By Fumio Oosawa
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Extra info for Polyelectrolytes
When the number o f molecules is increascd at a constant total = 24 volume of the gas, the condcnsation to liquid takes place at a certain critical value of the number or the density of gas molecules. Above this critical point the increase of the number of molecules does not i nc rease the number or the density in the gas phase. Only the amount of the liquid phase is increased (Fig. 1 0). In analogy to this gas-liquid condensation, the concentration of free co u nt er ions above the critical valuc of the charge density corresponds to the saturated vapor pressure coexisting with the liquid.
The charge density arises from charged groups of the macroions and counter ions. The former is assumed to be expressed as a uniform density i n the spherical volume, v, namely, III V • and Pm = 0 outside o f v (35) 3. 33 ELECTRIC POTENTIAL AND FREE ENERGY If the macroion is placed at the center of the spherical free volumc of radius R, the potential t/I is a function of the distance r from the center and the Poisson-Boltzmann equation for this system is difficult to solve. A method of approximation has been devised on the basis of the following fact.
Counter ions are distributed with a spherical symmetry and the system has no net electric dipole moment. If this system is placed in an external electric field, the distribution of counter ions is shifted relative to the charged groups of the macroion and an e1cctric dipole is produced. The average dipole moment (Ii) of a polyelcctrolyte moleCUle (a macroion with counter ions) in a static electric field E is' given by the following averaging proced ure. ([I) = S [I exp [-f(Ji)/kT+ fiE/kT] d p/S exp ( fCJI)/k T+ JIE/kTJdli - (77) where f(fi) is the free energy of the system in a state with a dipole Ii.