Microrheology of cross-linked polyacrylamide networks

Experiments investigating the local viscoelastic properties of a chemically cross-linked polymer are performed on polyacrylamide solutions in the sol and the gel regimes using polystyrene beads of varying sizes and surface chemistry as probes. The thermal motions of the probes are measured to obtain the elastic and viscous moduli of the sample. Probe dynamics are measured using two different dynamic light scattering techniques, diffusing wave spectroscopy (DWS) and quasielastic light scattering (QELS) as well as video-based particle tracking. Diffusing wave spectroscopy probes the short-time dynamics of the scatterers while QELS measures the dynamics at larger times. Video-based particle tracking provides a way to investigate the local environment of the individual probe particles. A combination of all the techniques results in a larger range of frequencies that can be probed compared to conventional bulk measurements while providing local information at the level of individual probes. A modified algebraic form of the generalized Stokes-Einstein equation is used to calculate the frequency-dependent moduli. A comparison of microrheological measurements with bulk rheology exhibits striking similarity, confirming the applicability of microrheology for chemically cross-linked polymeric systems.

Figure 1

(a) Acrylamide monomer units (open rectangles) are polymerized in the presence of the $N,N,N^{\prime },N^{\prime }$-tetramethylethylenediamine which is the initiator and ammonium persulfate which acts as a catalyst for the free radical polymerization reaction. (b) Cross-linking is obtained by adding methylenebisacrylamide (solid rectangles) to the mixture. The cross-linking molecules are two acrylamide monomer units bridged by a carbon atom attached to the two nitrogen atoms from two monomers.

Figure 2

(a) Absorption curves for $3\mathrm{wt}\%$ polymer gel with $0.05\mathrm{wt}\%$ cross-linker taken at after $13\min$ (solid line) and $76\min$ (dashed line). The decrease with time of the absorbance at $260\mathrm{nm}$ is the result of the decrease in the number of the $\mathrm{C}\mathrm{C}$ double bonds due to polymerization. (b) The filled circles denote the absorbance of the polymer solution at $260\mathrm{nm}$, measured over time. The onset of the plateau after approximately an hour is an indication of the completion of the polymerization reaction.

Figure 3

Plot of the scaled mean square displacements of probe particles measured by using multiparticle tracking in a $3\mathrm{wt}\%$ total polyacrylamide sample with $0.03\mathrm{wt}\%$ cross-linker concentration. The MSD’s are scaled with the corresponding probe sizes. The one-particle MSD for $0.48\mu \mathrm{m}$ (solid line) polystyrene spheres overlaps with that of the $1.0\mu \mathrm{m}$ (dashed line) particles. The scaled two-particle MSD’s for $0.48\mu \mathrm{m}$ (circles) and $1.0\mu \mathrm{m}$ (squares) show good agreement with each other and with the one-particle MSD. The line indicates a slope of 1.

Figure 4

Effect of different bead chemistry. Comparison of the mean square displacements probed with $1.0\mu \mathrm{m}$ negatively charged carboxylate-modified (solid line) and positively charged aldehyde-amidine (circles) polystyrene spheres in a $3\mathrm{wt}\%$ total polyacrylamide gel sample with $0.05\mathrm{wt}\%$ cross-linker concentration. The excellent overlap of the data suggests that there is not a significant effect due to the surface chemistry of these probe particles.

Figure 5

(Color online) Measure of heterogeneity in the polyacrylamide samples. Van Hove correlation functions for $0.1\mu \mathrm{m}$ diameter particles moving in a $3\mathrm{wt}\%$ total polyacrylamide solution with $0.05\mathrm{wt}\%$ cross-linker concentration. (a) The spread of the individual particle mean square displacements for several particles is an indicator of the difference in local elasticity measured by the probes. (b) Individual particle distributions, shown at a time lag of $0.1\mathrm{s}$, are each well described by a Gaussian, albeit of different widths indicating different local environments. (c) The ensemble-averaged distributions, shown at time lags of $0.033\mathrm{s}$ (diamonds) and $0.1\mathrm{s}$ (triangles) are non-Gaussian at large displacements.

Figure 6

(Color online) Comparison of the storage and loss moduli for a sol (a) and a gel (b) sample obtained from bulk rheology ($G^{\prime }$, solid line; $G^{″}$, dashed line) with diffusing wave spectroscopy (DWS) [(green) squares], quasielastic light scattering (QELS) [(red) triangles], and one-particle [(blue) circles] and two-particle [(magenta) diamonds] microrheology measurements. The solid and open symbols denote the storage and the loss moduli, respectively. For microrheology measurements every third data point is plotted for clarity.

Figure 7

(Color online) Moduli for $2\mathrm{wt}\%$ total polyacrylamide solution with $0.2\mathrm{wt}\%$ cross-linker concentration, which is near the sol-gel transition. Bulk rheology data measured on two separately prepared samples denoted by different line styles are compared with QELS (triangles) and one-particle (circles), and two-particle (diamonds) microrheology measurements. The solid and open symbols denote the storage and the loss moduli, respectively. The bulk data on one sample ($G^{\prime }$, solid line; $G^{″}$, dashed line) show a dominant elastic modulus. By contrast, an identical sample prepared on a different day ($G^{\prime }$, dot-dashed line; $G^{″}$, dotted line) with the same concentration shows a more viscouslike response for the same frequency interval. Both these data compare very well to microrheological measurements which are done using different techniques and where the samples are prepared separately. Every third data point is plotted for the microrheological measurements for purposes of clarity.