# Micro-Mechanical Viscoelastic Properties of Crosslinked Hydrogels Using the Nano-Epsilon Dot Method

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## Abstract

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## 1. Introduction

^{−1}strain rate) [18,20,21]. In this context, we recently proposed the nano-epsilon dot method (nano-$\dot{\epsilon}M$) to characterize the physiologically relevant micro-mechanical viscoelastic properties of soft tissues and (bio)materials through nano-indentation tests at different constant strain rates (𝜀̇) [30]. Using data from the loading portion of the indentation curve and accurately identifying the initial point of contact, the nano-𝜀̇𝑀 allows for the derivation of “virgin” material viscoelastic properties (i.e., instantaneous and equilibrium elastic moduli as well as characteristic relaxation times) at typical cell length scales in the absence of pre-stress, unlike classical nano-indentation methods based on the analysis of the unloading curve (e.g., the Oliver–Pharr method [31,32]) or dynamic nano-indentation [33,34,35].

## 2. Results

#### 2.1. Apparent Elastic Moduli and Actual Sample Indentation Strain Rate

#### 2.2. Maxwell Standard Linear Solid (SLS) Lumped Viscoelastic Constants

## 3. Discussion

- (i)
- the scale-dependency of a sample’s mechanical properties, i.e., surface micro-mechanical properties could be different from bulk volumetric ones [40];
- (ii)
- differences in testing and analysis methods, i.e., nano-indentation and unconfined compression techniques use different definitions of stress and strain, different models, etc., possibly affecting the mechanical properties obtained thereof [18]; and
- (iii)
- sample volumetric heterogeneity, i.e., GTA-crosslinking might be not uniform within the gelatin hydrogel volume due to the passive diffusion-reaction mechanism, which is established when submerging physically crosslinked gelatin hydrogels in GTA solution. This may lead to a highly crosslinked hydrogel shell and less crosslinked core, resulting in a lower increase of bulk mechanical properties with increasing GTA [41].

## 4. Materials and Methods

#### 4.1. Sample Preparation

#### 4.2. Nano-Indentation Measurements

^{−1}according to the following equation (Equation (1)) [30]:

#### 4.3. Data Analyses and Viscoelastic Parameters Identification

#### 4.4. Statistical Analyses

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Examples of indentation stress-strain curves obtained testing 25 mM of GTA-crosslinked gelatin hydrogels. Sample viscoelasticity is reflected in the increase of apparent elastic modulus (i.e., stress versus strain slope) with increasing strain rate.

**Figure 2.**(

**a**) Instantaneous (${E}_{inst}$) and (

**b**) equilibrium (${E}_{eq}$) elastic moduli as well as (

**c**) characteristic relaxation times ($\tau $) as a function of glutaraldehyde (GTA) concentration obtained by globally fitting experimental nano-indentation stress-time data recorded at different constant strain rates to a Maxwell SLS lumped parameter model, as per the nano-$\dot{\epsilon}M$. The error bars denote standard errors of estimation. Different letters indicate significant differences between samples (one-way ANOVA, p < 0.05), whereas the same letter means non-significant differences.

**Table 1.**Actual indentation strain rates (${\dot{\epsilon}}_{ind}$) and apparent elastic moduli (${E}_{app}$) obtained for GTA-crosslinked samples tested at different theoretical strain rates (${\dot{\epsilon}}_{t}$). Values are reported as mean ± standard error.

GTA (mM) | ${\dot{\mathit{\epsilon}}}_{\mathit{t}}$ (s^{−1}) | ${\mathit{E}}_{\mathit{a}\mathit{p}\mathit{p}}$ (kPa) | ${\dot{\mathit{\epsilon}}}_{\mathit{i}\mathit{n}\mathit{d}}$ (s^{−1}) |
---|---|---|---|

5 | 0.025 | 5.3 ± 0.3 | 0.021 ± 0.001 |

0.05 | 9.3 ± 0.8 | 0.047 ± 0.001 | |

0.10 | 12.4 ± 0.6 | 0.070 ± 0.001 | |

0.25 | 17.3 ± 1.1 | 0.150 ± 0.001 | |

25 | 0.025 | 27.5 ± 0.6 | 0.012 ± 0.001 |

0.05 | 30.9 ± 2.1 | 0.024 ± 0.001 | |

0.10 | 35.2 ± 0.8 | 0.044 ± 0.001 | |

0.25 | 37.3 ± 0.9 | 0.124 ± 0.001 | |

50 | 0.025 | 53.9 ± 0.8 | 0.008 ± 0.001 |

0.05 | 57.8 ± 0.6 | 0.016 ± 0.001 | |

0.10 | 62.9 ± 0.2 | 0.031 ± 0.001 | |

0.25 | 65.3 ± 1.9 | 0.098 ± 0.001 | |

100 | 0.025 | 76.7 ± 2.9 | 0.006 ± 0.001 |

0.05 | 79.7 ± 1.3 | 0.013 ± 0.001 | |

0.10 | 83.0 ± 1.3 | 0.025 ± 0.001 | |

0.25 | 84.8 ± 1.1 | 0.067 ± 0.001 |

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**MDPI and ACS Style**

Mattei, G.; Cacopardo, L.; Ahluwalia, A.
Micro-Mechanical Viscoelastic Properties of Crosslinked Hydrogels Using the Nano-Epsilon Dot Method. *Materials* **2017**, *10*, 889.
https://doi.org/10.3390/ma10080889

**AMA Style**

Mattei G, Cacopardo L, Ahluwalia A.
Micro-Mechanical Viscoelastic Properties of Crosslinked Hydrogels Using the Nano-Epsilon Dot Method. *Materials*. 2017; 10(8):889.
https://doi.org/10.3390/ma10080889

**Chicago/Turabian Style**

Mattei, Giorgio, Ludovica Cacopardo, and Arti Ahluwalia.
2017. "Micro-Mechanical Viscoelastic Properties of Crosslinked Hydrogels Using the Nano-Epsilon Dot Method" *Materials* 10, no. 8: 889.
https://doi.org/10.3390/ma10080889