## 5.1. Viscoelasticity of Covalently Crosslinked PAAm Gel

When AAm monomers are dissolved in water, they form an elastic covalently crosslinked PAAm hydrogel network (Fig. 3.a). However, upon the immersion of PAAm chains into a viscous alginate solution, the formed hydrogel network (Fig. 3.b) is expected to behave differently under compressive strains.

Upon the application of constant compressive strain, the covalently crosslinked PAAm gel experiences hyperplastic behavior with minimum relaxation over a long period of time as seen in the stress vs. time curves in Fig. 3. We expect this type of elastic behavior because the permanently crosslinked polymer network is immersed in water which has very low viscosity. But under the same compressive strain, and as more alginate solution is added into the network, obvious viscoelastic behavior is observed. At 5% compression (Fig. 3.c), the PAAm gel (0wt% Alg) has a maximum stress (*σ**max*) value of 0.178 kPa and an equilibrium stress (*σ**0*) value of 0.136 kPa confirming minimal relaxation behavior. However, and when exposed to the same compressive strain of 5%, the PAAm network immersed in 3wt% alginate solution starts with a *σ**max* value of 0.42 kPa and reaches an *σ**0* of 0.32 kPa at the end of the test.

In this case, the relaxed stress *Δσ* of 0.1 kPa is generated by the viscous alginate solution while the PAAm gel dissolved in water had *Δσ* of 0.037 kPa confirming how the two gels behave differently under constant compressive strain. As higher compressive strains are applied on the gels, the effect of alginate viscosity on the viscoelastic behavior of the gels becomes more apparent. For example, when the applied compressive strain is equal to 10% (Fig. 3.d), the PAAm gel still behaves elastically with a *σ**max* that is equal to 0.53 kPa and *σ**0* of 0.49 kPa (Δσ = 0.04 kPa). We compare this to the behavior of the PAAm gel immersed in 2wt% alginate under the same compression and find that the *Δσ* is equal to 0.102 kPa. This is also true when a 20% compressive strain is applied (Fig. 3.e) and the viscoelastic behavior is observed even with the PAAm gel is immersed in as little as 0.5wt% alginate where the *Δσ* is equal to 0.09 kPa, while the pure PAAm gel still exhibits elastic behavior and has a *Δσ* of 0.04 kPa.

To quantitatively study the dependence of the relaxation time on the applied strain and to illustrate the effect of solution viscosity on the gel’s viscoelastic behavior, we fit the experimental data with the theoretical predictions. Using Eqs. (7), (8) and (9) along with the assumptions discussed in section 4, we can plot the stress as a function of time as predicted by the viscoelastic Gent model where the theoretical curves are shown as dashed lines in Fig. 3.c, d, and e. We use the fitting results and plot relaxation time (*τ*) as a function of applied compressive strain in Fig. 3.f. Our calculations reveal that when PAAm is dissolved in water with 0wt% alginate, its relaxation time remains constant as higher compressive strains are applied. That is no longer the case when the PAAm is dissolved in viscous alginate solution where faster relaxation times are observed at higher compression. It’s interesting to note that the relaxation time is directly correlated to the alginate solution’s viscosity where faster relaxation time is achieved as the concentration of the alginate is increased (Fig. 3.f).

Additionally, it is noted that the equilibrium modulus of the gel \({\mu }_{1}\) also increases with the increased concentration of alginate polymer in the solution as shown in Table 1 and the stress vs. time curves in Fig. 3. Apparently, the increase of the equilibrium modulus of the gel cannot be caused by the increase of the viscosity of the solution in the gel. However, such modulus increase can be attributed to the interaction between the amine groups on the PAAm and carboxyl groups on the alginate which results in the formation of coordinated covalent bonds (illustrated by blue diamonds in Fig. 3.b) (sun et al. 2012; Agulhon et al. 2012; Fiorillo and Galbraith 2004). By increasing the amount of alginate solution in the gel, we statistically increase the interaction between the alginate and crosslinked PAAm which results in the formation of more coordinated covalent bonds within the gel and consequently increases its stiffness along with its equilibrium modulus.

Table 1

The shear moduli (\({\mu }_{1}\)) and (\({\mu }_{2}\)) along with the instantaneous modulus (\({\mu }_{1}+ {\mu }_{2}\)) for all the tested gels were obtained based on theoretical fitting of the rheological model using experimentally collected data. When the AAm monomers are immersed in viscous alginate solution, an increase in the values of \({\mu }_{1}\)is observed as the function of alginate solution viscosity which is directly correlated with the experimentally observed viscoelastic behavior **(**Fig. 3**)**. We also observe an increase in the values of \({(\mu }_{1}+ {\mu }_{2})\) when compared to single network PAAm (dissolve in water with 0wt% Alg) which can be attributed to the formation of coordinated covalent bonds and results in the experimentally observed increased stiffness **(**Fig. 3**)**. Finally, and when compared to single network gels, the double network gel has significantly larger moduli due to the formation of ionic, covalent, and coordinated covalent bonds within its networks.

Hydrogel | µ1 (kPa) | µ2 (kPa) | µ1 + µ2 (kPa) |

0wt% Alg + 8wt% PAAm | 1.26 | 0.13 | 1.39 |

0.5wt% Alg + 8wt% PAAm | 2.22 | 0.18 | 2.40 |

2wt% Alg + 8wt% PAAm | 2.83 | 0.37 | 3.2 |

3wt% Alg + 8wt% PAAm | 3.24 | 0.55 | 3.79 |

2wt% Alg | 2.94 | 2.40 | 5.34 |

PAAm-Alg Double Network | 4.94 | 4.51 | 9.45 |

## 5.2. Ionically Crosslinked Alginate Gel

When the alginate is mixed with water in the presence of divalent metal ions such as Ca2+, it results in the formation of egg-box structured crosslinkers (Agulhon et al. 2012). Upon the application of constant compressive strain, the ionic bonds experience unzipping under compression for a long period of time (Fig. 4.a). To determine the force-dependent relaxation behavior of the gel, we measure the stress relaxation of the gel under four different compressive strains (Fig. 4.b). To better reveal the force-dependent relaxation dynamics observed in the experiments, we normalize the relaxation stress as:

$${\sigma }_{Normalized }= \frac{\sigma \left(t\right)- {\sigma }_{0}}{{\sigma }_{max}- {\sigma }_{0}}$$

10

where \(\sigma \left(t\right)\) is the relaxation stress at time \(t\), \({\sigma }_{0}\) is the equilibrium stress which is obtained at *t* = 10,800 seconds and \({\sigma }_{max}\) is the initial stress when relaxation starts (*t* = 2 seconds) which is the value of the maximum compressive stress as defined in the previous section. The normalized stress as a function of time is plotted in Fig. 4.c and clearly illustrates accelerated relaxation behavior at higher strain.

As stated in section 5.1, we can plot the stress as a function of time as predicted by the Gent model where the theoretical curves are shown as dashed lines in Fig. 4.b. We also obtained the values of *µ**1*, *µ**2* (Table 1) and the relaxation time (*τ*) and according to fitting results of the rheological model, the relaxation time decreases with increased compressive strain (Fig. 4.d). At the lowest compressive strain of 5%, the gel’s relaxation time was 435 seconds whereas at the highest compressive strain of 20%, the gel undergoes relaxation within 207 seconds which is 52.4% faster.

We next correlate the measured stress relaxation kinetics to the ionic debonding process. Without the application of an external force, the time needed for the ionic debonding can be given by:

$$\tau =\frac{1}{\nu } \text{e}\text{x}\text{p}\left(\frac{{E}_{a}}{{k}_{B}T}\right)$$

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,

where \(\nu\) is the average atomic frequency, the typical value of which is 1014 Hz, *E**a* is the dissociation energy, \({k}_{B}\) is Boltzmann’s constant (\(1.38\times {10}^{-23}\)J/K) and \(T\) (300 K) is the absolute temperature.

We assume that the stress relaxation measured in the alginate hydrogel stems from the ionic debonding. When a polymer chain is subject to force *f*, the ionic debonding time can be modified as:

$$\tau =\frac{1}{\nu }\text{exp}\left(\frac{{E}_{a}}{{k}_{B}T}\right)\text{e}\text{x}\text{p}\left(\frac{\text{ƒ} {\varDelta }_{a}}{{k}_{B}T}\right)$$

12

,

where \({\varDelta }_{a}\)is the activation length. By re-arranging Eq. (12), we obtain the following linear equation:

$$\text{ln}\left(\frac{\tau }{{\tau }_{0}}\right)=\frac{{E}_{a}}{{k}_{B}T}-{\epsilon }$$

13

where \({\tau }_{0}= 1/\nu = {10}^{-14}sec\) and \(=\frac{\text{K} {\varDelta }_{a}}{{k}_{B}T}\), with the assumption \(\text{ƒ}=K\epsilon\).

In Fig. 4.e, we plot \(\text{ln}\left(\frac{\tau }{{\tau }_{0}}\right)\) as a function of the applied compressive strain where linear fitting is used to determine the values of \(\frac{{E}_{a}}{{k}_{B}T}=38.5\) and \(=4.78\) based on the y-intercept and slope. We can calculate the dissociation energy *E**a* to be 95.9 kJ/mol, which is consistent with previous studies (Agulhon et al. 2012; Hashemnejad and Santanu 2019; Fang et al. 2007).

We can link the force *f* that one chain experiences, the Young’s modulus and the applied strain based on eight-chain model as (Cioroianu et al. 2016):

$$\text{ƒ}=E{l}_{0}^{2}\frac{\sqrt{3}}{4}\epsilon$$

14

where \(E=0.013 MPa\) and is the Young’s modulus, \(\epsilon\) is the applied compressive strain and \({l}_{0}\) is the mesh size of the polymer network (Campbell et al. 2019). Based on the definition of \(,\) we have:

$$=\frac{\sqrt{3}E{l}_{0}^{2}{\varDelta }_{a}}{4{k}_{B}T}$$

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Built on the previous studies, we put the mesh size of the alginate gel to be 50 nm, and by using Eq. (15), we find that the activation length \({\varDelta }_{a}\)= 1 nm, and this is comparable to the size of a G-Block which forms an ionic bond with the Ca2+ in the alginate gel.

## 5.3. Crosslinked PAAm-alginate Double Network Hydrogel

Double network gels have been recently intensively explored to achieve superior mechanical properties. A representative double network gel is formed by combing these two polymers (Fig. 5.a) with covalent (green triangles) and ionic (red circles) bonds in addition to the formation of coordinated covalent bonds (blue diamonds) because of chain interaction as discussed in section 5.1. Under three different compressive strains (Fig. 5.b), the double network hydrogel behaves similarly to the single network alginate gel with noticeable change in relaxation behavior at higher strain as shown Fig. 5.c where normalized stress is plotted as function of time. The results indicate that the stress relaxation in a double network gel is also associated with the unzipping of ionic bonds within the gel’s network.

Like what we did previously, fitting data of the rheological model (dashed lines in Fig. 5.b) were used to determine the relaxation time as a function of the applied strain (Fig. 5.d) and the results confirm the presence of strong force-dependance behavior due to the alginate ionic bonds that exists within the hydrogel’s double network.

Using Eq. (13) and the concept of microscopic force sensitivity explained in section 5.2, the value of \(\)for the double network hydrogel was found to be equal to 5.44 (Fig. 5.e) which is 1.13 times larger than the value for single network alginate. Such difference is mainly because of the different environment and chain topology in ionically crosslinked alginate gel and the double network gel. Although the value of \(\)is different for the two gels, the dissociation energy *E**a* value remains unchanged and is also equal to 95.9 kJ/mol for the double network gel. This is expected since the source of energy comes from the ionic debonding of the alginate chains which are present in equal amounts in both hydrogels.

Furthermore, and based on the results obtained from the Gent model, we find that the double network instantaneous modulus (*µ**1**+ µ**2*) was equal to 9.45 kPa which is 40.4% higher than the addition of the single network moduli (Table 1) and this again can be explained by the formation of the coordinated covalent bonds between the alginate and PAAm polymer chain as explained previously.