A chemist polymerises a trifunctional monomer ( ) and finds
PHYSICS · Subjective · Fermi University Physics · Thermal, Statistical & Complex Systems · Condensed Matter, Materials & Soft Matter · Critical phenomena and the renormalisation group
Soft Matter at the Thermal Scale, Ch Rubber elasticity, swelling and gelation
Problem
A chemist polymerises a trifunctional monomer (f=3) and finds, by rheology, that the sample stops flowing at a measured conversion p=0.52, close to the Flory-Stockmayer prediction pc=1/(f−1)=1/2. She argues: "The measured gel point matches the mean-field value, so the simple picture works here. Just above the gel point the modulus should be proportional to the number of new bonds. Doubling (p−pc) from 0.02 to 0.04 doubles the number of new bonds, so the modulus should simply double too." Using the power law G∝(p−pc)t, find the factor by which G grows when (p−pc) doubles from 0.02 to 0.04, first with the Bethe (mean-field) exponent t=3 and then with the percolation estimate t≈1.9, and explain what is wrong with her reasoning.