Direct test of the critical exponents at the sol-gel transition Demet Kaya, O ¨ nder Pekcan, and Yas ¸ar Yılmaz* Department of Physics, Istanbul Technical University, Maslak, 34469 Istanbul, Turkey Received 10 March 2003; revised manuscript received 6 August 2003; published 29 January 2004 The steady state fluorescence technique was used to study the sol-gel transition for the solution-free radical cross-linking polymerization of acrylamide AAm, with N , N ' -methylenebis acrylamide as cross linker in the presence of ammonium persulfate as an initiator. Pyranine 8-hydroxypyrene-1, 3,6-trisulfonic acid, triso- dium salt is used as a fluoroprobe for monitoring the polymerization. Pyranine molecules start to bind to acrylamide polymer chains upon the initiation of the polymerization, thus the spectra of the bonded pyranines shift to the shorter wavelengths. Fluorescence spectra from the bonded pyranines allows one to monitor the sol-gel transition, without disturbing the system mechanically, and to test the universality of the sol-gel transition as a function of some kinetic parameters such as polymer concentration, cross-linker concentration, and temperature. Observations around the critical point show that there are three regimes for AAm concentra- tion in which the exponents differ drastically. The gel fraction exponent  and the weight average degree of polymerization exponent  agree best with the static percolation results for higher acrylamide concentrations above 1 M , but they cross over from percolation to mean-field Flory-Stockmayer values when the AAm concentration is lower than 2 M . For very low polymer concentrations, below which the system can not form the gel, the exponents differ considerably from both the percolation and the mean-field values. DOI: 10.1103/PhysRevE.69.016117 PACS numbers: 05.70.Jk, 64.60.Fr, 64.60.Ak I. INTRODUCTION The exact solution of the sol-gel transition was given first by Flory and Stockmayer 1,2 on a special lattice called a Bethe lattice on which the closed loops were ignored. An alternative to the chemical-kinetic theory is the lattice perco- lation model 3,4 where monomers are thought to occupy the sites of a periodic lattice. A bond between these lattice sites is formed randomly with probability p. For a certain bond concentration p c , defined as the percolation threshold, the infinite cluster is formed in the thermodynamic limit. This is called the gel in polymer language. The polymeric system is in the sol state below the percolation threshold p c . The predictions of these two theories about the critical exponents for the sol-gel transition are different from the point of the universality. Consider, for example, the expo- nents  and  for the weight average degree of polymeriza- tion D w pol and the gel fraction G average cluster size S a v and the strength of the infinite network P  , in percolation lan- guage near the gel point are defined as D w pol   p c - p  - , p →p c - , 1 G   p - p c   , p →p c + , 2 where the Flory-Stockmayer theory the so-called classical or mean-field theory gives  = =1, independent of the di- mensionality d, while the percolation studies based on com- puter simulations give  and  around 1.7 and 0.43 in three dimension 3–8. These two universality classes for gelation problem are separated by a Ginzburg criterion 9 that depends upon the chain length N between the branch points as well as the concentration of the nonreacting solvent. The vulcanization of long linear polymer chains large N belongs to the mean- field class. Critical percolation small N describes the poly- merization of small multifunctional monomers 3–6. Some realistic features like multiple bonding, reversibil- ity, and effect of solvent are generally not considered in static percolation 4. By the computer simulation studies, Pandey et al. 10a showed that the exponents  and  also,  of correlation length exponent change considerably for various solvent conditions, i.e., reversibility for physical gels, and the quality of solvent do affect the sol-gel transi- tion. They also argued that 10b the sol-gel transition for chemical gelation seems also nonuniversal with respect to quality of the solvent, degree of inhomogeneity depending on the quality of the solvent, and rate of reaction due to interplay between the phase separation and cross linking. No real experiment measuring directly the critical expo- nents  and  together with great sensitivity and accuracy has been performed so far, to our knowledge, at the sol-gel transition due to the experimental difficulties. Therefore, the result of the classical and percolation theories could not have been tested adequately with real experiments. In order to understand the physical nature of polymeriza- tion processes underlying the transitions from the sol to the gel state, one must follow the reaction kinetics, and compare results with experiments directly measuring some physical properties in the course of the polymerization reaction. Ex- perimental techniques used for monitoring this transition must be very sensitive to the structural changes, and should not disturb the system mechanically. Fluorescence technique is particularly useful for elucidation of detailed structural as- pects of the gels. This technique is based on the interpreta- tion of the change in anisotropy, emission and/or excitation spectra, emission intensity, and viewing the lifetimes of in- jected aromatic molecules to monitor the change in their mi- croenvironment 11–14. *Corresponding author. Email address: yyilmaz@itu.edu.tr PHYSICAL REVIEW E 69, 016117 2004 1063-651X/2004/691/01611710/$22.50 ©2004 The American Physical Society 69 016117-1