Rubberlike Elasticity Theory aims to provide a relationship between the stiffness of a hydrogel and its structural properties in terms of shear modulus. The relationship can be represented as follows:
\begin{align*} G = RT\left(1 – \frac{2}{f}\right)\left(1 – \gamma\right)\frac{\rho_{d}}{M_{r}N_{j}}\phi^{\frac{2}{3}}_{0}\phi^{\frac{1}{3}}_{s} \end{align*}Shear Modulus Calculator
Tip: Swollen polymer volume fractions can be estimated using equilibrium swelling theory.
Parameter Definitions and Ranges
Shear Modulus (G, kPa)
Shear modulus (G) represents a material’s stiffness under Neo-Hookean deformation conditions described on the stiffness page. While shear modulus values vary widely in hydrogels based on the formulation, typical values range from 0.1 kPa to 100 kPa.
Ideal Gas Constant (R, J/K⋅m)
The ideal gas constant (R) is a molecular constant related to the Botzmann constant by Avogadro’s number. It is typically represented as 8.314 J/K⋅m.
Temperature (T, °C)
The temperature (T) of the system during the deformation (assumed to be constant) is converted from degrees Celsius to Kelvin for calculations. We have currently only studied hydrogels at room temperature (25 °C), but studies that use varying temperature should be careful of the effects on polymer volume fraction and other temperature-dependent changes that are not immediately obvious from the rubberlike elasticity equation. Temperatures of interest are limited by water’s freezing (0 °C) and boiling (100 °C) temperatures and may be further limited by temperature-dependent polymer network degradation, glass transition temperatures (Wikipedia), or critical solution temperatures (Wikipedia).
Junction Functionality (f)
Junction Functionality (f) is a universal structural parameter of swollen polymer networks. It describes the number of chains that converge at each junction point, assumed to be consistent across all junctions in the network. The lower limit for a network is a junction functionality of three, but non-point-like junctions have achieved junction functionalities exceeding 100 [Beamish et al., 2010]. Junction functionalities of four, six, and eight produce networks with regular polyhedron structures that enable precise calculations of mesh radii from mesh sizes [Richbourg, Ravikumar, Peppas, 2021].
Frequency of Chain-End Defects (γ)
The frequency of chain-end defects (γ) is a universal structural parameter of swollen polymer networks. It describes the fraction of network chains that do not connect to two junctions. The frequency of chain-end defects was first estimated by Flory based on the molecular weight of the pre-network polymer, but his definition does not apply to end-linked and mid-linked networks. The lower limit of the frequency of chain-end defects is zero, which means the network is perfectly interconnected. The maximum value is likely dependent on the system’s integrity and junction functionality, but significant reduction in gel viability is expected around a frequency of chain-end defects of 0.5, which would mean that only half of the network chains are connected to two junctions.
Polymer Dry Density (ρd, g/mL)
The polymer dry density (ρd) is a polymer-specific parameter that can be evaluated independently from hydrogel properties. Typical values include 1.26 g/mL for PVA and 1.18 g/mL for PEG [Richbourg et al., 2021].
Molecular Weight of the Polymer Repeating Unit (Mr, g/mol)
The molecular weight of the polymer repeating unit (Mr) is a polymer-specific parameter than can be evaluated independently from hydrogel properties. The molecular weight of the polymer repeating unit can be calculated based on the chemical structure of the repeating unit. Typical values include 44 g/mol for both PVA and PEG [Richbourg et al., 2021].
Degree of Polymerization Between Junctions (Nj)
The degree of polymerization between junctions (Nj) is a universal structural parameter of swollen polymer networks. It describes the average number of repeating units in a chain between two network junctions. Low degrees of polymerization between junctions lead to chains that no longer behave like entropic springs, and high degrees can lead to highly entangled networks [Richbourg, Ravikumar, Peppas, 2021]. For PVA hydrogels, we observed deviations from linear swelling relationships at a lower bound of Nj = 20 [Richbourg et al., 2021]. Upper limits are system-dependent, but intact PEGDA hydrogels were formed with Nj = 589 [Richbourg et al., 2021].
Initial Polymer Volume Fraction (φ0)
The initial polymer volume fraction (φ0) is a universal structural parameter of swollen polymer networks. It describes the volumetric concentration of polymer in water when the hydrogel is formed. As we recently demonstrated, the initial polymer volume fraction is usually equivalent to the relaxed polymer volume fraction previously discussed by Peppas and colleagues [Richbourg et al., 2021]. The lower bound to initial polymer volume fraction is the critical overlap concentration, which varies by polymer, but for four-armed 10 kDa PEG it is around φ0 = 0.048 [Sakai et al., 2008]. The clear upper bound is φ0 = 1, which represents a dry polymer network, but most of the models described here for hydrogels are designed for φ0 < 0.5, and high polymer volume fractions may disrupt modeling assumptions, such as the use of the phantom-like deformation model.
Swollen Polymer Volume Fraction (φs)
The swollen polymer volume fraction (φs) represents the volumetric concentration of polymer after the hydrogel has reached equilibrium with an excess of water. If the hydrogel’s universal structural parameters and polymer-specific parameters are known, the swollen polymer volume fraction can be calculated using equilibrium swelling theory or it can be measured directly on a swollen hydrogel. Values for swollen polymer volume fractions can be lower or higher than the initial polymer volume fraction [Richbourg et al., 2021]. It is unclear if swollen polymer volume fractions can be below the critical overlap concentration. The broadest limits to swollen polymer volume fractions are therefore 0 < φs < 1. Here again, most of the models described here for hydrogels are designed for φ0 < 0.5, and high polymer volume fractions may disrupt modeling assumptions, such as the use of the phantom-like deformation model.
Key Assumptions
1. Mean-Field Approach
The model uses an approach based on mean-field theory (Wikipedia link), which simplifies and averages specific molecular interactions into effective interactions. This approach allows much simpler mathematical models to define a complex molecular system. Mean-field theory is a meta-model that influences every aspect of the swollen polymer network. For example, instead of considering each chain’s degree of polymerization between junctions and stitching together bulk properties from billions of chain-chain, chain-junction, and junction-junction interactions, we simplify degree of polymerization to the expected average value and assume that value is representative of bulk interactions. While this approach makes tractable calculations out of impossibly complex molecular systems, it also risks significant oversight. A comparable example is Newtonian physics vs special relativity. For many cases, it is okay to think of time and space as separate, fixed things, but Newtonian physics fail near the speed of light. Similarly, when you make “weird” hydrogels or want to make precise predictions of real hydrogel properties, you may need something more nuanced than mean-field-based models.
2. Worm-Like Polymer Chains
Each polymer chain in the network acts as an entropic spring and therefore must be long enough to have a significant variety of conformations that give the chain its entropic capacity. Implicit in the rubberlike elasticity model is the assumption that each chain is long and flexible enough to act as a “worm-like chain” which effectively acts as a fluid, dynamic chain instead of a series of discrete links wherein a change in one junction greatly affects the overall positioning of the chain. This assumption is partially linked to Gaussian Chain Lengths Distribution assumption below, since chains that are too short to acquire worm-like behavior are also likely to form skewed distributions when crosslinked into a network.
3. Phantom Network Deformation Model
Polymer networks can generally be described through two primary models: the affine network and the phantom deformation network. Both models aim to describe how nanoscale chain movements relate to large-scale macroscopic deformation, which impacts how stress is distributed within the network.
The affine model is the default assumption for networks, suggesting that all deformation is essentially accounted for by the network chains stretching or coiling. Therefore, the junctions move in proportion to the macroscopic deformation. Flory made the argument that in dense, rubbery polymer networks, the chains and junctions are highly entangled, causing junctions to be closely affected by the movements of all the chains around them, therefore justifying the affine deformation model [Flory, J Chem Phys, 1977].
The phantom deformation model starts with the unrealistic treatment of network junctions as phantoms that move independently of network chains. Junctions could then freely rearrange in space in whatever way minimizes the stress placed on the network chains. While this hypothetical freedom for junctions is mathematically convenient, it is impossible for the network junctions to move truly independently of the network chains. The forces each chain place on the junctions and any entanglements with other chains limit the motion of junctions. However, in highly swollen polymer networks, most junctions are likely to be unentangled and therefore only affected by the chains attached to the junction. This allows swollen polymer networks to behave under phantom-like deformation that become more affine-like with increasing junction functionality, or as more chains are tethered to the junction.
Affine and phantom deformation models are two extremes of a nanoscopic deformation behavior spectrum. Erman and Mark described a “constrained junction model” that provides more accurate analysis of the spectrum between the extremes, but it requires additional mathematical intricacies and knowledge of the network structure [Erman & Mark, 1997]. We use a phantom-like model as a simplifying approximation that is appropriate for hydrogels.
4. Gaussian Chain Length Distributions
Chain length can be defined as the number of polymer repeating units between junctions. Due to the nature of polymer networks, the distribution of chain lengths in a polymer network can take multiple forms including skewed, bimodal, as well as Gaussian (normal) distributions.
In order for rubberlike elasticity theory to be applicable for modeling stiffness in polymer networks, gaussian chain length distributions must be assumed because gaussian distributions produce a mean value that provides an accurate representation and insight to the chain length distribution, whereas the mean value in bimodal and skewed distributions provides very little meaning regarding the distribution behavior as seen from the figure above. This is a byproduct of the mean-field approach discussed above.
5. Incompressible Solvent and Polymer
Incompressible materials do not change volume when stretched or compressed. Instead, the stretching or compression in one direction results in a compensatory compression or stretching in other directions.
6. Point-Like Junctions
As a further consequence of the mean-field approach, network junctions are assumed to be point-like. Originally formulated around a tetrafunctional carbon atom as a junction point, point-like junctions refer to any network junctions that have negligible chain dynamics of their own and are comparatively much smaller than the network’s polymer chain. Examples of non-point-like junctions include using short PEGs as junctions, having a radically polymerized junction such as in PEGDA networks, having large crystallite junctions as in freeze-thawed PVA gels, and having junctions held together by large nanoparticles or dendrimers. Very small junctions such as glutaraldehyde in PVA gels and the cores of multi-arm PEGs can be treated as point-like. The mean-field rationale for requiring point-like junctions is that it allows the properties of the network to be associated primarily with the properties of the polymer chains.
7. Neo-Hookean Modeling & Shear Modulus
Rubberlike elasticity theory describes a special case of the Neo-Hookean hyperelastic model, which uses shear modulus to generalize a material’s stiffness. Learn more about shear modulus on the stiffness page. If hydrogel mechanical properties deviate from the assumptions of the Neo-Hookean model, such as by exhibiting viscoelastic behavior, then rubberlike elasticity theory is no longer applicable.
Additional Issues to Consider
Bulky Side-Groups and Chain Complexity
“Bulky” refers to a large group or collection of atoms that are not part of the original backbone in a polymer network. This results in the addition of various bonds originating as a result of the side chain interacting with various molecules, including ionic bonds and hydrogen bonds as well as hydrophobic/hydrophilic and steric interactions. Bulky side-groups might contribute to steric hindrance for changing conformations of the chain, effectively driving deviations from a freely jointed chain model. It is unclear how side-group bulkiness would affect the rubberlike elasticity model or if it can be addressed mathematically.
Complexity can also be associated the presence of alternative, variable, or copolymer repeating units which result in non-carbon atoms making up the backbone of the network as well as the presence of non-linear backbone structures, such as cyclic structures. These changes also deviate from a freely jointed chain model, and variable or copolymerized repeating units disrupt mean-field theory assumptions unless the copolymerization is regular enough (e.g., an alternating copolymer) that it could be incorporated into a mean-field unit.
Fatigue and Fracture
As hydrogels are subject to repeated loading, some of the chains may break under the strain. This results in changes in stiffness, or fatigue, that may ultimately lead to fracture [Tang et al., 2017]. Note: Fatigue is not the only mechanism of fracture in hydrogels. Fatigue and fracture are important properties of hydrogels, especially in tissue engineering applications, that are not addressed by rubberlike elasticity theory. Hydrogel fatigue and fracture is a growing field of research and will hopefully be incorporated into future iterations of the swollen polymer network model.
Poroelasticity and Viscoelasticity
Because rubberlike elasticity theory assumes a Neo-Hookean hyperelastic model, poroelasticity and viscoelasticity are not considered in the model. While viscoelasticity is likely irrelevant in covalently crosslinked hydrogels at biologically relevant time-scales, poroelasticity may be an important model for how hydrogels absorb or expel water under externally applied deformations, especially at longer time-scales [Kalcioglu et al., 2012]. Addressing viscoelastic and poroelastic effects will expand the time-scales for which rubberlike elasticity theory applies.
Theory History
The history of rubberlike elasticity theory closely matches equilibrium swelling theory history because the elastic free energy used in the equilibrium swelling theory energy balance comes from rubberlike elasticity theory and because both rubberlike elasticity theory and equilibrium swelling theory were initially used to estimate the average molecular weight between crosslinks (Mc), now generalized as the degree of polymerization between junctions (Nj).
In Flory’s seminal book, Principles of Polymer Chemistry (1953), “Rubber elasticity” is the topic of chapter 11, and Flory therein asserts the affine deformation assumption for polymer networks and introduces a correction based on polymer molecular weights for the effect of chain-end defects on the number of elastically effective chains. In chapter 11’s appendix B, Flory presents the tension expressed as the force per unit area of the swollen, unstretched sample:
\begin{align*} \tau = RT\frac{\nu_{e}}{V_{s}\upsilon^{\frac{2}{3}}_{s}}\left(\alpha – \frac{1}{\alpha^{2}}\right) \end{align*}In current notation but using Flory’s definition of the effective number of chains, we would write this equation as
\begin{align*} \tau = RT\left(1-\frac{2M_{c}}{M_{n}}\right)\left(\frac{\rho_{d}}{M_{r}N_{j}}\right)\phi_s^{\frac{1}{3}}\left(\alpha – \frac{1}{\alpha^{2}}\right) \end{align*}The change in the swollen polymer volume fraction comes from dividing the dry volume by the swollen volume. If we assume that Flory was using engineering stress and strain rather than true stress, then the relationship can be translated to a Neo-Hookean Shear Modulus:
\begin{align*} G = RT\left(1 – \frac{2M_{c}}{M_{n}}\right)\left(\frac{\rho_{d}}{M_{r}N_{j}}\right)\phi_{s}^{\frac{1}{3}} \end{align*}Which is not terribly far from the current rubber-like elasticity model for hydrogels. Notably, Flory assumed all swollen polymer networks were formed in their unswollen state, which leaves no difference between the dry state and the relaxed state. Also, junction functionalities are assumed to be four based on the simplest case of two pre-existing chains crosslinking along their lengths, and the frequency of chain-end defects is based on the length of the pre-existing chains (M_n).
Treloar’s summary textbook, The Physics of Rubber Elasticity (1975), provides further details on the development of a shear modulus equation for unswollen and swollen polymer networks, but reaches the equivalent conclusion as Flory, with the same equation in terms of shear modulus shown above, when combining equation 4.26, which focuses on swollen polymer networks, with equation 4.32, which considers the possibility of chain-end defects causing elastically ineffective chains.
Peppas and Merrill, “Crosslinked Poly(vinyl Alcohol) Hydrogels as Swollen Elastic Networks” in the Journal of Applied Polymer Science (1977), updated both the equilibrium swelling equation (using the correction of Bray and Merrill, 1973) and the rubberlike elasticity equation to account for network formation. Combining equations 5 & 6 in their paper yields
\begin{align*} \tau = RT\frac{C_{2, r}}{M_{c}}\left(1 – \frac{2M_{c}}{M_{n}}\right) \left(\frac{\upsilon_{2, s}}{\upsilon_{2, r}}\right)^{\frac{1}{3}}\left(\alpha – \frac{1}{\alpha^{2}}\right) \end{align*}which we can update to
\begin{align*} G = RT\left(1 – \frac{2M_{c}}{M_{n}}\right) \left(\frac{\rho_{d}}{M_{r}N_{j}}\right) \phi_{r}^{\frac{2}{3}}\phi_{s}^{\frac{1}{3}} \end{align*}Here, the influence if the relaxed polymer volume fraction is shown for the first time for a network formed in the presence of solution. Still, the authors assumed tetrafunctional junctions and frequencies of chain-end defects based on the lengths of the pre-crosslinking polymers.
While the edited book of Peppas, Hydrogels in Medicine and Pharmacy, Vol. 1 Fundamentals, (1986) did not directly update the rubberlike elasticity equation, they included an interesting correction to the elastic free energy of highly crosslinked swollen polymer networks that would also affect the rubberlike elasticity equation for such highly crosslinked hydrogels. However, since highly crosslinked hydrogels are not commonly studied as load-bearing structures, the modification was not extensively tested and has not been used frequently in the literature. We note it here, and explain it further in the equilibrium swelling theory history section, in case any are interested in further pursuit of the topic.
In a review paper, Anseth, Bowman, and Brannon-Peppas, “Mechanical properties of hydrogels and their experimental determination,” in Biomaterials, (1996) discussed a Mooney-Rivlin adaptation of the classic rubberlike elasticity equation of swollen polymer networks for high deformations, but the two phenomenological coefficients have yet to be associated with fundamental or structural influences.
\begin{align*} \tau = 2C_{1}\phi_{s}^{\frac{1}{3}}\left(\alpha – \frac{1}{\alpha^{2}}\right) + 2C_{2}\phi_{s}^{\frac{5}{3}}\left(1 – \frac{1}{\alpha^{3}}\right) \end{align*}Following the tradition of Flory and Treloar, Erman and Mark published Structures and Properties of Rubberlike Networks, (1997) which described the status of the field and explicitly discussed the ideas of affine and phantom deformation and their influences onswollen and unswollen polymer network models as they relate to rubberlike elasticity. Notably, they derive an equation for the phantom-like deformation of swollen networks that scales with cycle rank rather than purely the number of elastically effective chains, introducing a correction based on junction functionality in the Helmholtz free energy equation:
\begin{align*} \Delta A_{el, ph} = \frac{1}{2}\left(1 – \frac{2}{f}\right)\nu_{e}k_{b}T\left(I_1 – 3\right) \end{align*}Richbourg and Peppas, in “The swollen polymer network hypothesis,” Progress in Polymer Science, (2020) coordinated the phantom-like correction in Erman and Mark, the relaxed state correction in Peppas and Merrill, and several symbolic and practical simplifications, such as generalizing the term for frequency of chain-end defects, to create the current rubberlike elasticity equation for swollen polymer networks as described above. Notably, based on the low-entanglement argument for hydrogels, especially those formed in solution, the assumption of phantom-like networks is appropriate, but only for highly swollen hydrogels.
In a follow-up experimental validation of hydrogel swelling, Richbourg, et al. in “Precise control of synthetic hydrogel network structure via linear, independent synthesis-swelling relationships, “ Science Advances, (2021) demonstrated the equivalence of the initial polymer volume fraction and the relaxed polymer volume fraction. We therefore updated all reference to the relaxed polymer volume fraction to the initial polymer volume fraction to reduce the number of polymer volume fractions under consideration.