Thermodynamics of sol-gel transition as compared with Bose-Einstein condensation

3 Thermodynamics of sol-gel transition as compared with Bose-Einstein condensation 

At this stage, we recognize that our theory of thermoreversible gelation is mathematically analogous to those we encounter in the study of Bose-Einstein condensation (BEC) in ideal Bose gases [4,21]. The number density N/V and the pressure p of an ideal Bose gas consisting of N molecules confined in a volume V is given by 

The infinite summations on the right-hand side of these equations are known as Tmes-dell funetions [22] of order 3/2 and 5/2. Their singularity at the convergenee radius X = 1 was studied in detail [22]. Since the internal energy of a Bose gas is related to its pressure by (7 = 3pV/2, the singularity in the compressibility and in the specific heat have the same nature they reveal a discontinuity in their derivatives [21]. The transition (condensation of macroscopic number of molecules into a single quantum state) turns out to be a third-order phase transition [21]. 

We now show that a similar picture holds for our gelling solution a finite fraction of the total number of primary molecules condenses into a single state (gel network), which has no center of mass translational degree of freedom (no momentum), although there is no quantum effect. Since the solution is spatially uniform, gelation can be seen as a phase separation in the momentum space into the zero momentum phase (gel) and the finite momentum phase (sol). 

To find the nature of the singularity, we calculate the osmotic compressibility, defined by 

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