Osmotic pressure of suspensions comprised of charged microgels

We determine the osmotic pressure of microgel suspensions using membrane osmometry and dialysis, for microgels with different softnesses. Our measurements reveal that the osmotic pressure of solutions of both ionic and neutral microgels is determined by the free ions that leave the microgel periphery to maximize their entropy and not by the translational degrees of freedom of the microgels themselves. Furthermore, up to a given concentration it is energetically favorable for the microgels to maintain a constant volume without appreciable deswelling. The concentration where deswelling starts weakly depends on the crosslinker concentration, which affects the microgel dimension; we explain this by considering the dependence of the osmotic pressure and the microgel bulk modulus on the particle size.

Figure 1

(a) Sketch of a solute-solvent mixture whose osmotic pressure $\pi$ we want to know. (b) Sketch of the solute-solvent mixture in contact with a pure solvent reservoir via a semipermeable membrane. (c) Sketch of a piston compressing the solute-solvent mixture that is in contact with the pure solvent reservoir via a semipermeable membrane. The exerted pressure is the osmotic pressure $\pi$. (d) Sketch of the solute-solvent mixture in contact with the pure solvent reservoir via a semipermeable membrane in the absence of the piston in (c). In this case, equilibrium is achieved after pure solvent flows into the mixture to result in a hydrostatic pressure ${\rho }_{s}gh$, with ${\rho }_s$ the density of pure solvent, $g$ the acceleration of gravity, and $h$ the height change of the free surface of the solute-solvent mixture; this hydrostatic pressure exactly equals the osmotic pressure $\pi$ of the solute-solvent mixture.

Figure 2

(a) Sketch of two fluids comprised of identical particles (circles) separated by a partition (vertical line). There is a different number of particles to each side of the partition. (b) Same as in (a) but after removing the partition. The arrow represents the flow direction. In equilibrium, the intensive parameters match and thus $P_1=P_2$ and ${\mu }_1={\mu }_2$. (c) Sketch of two fluids comprised of different particles, stars, and circles. The two fluids are separated by a partition (vertical line). There are more circle particles on side 1 than on side 2, and more star particles on side 2 than on side 1. The total number of particles on either side is $N$. (d) Same as in (c) after removing the partition and once equilibrium is reached. In this situation, the pressure, and the chemical potential of both species, match. (e) Same as in (c), but now instead of removing the partition, we make it permeable to circle-particles only. (f) Equilibrium situation reached, where only the chemical potential of the circle particles matches at either side of the semipermeable membrane. The arrow represents the flow direction of these particles before equilibrium is established. In this case, ${\mu }_{\text{circle},1}={\mu }_{\text{circle},2}$ but $P_1\ne P_2$ and ${\mu }_{\text{star},1}\ne {\mu }_{\text{star},2}$.

Figure 3

Schematic of the membrane osmometer in the (a) setting the zero and (b) calibration stages. (c) Osmotic pressure readings after subsequent injections in a typical calibration. The dashed line indicates the stationary value, which is the one we take as the definite osmotic pressure value. (d) Schematic of the membrane osmometer in the measurement stage.

Figure 4

(a) Dialysis bag (white) immersed in a dextran solution reservoir. Red arrows indicate that solvent can be exchanged depending on the osmotic pressure ${\pi }_0$ of the sample in the dialysis bag and of the dextran solution ${\pi }_{dx}$. (b) Series of microgel suspensions immersed in dextran solutions. (c,d) Microgel suspensions synthesized with $c_X=4.0$ wt. % and $c_X=0.5$ wt. %. The microgel suspensions in the dialysis bags are in either a disordered or in a crystalline phase. On top of every image we indicate the generalized volume fraction, $\zeta$, and the osmotic pressure, $\pi$, of the microgel suspensions at equilibrium.

Figure 5

(a) Polymer weight fraction $c$ of a microgel suspension inside the dialysis tubing as a function of the time inside dextran reservoirs at osmotic pressures ${\pi }_{dx}=2.5$ kPa (circles) and ${\pi }_{dx}=50$ kPa (squares). The dashed lines indicate that equilibrium has been attained. (b) Suspension pH versus $c$ for samples (b) VP4, (c) VP5, and (d) VP6. Black solid lines represent the prediction from Eq. (6) using $f=0.8$ and ${\left[H^{+}\right]}_{\text{out}}=1$ mM. The dashed lines are the predictions for ${\left[H^{+}\right]}_{\text{out}}=2.9$ (upper curve) and ${\left[H^{+}\right]}_{\text{out}}=3.1$ (lower curve), corresponding to the error in our pH measurements.

Figure 6

(a-f) Normalized osmotic pressure versus $\zeta$ corresponding to samples VP1–VP6. Symbols represent experimental data from membrane osmometry (noncrossed symbols) and dialysis (crossed symbols). The solid line is the best fit to Eq. (8) for $\zeta <0.63$.

Figure 7

Illustration of the Donnan potential inside a swollen ionic microgel and its smooth decay at the outer regions of the particle.

Figure 8

Normalized osmotic pressure versus $\zeta$ for pNIPAM microgel suspensions comprised of particles with a diameter of 240 nm (circles) and 280 nm (squares). The lines are linear fits to the data according to Eq. (10).

Figure 9

(a) $\varphi -\zeta$ relation obtained from the mapping of the measured osmotic pressure and the model osmotic pressure described by Eq. (8) for VP1, VP4, and VP6 microgels. The solid line corresponds to $\varphi =\zeta$. (b) Relative volume change of individual microgels as function of $\zeta$. The solid line is the limiting case, $v\left(\zeta \right)\sim {\zeta }^{-1}$, when only isotropic shrinking happens. (c) Relative volume change as function of $\varphi$. In all panels: (circles) VP1, (down triangles) VP4, and (squares) VP6 microgels.

Figure 10

(a) Packing fraction ${\varphi }_s$ where microgel shrinking begins as a function of the microgel swollen diameter $d$. (b) Ratio between the osmotic pressure at which microgels begin to shrink ${\pi }_S=\pi \left({\varphi }_s\right)$ and the prefactor, $\mathrm{\Gamma }Q/v_0$, of ${\pi }_c$.

Figure 11

(a) Normalized fraction of unbounded counterions times the microgel volume versus $\zeta$ after two iterations using Eq. (12). The dashed line corresponds to the initial condition for the iterative model. The triangles are the results after the first iteration while the squares correspond to the results after the second iteration. (b) Osmotic pressure for suspensions comprised of VP5 microgels and fits obtained after each iteration. (c) $\varphi -\zeta$ mapping for each iteration. (d) Relative volume change as a function of packing fraction for each iteration. In panels (c) and (d), circles, triangles, and squares correspond to the initial condition, and the first and second iterations, respectively.