Mean-field theory osmotic pressure

As X is lowered, the polymer likes the solvent more, increasing the osmotic pressure. However, the mean-field theory that is the basis of Eq. (4.71) is only valid close to the 0-temperature, where chains interpenetrate each other freely [Eq. (3.102)]. Far above the 0-temperature (in good solvent), the second virial coefficient A2 is related to chain volume [Eq. (3.104)] rather than monomer excluded volume v. Recall that the second virial coefficient can also be determined from the concentration dependence of scattering intensity [Eq. (1.91)]. 

The Flory-Huggins mean-field theory recovers the van t Hoff Law 4Fq. (1-72)] in the dilute limit (as 0—>0). At higher concentrations, the mean-field theory predicts that two-body excluded volume interactions make osmotic pressure proportional to the mean-field probability of monomer-monomer contact (0 ) ... 

The mean-field theory correctly predicts that the osmotic pressure is independent of molar mass in semidilute solution. However, the mean-field theory does not take into account the correlations between monomers along the chain, but instead assumes that they are distributed uniformly as in a solution of monomers. This is the reason why the two-body interac-Tioiis [Eq. (5.41)] become important at volume fraction --------------------... 

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 ... 

Figure 4. Adnesion energy (erg/cm ) for SUPC biiayers in 0.1 M salt (PBS) plus dextran polymers. Number average polymer indices - Np (number of glucose monomers). Solid and dasned curves - predictions from mean field theory with first and second virial coefficients from osmotic pressure measurements (14).

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). 

This is not in accordance with a pure electrosorption according to mean-field theory [12] but it is in accordance with the build-up of an osmotic pressure due to ntm-neutralized charges of the adsorbed PEC on the solid, in combination with the dimensions and the composition of the PEC, as described above. 

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)] ... 

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