Similarity between Gel and Semi-Dilute Solution
Authors: Fujinaga Ikuo , Sakai Takamasa 
Published: February 2020

A polymer gel in the air is in equilibrium state, as a closed system. However, when the gel is immersed in a solvent, the volume changes automatically toward the new equilibrium as an open system. The equilibrium is determined as the balance of elastic pressure and osmotic pressure according to Flory-Rehner theorem. Conventional polymer gels swell in good solvent because osmotic pressure is larger than elastic pressure. The swelling brings some drawbacks in actual applications of polymer gels. For example, swelling of hydrogels in vivo compress the surrounding tissues, and causes severe adverse reactions. Thus, understanding and control over swelling is an urgent issue in material science.
On the other hand, gel swelling has been investing in the field of polymer physics, especially focusing on the equilibrium-swollen condition. Based on the scaling rules on elastic pressure (Πel) and osmotic pressure (Πos), both pressures balance in the equilibrium-swollen state as, Πel ≈ Πos ~ C
3ν/3ν-1. Here, ν is a parameter indicating the interaction between polymer and solvent. Previous researches have shown that the scaling relationship is consistent with experiments. Based on the consistency, some papers concluded the validity of C* theorem which predicted that a polymer gel maintains a concentration proportional to the overlapping concentration (C*) of polymer solution of the corresponding network strand of the gel. However, these agreements only suggest thatΠos of polymer gel is the same form with that of semi-dilute solution, as pointed out by Urayama. Polymer gels swell toward equilibrium keeping semi-diluted, even if the concentration of the polymer gels is below C* of the network strands. Once polymer gels are fabricated, the networks of the polymer gels behave as polymers in the semi-dilute solution.
In the case of gels fabricated from prepolymers such as tetra-PEG gel, sol-gel transition can occur even if the initial polymer concentration (C
0) is far below C*. As for conventional gelling system from small molecular monomer and crosslinker, the initial condition is inherently below C*. Then, whether or not polymer gels are “universally” semi-diluted is an inevitable question. If any polymer gels are semi-diluted system, it is important to understand how the gel from diluted prepolymer solution reaches semi-diluted system during gelation. Despite of the importance, these issues have never been investigated. This situation is in good contrast with viscoelastic change during gelation, which is fully understood based on Winter-Chambon’s criterion.
In this research, we measured the Πos of gelling system from diluted prepolymer solutions. Gelation process was reproduced statically using the tetra-PEG system that can control the reaction conversion by controlling mixing ratio of AB-type prepolymer solutions, and Πos was measured sequentially.
As a result, we discovered that Πos of critical clusters, which are the polymeric clusters just before the sol-gel transition, determined that of corresponding gels. In addition, the molecular weights of critical clusters depend on C
0 when gelling from the diluted systems which simply suggested that sol-gel transition requires more growth of polymeric clusters at lower C
0. By integrating the result of previous study about fractal dimension of the critical clusters, similarity between gelation thresholds and the C* condition was derived. Based on this suggestion, the picture of gelation from the diluted solution is as follows.

Given that the system is initially diluted, overlapping of prepolymers is not preferred at beginning (FIG. a). As the branching reaction proceeds (FIG. b), polymeric clusters grow, and C* of the system decreases. As a result, although C
0 is constant, the system approaches the overlapping condition. In this process, Πos decreases due to the decrease in translational entropy of clusters. The system reaches the overlapping condition, just before sol-gel transition (FIG. c). At the sol-gel transition point, viscosity and stiffness show drastic changes, because they strongly relate to connectivity. In contrast, Πos does not show any apparent changes most likely due to the concentration fluctuation at their overlapping condition is not strongly influenced by further crosslinking. As a result, Πos of gels corresponds to the osmotic pressure just before sol-gel transition point.

Figure 1