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Journal Club Theme of September 2012: Fracture of polymeric gels

Wei Hong's picture

A polymeric gel consists of a polymer network swollen by small solvent molecules. It could be synthesized from a monomer solution through gelation or from a dry elastomer directly through swelling. The swelling capability attaches unique attributes to polymeric gels, such as 1) the extremely low stiffness, comparable to that of biological tissues, and 2) the coupling between deformation and solvent migration. However, swelling is also often accompanied with the reduction of both toughness and strength (Tsunoda 2000; Tanaka, 2000, Miquelard-Garnier, 2009), which could occur in gels obtained from either synthesis method. The fragility and weakness of gels hinder potential applications such as tissue replacement, tissue scaffold or as stimulus responsive material for sensors and actuators. In addition to promoting structural applications of gels, understand the fracture process of gels may also improve the handling and mouthfeel of many gel ingredients in foods (Lillfor, 2001; Foegeding, 2007) - an area less known to mechanical engineers.

Regular gels that are weak and brittle are often built upon a single type of polymer network. Some novel gels with modified network architecture show significant improve in strength, such as the double network gel (DN gel, Gong et al, 2003), the nano-composite gel (Haraguchi & Takisha, 2002), the topological gel (Mayumi & Ito, 2001), and the model network gel (Malkoch et al, 2006, Ossipov & Hilborun, 2006). This review aims to elucidate the current understanding of 1) the origin of fragility of simple gels 2) general toughening mechanisms of DN gel and nano-composite gel 3) the special role of swelling/diffusion coupling in fracture process of gels.

Swelling embrittlement
The rule of thumb for the strength requirement of polymeric gels in most biomedical applications is a level comparable to the load carrying capacity of nature tissues. For example, cartilage (Kempson 1982), aortic wall (Mohan & Melvin, 1982) and skin (Edwards & Marks 1995), could carry a load on the order of 10 MPa. However, the strength of a gel is usually dependent on the loading condition, sample geometry, and constraints, and thus may not be an objective measure.
The failure mechanisms of gels vary from localized flow (Moller, et al 2008), softening and yielding (Na et al, 2006; Webber et al 2007; Haraguchi 2006), cavitation (Kundo and Crosby, 2009), to fracture (Tanaka, 2000). If we limit the discussion to the macroscopic propagating cracks (with a length scale much larger than the mesh size of the polymer network), a suitable characteristic quantity is the fracture energy, namely the energy required to create unit area of fresh crack face.
To understand the reduction of fracture energy due to swelling, let us start from a generic conceptual picture: a mixture consists of homogeneous rubbery network with a viscous liquid. The cohesion of the material is provided solely from the polymer network, and a crack propagates by scissoring the polymer chains along its path. Besides a small amount of the excess energy of the surface itself, the fracture energy is mainly composed of two dissipation mechanisms. Firstly the energy of stretching an entire polymer chain to its elastic limit is dissipated irreversibly after subsequent chain rupture (Lake & Thomas 1967). Its magnitude is typically ~50 J/m2, and is often referred to as the intrinsic fracture energy. For dry polymers, the second means of dissipation is crystallization and viscoelastic dissipation of the highly stretched polymer network in front of crack tip due to interaction between closely packed chain strands, which is strongly rate dependent with the magnitude ranging from 0 to 105 J/m2 (Gent, 1996). The effect of this dissipation mechanism diminishes in a swollen gel when chains are diluted, and the fracture energy is shown to be strongly correlated with the dynamic loss modulus of a swollen elastomer (Tsunoda et al, 2000). At the high swelling limit (low elastomer fraction), the fracture energy becomes almost rate independent and reduced to the intrinsic fracture energy (Tsunoda et al, 2000; Tanaka 2007). The effect of dilution of chain density due to swelling could be easily corrected by converting the fracture energy into a value measured with respect to the corresponding area in the dry state.


Toughening of novel gels through distributed damage
Although simple polymer network tend to become very brittle when swollen, it has been demonstrated that by modifying the molecular and micro- structure of a gel, the deformation and energy dissipation mechanism could be changed, and the toughness may be increased by orders of magnitude (Tanaka et al, 2005; Lin et al 2010). The toughening mechanism in two tough gels, the DN gel and the nano-composite gel, are explained from a multi-scale perspective.
A DN gel consists of interpenetration networks of a highly extended 1st network of crosslinked polyelectrolyte, and a coiled 2nd network of neutral chains (Gong et al, 2003). The 1st network ruptures with microcracks prior to the 2nd network at finite deformation. Under an increasing deformation, the multiplication of microcracks or damage in 1st network proceeds stably, while the 2nd network remains almost intact, due to the double network topology, even after the 1st network has been shattered into small islands with percolated microcracks (Nakajima et al, 2008). Recently, a new type of DN gel is developed with the 1st network crosslinked by reversible ionic crosslinks (Sun et al,2012). This new DN gel achieves a much higher toughness and has a self-healing capability due to the reforming of the ionic crosslinks. A nano-composite gel undergoes a different damage process. A nano-composite gel consists of a swollen single network imbedded with nano-clay particles (Haraguchi & Takehisha, 2002). The polymer chains absorb and form physical bonds with clay particle, i.e. the clay acts as weak physical crosslinking points attached with a large amount of short chains, and breaks easily before the final failure of the main backbone (Nishida et al 2009). A signature for the damage of stress-active chains in the network for both materials is the significant hysteresis and softening in simple mechanical tests such uniaxial tension and compression (Webber et al, 2007; Zhu et al, 2006; Xiong et al 2008). Both materials are highly heterogeneous before and after damage. For example, the typical length scale for the island structure a damaged DN gel (Nakajima et al, 2008) (and other derivatives based on this structure such as micro-gel reinforced gels (Hu et al, 2012), and micro-voids reinforced gels ((Nakajima et al, 2011)) is on the order of microns, and the typical size of a nano-composite-reinforced gel is in the submicron region (Haraguchi et al, 2006).
For the problem of an advancing macroscopic crack, the microstructural damage leads to the formation of an extended damage zone ahead of the crack tip. The fracture energy has the additional contribution from the dissipation in the damage zone. While the intrinsic fracture energy corresponds to the dissipation at a region of size comparable to merely the mesh size of polymer network (Lake & Thomas, 1967; Hui, 2003), the damage of sacrificial bonds occurs on a much larger length scale, typically several hundreds of microns, which has been observed directly under optical micro scope for double network gels (Yu et al, 2009; Liang et al, 2011). Thus comparing to the rupture a single layer of chains in a simple gel, the fracture energy is amplified by orders of magnitudes.
The toughening effect is determined form the competing process of an extending damage zone against the propagation the actual crack. For a DN gel, it has been shown experimentally that the fracture energy increases with the average chain length of the 2nd network (Nakajima et al, 2009). On the other hand, it is also found that that the fracture energy could be further increased by increase the heterogeneity and weakening of the 1st network, which enables the formation of damage zone at lower strain (Nakajima et al, 2011). Simple models have been developed to relate the energy dissipation in the hysteresis of a simple test to that in the damage zone (Brown 2007; Tanaka 2007). The general mechanism of toughening in all these gels are the same: to sacrifice part of the gel structure (1st network, crosslinks, clay particles, etc) as much as possible in a stable way. The weaker the sacrifice, the larger the damage zone, the tougher the composite gel would be.


Weakening effect of heterogeneity
A simple gel, even without any visible flaw, often breaks easily due to the large amount of defects and heterogeneities in the network, especially for those formed by regular gelation methods such as radical polymerization (Cohen et al,1992; Ikkai & Shibayama 2004). It is found that the magnitude of heterogeneity tends to be amplified by swelling or deformation (Mendes et al, 1996; Basu et al, 2011, Shibayama 2011), which eventually leads to the formation of microcracks.
On the other hand, both the topological network defect and heterogeneous crosslinking problem could be avoided by producing a nearly-homogeneous network using methods such as ‘click chemistry' (Malkoch et al, 2006, Ossipov & Hilborun, 2006), and combining star polymers of the same size (Sakia et al, 2008). Alternatively, one can also enable the network crosslinking point to slide with deformation by producing a ‘topological gel'. Both types of gels could survive much larger stretch than a regular single gel, to almost the ultimate stretch of each chain. These homogeneous gels are, however, not much tougher, and are expected to fracture at a much lower stretch when a notch in the geometry is present.


Effect of solvent
The effect of the solvent and solvent-network interactions to the strength and toughness of gels is relatively less studied. Firstly, solvent may induce a frictional drag on the polymer chains, especially for those dangling chains formed during damage. Secondly, the highly concentrated hydrostatic tension near a crack tip lower the local solvent chemical potential, and causes a converging solvent flow (Wang & Hong, 2012). Despite the diffusional origin of the second effect, the redistribution of solvent takes as short as milliseconds near a microcrack. While the first effect generally stabilizes the fracture process, the second softens the crack tip and promotes fracture (at constant load).
Experiments on highly viscous physical gel has shown that the wetting of propagating crack in dry air could lead to a decrease in fracture energy (Baumberger et al, 2006), increase in propagation velocity of steady state cracks (Baumberger & Ronsin, 2009, 2010), and the initiation of secondary cracks (Baumberger & Ronsin, 2010). Simple analytical model shows that these effects are partly due to the stress-assisted dissociation of physical crosslinks, and the friction between polymer chains and solvent (Baumberger & Ronsin, 2009). While solvent friction is also present in a permanently crosslinked gel (Tanaka et al, 2000), similar observation has not been reported on chemical gels.
The effect of solvent on a weak simple gel may seem minor. However, it could lead to a large improvement on the performance of tough gels by stabilizing crack propagation and increase the damage-zone size. For example, it has been shown that by increasing the viscosity of the swelling medium, the fracture energy of a DN gel could be further improved (Liang et al, 2012).


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Comments

Lihua Jin's picture

Thanks Wei Hong for the wonderful review!

One question I have is why the fracture energy of natural rubber is so high, on the order of 10,000J/m2, much higher than the sum of two fracture mechanisms you mentioned.

Wei Hong's picture

According to the classic Lake-Thomas picture, the major part of the fracture energy of rubber is due to the viscoelsticity in rubber (chain entanglement, etc.).  Actually, the fracture energy of rubber is rate-dependent.  At an extremely low rate of crack propagation, it is believed that the fracture energy will approach the thredshold, i.e. the intrinsic fracture energy, on the order of 50J/m^2.

Zhigang Suo's picture

Readers of this issue of the Journal Club may wish to take a look at the Themed Issue of Soft Matter (2012, Issue 31, page 7991-8242) on the Mechanics and Physics of Hydrogels, eidited by Wei Hong and Jian Ping Gong.  The Issue contains a large collection of papers focussing on the mechanical and thermodynamic behavior of hydrogels. 

Congratulations to Wei and Jian Ping for a job well done.

Zhigang Suo's picture

Thank you, Wei, for this timely and excellent review.  Your own papers  in this field have been thought-provocative.   

Fracture of gels is a less established field of study--much less so than that of metals, ceramics and polymers.  The science of fracture, especially with respect to toughening mechanisms in ceramics, reached a great level of sophistication in 1980s. Here is a review article by A.G. Evans:  Perspective on the development of high-toughness ceramics, J. Am, Ceramic Soc. 73, 187-206 (1990).  The article is free online.

Knowledge of fracture in metals, ceramics and polyemrs provides a useful backgroud as we move on to study fracture of gels.  In our paper published in Nature this week, our experiments can be interpreted in terms of the synergy between two mechanisms:  crack bridging and background hystereisis (Supplementary Fig. 11).  Both mechanisms have long been well documented for metals, ceramics and polymers. 

Compared to metals and ceramics, gels have markedly different, and much richer, chemistry.  A lot of more knobs to crank up toughness.  

Fracture of gels is yet another field where mechanics meets chemistry.  The field is very young and very promising, full of opportunities for scientific discovery and useful invention.  The time is ripe, given the diverse applications of gels emerged in recent decades. 

Thank you, Wei and Zhigang, for launching this stimulating Journal Club.

The chemistry of hydrogels of polymers, either natural or synthetic, is so rich that some classification is required to rationalize the design of complex systems. A first level of taxonomy consists in distinguishing between "chemical" and "physical" gels.

In chemical gels, the archetype of which is polyacrylamide gels, cross-linking is due to covalent bonds whose energy is so large compared to kT that thermal activation is precluded. These gels break in a "brittle" way via scission of some C-C bonds of the polymer backbone [A. Livne et al. Highly stretchable and tough hydrogels, Phys. Rev. Lett. 101, (2008)264301].

Physical gels, e.g. self-assembled biopolymer gels such as gelatin, are cross-linked via H- or ionic bonds, usually involving extended portions of the polymer chains. Thermal activation is relevant at room temperature (e.g. gelatin gels are thermoreversible and melt slightly above 30°C). Physical gels exhibit poor structural properties, since they relax stress via thermally activated bond rearrangements [O. Ronsin et al. Interplay between Shear Loading and Structural Aging in a Physical Gelatin Gel Phys. Rev. Lett. 103, (2009) 138302]. The same bond weakness is responsible for the "ductile" nature of the rupture process : the crosslinks act as mechanical fuses which prevent chain scission. Fracture proceeds via "unzipping" of the loaded extended crosslinks and pulling polymer chains out of the gel matrix, at the cost of viscous friction against solvent molecules. The fracture energy of physical gels thus increases markedly with rate and solvent viscosity. [T. Baumberger et al. Fracture of a biopolymer gel as a viscoplastic disentanglement process, Eur. Phys. J. E 21 (2006) 81, T. Baumberger and O. Ronsin, From thermally activated to viscosity controlled fracture of biopolymer hydrogels, J. Chem. Phys. 130 (2009), 061102].

Gel toughening can be thought about along these lines. I can evoke here two hot topics:

(i) The transition from brittle to ductile gels upon adding physical bonds to a chemical network [H. J. Kong et al Macromolecules, 36 (2003), 4582, G. Miquelard-Garnier et al. Large strain behaviour of nanostructured polyelectrolyte hydrogels, Polymer 50 (2009) 481].

(ii) Basically, double network gels are made of a chemical network which provides structural integrity and a physical one which dissipates energy [Gong, Why are double network hydrogels so tough? Soft Matter (2010), Sun et al. , Highly stretchable and tough hydrogels Nature, 489 (2012) 133]. However, this is not yet the whole story since the nature of the synergistics between both networks remains an exciting open issue [T. Nakajima et al. True Chemical Structure of Double Network Hydrogels Macromolecules, 42 (2009) 2184].

Wei Hong's picture

Dear Tristan, Thank you for bringing up the fracture of physical gels!  The inelasticity/damage/fracture of physical gels is indeed an intriguing topic.

As you pointed out, a chemical gel would be more ductile if we replace some chemical crosslinks by reversible physical crosslinks.  However, if we keep increasing the fraction of chemical crosslinks, all the way till it entirely becomes a physical gel, it probably would not become even tougher.  It seems to me that the fracture of physical gel may have a different mechanism.  I would guess instability plays an important role in a physical gel.  Zhigang's gel with a second network with chemical crosslinks, or those gels with both physical and chemical crosslinks may help stablizing the localized deformation, and thus make the already ductile physical gel tougher.

Paul Calvert's picture

Cartliage can be view as a collagen-fiber reinforced composite with a gel matrix.  It would be nice to mimic this structure but it is difficult because short fibers tend to clump in gel rather than disperse.  We made a crude mimic based on rubber fibers in a gel matrix (published here: Mater. Res. Soc. Symp. Proc. Vol. 1420 DOI: 10.1557/opl.2012.686), look for a copy on my website http://bngumassd.org/faculty/calvert/calvert.html.  We got high strength and toughness but would not pretend that this is the answer for cartilage replacement.  Our approach is too clumsy.  If we could think of some way of doing it with high aspect ratio, short rubber fibers dispersed in a gel, that might be one answer.

This is not instead of other ways of toughening gels.  We also have some recent work on ionic/covalent double networks in this month's Soft Matter. Like with other classes of materials, we should think of making tough homogeneous materials and then improving them further by making composites. 

Derek Hull wrote a concise text on theory of composites.  How one would apply these arguments to rubber fibers in a gel matrix is not at all clear but there should be some very interesting science to be explored.

 

Zhigang Suo's picture

Dear Paul:  Thank you so much for adding your thoughts to this thread of discussion, as well as for your suggestion on fiber reinforced tough gels.  We have been thinking about using fibers to reinforce the tough gels, and will look at your earlier work on the subject.

I am grateful to you for getting us thinking about tough gels in the first place.  I was deeply impressed by your review article on Hydrogels for Soft Machines, which I first read from a book chapter you wrote.  In August 2009, you organized an informal workshop in Woods Hole, and invited me to give a talk on fracture of gels.   I did do a lot of fracture mechanics, but my past works on fracture were all on metals and ceramics.  I had to learn about fracture of gels to give the talk. Through your review article, I went back to Gong's work, and saw the connection between her work with transformation-toughened ceramics and ceramic matrix composite.  I gave the talk along the line, but perhaps nobody in the audience was really impressed.  People were thinking about how to go beyond Gong's work.  They were thinking about replacing one network with one with weak bonds.  

By pure coincidence, we had worked on alginate gel (Xuanhe Zhao, Nathaniel D. Huebsch, David J. Mooney, Zhigang Suo, Stress-relaxation behavior in gels with ionic and covalent crosslinks. Journal of Applied Physics 107, 063509 (2010)).  In that work we learned about viscoplastic behavior of alginate.  But your workshop was instrumental to get us to focus on the fracture mechanics of hydrogels, and to explore hydrogels of exceptional toughness

Thank you, Paul, for the inspiration.

Chuancheng Duan's picture

Dear Wei,

 

Thanks for your  information. It is very useful. The research of gel is very interesting. Now, I am learning some knowledge about it. 

 

Chuancheng Duan 

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