Negative energy elasticity in a rubber-like 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 $G$ is approximately equivalent to the entropy contribution $G_S$. Similarly, polymer gels containing a large amount of solvent have also been assumed to exhibit $G\simeq G_S$, but this has yet to be verified experimentally. Here, we measure the temperature dependence of the shear modulus in a hyperelastic polymer gel and find that the energy contribution $G_E=G-G_S$ can be a negative value, reaching up to double the $G$ (i.e., $\left|G_E\right| \simeq 2G$), although the elasticity of stable materials is generally bound to be positive. We further argue that $G_E$ 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 would stimulate a re-examination of a vast amount of research on gel elasticity.

Rubber elasticity is the archetype of the entropic force emerging from the second law of thermodynamics [1][2][3]; numerous experimental [4][5][6][7][8] and theoretical [9,10] studies on natural and synthetic rubbers have shown that the elasticity G is approximately equivalent to the entropy contribution G S . Similarly, polymer gels containing a large amount of solvent have also been assumed to exhibit G G S [11][12][13][14], but this has yet to be verified experimentally. Here, we measure the temperature dependence of the shear modulus in a hyperelastic polymer gel and find that the energy contribution G E = G − G S can be a negative value, reaching up to double the G (i.e., |G E | 2G), although the elasticity of stable materials is generally bound to be positive. We further argue that 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 would stimulate a re-examination of a vast amount of research on gel elasticity.
We can experimentally determine the entropy contribution σ S and the energy contribution σ E by measuring the (shear) stress σ as a function of temperature T in a constant-volume condition [1,3] (the van't Hoff isochore [15]). We consider an incompressible elastomer and give an external (shear) strain γ. In an isothermal process, the corresponding stress σ = σ(T, γ) is related to the Helmholtz free energy density f = f (T, γ) as σ = ∂f /∂γ. On the basis of f = e − T s, where e is the internal energy density and s is the entropy density, we can separate the entropy part σ S and the energy part σ E of the stress σ = σ S + σ E as σ S ≡ −T ∂s/∂γ and σ E ≡ ∂e/∂γ. According to the Maxwell relation ∂s/∂γ = −∂σ/∂T , we have By using equation (1), we can determine σ S by measuring the T dependence of σ when γ is fixed. Then, we can obtain σ E as σ E = σ − σ S . Figure 1 demonstrates how to determine σ S and σ E from experimental data (σ(T )) for natural rubber (Fig. 1a) and polymer gel (Fig. 1b). Both are commonly highly stretchable as a result of network structures formed by chemically crosslinked polymer chains. As shown in Fig. 1a, it was confirmed that σ σ S in the case of natural and synthetic rubbers [4][5][6][7][8]; |σ E | was less than a quarter of σ. Thus, the elasticity of rubber-like (i.e., hyperelastic) polymer materials has been widely considered to be described primarily as entropy elasticity [1,3]. For example, in polymer gels containing solvents, σ σ S has also been assumed [11][12][13][14]; nevertheless no experimental verification has been conducted. In this study, to examine this conventional assumption, we use a tetra-functional hydrogel (tetra-PEG gel) [16] as a model rubber-like polymer gel. As shown in Fig. 1b, this gel is synthesized by AB-type cross-end coupling of two kinds of precursors: tetra-armed polymer chain units of the same size. Remarkably, we find that σ E can be a negative value as large as σ, i.e., |σ E | ∼ σ (Fig. 1b).
To determine σ S and σ E , a constant-volume condition is required; hence, we use a dynamic shear rheometer (Fig. 2a) in the range of 278 K ≤ T ≤ 298 K. Shear deformation can suppress volume change caused by a decrease in internal pressure [1,6]. In addition, setting 278 K ≤ T ≤ 298 K guarantees a constant-volume condition (see methods). Figure 2b shows the T dependence of σ under fixed strain γ for a polymer gel. Figure 2b demonstrates that (i) σ is a nearly linear function of T within the measured range and (ii) all linear extrapolations of σ = σ(T ) for γ ≤ 140% (solid gray lines) pass through the one point (T 0 ) on the T axis. We call this proper temperature T 0 the "vanishing temperature". We emphasize that the "actual" stress σ does not follow the extrapolation lines (solid gray lines) at low temperatures away from the measured temperature (< 278 K) and certainly does not vanish at T 0 . We extrapolate the σ-T relation to calculate the energy and entropy contributions of σ.
Irrelevant energy elasticity in natural rubber and negatively relevant energy elasticity in polymer gel. a, Temperature (T ) dependence of the stress σ (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 is extrapolated to T = 0 K. According to equation (1), we have the entropy contribution σS (blue dashed line) and the energy contribution σE (red dashed line) that corresponds to the intercept of the solid gray line. In the measured temperature range (black symbols), the ratio of σE to σ is less than 15%. Similarly, small energy contributions to elasticity were observed in many rubber materials [4,[6][7][8].
b, A typical result of the T dependence of the shear stress σ (black symbols) of a polymer gel through rheological measurements under 60% shear strain γ and a schematic illustration of tetra-PEG gel. We use a highly stretchable rubber-like hydrogel (tetra-PEG gel), which 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 (see Methods). In b, the gel sample was synthesized by equal weight mixing of the two precursors whose molar mass M and concentration c are 20 kg/mol and 60 g/L, respectively. The gray solid, blue dashed, and red dashed lines are obtained in the same way as in a. Notably, we find that σE is a negatively relevant value.  c, Stress (σ)-strain (γ) curve (black symbols) with σS (blue symbols) and σE (red symbols). The data are extracted from b at T = 288 K. The σ, σS and σE show linear elasticity over a wide range up to γ = 140%. The linearity of σS and σE to γ corresponds to the γ-independence of T0. d, Frequency dependence of the storage modulus G and loss modulus G . Since the (static) modulus is given by G = limω→0 G (ω) and G is independent of the frequency (ω/2π) below 10 Hz, we regard G at 1 Hz as G. At 1 Hz, the loss tangent tan δ = G /G is at most 10 −2 in all samples. The greater the connectivity p is, the lower tan δ is. (p is defined in the main text.) For example, when p 0.95 (a nearly perfect network structure), G is underdetected.
The polymer gel (tetra-PEG gel) is an ideal rubber-like (i.e., hyperelastic) material in the sense that the stressstrain relation (the black symbols in Fig. 2c) exhibits a wide range of linear elasticity (γ 140%). Thus, the (shear) modulus G describes the elasticity of the polymer gel in a wide range of the strain (γ 140%), where This ideal linear elasticity of the polymer gel implies that the volume is certainly constant under shear deformation and is an advantage for investigating elasticity compared to natural and synthetic rubbers. To be precise, we regard G as the storage modulus G at 1 Hz because G is independent of the frequency (below 10 Hz) and is much greater than G , as shown in Fig. 2d.
The blue and red symbols in Fig. 2c show σ S and σ E , respectively, calculated from equation (1) and σ(T ) in Fig. 2b. Figure 2c (and Fig. 1b above) demonstrates that σ E is comparable to σ and that σ S and σ E are also linear with respect to γ for γ ≤ 140%. Thus, the entropy and energy contributions of the shear modulus, which are defined by G S ≡ lim γ→0 ∂σ S /∂γ and G E ≡ lim γ→0 ∂σ E /∂γ, respectively, describe the elasticity in a wide range of the strain (γ 140%). In the same manner as equation (1), we can determine To investigate the relationship between the negative energy elasticity of the polymer gel and the microscopic structure of the polymer network, we independently and systematically control three parameters of the precursors: the molar mass M , the concentration c, and the connectivity p. Here, p (0 ≤ p ≤ 1) is defined as the fraction of the reacted terminal functional groups and is controlled by mixing two kinds of precursors nonstoichiometrically (see Methods). In the polymer network after completion of the chemical reaction, M and c correspond to double the molecular weight between crosslinks and the polymer (network) concentration, respectively. From the experimental results shown in Fig. 3a, we find two features: (i) G is a nearly linear function of T in the measured range, and (ii) the vanishing temperature T 0 is independent of p. These features indicate that where we introduce a prefactor a = a(M, c, p). According to equation (3), in the measured range, G S = aT , i.e., the entropy contribution is a linear function of T , and G E = aT 0 , i.e., the energy contribution is independent of T and governed by T 0 .
By analyzing the systematic results shown in Fig. 3a, we reveal a law governing T 0 . Figure 3b demonstrates that all the results at different values of M and c collapse onto a single master curve, which means Here, c * = c * (M ) is the normalization factor chosen to construct the master curve. It is notable that c * (M ) is in close agreement with the overlap concentration of the precursors c * vis (M ) obtained by a viscosity measurement [13]. This fact and equation (4) invoke the osmotic pressure in the polymer solution, which is represented by a universal function of c/c * vis in the dilute and semidilute regimes [17].
We consider the dilute and dense regimes.
In the dilute regime (c/c * < 1), we find a scaling law Fig. 3b. Since c −1/3 seems to be proportional to the linear distance between crosslinks l, we have T 0 ∼ l. As discussed below, this fact is important in conjecturing the molecular interpretation of negative energy elasticity. If T 0 follows the scaling law (T 0 ∼ (c/c * ) −1/3 ) below the measured c/c * range, T 0 reaches the measured temperature (T 280 K) at c/c * 0.12. Thus, if c < 0.12c * , tetra-PEG gels are expected to be mechanically unstable because G < 0. This expectation is consistent with a previous study that reported that tetra-PEG gels cannot be formed below c/c * 0.1 around 300 K [18].
In the dense regime (c/c * 1), Fig. 3c shows that as (c/c * ) −1 → 0, which means that the solvent is removed, T 0 tends to decrease to nearly zero. This result agrees with previous studies on natural and synthetic rubbers without solvent; the absolute value of energy elasticity (aT 0 ) is much smaller than the value of entropy elasticity (aT ) [4][5][6][7][8].
Using dimensional analysis, we determine the functional form of a = a(p, M, c). Since a has the same dimension as cR/M , the dimensionless combination is g ≡ aM/(cR), where R is the gas constant. Then, g depends on the dimensionless parameters composed of p, M and c, i.e., g = g(p, c/c * ). Figure 3d validates this dimensional analysis. Substituting g = g(p, c/c * ) and equation (4) into equation (3), we have Although g(p, c/c * ) and T 0 (c/c * ) depend on the kinds of polymer chains and solvents that constitute polymer gels, equation (5) generally represents G in homogeneous polymer gels. For example, Fig. 3c gives T 0 (c/c * ) for the tetra-PEG gel. Since the measured region is limited to 278 K ≤ T ≤ 298 K, it does not necessarily mean that equation (5) is applicable for other T regions. GS for rubber elasticity given in previous studies [4][5][6][7][8]. 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. [18].
The behavior of the contour lines of g(p, c/c * ) in Fig. 3d is consistent with recent experiments on the polymer gels for the dilute (c/c * < 1) and semidilute (c/c * > 1) regimes. In the dilute regime (c/c * < 1), Fig. 3d shows that the contour lines of g(p, c/c * ) are nearly parallel to the sol-gel transition line [18] corresponding to g(p, c/c * ) = 0. Thus, the contour lines are consistent with previous experiments [18,22]. There seems to be no theory (e.g., percolation theories and mean-field theories) that quantitatively explains the dependence of g on c/c * [22]. The behavior of g(p, c/c * ) is considered to be caused by elastically ineffective connections such as intramolecular bonds and loops [14,20,22].
In the semidilute regime (c/c * > 1) , Fig 3d shows that g is almost independent of c/c * , and we find g(p, c/c * ) 2.4 ξ(p) from the experimental data (see Supplementary Fig. S1). Here, the number per precursor of elastically effective cycles ξ(p) satisfies 0 ≤ ξ(p) ≤ 1 and is calculated by the Bethe approximation [20,23,24] (see Supplementary). Equation (5) with g(p, c/c * ) 2.4 ξ(p) leads to G(p) ∼ ξ(p), which is consistent with previous experiments [20,21]. We remark that the entropy contribution obtained in our study, G S = cRg(p)T /M , is 2.4 times as large as G S = cRξ(p)T /M , which is predicted by the phantom network model [3,19]. Figure 4a reveals that the conformational energy change cannot explain the microscopic origin of negative energy elasticity in a rubber-like gel. Previous studies on a rubber [7,8,25] have shown that σ E /σ = T d dT ln r 2 0 , where r 2 0 is the unperturbed mean-square end-to-end distance. This equation is interpreted as the conformational energy change with deformation by the rotational isomeric state (RIS) model [26]. The RIS model explains that the experimentally determined small and positive energy contribution in the PEG rubber (σ E /σ = 0.07 ± 0.01) originates from the conformational change around the C-C bond (Fig. 4b) [7]. However, this result is considerably different from our result, as shown in Fig. 4a; σ E /σ for tetra-PEG gel is large and negative (σ E /σ = G E /G = −1.81 ± 0.04 at the maximum) and has the strong dependence on c. Thus, we suggest that the internal energy change with deformation originates mainly from some kind of intermolecular (i.e., polymer-polymer, solvent-polymer, and solvent-solvent) interactions. In addition, since the components remain constant, the Flory-Huggins theory (and the χ parameter) [1][2][3] cannot explain the interaction energy change with deformation. Therefore, we need a new molecular interpretation of negative energy elasticity.
We propose a possible molecular interpretation of negative energy elasticity in a rubber-like gel, as shown in Fig. 4c. The negative G E /G value in the gel origi-  Fig. 3a, and the symbols are the same as those in Fig. 3b and c. The energy contribution (σE/σ) of the polymer gel is considerably different from that of poly(ethylene glycol) (PEG) rubber (σE/σ = 0.07 ± 0.01 [7], green line). On the other hand, σE/σ tends to approach that of PEG rubber in the dense limit (c/c * → ∞). b, 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) [7]. When the PEG chain is stretched, the trans-gauche-trans (tgt) conformation around the successive O-C-C-O bonds is transformed to the trans-trans-trans (ttt) conformation. Since the conformational energy of trans around the C-C bond is higher than that of gauche originating from the dispersion interaction between the adjoining oxygen atoms [7], the energy elasticity is positive. c, Energy change accompanying attractive solvent-polymer interaction, which would cause "negative" energy elasticity in the polymer gel. In the dilute regime, the solvent-polymer interaction region of a chain (blue curves) between crosslinks (red points) can be roughly described as a cylinder shape (light blue region) with radius r and height l. We assume that the interaction energy is proportional to the interaction region, i.e., T0 ∼ r 2 l. Since r is constant with concentration changes and small deformations [2], the c dependence of T0 is T0 ∼ l ∼ c −1/3 , which agrees with the scaling law in Fig. 3b.
nates from the attractive solvent-polymer interaction.
In addition, this interpretation with a fixed M (and a fixed c * (M )) is consistent with the experimental results in each regime. In the dilute regime (c/c * < 1), this interpretation shows that T 0 is proportional to the linear distance between crosslinks l (see the legend of Fig. 4c). Assuming l ∼ c −1/3 , we have T 0 ∼ c −1/3 , which agrees with the scaling law shown in Fig. 3b. In the semidilute regime (c/c * > 1), the light blue regions in Fig. 4c overlap with each other. Thus, the dependence of T 0 on c falls below the scaling law (T 0 ∼ c −1/3 ) as shown in Fig. 3b. In the dense regime (c/c * 1), the solvent diminishes, and the effect of the solvent-polymer interaction on energy elasticity becomes negligible compared to that of the conformational energy change (Fig. 4b), which is consistent with the results shown in Figs. 3c and 4a.
In conclusion, we have discovered that the energy contribution of elasticity (G E ) can be a significant negative value in rubber-like polymer gels with various network structures. Further systematic experiments have revealed that the shear modulus G is simply described by equation (5), and G E is governed by a vanishing temperature T 0 that is a universal function of the normalized polymer concentration c/c * . The vanishing temperature T 0 exhibits a scaling law of T 0 ∼ (c/c * ) −1/3 in the dilute regime (c/c * < 1). Based on this scaling law, we have suggested the microscopic origin of negative energy elasticity: it emerges from the interaction between the polymer chain and the solvent, as shown in Fig. 4. To establish this origin from a molecular description and to verify whether our findings are universal in other polymer gels, further studies are needed.
Our findings have essential implications for past research on gel elasticity. For example, in a previous paper using tetra-PEG gels [13], G G S , i.e., G E 0, was assumed, and the dependence of the shear modulus G on the polymer-concentration c was interpreted as the crossover between the phantom [3,19] and affine [1] net-work models. However, our results ( Fig. 3c and equation (5)) point out that the above assumption is invalid, and "the crossover" does not mean the phantom-affine crossover but originates from the dependence of T 0 on c. As this example shows, our study provides a new perspective on gel elasticity and urges re-examinations of a vast amount of previous research on gel elasticity. FIG. S1. Comparison between the prefactor g(p, c/c * ) given in equation (5) and the dimensionless structure parameter ξ = ν − µ. Here, ν and µ are the numbers per precursor of elastically effective chains and crosslinks, respectively. Each symbol represents one sample that is characterised by the molar mass M , the concentration c, and the connectivity p.
The blue diamonds, red circles and black squares represent M = 10, 20, and 40 kg/mol, respectively. The data are taken from Fig.3a. To calculate ξ = ξ(p), we use the Bethe approximation with p estimated by equation (6). The green line represents g = 2.4 ξ, which means that the entropy elasticity in tetra-PEG gel is 2.4 times as large as that calculated by the phantom network model [19]. The four blue diamonds enclosed by the gray curve represents the samples with the lowest normalized polymer concentrations (c/c * 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 [14,20,22], which causes overestimation of ξ.

BETHE APPROXIMATION TO CALCULATE THE STRUCTURAL PARAMETERS
The dimensionless structural parameters ν, µ and ξ cannot be experimentally observed but be theoretically estimated as a function of connectivity p by using the Bethe approximation [23,24] (also called the tree approximation [2] and the mean-field approximation [3]). The number per precursor of the elastically effective cycles ξ is obtained by ξ = ν − µ, where ν and µ are the numbers per precursor of elastically effective chains and crosslinks, respectively. Here, the elastically effective chain is defined as the chain whose both ends connect to crosslinks.
In the network structure formed by two kinds of tetrafunctional precursors (A 4 or B 4 ), the Bethe approximation assumes that the probability that one arm of A 4 or B 4 does not connect to the infinite-sized network (P (F out A ) or P (F out B ), respectively) as [23,24] P (F out A ) = p A P (F out B ) 3 + 1 − p A , P (F out B ) = p B P (F out A ) 3 + 1 − p B . (S1) Here, p A and p B are the fractions of reacted A and B groups, respectively. By using P (F out A ) and P (F out B ), we can calculate the probabilities that A 4 or B 4 becomes a f -functional crosslink for f = 3 or 4 (P (X Af ) or P (X Bf ), respectively) as Note that A 4 and B 4 cannot play a role as a crosslink for f = 1 and 2. If a molar ratio of A and B groups is [A] : [B] = q : 1 − q (0 < q < 1), we have ν = q 3 2 P (X A3 ) + 2P (X A4 ) In this study, we calculate ν and µ as a function of the connectivity p = qp A + (1 − q)p B [20].