Dynamics of Swelling and Drying in a Spherical Gel

Swelling is a volumetric-growth process in which a porous material expands by spontaneous imbibition of additional pore fluid. Swelling is distinct from other growth processes in that it is inherently poromechanical: local expansion of the pore structure requires that additional fluid be drawn from elsewhere in the material, or into the material from across the boundaries. Here, we study the swelling and subsequent drying of a sphere of hydrogel. We develop a dynamic model based on large-deformation poromechanics and the theory of ideal elastomeric gels, and we compare the predictions of this model with a series of experiments performed with polyacrylamide spheres. We use the model and the experiments to study the complex internal dynamics of swelling and drying, and to highlight the fundamentally transient nature of these strikingly different processes. Although we assume spherical symmetry, the model also provides insight into the transient patterns that form and then vanish during swelling as well as the risk of fracture during drying.

Figure 1

A polymeric hydrogel is a cross-linked network of polymer chains saturated with water. Swelling occurs due to the spontaneous imbibition of additional water, stretching the polymer chains; drying or deswelling is the reverse. Here, we show the evolution of the mean radius of beads with a dry radius $a_d=0.76\mathrm{mm}$ and a fully swollen radius $6.4a_d$ during (a) swelling and (b) drying with sketches illustrating the composition.

Figure 2

Free swelling: Spatial distributions of (a) porosity ${\varphi }_f$, (b) radial effective stress ${\tilde{\sigma }}_r^{\prime }$, (c) pressure $\tilde{p}$, (d) displacement ${\tilde{u}}_s$, (e) azimuthal effective stress ${\tilde{\sigma }}_{\theta }^{\prime }$, and (f) chemical potential ${\tilde{\mu }}_f$ (all dimensionless) at $\tilde{t}=0$ and then several times logarithmically spaced between $\tilde{t}={10}^{-6}$ and ${10}^{-1}$ (light to dark red). The arrows guide the eye through the time evolution, which is in many cases nonmonotonic. These results are for material properties ${\mathrm{\Omega }}_f/{\mathrm{\Omega }}_p=1.28\times {10}^{-4}$, $\alpha =250$, and $\chi =0.4$. The initial state is nearly dry (${\tilde{\mu }}_{f,0}^{\star }=-5\times 10^3$ and ${\tilde{a}}_0=1.067$) and the final state is fully swollen (${\tilde{\mu }}_f^{\star }=0$ and ${\tilde{a}}_{\mathrm{eq}}=6$).

Figure 3

The swelling of a spherical gel. (a) Time evolution of the radii of three hydrogel spheres after immersion in water, showing experimental data (orange, blue, and yellow, shifted vertically by 0, 0.5, and 1, respectively, for clarity) and the predictions of the model (dashed gray, also shifted by the same amounts). The inset shows $a/a_d-1$ against $t$ for the data and the model on a logarithmic scale to highlight the power-law behavior at early times (same colors, and scaled vertically by factors of $2/3$, 1, and $3/2$, respectively, for clarity). (b) Time evolution of the number of lobes around a circumference of the swelling sphere for four different experiments. (c) Photographs of a swelling gel at different times, as indicated, where the initial radius is $a_d\approx 1.5\mathrm{mm}$ and the final radius is $\sim 6.7a_d$.

Figure 4

Free drying: Spatial distributions of (a) porosity ${\varphi }_f$, (b) radial effective stress ${\tilde{\sigma }}_r^{\prime }$, (c) pressure $\tilde{p}$, (d) displacement ${\tilde{u}}_s$, (e) azimuthal effective stress ${\tilde{\sigma }}_{\theta }^{\prime }$, and (f) chemical potential ${\tilde{\mu }}_f$ (all dimensionless) at $\tilde{t}=0$ and then several times logarithmically spaced between $\tilde{t}={10}^{-12}$ and ${10}^{-3}$ (light to dark red). These results are for the same material properties used in Fig. 2, but the ambient conditions and the initial and final states are precisely reversed. The arrows guide the eye through the time evolution, which is strikingly different from swelling (cf. Fig. 2).

Figure 5

The drying of a spherical gel. (a) Time evolution of the radii of three different gel beads after removal from water to air, showing experimental data (blue, orange, and yellow, shifted vertically by 0, 0.66, and 1, respectively, for clarity) and the prediction of the model (dashed gray). (b) Time evolution of the drying flux from the experiments and the model. The experiments exhibit a drying flux that is nearly constant in time, $F_d^{\star }\approx 0.284$, 0.234, and $0.232\mathrm{mm}{\mathrm{h}}^{-1}$ (${\tilde{F}}_d^{\star }\approx 50.8$, 38.6, and 48.2, respectively), with some small variations that may be due to variation in the ambient relative humidity over the long duration of drying, or because the assumption of a constant drying rate is a crude approximation to the true dynamics of water transport in the room. Swelling data for these beads is shown in Fig. 3 (same colors) and we determine the best-fit material properties from the swelling results.

Figure 6

Using the model, we study evaporation-limited drying: (a) Evolution of the outer radius $\tilde{a}$, (b) drying flux ${\tilde{F}}_d$, (c) porosity at the outer radius ${\varphi }_f(\tilde{a},\tilde{t})$, and (d) maximum azimuthal stress ${\max }_r\{{\tilde{\sigma }}_{\theta }^{\prime }\}$ for ${\tilde{F}}_d^{\star }\to \infty$ (free drying, black line) and then for nine values logarithmically spaced between ${\tilde{F}}_d^{\star }=1\times 10^5$ and $1\times 10^3$ (dark to light colors). Note that free drying exhibits a maximum drying rate of about $1.5\times 10^5$ for these parameters, so any value of ${\tilde{F}}_d^{\star }$ greater than this would be equivalent to free drying.

Figure 7

Fracture during drying. Dimensional overall maximum azimuthal stress experienced by the bead during drying ${\max }_{r,t}\{{\sigma }_{\theta }^{\prime }\}$ as a function of the dimensionless maximum evaporation rate ${\tilde{F}}_d^{\star }$. The black dash-dotted line represents the overall maximum azimuthal stress level in quasistatic drying for ${\tilde{F}}_d^{\star }\ll 10^2$. For ${\tilde{F}}_d^{\star }\ge 1.2\times 10^5$, ${\max }_{r,t}\{{\sigma }_{\theta }^{\prime }\}$ jumps to its free-drying value. We plot as horizontal black dotted lines typical values of fracture stresses for polyacrylamide hydrogels [56].

Figure 8

The size of a hydrogel sphere in air is effectively independent of ${\mathrm{\Omega }}_f/{\mathrm{\Omega }}_p$. Top: Properties of the equilibrium state for a wide range of ambient conditions (${\tilde{\mu }}_f^{\star }$) with material properties ${\mathrm{\Omega }}_f/{\mathrm{\Omega }}_p=1.28\times {10}^{-4}$, $\alpha =250$, and $\chi =0.4$. Middle: Actual equilibrium size in air ${\tilde{a}}_{\mathrm{air}}$ as a function of ${\mathrm{\Omega }}_f/{\mathrm{\Omega }}_p$ for several values of RH (colors) compared with the value of ${\tilde{a}}_{\mathrm{air}}$ for the same RH for ${\mathrm{\Omega }}_f/{\mathrm{\Omega }}_p\to 0$ (dashed gray). Bottom: The relative error between the colored and gray curves from the middle figure.

Figure 9

Space-time evolution of (top) the mean total stress $\overline{\tilde{\sigma }}=({\tilde{\sigma }}_r+{\tilde{\sigma }}_{\theta })/2$ and (bottom) the shear stress $\tilde{\tau }=|{\tilde{\sigma }}_r-{\tilde{\sigma }}_{\theta }|/2$. The colors show $\mathrm{sign}(\tilde{\sigma })\log |\tilde{\sigma }|$, where blue tones are compressive, red tones are tensile, and the dashed black line in the top panel indicates the contour of zero mean total stress.

Figure 10

We image the swelling process using a shadowgraph technique, revealing two distinct regions in the internal structure: A dark, low-porosity core surrounded by a light, high-porosity shell (inset). Thresholding this image provides the time evolution of the outer radius of the core $a_i$ and of the sphere $a$, which together define the shell. We shade the core and shell regions in orange and blue, respectively.

Figure 11

The positions of several isolines of porosity during swelling. The black line marks the outer radius of the sphere and the inset highlights the early-time evolution.

Figure 12

Evolution of the outer radius ${\tilde{a}}_{\mathrm{qs}}$ and the effective stress ${\tilde{\sigma }}_{\mathrm{qs}}^{\prime }$ during strongly limited drying from the full model (solid blue) and from the quasistatic model (dashed yellow). Parameters are the same as Fig. 4, but with ${\tilde{F}}_d^{\star }=1$.

Figure 13

Free swelling for a small change in size, from ${\tilde{a}}_0=1.067$ to ${\tilde{a}}_{\mathrm{eq}}=1.078$ (${\tilde{\mu }}_{f,0}^{\star }=-5\times 10^3$ to ${\tilde{\mu }}_f^{\star }=-4.3\times 10^3$). Same material properties as Fig. 4.

Figure 14

Free drying for a small change in size. Same material properties as Fig. 13, but with initial and final states reversed.

Figure 15

Evaporation-limited drying for ${\tilde{F}}_d^{\star }=10^4$. Same material properties and conditions as Fig. 4.

Figure 16

Free drying for the same parameters and conditions as Fig. 5, but taking $F_d^{\star }\to \infty$.