A hyperelastic theory for nonlinear hydrogel diffusiophoresis

Abstract

Hydrogel diffusiophoresis is the deformation of a hydrogel due to a solute gradient that leads to a gradient of pairwise interactions between the solute particles and the hydrogel polymers to trigger osmotic flux. Unlike typical osmosis, it occurs without any interface selectivity of the gel to the solute and can overcome the diffusive swelling without any structural modifications to the gel. We have recently shown this effect for linear deformations of a chemically responsive polyacrylic acid (PAA) hydrogel that releases ions upon arrival of a stimulus (acid), thus internally generating the solute gradient required for diffusiophoresis [Phys. Rev. Lett. 132, 208201 (2024)]. Here we develop a nonlinear poroelastic theory for large diffusiophoretic gel strains in two models: Model I considers deformations of a generic gel when an external solute gradient is imposed. In Model II, the gel generates the solute gradient internally, motivated by the coupled PAA gel, solute (copper), and stimulus (acid) system. In Model II, we investigate the nonlinear deformations for high stimulus concentrations or by changing the solute particle size to boost steric polymer-solute interactions, as well as under a stimulus flow through the gel driven by a pressure drop across the domain. Model I indicates that deformations can be stored while the stimulus gradient persists. Compared to the experimental strain rates in Katke [Phys. Rev. Lett. 132, 208201 (2024)], Model II demonstrates that varying the stimulus concentration can increase the strain rate up to four times, changing the solute particle size up to $\sim 25$ times, and imposed flow up to $\sim 40$ times. Our theory couples nonlinear poroelasticity, polymer-solute interactions, and reaction-transport dynamics to predict large and fast diffusiophoretic gel deformations, which may find applications in hydrogel-based soft robotics and drug delivery.

Figures (10)

• Figure 1: Colloidal versus gel diffusiophoresis.(a), (c) A colloidal particle (gray) with radius $R_c$ surrounded by a smaller particle species (dark green disks) with radius $r_s$ and volume fraction $\phi\,.$ A gradient $\nabla\phi$ is externally imposed. Blue arrows: body forces. Magenta color scale: potential energy variation for (a) pairwise repulsion, (c) pairwise attraction between the colloidal particle and small particles. Black arrows: Diffusio-osmotic flow of the solution (velocity: $\mathbf{v}_{DO}$) and counteracting diffusiophoretic flow of the colloidal particle (velocity: $\mathbf{v}_{DP}$). The normal vector to the colloidal particle surface $\mathbf{\hat{N}}$ points upward in the vertical direction when $R_c\gg r_s\,.$(b), (d), (e), (f) Black lines: polymer network of a hydrogel. The hydrogel is adhered to a substrate (gray slab). Black arrows: Diffusio-osmotic flow of the solution (velocity: $\mathbf{v}_{DO}$) and counteracting diffusiophoretic flow of the gel (velocity: $\mathbf{v}_{DP}$). In (d)--(f), depending on the combination of the pairwise interactions (repulsion or attraction) and the gradient direction, diffusiophoretic gel expansion or contraction is generated.
• Figure 2: Models I and II for gel diffusiophoresis.(a) Model I: Gel response to an external solute gradient. A solute gradient (shown by the triangle with a dark blue color gradient) is maintained across a hydrogel, where the solute particles (dark blue) and the gel polymers interact through steric repulsion. In the frame of the gel, the interaction gradient generates diffusio-osmotic solvent flow from the supernatant domain with a velocity $\mathbf{v}_{DO}$ that is counteracted by the diffusiophoretic swelling of the gel with a velocity $\mathbf{v}_{DP}$ (Fig. \ref{['fig:diffusiophoresis']}). (b)--(d) Model II, Scenarios 1 and 2: PAA gel response to competing stimuli. (b) Acid (red, volume fraction $\phi^{(a)}_+$) is delivered from the supernatant solution into a copper-laden PAA hydrogel attached to a substrate with a contracted initial height $h(0)<H$ due to the chelation between COO$^-$ and Cu$^{2+}$ (blue, volume fraction $\phi^{(b)}$), which turns the gel blue. (c) The formation of COOH groups (volume fraction $\phi^{(b)}_+$) releases Cu$^{2+}$ with a volume fraction $\phi^{(0)}$ into the gel solution. The gel swells with a time-dependent height $h(t)>h(0)$ and loses blue color while a gradient $\nabla\phi^{(0)}$ along the $-z$ axis emerges Korevaar2020. The diffusiophoretic swelling velocity $\mathbf{v}_{DP}$ negates the diffusio-osmotic solvent velocity $\mathbf{v}_{DO}$ (Fig. \ref{['fig:diffusiophoresis']}). (d) The copper gradient, $\mathbf{v}_{DO}\,, \mathbf{v}_{DP}$ eventually vanish due to Cu$^{2+}$ diffusion, and the gel relaxes to the COOH-induced final height $h(\infty)\approx h(0)\,.$ The same dynamics is observed when Cu$^{2+}$ is replaced by the calcium ion Ca$^{2+}$Korevaar2020. (e), (f) Model II, Scenario 3: PAA gel response to acid flow. (e) Acid flows into the hydrogel through the semipermeable substrate at $z=0\,.$(f) Acid decomplexes the bound copper and releases it into the fluid phase of the gel. Due to the diffusiophoretic interactions between the gradient of free Cu$^{2+}$ and the polymer network, the hydrogel transiently swells and returns to the height imposed by the flow when the Cu$^{2+}$ gradient vanishes. Figure panels (b)--(d) are reproduced from Ref. Katke_2024.
• Figure 3: Model I: Hydrogel response to imposed agent gradient.(a) Gel height $h/H\equiv 1+u|_{Z=1}$ versus unitless time for different unitless supernatant domain sizes $\alpha\,,$ with $\alpha=770$ corresponding to the experimental setup in Ref. Katke_2024. (b) For $\alpha=20\,,$ time evolution of the agent true volume fraction $\phi^{(0)}$ with fixed boundary conditions (Eq. \ref{['eq:BC_model1']}) across the gel and the supernatant domain.
• Figure 4: Model II, Scenario 1: Hydrogel response to acid. Hyperelastic model results are shown by the full curves for $\phi_{+,i}^{(a)}=0.006\,, 0.012\,, 0.03\,,$ equivalent to 1M, 2M, and 5M acid, respectively. The dotted data points correspond to the experiments for 1M acid addition, taken from Refs. Katke_2024Korevaar2020. (a) Gel height $h/H\equiv 1+u|_{Z=1}$ versus unitless time $t/\tau$. (b) Time dependence of the true volume fraction of the total bound agent $\phi^{(b)}_{total}\equiv\int^1_0 \Phi^{(b)}dZ$ for each curve in (a).
• Figure 5: Model II, Scenario 2: Gel response to varying polymer-agent interaction strength.(a) Upon adding 1M acid ($\phi^{(a)}_{+, i}=0.006$), gel height $h/H\equiv 1+u|_{Z=1}$ versus unitless time $t/\tau$ for changing polymer-agent steric repulsion strength when the unitless exclusion radius is varied between $R_e/R=1-5$ where $R=4.24\times 10^{-10}$m is a reference length that corresponds to the exclusion radius between copper divalent ions and PAA polymers (Table \ref{['table:simulation_parameters']}). (b) Average strain rates $s\equiv(h_{max}-h_0)/h_0 t_{max}$ extracted from the data in (a) as a function of $R_e/R$ ($h_0:$ initial gel height, $h_{max}:$ height of maximum swelling, $t_{max}:$ time of maximum swelling).