions < 0 solution properties are dictated by overlapping chains. 

On length scales larger than the correlation length, the excluded volume interactions are screened by the overlapping chains. The semidilute solution on these length scales behaves as a melt of chains made of correlation blobs and the polymer conformation is a random walk of correlation blobs ... 

Outside the dilute regime, chain overlap occurs, which implies that polymer molecules are mutually entangled. This greatly increases viscosity, as well as the dependence of viscosity on concentration and the extent of strain rate thinning. Moreover, the solution shows elastic besides viscous behavior, and if intermolecular cross-links are formed, a gel is obtained. The chain overlap concentration decreases with increasing molecular size, [j value, and stiffness. For overlapping chains, the solution is characterized by a correlation length, which does not depend on molecular size, and which is... 

In fact, the decisive experimental proof of the macromolecular hypothesis has been given by osmotic pressure measurements in very dilute solutions. A second fundamental contribution is more recent. It deals with the study of systems of strongly overlapping chains. The observations made by Noda and his collaborators (1980) definitely showed that, in this case, the osmotic pressure dependence of the solute concentration follows a universal law the crossover with the dilute state is also universal. These results gave an experimental proof of the principles which constitute the basis of the modern theory introduced by de Gennes and des Cloizeaux. 

The value 1/(6) is determined by linear extrapolation of the inverse intensity at q = 0, and 2 by measuring the slope of the inverse intensity at q = 0. This method does not differ much from the method we used to determine the screening length e for good solutions of overlapping chains [see (15.4.3.3)]. In the present case, it is actually difficult to make a distinction between e and , the inverse intensity being practically linear with respect to q2 for small q2. Let us also note that, in the situation described here, we have /°RG < 1. 

Noda et a/.21 pursued this experimental study, but at higher concentrations. They measured the osmotic pressure of polymer chains in solution, for T> TF and > c so as to remain always in the poor solvent state with a strong chain overlap (see Fig. 13.26, p. 642). In this physical situation, the volume fractions of polymer are high for instance, it may occur that

0.3. Thus, one gets out of the theoretical framework fixed in Chapters 13 and 14. To interpret the pressure measured by the authors quoted above, a theory for the liquid polymer state is needed. However, we present here their results without referring to any theory of this sort because, per se, these results manifest properties which are those of solutions of overlapping chains with (p < 1 (they were studied in Chapter 15 and at the beginning of this section). 

Thus for times of or shorter, and only for these times, an assembly of mutually overlapping chains can act as a mechanically connected network. 

The weak, repulsive interactions between overlapping chains also give rise to corrections to the single-chain stmrture factor, " which deviates from the Debye-function that characterizes the statistics of Gaussian chain molecules on the length scale of R. These effects have been analytically predicted by renormalization group calculations in semidilute solutions and by perturbation calculations in melts. For (JR I1 and long chains. 

Finally, a brief mention must be made of the work by Adam and Delsanti, who studied diffusion of polystyrene in benzene as a function of concentration. They show that, above c, a co-operative diffusion coefficient is found dependent only on c, and the overlapping chains behave like gels with a finite lifetime. Dynamic scaling laws recently proposed by De Gennes are compared and largely verified. 

In a solution of highly overlapped chains, the thermodynamic properties do not explicitly depend on N but on p only, as the result of the blob model indicates. From the condition that W/k Tbe independent of N, we determine m as m = l/(3v — 1) or m - 5/4, and Eq. 4.20 becomes identical to Eq. 4.12. Without assuming the blobs, we derived an expression for II that is identical to the one we obtained earlier in the blob model. This identity is already a proof for the blob model. 

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