Thermodynamic Analysis of Polymer Gel Elasticity
Authors: Yoshikawa Yuki , Sakai Takamasa 
Published: February 2020

Elastomer and polymer gel have a three-dimensional polymer network structure and the characteristic of being soft and well extended (rubber elasticity). Many researches on rubber elasticity have been carried out mainly with elastomer since 1950's. Entanglement of network chains dominates the physical properties because elastomer contains almost no solvent, thus it is difficult to investigate the correlation between the structure and the physical properties in detail. On the other hand, polymer gel is a dilute system coexisting with solvent, thus it is possible to design various network structures with controlled entanglement. Recently, we developed Tetra-PEG gel, which is formed by AB-type polycondensation of tetra-armed prepolymers. Tetra-PEG gel has a highly homogeneous network structure, which is suitable for verifying the theory of polymer gel. In the previous study, Phantom network model, which is a rubber elasticity model, was verified. Phantom network model assumes that the crosslinks fluctuate with deformation, and elastic modulus (G) is described as

G = ξk
B
T

Here, k
B is the Boltzmann constant, T is absolute temperature, and ξ is the number densities of independent cycles surrounded by the mesh chain. In tree-like approximation, ξ can be calculated from polymer concentration, polymer molecular weight, and connection probability between prepolymers. This equation indicates that one cycle has the influence of the thermal energy k
B
T on the elasticity. As a result of the verification using Tetra-PEG gel, it was confirmed for the first time in the world that the connection probability dependence of the elastic modulus and the calculated ξ (ξ
th) was the same. This indicates that Phantom network model can describe the elasticity of polymer gel. However, the polymer concentration (C) dependence of them was different as shown in Figure 2. Here, the polymer concentration was normalized by the overlapping concentration (C*). The polymer concentration is a concept inherent to polymer gel, and it has not been considered so much in conventional researches. Therefore, we returned to thermodynamics, which is the origin of rubber elasticity models, and experimentally verified it.
The elastic modulus from Helmholtz free energy is described as the following equation.

G = U” + S”T

The first term (U”) is referred to as energy elasticity, and the second term (S”T) as entropy elasticity. From this equation, it is shown that elastic modulus can be separated into energy elasticity and entropy elasticity by measuring temperature dependence of elastic modulus. Classical rubber elasticity theory ignores the energy elasticity and predicts that the elastic modulus is proportional to temperature (G ~ T). It has been experimentally confirmed that the proportional relationship is roughly correct in the case of elastomer. On the other hand, the elastic modulus of Tetra-PEG gel was not proportional to temperature, and the negative energy elasticity whose value cannot be ignored was measured (Figure 1). This result may indicate an essential difference between elastomer and polymer gel. Therefore, we further investigated the energy elasticity and entropy elasticity using Tetra-PEG gel with various network structures.
When only the connection probability was changed, energy elasticity was proportional to entropy elasticity (U”~ S”T). Since in previous study it was shown that the connection probability dependence of the elastic modulus and ξ was the same, we found that the dependence of the entropy elasticity and the energy elasticity was also the same as ξ. Furthermore, the entropy elasticity had the same polymer concentration dependence as ξ, which shows that the entropy elasticity follows Phantom network model except the value. Therefore, this result also shows the different concentration dependence of the elastic modulus and ξ is derived from the energy elasticity (Figure 2).
In the poster presentation, I will talk about the cause of energy elasticity and the effect of solvent quality of polymer gel, too.

Figure 1