Negative Energy Elasticity in a Rubberlike Gel

Rubber elasticity is the archetype of the entropic force emerging from the second law of thermodynamics; numerous experimental and theoretical studies on natural and synthetic rubbers have shown that the elasticity originates mostly from entropy change with deformation. Similarly, in polymer gels containing a large amount of solvent, it has also been postulated that the shear modulus (the modulus of rigidity) $G$, which is a kind of modulus of elasticity, is approximately equivalent to the entropy contribution $G_S$, but this has yet to be verified experimentally. In this study, we measure the temperature dependence of the shear modulus $G$ in a rubberlike (hyperelastic) polymer gel whose polymer volume fraction is at most 0.1. As a result, we find that the energy contribution $G_E=G-G_S$ can be a significant negative value, reaching up to double the shear modulus $G$ (i.e., $|G_E|\simeq 2G$), although the shear modulus of stable materials is generally bound to be positive. We further argue that the energy contribution $G_E$ is governed by a vanishing temperature that is a universal function of the normalized polymer concentration, and $G_E$ vanishes when the solvent is removed. Our findings highlight the essential difference between rubber elasticity and gel elasticity (which were previously thought to be the same) and push the established field of gel elasticity into a new direction.

Popular Summary

Our study reveals a secret of the softness of polymer gels: networks of cross-linked flexible polymer chains that contain a large amount of solvent. Polymer gels and rubbers (similar polymer networks without the solvent) are softer than metals and ceramics by several orders of magnitude. It has been believed for nearly a century that this softness could be explained by entropy elasticity, where an elastic deformation produces a decrease in the specific entropy. Here, we experimentally show that this common belief is false for polymer gels. Instead, their softness is determined by negative energy elasticity that coexists with entropy elasticity.

We systematically measure how the rigidity of polymer gels with various network structures varies with temperature. We find that a relative temperature change in rigidity is several times greater in gels than that of rubbers because of the negative energy elasticity. We further argue that the negative energy elasticity is governed by a universal function, and it vanishes when the solvent is removed. Because the solvent is the critical factor in differentiating gels from rubbers, the negative energy elasticity is considered to be a distinct characteristic of gels.

Our findings are of great practical importance because gels are used for medical applications at various temperatures. Additionally, our findings offer a new perspective and stimulate further research on gel elasticity as well as other fields where entropic force plays an important role, such as in entropic gravity theory.

Figure 1

Irrelevant energy elasticity in natural rubber and relevant negative energy elasticity in a rubberlike polymer gel. (a) Temperature ($T$) dependence of the tensile stress $\sigma$ (black symbols) of vulcanized rubber through stretching measurements under 60% strain. The data are taken from Ref. [5]. The gray solid line is obtained from a least-squares fit and extrapolated to $T=0\mathrm{K}$. According to Eq. (1), we have the entropy contribution ${\sigma }_S$ (blue dashed line) and the energy contribution ${\sigma }_E$ (red dashed line), which corresponds to the intercept of the gray solid line. In the measured temperature range (black symbols), the ratio of ${\sigma }_E$ to $\sigma$ is less than 15%. Similarly, small energy contributions to elasticity are observed in many rubber materials [6, 7, 8, 9, 10]. (b) Typical result of the $T$ dependence of the shear stress $\sigma$ (black symbols) of a polymer gel through rheological measurements under 60% shear strain $\gamma$. The gel sample is synthesized by equal-weight mixing of the two kinds of precursors whose molar mass $M$ and concentration $c$ are $20\mathrm{kg}/\mathrm{mol}$ and $60\mathrm{g}/\mathrm{L}$, respectively. The gray solid, blue dashed, and red dashed lines are obtained in the same way as in (a). Notably, we find that ${\sigma }_E$ is a significant negative value.

Figure 2

Highly stretchable rubberlike hydrogel with a homogeneous model network (tetra-PEG gel). This gel is synthesized by $AB$-type cross-end coupling of two kinds of precursors of the same size. These precursors are tetra-armed poly(ethylene glycol) (PEG) chains whose terminal functional groups ($A$ and $B$) are mutually reactive.

Figure 3

Measurement of shear modulus. (a) Schematic illustration of the double-cylinder dynamic shear rheometer. Inside the rheometer, a solution of the precursors reacts to form a polymer gel. After the completion of the chemical reaction, the polymer gel is sheared cyclically in the gap between the inner cylindrical jig and the outer cup. (b) Frequency ($\omega /2\pi$) dependence of the storage modulus $G^{\prime }$ and loss modulus $G^{\prime \prime }$. The molar mass $M$ and concentration $c$ of the precursors are $20\mathrm{kg}/\mathrm{mol}$ and $60\mathrm{g}/\mathrm{L}$, respectively. The greater the connectivity $p$ is, the lower $\tan \delta$ is ($p$ is defined in the main text). For example, when $p\simeq 0.95$ (a nearly perfect network structure), $G^{\prime \prime }$ is underdetected.

Figure 4

Polymer gel exhibiting a wide range of linear elasticity under shear deformation. (a) Temperature ($T$) dependence of (shear) stress $\sigma$ under fixed (shear) strain $\gamma$ (1%, 20%, 40%, 60%, 80%, 100%, 120%, 140%, 160%, and 180%) within $278\mathrm{K}\le T\le 298\mathrm{K}$. The data for $\gamma =60\%$ correspond to Fig. 1. The straight lines are obtained from the least-squares fit and extrapolated to $T=0\mathrm{K}$. Within the measured range, $\sigma$ is a nearly linear function of $T$ at each $\gamma$ (inset). The gray solid extrapolation lines ($\gamma \le 140\%$) pass through the vanishing temperature $T_0$ on the $T$ axis. (b) Stress ($\sigma$)–strain ($\gamma$) curve (black symbols) with ${\sigma }_S$ (blue symbols) and ${\sigma }_E$ (red symbols). The data are extracted from (a) at $T=288\mathrm{K}$. The $\sigma$, ${\sigma }_S$, and ${\sigma }_E$ show linear elasticity over a wide range up to $\gamma =140\%$. The linearity of ${\sigma }_S$ and ${\sigma }_E$ to $\gamma$ corresponds to the $\gamma$ independence of $T_0$.

Figure 5

The existence of a universal function that governs the energy contribution of gel elasticity. (a) Temperature ($T$) dependence of shear modulus $G$. We obtain each gray line from a least-squares fit of each sample, which is characterized by the three parameters of the precursors: the molar mass $M$, the concentration $c$, and the connectivity $p$. All gray lines that have the same $M$ and $c$ pass through a vanishing temperature $T_0$ on the $T$ axis, which leads to Eq. (7). The value of $T_0$ in each graph is the average of the four samples with different values of $p$, and the values in parentheses represent the standard deviation. (b,c) Normalized concentration ($c/c^{*}$) dependence of $T_0$. We set $c^{*}=80$, 60, 40, and $30\mathrm{g}/\mathrm{L}$ for $M=5$, 10, 20, and $40\mathrm{kg}/\mathrm{mol}$, respectively, to construct the master curve. The orange triangles, blue diamonds, red circles, and black squares represent $M=5$, 10, 20, and $40\mathrm{kg}/\mathrm{mol}$, respectively. Each filled symbol represents the average of four samples taken from (a). Additionally, each open symbol represents the value of one equal-weight mixing sample ($p\simeq 1$). (b) Log-log (main panel) and linear (inset) plots of $c/c^{*}$ vs $T_0$, which demonstrate the scaling law $T_0\sim (c/c^{*}{)}^{-1/3}$ in the dilute regime ($c/c^{*}<1$). (c) A linear plot of the inverse normalized concentration $(c/c^{*}{)}^{-1}$ vs $T_0$, which demonstrates that $T_0$ becomes nearly zero in the dense limit [$(c/c^{*}{)}^{-1}\to 0$]. This result agrees with $|G_E|\ll G_S$ for rubber elasticity as given in previous studies [5, 6, 7, 8, 10]. (d) Connectivity $p$ and $c/c^{*}$ dependence of $g$. We obtain the contour plot from data points (white circles) that represent samples shown in (a). The sol-gel transition line (gray thick line) is the interpolation of data (black crosses) taken from Ref. [21].

Figure 6

Strong dependence of the energy contribution for polymer gel elasticity (${\sigma }_E/\sigma =G_E/G$) on the normalized polymer concentration $c/c^{*}$. The data are taken from Fig. 5, and the symbols are the same as those in Figs. 5 and 5. The energy contribution (${\sigma }_E/\sigma$) of the polymer gel is considerably different from that of PEG rubber (${\sigma }_E/\sigma =0.07\pm 0.01$ [8], green line). This discrepancy suggests that the explanation of energy contribution in previous studies of rubber elasticity [e.g., the RIS model [30] as shown in Fig. 7] is not applicable to that in this study of gel elasticity. On the other hand, ${\sigma }_E/\sigma$ tends to approach that of PEG rubber in the dense limit ($c/c^{*}\to \infty$).

Figure 7

Possible scenarios for the microscopic origin of negative energy elasticity. (a) Energy change accompanying the conformational change with deformation of a PEG chain, which has been considered to be the origin of “positive” energy elasticity for PEG rubber [the green line in (a)] [8]. When the PEG chain is stretched, the trans-gauche-trans (tgt) conformation around successive $\mathrm{O}─\mathrm{C}─\mathrm{C}─\mathrm{O}$ bonds is transformed to the trans-trans-trans (ttt) conformation. Because the conformational energy of trans around the $\mathrm{C}─\mathrm{C}$ bond is higher than that of gauche originating from the dispersion interaction between the adjoining oxygen atoms [8], the energy elasticity is positive. (b) Energy change accompanying the attractive solvent-polymer interaction with deformation of a polymer (PEG) chain between cross-links, causing “negative” energy elasticity in the polymer gel. In the dilute regime ($c/c^{*}<1$), the “territory” of a chain (blue curves) between cross-links (red points) can be roughly described as cylindrical in shape (light blue region) with radius $r$ and height $l$. When the chain is mechanically stretched at a stretch ratio $\lambda$, the number of solvent molecules interacting with the chain increases, decreasing the total energy of the solvent-polymer interaction.

Figure 8

Comparison between the experimentally obtained $g=g(p,c/c^{*})$ [defined in Eq. (12)] and the dimensionless structure parameter $\xi =\nu -\mu$. Here, $\nu$ and $\mu$ are the numbers per precursor of elastically effective chains and cross-links, respectively. We calculate $\nu$, $\mu$, and $\xi$ using the Bethe approximation with $p$ estimated from Eq. (a1). We determine $g$ from the data in Fig. 5 with Eq. (12). Each symbol represents one sample that is characterized by the molar mass $M$, concentration $c$, and connectivity $p$. The blue diamonds, red circles, and black squares represent $M=10$, 20, and $40\mathrm{kg}/\mathrm{mol}$, respectively. The green line represents $g=2.4\xi$. The four blue diamonds enclosed by the gray curve represent the samples with the lowest normalized polymer concentrations ($c/c^{*}\simeq 0.5$) in this experiment. They deviate from the green line because they have many elastically ineffective connections such as the intramolecular bonds and loops [16, 23, 24], causing the overestimation of $\xi$.