# Chemically Responsive Hydrogel Deformation Mechanics: A Review

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## Abstract

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## 1. Introduction

## 2. Hydrogel Swelling Theory

#### 2.1. Mixing Energy

#### 2.2. Ionic Energy

#### 2.3. Elastic Energy

#### 2.3.1. Statistical Mechanics: Flory–Rehner Theory

#### 2.3.2. Continuum Mechanics: Mixture Theory

**F**), which is related to the strain through the right Cauchy–Green strain (

**C**):

**T**and

**S**, respectively) are, then, given by

**T**is non-symmetric and

**S**is symmetric. Typically, the second Piola–Kirchhoff stress tensor is used in the Lagrangian description involving large deformations [59]. The pore pressure is equal to the summation of the chemical potential of the fluid inside the porous solid (${\mu}_{f,gel}$) and the osmotic swelling pressure (${\Pi}_{swelling}$ = ${\Pi}_{ionic}$ + ${\Pi}_{mixing}$):

**T**is the first Piola–Kirchoff stress tensor, ${\nabla}_{X}$ is a divergence operator applied to the Lagrangian variable X, and

**Q**is a Lagrangian vector with its Eulerian counterpart as

**q**= J${\mathbf{F}}^{-1}$

**Q**.

## 3. Experimental Analysis

#### 3.1. Deformation Measurements

#### 3.2. Mechanical Behavior

## 4. Transient Surface Instabilities

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

2D | Two-Dimensional |

3D | Three-Dimensional |

CCD | Charged Couple Device |

CMOS | Complementary Metal Oxide Semiconductor |

CRC | Centrifuge Retention Capacity |

DIC | Digital Image Correlation |

FEM | Finite Element Method |

MHFEM | Mixed Hybrid Finite Element Method |

MRI | Magnetic Resonance Imaging |

NMR | Nuclear Magnetic Resonance |

PTV | Particle Tracking Velocimetry |

SAP | Superabsorbent Polymer |

− (subscript) | Negative Charge |

+ (subscript) | Positive Charge |

${N}_{a}$ | Avagadro’s Number |

${k}_{B}$ | Boltzmann Constant |

C | Capacity |

$\Delta \u03f5$ | Change in energy for the formation of a polymer–solvent interaction |

$\mu $ | Chemical Potential |

n and c | Concentration (amount-of-substance and molar) |

$\alpha $ (subscript) | Constituent |

CI | Counter-ion |

$\alpha $ | Deformation of the system (Statistical Mechanics) |

F | Deformation Tensor |

$\rho $ | Density |

∇ | Divergence operator |

S | Entropy |

T | First Piola–Kirchhoff stress tensor |

$\chi $ | Flory–Huggins Parameter |

Q | Fluid Flux |

$\rho $ | fluid specific mass |

G | Gibbs Free Energy |

F | Helmholtz Free Energy |

I | Identity Tensor |

U | Internal Energy associated with Enthalpy |

J | Determinant of the deformation tensor (Volume change) |

Q | Lagrangian vector |

$\lambda $ | Principal stretch |

$\chi $ | Mixture theory mapping function |

M | Mass |

G | Moduli |

$\overline{\mu}$ | Molar chemical potential |

$\overline{V}$ | Molar Volume |

${\mu}_{el}$ | Number of elastic junctions |

z | Number of lattice connections |

N | Number of molecules |

$\Pi $ | Osmotic pressure |

k | Permeability |

p (subscript) | polymer |

p | pore pressure |

r | Radius |

r | repeat units |

C | right Cauchy–Green strain tensor |

S | second Piola–Kirchhoff stress tensor |

s (subscript) | solvent |

$\sigma $ | Stress |

T | Temperature |

R | Universal Gas Constant |

V | Volume |

$\varphi $ | Volume Fraction |

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**Figure 1.**(

**A**) Material structure of a superabsorbent polymer (sodium neutralized polyacrylic acid). (

**B**) Response classification of hydrogels, separated into physically and chemically responsive gels and including the stimuli for each. (

**C**) Schematic of dry gel (left) exposed to a swelling solution, showing the transience of the process (including surface instabilities) and finally coming to equilibrium (right).

**Figure 2.**Visual representation of Equation (4) showing a three-dimensional lattice of a polymer solution mixture. ${N}_{0}$, total number of lattice sites (all cubes): ${N}_{s}$, number of solvent molecules (empty cubes); and ${N}_{p}r$, number of polymer repeat units (cubes with black dots connected with black lines) [30].

**Figure 3.**Schematic of a particle tracking velocimetry (PTV) setup used to quantify the transient swelling of hydrogels in three dimensions [97]; adapted from [96]. Placing a cylindrical lens in front of the CCD (charge-coupled device) camera creates an anamorphic imaging system, resulting in different optical properties in the x and y directions. Combining the x and y projections allows for the information of the z-dimension to be calculated [102]. (BS = beamsplitter)

**Figure 4.**(

**A**) A circular core (blue)–shell (grey) hydrogel, before and after exposure to a swelling solution, showing the anisotropy in the buckling formation. (

**B**) Surface-attached hydrogel showing the wavelength, $\lambda $, and layer thickness (H).

**Table 1.**Computational models implemented to replicate deformation in porous structures defining the material, strain energy density function theoretical framework, and the dimension reached in each study.

Study | Material | Framework | Dimension | Reference |
---|---|---|---|---|

Tanaka and Fillmore 1979 | Hydrogel | Statistical | 2D | [47] |

Bowen 1980 | Hydrogel | Continuum | 2D | [62] |

Lanir 1987 | Biological Tissue | Continuum | 2D | [66] |

Lai et al., 1991 | Articular Cartilage | Continuum | 2D | [67] |

Huyghe and Janssen 1997 | Porous Media | Continuum | 2D | [68] |

Oh et al., 1998 | Hydrogel | Statistical | 2D | [83] |

Van Loon et al., 2003 | Biological Tissue | Continuum | 3D | [72] |

Dolbow et al., 2005 | Hydrogel | Statistical | 2D | [84] |

Malakpoor et al., 2007 | Articular Cartilage | Continuum | 2D | [81] |

Hong et al., 2008 | Hydrogel | Statistical | 2D | [17] |

Hong et al., 2009 | Hydrogel | Statistical | 2D | [70] |

Kang and Huang 2010 | Hydrogel | Continuum | 2D | [71] |

Chester and Anand 2010 | Hydrogel | Statistical | 2D | [49] |

Duda et al., 2010 | Hydrogel | Statistical | 2D | [50] |

Bouklas et al., 2012 | Hydrogel | Statistical | 2D | [85] |

Bouklas et al., 2015 | Hydrogel | Statistical | 2D | [73] |

Bertrand et al., 2016 | SAP | Statistical | 3D | [10] |

Yu et al., 2018 | SAP | Continuum | 3D | [11] |

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Fennell, E.; Huyghe, J.M. Chemically Responsive Hydrogel Deformation Mechanics: A Review. *Molecules* **2019**, *24*, 3521.
https://doi.org/10.3390/molecules24193521

**AMA Style**

Fennell E, Huyghe JM. Chemically Responsive Hydrogel Deformation Mechanics: A Review. *Molecules*. 2019; 24(19):3521.
https://doi.org/10.3390/molecules24193521

**Chicago/Turabian Style**

Fennell, Eanna, and Jacques M. Huyghe. 2019. "Chemically Responsive Hydrogel Deformation Mechanics: A Review" *Molecules* 24, no. 19: 3521.
https://doi.org/10.3390/molecules24193521