The Sol-Gel Transition

We now restrict our attention to strong gelation processes and discuss the area of the threshold. The classical picture for this transition - is based on a tree approximation, where the growing clusters are represented in Fig. V.8. The essential simplification underlying this approach is 

However, the tree approximation has been accepted by polymer scientists because it gives good values for the threshold (i.e., for the fraction of reacted bonds Pc which is reached when the sol-gel tranation takes place). It has been pointed out only recently (by D. Stauffer and by the present author ) that despite this agreement on Pc, the behavior near the threshold (p — Pc) must differ widely from the predictions of classical theory. In what follows, we insist on these aspects, which are not appreciated enough. 

A simple model for gelation without solvent is again a lattice model, where each lattice site (with z neighbors) represents one polyfimctional unit with z reacting arms. Two neighboring units can react, and we represent such a reacted bond by a heavy line, as in Rg. V.9. This 

This type of problem was first introduced by Hammersley, and he coined the word percolation to describe it. Since there are many reviews of percolation theory here we give only selected results concerning the behavior near the threshold. 

We consider the regime where p is slightly smaller than pc (/ = 

---

Olivares et al. (2006), studies performed viscometers very dilute gelatin solutions with concentrations between 10-5 and 10-3 g/cm3, where either intermolecular aggregation or intramolecular folding are possible, respectively, and the sol-gel transition is not observed. 

Djabourov M., Leblond J., Papon P. Gelation of aqueos gelatin solutions. II. Rheology of the sol-gel transition. J. Phys. France 49 (1988b) 333-343. 

Djabourov M., Grillon Y., Leblond J. The sol-gel Transition in gelatin viewed by Diffusing colloidal probes. Polymer Gels and Networks 3 (1995) 407-428. 

Then, the ionic selectivity is discussed and related to the mechanism of crosslinking with divalent counterions. The sol-gel transition is then examined for LM and HM pectins and the mechanisms described in these two cases. The physical properties of the gels are related to the microstructure of the polymers and few data are examined. 

In the one phase region, when the sample was seen to flow easily, it was said that the system was still a sol. When the meniscus was seen not to deform under it own weight, the system was considered a gel. The sol-gel transition was taken at the onset of meniscus deformation when the tube is held horizontal. Syneresis and precipitation were detected by the presence of water at the gel surface or by the existence of large turbid aggregates which could be centrifugated. 

The sol-gel transition has been determined visually, with calcium and copper, for different pectins under different external conditions. As shown in Figure 5 for sample C44 the homogeneous gel phase is situated between the two transition lines. The extension of this phase was found to depend mainly on the DE, temperature and nature of the cation. With calcium the amount of cation required to get a gel increased with the degree of esterification and above 50% it became impossible to get a gel [8]. 

Contrary to the phase separation curve, the sol/gel transition is very sensitive to the temperature more cations are required to get a gel phase when the temperature increases and thus the extension of the gel phase decreases [8]. The sol/gel transition as determined above is well reproducible but overestimates the real amount of cation at the transition. Gelation is a transition from liquid to solid during which the polymeric systems suffers dramatic modifications on their macroscopic viscoelastic behavior. The whole phenomenon can be thus followed by the evolution of the mechanical properties through dynamic experiments. The behaviour of the complex shear modulus G (o)) reflects the distribution of the relaxation time of the growing clusters. At the gel point the broad distribution of... 

Statistical analysis did not reveal any significant interactions between Na and K. Three-dimendonal plots for each deagn showed that the data of each mixtiu e design could be easily described by the data belonging to the border lines of the defined fields, because minimum and maximum values were located there. Thus, without loss of information, figure 1 gives a representative idea of the influence, which the studied cations exert on the sol / gel transition of the three pectins. The curvature in the dotted curves reveal the interactions between Ca and a monovalent cation. 

Despite the importance of initiators, synthesis conditions, and diluents on the properties of a gel, composition is, of course, the most important variable. When growing polymeric chains are first initiated, they tend to grow independently. As the reaction proceeds, different chains become connected through cross-links. At a critical conversion threshold, called the gel point or the sol-gel transition, enough growing chains become interconnected to form a macroscopic network. In other words, the solution gels. The reaction is typically far... 

During our early experiments on chemical gels, when first observing the intermediate state with the self-similar spectrum, Eq. 1-5, we simply called it viscoelastic transition . Then, numerous solvent extraction and swelling experiments on crosslinking samples showed that the viscoelastic transition marks the transition from a completely soluble state to an insoluble state. The sol-gel transition and the viscoelastic transition were found to be indistinguishable within the detection limit of our experiments. The most simple explanation for this observation was that both phenomena coincide, and that Eqs. 1-1 and 1-5 are indeed expressions of the LST. Modeling calculations of Winter and Cham-bon [6] also showed that Eq. 1-1 predicts an infinite viscosity (see Sect. 4) and a zero equilibrium modulus. This is consistent with what one would expect for a material at the gel point. 

The first phase in the process is the formation of the sol . A sol is a colloidal suspension of solid particles in a liquid. Colloids are solid particles with diameters of 1-100 nm. After a certain period, the colloidal particles and condensed silica species link to form a gel - an interconnected, rigid network with pores of submicrometer dimensions and polymeric chains whose average length is greater than one micrometer. After the sol-gel transition, the solvent phase is removed from the interconnected pore network. If removed by conventional drying such as evaporation, so-called xerogels are obtained, if removed via supercritical evacuation, the product is an aerogel . 

In t-DVB/S copolymerization, Antonietti and Rosenauer isolated microgels slightly below the macrogelation point [221]. Using small angle neutron scattering measurements they demonstrated that these microgels exhibit fractal behavior, i.e. they are self-similar like the critically branched structures formed close to the sol-gel transition. 

Several groups used sol-gel transition to immobilize the beads packed in a capillary. For example, Dulay et al. [102] packed a slurry of ODS beads in tetraethylorthosilicate solution and heated it to 100 °C to achieve the sol-gel transition and create the monolithic structure shown in Fig. 17. This technology is extremely sensitive and even a small deviation from the optimal conditions leads to cracks in the monoliths and a rapid deterioration in the column performance. However, even the best efficiency of 80,000 plates/m achieved with these column was relatively low. Henry et al. modified the original procedure and increased the efficiencies to well over 100,000 plates/m [103,104]. 

While only a few reports concern the in situ preparation of monolithic CEC columns from silica, much more has been done with porous polymer monoliths and a wide variety of approaches differing in both the chemistry of the monomers and the preparation technique is currently available. Obviously, free radical polymerization is easier to handle than the sol-gel transition accompanied by a large decrease in volume. 

The hypothesis can be checked by the mathematical experiment, if one takes into consideration the probability of the intemolecular cross-linking. Let s assume that sol fraction contains only intramolecularly cross-linked chains while the formation of even one intermole-cular cross-linkage leads to the sol-gel transition. Because only the properties of sol fraction are of our interest we don t need to follow the intermolecularly cross-linked chains. It is rather natural to assume that the probability of the transition into the gel is proportional to the dimensions of the macromolecular coil. 

We have concentrated our studies on the influence of thermal treatments both on the structure of the gels and on their mechanical properties, in particular around the sol-gel transition. 

The rheological properties, during the sol-gel transition, are investigated with a Weissenberg rheogoniometer in oscillatory shear (linear behaviour). 

The relation with the microscopic structure is pointed up. An analysis of the sol-gel transition in terms of a critical phenomenon is proposed. 

We show in the next section how we analyzed the sol-gel transition of this system. 

When the helix amount increases the medium changes from a viscous liquid (sol) to an elastic solid (gel). The kinetics of gelation depends strongly on the quenching temperature. The rheological measurements that we performed are particularly focused on the sol-gel transition and on the definition of the "gel point". The greatest difficulty encountered is due to the weakness of the bonds which can easily be destroyed by the mechanical stress. 

All these arguments led us to analyze the sol-gel transition as a critical phenomenon, following the ideas put forward by the percolation models (11) (12). 

The rheology of the sol-gel transition was undertaken with special care in order to avoid gel disruption. A critical behaviour for the shear modulus with respect to the helix amount, is noticed. A simple relation between the rheological parameters and the degree of helix formation is pointed out in a limited range of helix amounts (X<15%). These experiments will continue on the fully matured gels. 

The anion as is to be expected in the case of a negatively charged colloid evidently exerts but little effect. The sol-gel transition is generally examined by means of observation of the change in viscosity of the solution. The changes in viscosity as measured by the flow method in an Ostwald or similar apparatus differ as a rule... 

Paulusse JMJ, van Beek DJM, Sijbesma RP. Reversible switching of the sol-gel transition with ultrasound in rhodium(I) and iridium(I) coordination networks. J Am Chem Soc 2007 129 2392-2397. 

In Ref. 76 polymer solutions at the sol-gel transition point were treated, where it holds... 

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. 

---