Scaling theory osmotic pressure

This article reviews the following solution properties of liquid-crystalline stiff-chain polymers (1) osmotic pressure and osmotic compressibility, (2) phase behavior involving liquid crystal phasefs), (3) orientational order parameter, (4) translational and rotational diffusion coefficients, (5) zero-shear viscosity, and (6) rheological behavior in the liquid crystal state. Among the related theories, the scaled particle theory is chosen to compare with experimental results for properties (1H3), the fuzzy cylinder model theory for properties (4) and (5), and Doi s theory for property (6). In most cases the agreement between experiment and theory is satisfactory, enabling one to predict solution properties from basic molecular parameters. Procedures for data analysis are described in detail. 

Now we compare the above osmotic pressure data with the scaled particle theory. The relevant equation is Eq. (27) for polydisperse polymers. In the isotropic state, it can be shown that Eq. (27) takes the same form as Eq. (20) for the monodisperse system though the parameters (B, C, v, and c ) have to be calculated from the number-average molecular weight M and the total polymer mass concentration c of a polydisperse system pSI in the parameters B and C is unity in the isotropic state. No information is needed for the molecular weight distribution of the sample. On the other hand, in the liquid crystal state2, Eq. (27) does not necessarily take the same form as Eq. (20), because p5I depends on the molecular weight distribution. 

Fig. 2. Comparison between the scaled particle theory (solid curves) and experiment (circles and triangles) for osmotic pressure II of PBLG-DMF [56,57], For the samples with M = 6.6 x 104 and 15.5 x 104, the data at T = 15, 30, and 45 °C are plotted with the same symbols...

This mean-field prediction can be substantiated by constructing a simple de Gennes scaling theory. A similar scaling form for the osmotic pressure is assumed in 0-solutions as was used in good solvent [Eq. (5.43)] ... 

The scaling argument leads to the same prediction in semidilute 0-solutions as the mean-field theory [Eq. (5.57)]. The osmotic pressure in semidilute solutions is again of the order of the thermal energy kT per correlation volume ... 

In the scaling theory, introduced by de Gennes [15], the osmotic pressure behaves like some powers (m) of concentration and becomes independent of the degree of polymerization N). For semidilute solutions (large x). 

The dependence of k on pin the mean field theory (p ) is different from that in the scaling theory by a factor of Such difference is related to a correlation effect given by the number of contacts between monomers. The reduced osmotic pressure (K/Kjdeai) plotted as a function of the reduced concentration pip ), in a double logarithm scale, displays a curve that shows the change in slope to 5/4 for pip > 1 (Fig. 25.9). 

Kosmas and Freed [41] presented another approach to scaling laws. Differing from the theory described above, it starts from the partition function for a solution of continuous chains which interact subject to the binary cluster approximation. For example, their theory derives for osmotic pressure (in three dimensions) a general scaling law which, in our notation, may be written... 

Renormalization group theory (see, e.g., [35]) lies at the heart of this theory, justifying the use of scaling laws in the asymptotic limit, i.e., for infinitely long polymer chains and for dilute solutions. For semidilute solutions, however, this criterion is not so crucial because the polymer chains are overlapping and many properties, e.g., osmotic pressure, are independent of the chain length. 

The osmotic response of swollen polymeric networks was studied on the basis of the scaling theory by Horkay et al. [17-19,22,23,133]. They measured both the swelling pressure, CO, and the shear modulus of gels, G, at different stages of dilution. The swelling pressure vs. polymer volume fraction data were analyzed according to the equation [22]... 

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