Abstract
When the polymer chains are cross-linked by physical bonds having a finite lifetime, the relaxation time and viscosity do not diverge at the gel point though percolation occurs. These undivergent quantities are related to the finite-sized “largest relaxed cluster,” which can relax before it breaks. Its size is the key rheological parameter characterizing of the critical physical gels. In order to evaluate this characteristic size, we propose here a generalized phenomenological model for the viscoelasticity of critical physical gels. We apply the theory to the previously reported experimental result for the physical gel consisting of polyacrylamide-derivative associating polymers containing imidazole groups cross-linked by coordination bonds with Ni ions. We successfully estimate the size of the largest relaxed cluster and the fractal dimension. The size is in good agreement with that estimated from the mean-square displacement of probe particles at the gel point by microrheological measurement. We also compare this system with the poly(vinyl alcohol) hydrogel cross-linked by borate ions, and show that the difference in the cluster structures is originating from the differences of precursor chain properties such as overlap concentration and radius of gyration and of the cross-linking states in these systems.
Highlights
In a polymer solution in which polymer chains can be cross-linked to each other, a three-dimensional network structure, or gel, is formed as the extent of cross-linking increases [1,2,3]
In the microrheology measurement, n corresponds to the exponent of mean-square displacement (MSD) of probe particles in the long-time region [30]
We previously distinguished these modes by demonstrating the time–cure superposition showing the change in direction of shifting the MSD curves at the gel point [7]
Summary
In a polymer solution in which polymer chains can be cross-linked to each other, a three-dimensional network structure, or gel, is formed as the extent of cross-linking increases [1,2,3]. We have comprehensively studied dynamic-mechanical and structural properties of polyacrylamide-derivative associating polymers containing imidazole groups (PAAmVIm) across the gelation point using diffusing-wave spectroscopy (DWS) microrheology, macrorheology, and dynamic light scattering (DLS) [7]. This physical gel is cross-linked by coordination bonds between imidazole groups on the PAAmVIm chains and Ni ions added into the PAAmVIm solution. The gelation is physical in a sense that the cross-links are due to noncovalent bonding but the resulting gel flows in a typical observation time (e.g., on the order of seconds) This kind of physical gels formed by metal–ligand bonds shows controllable viscoelastic properties by changing the metal ion, the ligand, and pH [8,9]
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