From polymer physics to the toughening of hydrogels
Polymeric materials are used in a large variety of applications, in particular because of their interesting and easily tunable mechanical properties. By playing on the molecular composition of the polymer, on its supramolecular architecture and on its microstructure, it is possible to reach a large array of properties. The challenge in this process is to relate the molecular properties of the polymer, what chemistry imposes, to the macroscopic (mechanical, electrical, optical…) properties of the material.
In this chapter, we will detail first the physics of an isolated chain of polymer and, from this basis, try to understand the mechanical behavior of a network of polymer chains. We will then study the fracture of a soft material, and review some methods of toughening soft matter (elastomers or hydrogels). Finally, we will focus on a specific strategy to toughen materials: the use of a transient and reversible crosslink.
Polymer network physics
In this part, the models presented to understand the behavior of polymeric networks have been taken from a reference textbook from Colby and Rubinstein .
a. Physics of an isolated polymer chain
In reality, a relevant characteristic length of this chain is the average end-to-end distance, which represents the average size of the polymer chain at thermodynamic equilibrium. Maximizing the conformational entropy imposes the polymer chain to be a random walk of monomers, in the ideal case where the chain is made of independently linked and randomly oriented segments, this distance is given by: 〈 〉 = × Eq. 1.3
In real chains, the segments are not randomly oriented. The characteristic ratio accounts for this correlation. It is a material parameter, dependent on the material used, and corresponds to the rigidity (or flexibility) of the bonds: 〉=×× = 〈 Eq. 1.4
The polymer chain can then be extended, in theory up to its maximal length , by applying a force changes in conformation, on one end of the chain. When applying such a force, the chain unfolds and which implies a low energy cost. This low energy cost explains why polymeric materials are easier to deform than crystalline materials, where deformation implies breaking of bonds.
The mechanical characterization of a single polymer chain has been carried out using an AFM tip. In particular, Oesterhelt et al. studied a poly(ethylene glycol) polymer chain . In Figure 1.1, two different regimes can be seen:
– At first, while the force value is low, the chain is unfolding. The response is not specific of the properties of the monomer, but of the statistics of the chain itself. The main drive for the force here is the loss of entropy that one induces when extending a chain – the loss of accessible conformations increases the entropy of the system, and the polymer chain tends to go back to its initial conformation. Force-displacement plot is linear;
– When reaching a critical deformation, dependent on the length of the chain, the force increases steeply. The chain has reached its extended state, so further elongation implies the distortion of the bonds of the chain (decrease of the valence angle ). This regime is enthalpy-controlled, and depends mainly on the properties of the bonds.
At a certain level of elongation, the energy stored in the polymer chain (area under the force-displacement curve) reaches a critical level, dependent on the nature of the monomers: the chain breaks.
Figure 1.1: Schematic of a single molecule force spectroscopy experiment. Upon retraction of the cantilever the polymer chain is stretched while force and elongation are recorded. Figure reprinted from .
b. Rubber elasticity of a network
The physics of polymer networks derives from the physics of isolated chains. However, in this particular case, one needs to consider how the different polymer chains interact with each other, whether by chemical covalent bonds or physical interactions.
In the case of “standard” hydrogels, covalently crosslinked networks swollen in water, the only interaction between the different polymer chains is through the chemical crosslinks. There is no physical interaction between the chains, friction is negligible.
To describe the mechanical properties of such a network, Kuhn, Wall and Flory  developed an affine model. The main assumption of this model is that the relative deformation of each strand of the network is the same as the macroscopic deformation of the material. As long as there is no bond breakage, any deformation applied to this material is reversible, because of the entropic elasticity of the strands.
With this kind of behavior, it is possible to extract three common values:
the minimal value that the reduced stress reaches. In an experiment realized infinitely slowly, with no visco-elastic behavior, one would find the elastic modulus to be equal to However, is usually dependent on the stretch-rate (experiments are never realized infinitely slowly);
– , the value of stretch of onset of strain-hardening. This stretch is related to the finite
– ∗ , which is characteristic of the softening ∗ ) by ≈ ( / ∗ ) / ; extensibility of the crosslinked polymer chains (and hence to = − process due to the different visco-elastic relaxations taking place in the material during the stretching.
These three parameters will depend on the stretch-rate in a visco-elastic material, or in a network with entanglements.
Fracture of soft matter
“Bulk” vision of fracture
a. Linear elastic fracture mechanics
The study of fracture consists in determining how much mechanical energy a material needs to propagate a crack. This fracture energy is generally noted Γ. In order to simplify this study, a major assumption is that the behavior of the bulk of the material remains linearly elastic everywhere, except in a small region around the tip of the crack – which is the basis of the Linear Elastic Fracture Mechanics (LEFM).
During the propagation of a crack, two new surfaces are created. The so-called “strain energy release Griffith  stated that a crack from a pre-existing defect would propagate in the material when this strain energy release rate reaches a critical value . In theory, this energy corresponds to the energy needed to create two new surfaces of the material, also called Dupré energy of adhesion. The crack will only propagate when there is enough energy to create these two new surfaces. In practice, this critical value , also noted Γ, includes two components. The first is this energy needed to create two new surfaces of the material. The second component is the energy dissipation linked to the propagation.
Irwin  expanded this model by studying the stress concentration at the tip of a crack, expressing a singularity at the tip of the crack in the form of an inverse square root dependence of on the distance to the crack tip: ( ) ~ √ Eq. 1.12 where (in . / ) is called the stress intensity factor. Here, the crack propagates when ≥ , with a material property called material toughness. In LEFM, ≥ and ≥ Γ are equivalent parameters.
However, these results are valid in the case of the LEFM, which is not a suitable tool to describe soft materials such as elastomers or hydrogels. In those cases, the propagation of a crack is very different from what happens in a brittle material, where only the bonds near the tip of the crack break. The energy needed to propagate a crack is dissipated by:
– Bond breakage to propagate the crack;
– Possible visco-elastic or plastic deformations in the bulk of the material, which decrease the energy available to propagate the crack;
– Blunting of the crack tip. Elastomers and gels being very soft, the deformations at the crack tip can reach very high values, leading to a removal of the stress concentrations in this region;
– Visco-elastic or plastic dissipation in a region around the crack tip, that are not beneficial to the crack propagation.
The fracture energy Γ takes then into account all these sources of energy dissipation, including the breaking of bonds for propagation of the fracture .
b. Greensmith approximation
The measurement of the fracture energy has been extensively studied in the literature, and some reliable experimental protocols have been developed. The general idea is always the same: pre-notching a sample, and stretching it until it breaks.
Based on a series of experiments with different notch lengths, Greensmith  proposed a simple empirical expression to quantify the strain energy release rate for neo-Hookean elastomers, in a single-edge notch test:
This approach was first developed for adhesion problematics, but also applied to fracture tests later on. Persson estimated the contribution ( , ) from the linear viscoelastic properties of the material. However, this contribution does not take into account the non-linear processes happening during crack propagation (cavitation, chain pullout, bond breaking…). Those processes cannot yet be calculated theoretically.
Γ , on the other hand, has been estimated quite successfully by Lake and Thomas as we will now see.
In order to compute the intrinsic fracture energy of an elastic material Γ , Lake and Thomas  studied materials at temperatures and stretches where the viscoelasticity could be neglected. To be able to compute and predict accurately this Γ , they decided to take a “bottom-up” approach by considering the molecular mechanics taking place when a crack propagates.
Let us consider a notched material under tension. The propagation of this notch requires the breaking of all the chains in front of the crack, in the plane normal to the tension direction. Each C-C bond of these chains carries an energy , determined by the level of stretch at which they are. Once this energy reaches a critical level , one of the bonds of the chain will break; this will allow the chain to release its stored energy in all bonds, moving back to an un-stretched conformation. Lake and Thomas proposed that this energy would simply be the bond energy (around 350 kJ/mol for a carbon-carbon bond). The fracture energy Γ would then be, with no source of dissipation: Γ=2 Σ Eq. 1.17
Where is the average number of C-C bonds between two crosslinks and Σ is the areal density of chains in the fracture plane.
A recent article from Craig and coworkers  suggests that the average energy per bond needed to break a chain is much lower than the energy of the carbon-carbon bond, around 60 kJ/mol instead. Indeed, the previous assumptions from Lake and Thomas neglected the stochastic nature of bond scission and the fact that the application of a the force on the chains during the deformation, increases the probability to break the bond so that at realistic strain rates the most probable energy stores at breakage is 60 kJ/mol. This Lake and Thomas theory has been used for 50 years because although it is unrealistic in terms of localization of fracture in a plane it predicts the correct scaling with Nx and the right order of magnitude in Γ .
Reinforcement of hydrogels
Hydrogels, polymeric networks swollen in water, are interesting materials, with various potential applications. However, the use of conventional chemically crosslinked hydrogels is limited due to their poor mechanical properties. Being neither stiff nor tough, it is then no surprise that the improvement of their mechanical properties has been intensively studied.
The low resistance to fracture of hydrogels comes from two main considerations:
– A heterogeneity of the network, linked to the polymerization process. This leads to the existence of weaknesses and defects in the network, where fractures can nucleate;
– The absence of energy dissipation inside the hydrogel. As seen before, a mechanism of energy dissipation is a good way to increase the fracture energy of a material. Conventional hydrogels, show none of the friction between chains that can be observed in elastomers, thus having no inherent mechanism to dissipate mechanical energy.
Several strategies have been developed, following these two main trends: increasing the homogeneity of the material, or incorporating dissipative mechanisms inside the hydrogel.
a. Creating a “perfect” network
Sakai et al.  developed a homogeneous hydrogel, with a well-defined mesh size and a minimal number of imperfections. To do so, they used tetra-PEG, a four arms star-shaped macro-monomer, terminated with either amine (TAPEG) or NHS-glutarate (TNPEG) functions. The polymerization process is simply a reaction between the amines and the glutarate functions. This ensures a very small distribution of mesh sizes inside the hydrogel, and a good consequent homogeneity.
Figure 1.3: Stress-strain curves of agarose gel (squares), acrylamide gels (triangles) and tetra-PEG gel (circles). Figure reprinted from .
The obtained hydrogels had a high compression stress at break and high extensibility in traction (up to 12 times its initial size). The very regular structure of this network prevents the early nucleation of a crack inside the hydrogel, which explains its high extensibility and stress at break. However, this material is not intrinsically tough: once a crack appears, it propagates very fast. The measured fracture energy is in good agreement with the Lake-Thomas theory.
b. Are homogeneous networks the best solution?
Recent work from Yamaguchi et al.  on model sparse elastic networks suggests that, contrary to what has been said just before, the existence of heterogeneities in the chain lengths contributes to the introduction of sacrificial bonds in the network. This delays the final rupture of the material, as shown in Figure 1.4. In this work, they created a chain length heterogeneity such as half of the chains are 2 longer than the other half, being the mean chain length of the network. It is clear on the figure that, when increasing the heterogeneity of the network, one creates energy dissipation mechanisms that prevent the complete breaking of the network.
Figure 1.4: Fracture behavior for samples with different degrees of link length heterogeneity δ. (a) Force-displacement curves,(b) fracture energies at initial break and final rupture, and(c) snapshots of the deformations, where numbers (i)–(vi) correspond to those shown in Figure 1.4(a). Figure reprinted from .
This study also demonstrates that increasing the functionality of the crosslinker increases the toughness of the material. It consequently implies the possibility of an optimization of the network, maybe giving better performances than the “simple” tetra-PEG network previously presented.
c. Creating a perfect network is very challenging – how about erasing the imperfections?
Okumura and Ito, in 2001, [16-20] first developed what they called topological hydrogels, or polyrotaxane gels: instead of using fixed crosslinking points between the polymer chains, they introduced a moving crosslink. The crosslink acts as a pulley, and is able to homogenize the tension along the chains of the network.
Figure 1.5: Left: Schematic comparison between chemical and slide-ring gels. Right: Stress-strain curve of the slide-ring gel with different concentrations of cross-linker. Figure reprinted from .
These gels show some resistance to fracture propagation, linked to the dynamic movement of those slide-rings when a fracture propagates.
Introducing an energy dissipation mechanism
a. Double network hydrogels
One example of dissipation mechanism is given by the “double-network” hydrogels developed by Gong and coworkers, in 2003 [21,22]. This material is based on inter-penetrated networks, with a first network of highly crosslinked polyelectrolytes poly(2-acrylamido-2-methylpropane sulfonic acid) which is diluted and stretched, and a second network of a concentrated and loosely crosslinked neutral polyacrylamide. In large deformations, the first network acts as a sacrificial network: being more stretched than the second network, it is the one storing the more energy during deformation. When it breaks, all the energy it was bearing is dissipated. The deformable second network keeps the integrity of the material by preventing the correlated fracture propagation in the first network. None of the mechanisms involved are of viscoelastic nature, which makes the material a strain-rate independent soft and tough hydrogel.
Figure 1.6: Left: Schematic representation of a double network hydrogel. Right: Mechanical resistance of a double network hydrogel under compression, compared to its fragile single network counterpart. Figure reprinted from .
This sequential synthesis method has been successfully applied to elastomers, with double, triple and quadruple networks . Millereau et al.  showed that the mechanical properties of the so-called “multiple network” elastomers are directly dependent on the swelling (and thus the pre-stretch) of the first sacrificial network, and not on the number of networks used. Figure 1.7 shows very well this phenomenon: by normalizing the stress with the dilution of the filler network, and shifting the stretch by the initial elongation of this same network, it is possible to obtain a master curve of the behavior of a multiple network elastomer, independent on the number of networks used (1, 2, 3 or 4 here).
Figure 1.7: Left: Nominal stress as a function of the stretch for a single (black), double (red), triple (blue) and quadruple (green) network elastomer. The numbers represent the elongation of the filler network. Right: Nominal stress renormalized by the concentration of the filler network, versus the elongation of the filler network. Figure reprinted from .
b. Composite hydrogels
The double network previously presented can be seen as composite materials, composed of a soft matrix filled with a hard and brittle network. ‘Classical’ composite hydrogels have been developed in the literature, with excellent fracture resistance properties. For example, silica nanoparticles have been introduced in a poly(N,N-dimethylacrylamide) network [25,26]. By increasing the amount of nanoparticles, it is possible to increase dramatically the stiffness, strength and the toughness of the material.
Figure 1.8: Left: Schematic representation of hybrid hydrogels combining covalent crosslinks (orange) and physical interactions sketched by polymer chains absorbed onto the surface of silica nanoparticles. Center: Strain rate effect on hysteresis of hybrid hydrogels. Right: Tensile stress-strain curves with for hydrogels with different silica contents. Figure reprinted from .
In this loosely crosslinked hydrogel, the polymer chains are adsorbed on the surface of the silica nanoparticles. This adsorption process is reversible and consequently possesses a definite dynamic of sorption and desorption. The resulting hydrogels show then a clear visco-elastic behavior, with energy dissipation and recovery processes. When stretching the hydrogel, this reversible bonding of the polymer on the surface of the nanoparticle will provoke energy dissipation, toughening the hydrogel.
c. Ionically crosslinked
Another mechanism of dissipation used in the literature is based on ionic interactions between a polyelectrolyte, and oppositely charged ions in the solution. This creates dynamic physical bonds. The mechanism of energy dissipation is reversible here, and the material can be self-healing.
A well-known example of this particular bond is the PAAm/Alginate network developed by Suo and coworkers . The polyacrylamide network is covalently crosslinked, and serves as a ‘frame’ of the material, onto which is branched a secondary, physically crosslinked network of alginate and Ca2+ ions (Figure 1.9). This second network forms a special “zip” structure by physical ionic bonds, which is the key point of the reinforcing mechanism. The unzipping of the crosslinks causes a dissipation of energy, which is reversible by “re-zipping”. The stretch-at-break of such a hydrogel can reach 21 (Figure 1.10), with fracture energies up to 3000 J/m².
Figure 1.9: Schematics of three types of hydrogel. a, In an alginate gel, the G blocks on different polymer chains form ionic crosslinks through Ca2+ (red circles). b, In a polyacrylamide gel, the polymer chains form covalent crosslinks through N,N-methylenebisacrylamide (MBAA; green squares). c, In an alginate–polyacrylamide hybrid gel, the two types of polymer network are intertwined, and joined by covalent crosslinks (blue triangles) between amine groups on polyacrylamide chains and carboxyl groups on alginate chains. Figure reprinted from .
Figure 1.10: Mechanical tests under various conditions. a, Stress–stretch curves of the three types
of gel, each stretched to rupture. The nominal stress, s, is defined as the force applied on the deformed gel, divided by the cross-sectional area of the undeformed gel. b, The gels were each loaded to a stretch of 1.2, just below the value that would rupture the alginate gel, and were then unloaded. c, Samples of the hybrid gel were subjected to a cycle of loading and unloading of varying maximum stretch. Figure reprinted from .
Using dynamic coordination crosslinks
Coordination complexes have been widely used in the literature as a very versatile physical bond, as we will now see. Its dynamic is simply controlled by the kinetics of association and dissociation of the coordination complex.
These coordination crosslinks are found in nature, particularly in the mussel byssus (the organ with which the mussel attaches to a surface), where catechol functions create coordination crosslinks with Fe3+ ions, which reinforces the byssus. Holten-Andersen et al.  have shown that this bond can be integrated in a synthetic model polymer solution, creating a viscoelastic solution with high energy dissipation and a characteristic dynamic. This idea has been applied to multiple types of complexes, in polymer solutions and polymer networks.
Table of contents :
Chapter 1: From polymer physics to the toughening of hydrogels
1.1. Polymer network physics
1.2. Fracture of soft matter
1.2.1. “Bulk” vision of fracture
1.2.2. Molecular vision
1.3. Reinforcement of hydrogels
1.3.1. Homogeneous networks
1.3.2. Introducing an energy dissipation mechanism
1.4. Using dynamic coordination crosslinks
1.4.1. Transient networks with versatile dynamics
1.4.2. Toughened gels
1.5. Objectives of this project
Chapter 2: Preparation and microstructure of a dual-crosslink hydrogel
2.1. Gel synthesis
2.1.1. Synthesis of the P(AAm-co-VIm) chemical gel
2.1.2. Synthesis of the P(AAm-co-VIm)/Ni2+ dual crosslink gels
2.2. Absorption of Ni2+ ions
2.2.1. Absorption kinetics of Ni2+ ions
2.2.2. Influence of [NaCl]
2.2.3. Absorption isotherms – influence of [Ni2+]
2.3. Stress-free dynamics – study by dynamic light scattering (DLS)
2.3.1. Dynamic Light Scattering – Principles and protocol
2.3.2. Results for a bare chemical gel
2.3.3. Results for a “One-pot” dual-crosslink gel
2.4. Linear mechanical behavior at small deformations
2.4.1. Rheology of a chemical gel
2.4.2. Rheology of a “One-Pot” dual-crosslink hydrogel
2.4.3. Comparing “One-pot” and “Diffusion” dual-crosslink hydrogels
2.4.4. Fitting with fractional model
2.4.5. Relaxation experiments
2.5. Small-Angle X-Ray scattering experiments
2.5.1. SAXS – Principle and protocol
2.5.2. Qualitative analysis
2.5.3. Attempt at quantitative analysis
2.5.4. Conclusions from X-ray scattering
2.6. Conclusion on microstructure and dynamics
Chapter 3 : Large deformations of a dual-crosslink hydrogels with Ni2+ ions
3.1. Standard uniaxial tensile tests: continuous loading
3.1.1. Comparison of “One-pot” and “Diffusion” gels with [Ni2+] = 100 mmol/L
3.1.2. Continuous loading
3.2. Cyclic testing
3.2.1. Influence of stretch and stretch-rate
3.2.2. Damage during cycles?
3.4. Fracture of Ni2+ dual-crosslink hydrogels
3.4.1. Single-notch fracture under continuous stretching
3.4.2. Delayed fracture of “pure-shear” samples
3.5. Conclusions and discussions
Chapter 4: Dual-crosslink hydrogels with fast dynamics
4.1.1. Cu2+ dual-crosslink hydrogel
4.1.2. Zn2+ dual-crosslink hydrogels
4.1.3. Absorption isotherms
4.2.1. Cu2+ and Zn2+ dual-crosslink hydrogels
4.2.2. Zn2+ dual-crosslink hydrogel – “One-pot” vs. “Diffusion”
4.3. Small-angle X-Ray scattering
4.3.1. General observations
4.3.3. Scattered intensity normalization method
4.3.4. Normalized results – comparison between Ni2+ and Zn2+ dual-crosslink hydrogels
4.4. Large deformations
4.4.1. “One-Pot” vs “Diffusion” Zn2+ hydrogels
4.4.2. Tensile tests on Zn2+ and Cu2+ dual-crosslink hydrogels
4.5. Cyclic tests
4.5.1. Influence of stretch-rate and stretch
4.5.2. Damage during cycles?
4.6. Fracture of a fast dual-crosslink hydrogel
4.6.1. Single-notch fracture under continuous stretching
4.6.2. Delayed fracture of “pure-shear” samples of Zn2+ dual-crosslinked hydrogel
Chapter 5: Slowing down the dynamics
5.1. P(AAm-co-VIm) – Hg2+ hydrogels
5.1.1. Preparation and absorption isotherms
5.1.2. Linear rheology
5.1.3. Small angle X-Ray scattering
5.1.4. Tensile tests
5.1.5. Conclusions on P(AAm-co-VIm)-Hg2+ dual-crosslink hydrogel
5.2. Synthesis of P(AAm-co-tPy) hydrogels
5.3. Rheology of the P(AAm-co-tPy) chemical hydrogel
5.4. Incorporation of physical bonds
5.5. Rheology of the slow Terpyridine-Zn2+ hydrogel
5.6. Behavior in large deformations
5.6.1. Tensile tests
5.6.3. Fracture – Single notch test
5.7. Discussion and conclusions
Chapter 6: One bond to rule them all
6.1. Linear rheology
6.2. Large deformations
6.2.1. Tensile tests
6.3. General remarks on the results in large deformations
6.4.1. Single edge notch
6.4.2. Delayed fracture of pure shear samples
6.5. On the dynamic behavior of the dual-crosslink network
Chapter 7: Mapping the damage in dual-crosslink hydrogels with mechanophores
7.1. Mechanophores in hydrogels
7.2. A hydrosoluble dioxetane derivative to obtain time-resolved fracture information
7.2.1. Synthesis of the mechanophore and the hydrogel
7.2.2. Results and discussion
7.3. Anthracene Diels-Alder adduct as a crosslinker
7.3.1. Synthesis of the mechanophore
7.3.2. Synthesis of the chemical network with mechanophore
7.3.3. Mechanical properties of the dual-crosslinked hydrogel with mechanophores
7.3.4. Imaging the fracture front
7.4. Conclusion and perspectives