Gelation threshold

Near the gelation threshold (i.e., if g l) one finds the yet simpler form... 

The results of the friction measurements are shown in Fig. 8. The friction coefficient of the poly(acrylamide) gel decreases with an increase of the crosslinker concentration in the major part of the concentration range studied, except that it sharply decreases at a mole fraction of 0.2% of the cross-linker concentration. The measurements cannot be performed below this concentration because the gel becomes too soft to be used in the present experimental setup. It suggests that the system is dose to the gelation threshold. The sharp decrease of the friction observed in the gel with a mole fraction of 0.2% may be due to the effect of the sol-gel transition. It is also found from visual inspection that the gels becomes opaque above a cross-linker concentration of 3 mole-% fraction. 

Both the Flory-Stockmayer mean-field theory and the percolation model provide scaling relations for the divergence of static properties of the polymer species at the gelation threshold. 

The formulas discussed above assume that there exists a point at which oo, since IVIw - oo where gelation threshold is achieved. Since the molecular mass is a function of conversion the expression for ] a) dependence (Eq. (7)) has be suggested employing the following boundary condition a = 0, = 1, and... 

Equations (24) and (27) must have one common point which corresponds to the gel point. For the condition of reaching the gelation threshold, Eq. (28) can be rewritten as follows ... 

Percolation describes the geometrical transition between disconnected and connected phases as the concentration of bonds in a lattice increases. It is the foundation for the physical properties of many disordered systems and has been applied to gelation phenomena (de Gennes, 1979 Stauffer et al., 1982). At just above gelation threshold, denoting the fraction of reacted bonds as p and p=Pc + A/ , pc the critical concentration (infinite cluster), the scaling laws (critical exponents) for gel fraction (5oo) and modulus E) are ... 

Figure 6-7 Influence of High-Methoxyl Pectin Concentration on the Gelation Rate (inverse of the apparent gel time, tc) (A) and on the Rate of Change in G at the Gelation Threshold (B), The slope of the straight line in (A) was 3.6 = 0.981) least squares fit of data in (B) gave a slope of 4.1 and = 0.986.

Adam, M. 1991. Growth process of polymers near the gelation threshold. Die Makromolekulare Chemie, Macromol. Symp. 45 1-9. 

Durand, D., Naveau, F., Busnel, J. P., Delsanti, M., and Adam, M. 1989. Evolution of polyurethane gel fraction near the gelation threshold. Macromolecules 22 2011-2012. 

Yang, Y, and Sigmund, W., Rheological properties and gelation threshold of temperature induced forming (TIE) alumina suspensions with variation in molecular weight of polyacrylic acid, J. Mater. Synth. Process., 10, 249, 2002. 

Floes pack in a space-filling, Euclidean fashion hence, at the floe level of structure, the material can be considered as an orthodox amorphous substance. Within the floes, however, particles pack in a non-Euclidean, fractal fashion. For such a structural arrangement, the volume fraction of particles in a floe (O ) is equivalent to the volume fraction of particles in the entire system (O), namely = O. This well-known relation of polymer physics (11) has been experimentally shown to also apply to colloidal aggregates above their gelation threshold (12). 

Figure 2.16 (Stanley, 1985) shows the analogy between (a) the gelation threshold and (b) the percolation thershold, and R a) is the probability that there exists an infinite cluster.

The sol-gel transition (Winter and Mours 1997 Stauffer 1998 Stauffer et al. 1982 Adam and Lairez 1996) of polymers is a very important branch of critical phenomena. Its kinetics characterizes the chemistry and physics of gelation. The determination of the gelation threshold is an important parameter to control the mechanical properties of a gelling system in polymer industry. In the sol-gel phase transition, an infinitely large macromolecule is formed, which can only swell but cannot be dissolved in a solvent. Critical phenomena are those which occur exactly in the phase transition or asymptotically close to it. 

At the gelation threshold, a clear power law behaviour for the TCE occurs ... 

On the other hand, a power law relaxation in rheology cannot be a sure indication for a gelation threshold. Self-similar relaxations has been associated with selfsimilar structures on the molecular and supermolecular level as well as for suspensions and emulsions (Winter and Mours 1997). Similarities between gelation and long-chain branching viscoelastic behaviour have been also discussed (Garcia-Franco et al. 2001). 

Critical Dynamical Exponents for the Gelation Threshold of Gelatin... 

Takeda M, Norisuye T, Shibayama M (2000) Critical dynamics of cross-linked polymer chains near the gelation threshold. Macromolecules 33 2909-2915 Te Nijenhuis K (1997a) Thermoreversible networks. Introduction. Adv Polym Sci 130 1-12 Te Nijenhuis K (1997b) Thermoreversible networks. Gelatin. Adv Polym Sci 130 160-193 Theiss D, Schmidt T, Dorschner H, Reichelt R, Arndt K-F (2005) Filled temperature-sensitive poly(vinyl methyl ether) hydrogels. J Appl Polym Sci 98 2253-2265 Toomey R, Freidank D, Riihe J (2004) Swelling behavior of thin, surface-attached polymer networks. Macromolecules 37 882-887... 

Adam, M. Delsanti, M. Durand, D. Hild, C. Munch, J. P. Mechanical properties near gelation threshold, comparison with classical and 3d percolation theories. Pure... 

The gelation threshold at this temperature coincades with disappearance of birefringence in the sample. This could be attributed to the formation of micro-crystallites of the polyster during gelation or to the formation of crystal solvates. 

Sites are the unreacted monomers, bonds are the reacted functionalities. Clusters are the polymers, and the infinite cluster is the gel. We recall very briefly some results of percolation. The main result concerns the distribution of cluster sizes. This corresponds to what we called polydis-persity. The distribution is very broad. If we call p the probability that a bond is reacted, pc is its value at the gelation threshold, the average probability P(N, b) that a randomly selected monomer belongs to a large polymer with N monomers each at a small distance s=p pc from the threshold is... 

Fig. 2.13. Evolution of the mechanical properties of a reaction mixture as a function of the distance from gelation threshold (see Box IV entitled The Sol-Gel Transition). Before gelation (p < pc), the medium is viscous and the viscosity (s) increases as the threshold is approached. Beyond the gelation threshold (p > pc), the medium is elastic, and its elastic modulus G) increases with p. Both phenomena obey power laws (the scales here are logarithmic). These experiments were carried out by Adam, M., Delsanti, M. in Saclay, France...

---