Does size matter? Elasticity of compressed suspensions of colloidal- and granular-scale microgels

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Microgels are particles consisting of a cross-linked polymer network swollen with solvent. Most colloids consist of incompressible solids suspended in solution, whereas microgels can be compressed and deformed. The degree of microgel deformation is governed by the cross-link density of the polymer network. Due to their compressibility, microgels can achieve much higher packing densities (jammed packing) compared to the hard spheres inherent to many typical colloidal solutions. When microgel particles become jammed, they are unable to rearrange in response to an applied stress, giving rise to an effective glass transition. However, it has been noted that thermally induced particle rearrangement can occur in these systems. It is difficult to describe the nature of these interactions because particle properties are not easily isolated from interparticle interactions.

In an effort to elucidate the thermal effects on elasticity of microgels, the author explores the elasticity and rheology of colloidal and granular microgels. By comparing microgels of widely varying sizes the effects of interparticle interactions can be accounted for. Due to the small size of colloidal microgels, Brownian interactions play a significant role in the stress response in the particle - hence their stress response will be a combination of intraparticle and interparticle interactions. However, for the larger granular microgels, it is hypothesized that Brownian interactions will play a negligible role in the stress response, eliminating the interparticle component of the stress response. By comparing the stress response of the colloidal microgels to the granular microgels in both compression and shear, the stress responses arising from intraparticle and interparticle interactions will be separated and evaluated.


Colloidal microgels where made of poly(N-isopropylacrylamide-co-acrylic acid) (p(NIPAAm-co-AAc)) via precipiation polymerization. Three different cross-link densities were created in the microgels - 0.1, 1.0, and 10 wt% relative to monomer concentration. Granular microgels were made from p(NIPAAm-co-AAc) as well, except particle size was maintained between 100-400 microns via emulsion polymerization. A second type of granular microgel of similar size was made from polyacrylamide again via emulsion polymerization.

Isotropic compression was performed on the colloidal and granular microgels by way of osmostic pressure. The microgels were emmersed in a solution of polyethylene glycol (PEG) of known osmotic pressure and stressed between 1-200kPa. Shear stress was inflicted on all specimen types via a plate-plate rheometer.

Effect of Particle Size on Compressibility

Equation 1 - The Flory-Rehner Equation
Figure 1. Osmotic pressure, P, as a function of the polymer concentration, cp, in compressed microgel suspensions. Full symbols: colloidal-scale particles; empty symbols: granular-scale particles. Circles represent p(NIPAAm-co-AAc) microgels at cross-linker contents of 0.1 (green), 1 (blue), and 10 wt% (dark red); squares represent pAAm microgels at cross-linker contents of 1 (blue) and 10 wt% (dark red). The black dashed line represents the scaling prediction for an uncross-linked, semi-dilute polymer solution. The curved lines are calculated with the Flory–Rehner theory for macroscopic gels.

The osmotic pressure for a macrogel is described by Flory–Rehner theory: where the left side of the Equation 1 represents the total osmotic pressure, and the right side represents the mixing and elastic contributions to osmostic pressure, respectively.

As the particles are compressed by the applied pressure, the particles become squeezed together, forming a tightly packed network. As the network compresses, the elastic contribution to osmotic pressure dominates the Flory-Rehner equation giving rise to the sharp increase in osmotic pressure for low polymer concentrations. However, as the pressure is increased further, the polymer matrix becomes fully compressed, and the only way for the matrix to further contract is through the expulsion of entrapped solvent (raising the polymer concentration). As the solvent escapes the gel matrix and begins to mix with the rest polymer of the polymer, the osmotic pressure contribution as a result of mixing increases. Thus, at high polymer concentration when the polymer is fully compressed, the elastic contribution to osmotic pressure is negligible and the Flory-Rehner equation is dominated by the mixing term.

Based on this reasoning, all microgels regardless of size, should behave similarly for a given cross-link density. Moreover, while the elastic contributions to osmotic pressure will differ for small polymer concentrations in the elastic regime, all microgels should converge to a singular concentration-osmotic pressure relationship in the mixing regime. From Figure 1, we see this is indeed the case. From Figure 1, we can conclude that microgels follow similar principles as macrogels when subjected to hydrostatic pressure.

Effect of Particle Size on Shear Response

Modulus Behavior

Figure 2. Rheological behavior of a suspension of colloidal-scale p(NIPAAm-co-AAc) microgels cross-linked with 1 wt% BIS. (a) Selected frequency sweeps of the elastic (G', full symbols) and viscous (G", open symbols) part of the complex shear modulus at increasing particle compression, and hence, increasing polymer concentration (from bottom to top).
Figure 3. Osmotic pressure, P, vs. polymer concentration, cp, in compressed microgel suspensions. Full symbols: colloidal-scale particles; empty symbols: granular-scale particles. Circles represent p(NIPAAm-co-AAc) microgels at cross-linker contents of 0.1 (green), 1 (blue), and 10 wt% (dark red); squares represent pAAm microgels at cross-linker contents of 1 (blue) and 10 wt% (dark red). The curved lines represent Flory–Rehner theory predictions.

Figure 2 shows the Storage Modulus G' and Loss Modulus G" as a function of oscillatory frequency. G' is nearly constant across all frequency ranges and increases with crosslink density. Indicating that the degree of networking in the microgel is the only discriminating factor in G'. If G' is dependent solely on crosslink density, G' should be independent of particle size. Normalizing all of the data according to average length of the elastically active chains such that all the G' data can be plotted on the same plot shows this is indeed the case. The dashed line in Figure 3 shows the normalized G' expected for macrogels. As stress increases, the microgel deforms, and approaches the predicted normalized G' value for macrogels. This demonstrates that at low polymer concentration, when the particles are free to move in response to an applied stress, relaxation is attained by particulate rearrangement; however, at high pressures when particulate rearrangement is not possible, microgel shear response is nearly identical to behavior predicted by macrogel models.

The loss modulus behavior in Figure 2 can also be explained according competing dissipation mechanisms. At low frequencies, stress is readily dissipated by particulate shearing, leading to large losses. This is shown by the definitive minimum attained in the middle frequency range. As frequency increases, dissipation is limited to the viscous regime of solvent moving past the particles rather than particle-particle dissipation. The rapid oscillation kinetically limits particulate motion, leading to increased G.

Yield Behavior

As described above, microgels can be described by macrogel continuum mechanics for relatively small strains. However, at strains near the yield point, microgels perform fundamentally different compared to macrogels. At high strains in macrogels the covalent polymeric bonds govern yield response, whereas in microgels, weak interparticle interactions dictate material behavior. Thus, when subjected to near yield shear, microgels approach a fluid like state. Figure 4 shows the modulus response as a function of shear strain. For high strains, modulus drops drastically indicating particulate slip, confirming the dominant interparticle interactions. Further examination of Figure 4 shows that all colloidal systems behaved identically differing only based on degree of crosslinking, but granular systems were chemistry dependent. This shows that for small particle sizes, Brownian motion is the main dissipating mechanism, but for larger particles, solvent-polymer interactions are much more important.

Figure 5 again highlights the effects of Brownian motion in colloidal systems and chemical interactions in granular systems. From the plot of strain against particle packing fraction, all colloidal specimens fell along the same linear regime; however granular systems scattered widely depending on system chemistry. This result shows that Brownian motion dominates particle response for small particle sizes, while solvent-polymer interactions are the main contributor to granular system response.

Figure 4. Normalized G' (full symbols) and G" (open symbols) of compressed microgel suspensions measured as a function of the deformation. Colors represent different degrees of microgel cross-linking: 0.1 (green), 1 (blue), and 10 wt% (dark red). (a) Data for colloidal particles with different cross-link densities and different levels of osmotic pressure. (b) Similar data obtained for granular-scale particles consisting of pAAm (squares) and p(NIPAAmco-AAc)(circles) after scaling.
Figure 5. Yield strain versus microgel packing fraction for suspensions of colloidal- and granular-scale microgels. Full symbols: colloidal scale particles of p(NIPAAm-co-AAc) at cross-linker contents of 0.1 (green), 1 (blue), and 10 wt% (dark red). Empty symbols: granular-scale particles of p(NIPAAm-co-AAc) (circles) and pAAm (squares) at cross-linker contents of 1 (blue) and 10 wt% (dark red). The dashed black line is a guide to the eye with a slope of 9.


Does size matter in the elastic properties of microgels? It depends. In compression, both colloidal and granular microgels follow macrogel models for mechanical and osmotic response. However, in shear, especially near the yield point, differences in response based on size are evident. Brownian motion is the dominant factor in determining the shear response of colloidal systems. As a result, properties of colloidal microgels are chemistry independent and depend only on degree of cross-linking. In granular systems, particle sizes are too large for Brownian motion to significantly effect the system. As a result, solvent-polymer interactions dominate microgel performance. Therefore, chemistry is the major contributor for larger systems.