Solvent-Induced Negative Energetic Elasticity in a Lattice Polymer Chain
Abstract
The negative internal energetic contribution to the elastic modulus (negative energetic elasticity) has been recently observed in polymer gels. This finding challenges the conventional notion that the elastic moduli of rubberlike materials are determined mainly by entropic elasticity. However, the microscopic origin of negative energetic elasticity has not yet been clarified. Here, we consider the -step interacting self-avoiding walk on a cubic lattice as a model of a single polymer chain (a subchain of a network in a polymer gel) in a solvent. We theoretically demonstrate the emergence of negative energetic elasticity based on an exact enumeration up to and analytic expressions for arbitrary in special cases. Furthermore, we demonstrate that the negative energetic elasticity of this model originates from the attractive polymer–solvent interaction, which locally stiffens the chain and conversely softens the stiffness of the entire chain. This model qualitatively reproduces the temperature dependence of negative energetic elasticity observed in the polymer-gel experiments, indicating that the analysis of a single chain can explain the properties of negative energetic elasticity in polymer gels.
Since the widespread acceptance of the macromolecular hypothesis [1], the entropic elasticity originating from flexible polymer chains in rubberlike materials has been investigated experimentally and theoretically [2, 3, 4]. The simplest theoretical explanations for entropic elasticity are provided by statistical models of ideal chains. For example, the random walk, freely jointed chain, and freely rotating chain models are described in textbooks on statistical mechanics [5, 6], polymer physics [7, 8], and soft matter physics [9].
Rubberlike materials composed of polymer chains exhibit an interplay between entropic () and energetic () contributions to the elastic modulus (). In the case of conventional natural and synthetic rubbers, is significantly smaller than [2, 3, 4]. A small is considered to originate from the conformational change of polymer chains, which has been theoretically modeled such as the rotational isometric state model [10]. By contrast, the significant negative was recently observed in a chemically crosslinked polymer gel, i.e., a polymer network containing a large amount of solvent [11, 12, 13]. In this observation, is significantly larger than that of the energetic elasticity originating from conformational changes, and reaches the same order of magnitude as , as shown in Fig. 1(a).

Although two previous studies [11, 13] measured the shear modulus over a narrow temperature () range where can be approximated as a linear function of , the combination of both results shows that is an increasing convex function of , as shown in Fig. 1(a). Here, dependence of is fitted by quadratic function (black solid curve), and the tangent of the fitted curve (dotted line) at the reference temperature intersects with the horizontal axis at and vertical axis at . These two experiments had the same conditions, excluding the temperature ranges; Ref. [11] used – , whereas Ref. [13] used – . By shifting from (Ref. [11]) to (Ref. [13]), the values of increased. These results imply that (i) is an increasing convex function of , (ii) depends on the reference temperature , and (iii) is a monotonically decreasing function of .
In this Letter, focusing on a subchain (i.e., a chain between adjacent crosslinks) in the polymer network of polymer gels, we theoretically demonstrate the emergence of negative energetic elasticity in a single polymer chain model on a lattice. Notably, the previous experimental studies [11, 13] show that is described as in a narrow temperature range, where depends on the polymer network topology, but does not [see Eqs. (7) and (9) in Ref. [12]]. Thus, network topology does not contribute to the proportion of energetic elasticity to entropic elasticity (), suggesting that a subchain is sufficient to investigate . As shown in Figs. 1(a) and 1(b), this polymer chain model explains as a monotonically decreasing function of temperature in polymer gel experiments. Furthermore, we provide a microscopic mechanism for the emergence of negative energetic elasticity by examining the polymer–solvent interaction strength in this model.
Lattice polymer chain model.—We consider a single polymer chain model surrounded by solvent molecules on a simple cubic lattice (three dimensions). This model was first introduced by Orr for highly dilute polymer solutions [14] and is one of the simplest ways to express the interaction between a polymer chain and solvent molecules. In Figs. 2(a) and 2(b), we use a square lattice (two dimensions) for illustration. As indicated in Fig. 2(a), this model consists of solvent molecules and a polymer chain represented by an -step self-avoiding walk (SAW) [15], which has bonds connecting consecutive and distinct lattice sites (i.e., the polymer segments). The energy function of the model is given by
(1) |
where denotes the configuration of SAWs, and , , and are the numbers of the polymer–polymer, polymer–solvent, and solvent–solvent contact pairs, respectively. Here, the interaction energies acting between each pair are , , and , respectively. In Eq. (1), we do not consider the bending energetic terms, which are essential for semiflexible polymers [16], to focus on the solvent-induced energetic elasticity.

Once the entire lattice size is given, the total number of contact pairs, , is constant. In addition, we derive by counting solvent molecules surrounding . Here, is the coordination number of a lattice, e.g., and for the square and cubic lattices, respectively. Thus, when is constant, Eq. (1) is rewritten [14, 9] as
(2) |
where a constant term has been omitted, and and . Here, the original model illustrated in Fig. 2(a) was reduced to the so-called interacting SAW [17] shown in Fig. 2(b), which is a single-chain system with the intrachain interaction. The interacting SAW reduces to the (noninteracting) SAW at . Notably, many studies on the interacting SAW have focused on the self-attractive condition to investigate the collapsing transition [18, 19, 20, 21, 22]. By contrast, this study focuses mainly on the self-repulsive condition () to investigate the effect of attractive polymer–solvent interactions.
Energetic and entropic elasticities in lattice polymer chain.—To calculate the stiffness (a single-chain counterpart of the elastic modulus) of the lattice polymer chain model, we impose an on-axis constraint on the end-to-end vector of . Here, the length of the vector is , and the direction of the vector is the same as the axis, as shown in Fig. 2(c). The partition function with the on-axis constraint using Eq. (2) is given by
(3) |
where is the Boltzmann constant, is the absolute temperature, and is the number of possible for a given set of , , and . In Eq. (3), is an upper bound of summation [i.e., for ], which is discussed in Supplemental Material, Sec. S1 [23]. The corresponding free energy is .
We define the stiffness of the lattice polymer chain model with the on-axis constraint. In the continuum limit ( and lattice spacing ), the stiffness is defined as the second derivative of the free energy:
(4) |
Thus, we define the finite difference form of stiffness as
(5) | |||||
where the first- and second-order differences of are given by , and , respectively. Here, because exists only for odd for odd and only for even for even (see Supplemental Material, Sec. S11 [23]).
We decompose the elasticity into its energetic and entropic contributions as in the same way as in Refs. [11, 12]. According to thermodynamics, , where is the internal energy, and is the entropy. Thus, in the continuum limit, the energetic and entropic contributions are and , respectively. From Maxwell’s relation, we have . Thus, we calculate and in the lattice polymer chain model using Eq. (5).
Exact enumeration and derivation of polynomial functions.—We exactly enumerate for using the simplest recursive algorithm [27] with two pruning algorithms, considering the octahedral symmetry of the simple cubic lattice and the reachability of to a specific endpoint (see Supplemental Material, Sec. S2 [23]). The complete lists of are provided in Sec. S11 of Supplemental Material [23]. Those lists are consistent with the results reported in Refs. [14, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37] (see Supplemental Material, Sec. S3 [23]). Using the lists of with additional results up to for , we exactly derive the polynomial functions , , , and for arbitrary integers and , which are provided in Sec. S12 of SM [23].
Emergence of negative energetic elasticity.—From the lists of , we can analytically calculate , , and . Figure 1(b) shows a representative result for and . Here, we introduce the dimensionless quantities , , , and . Figure 1(b) demonstrates the emergence of the solvent-induced negative energetic elasticity () in the model. Figure 2(c) displays an example of for . In this Letter, we use for the illustration [e.g., Fig. 1(b)] because the maximum value of is larger than . Although the extent of depends on , the negative can be observed for different and . Notably, we find that for arbitrary , , and positive finite using the polynomial functions , , , and (see Supplemental Material, Sec. S13 [23]).
Figures 1(a) and (b) demonstrate the qualitative consistency between the previous experimental results for the shear modulus of the poly(ethylene glycol) (PEG) hydrogel [11, 13] and our results. These results suggest that negative energetic elasticity () in the polymer gel originates from a single chain ().
To examine the effect of the sign of on the energetic elasticity, we show the dependence of the stiffness on in Fig. 3. [Note that and at (noninteracting SAW), whereas and .] Figure 3 depicts that the energetic contributions are negative, zero, and positive for , , and , respectively, for .

As depicted in Fig. 1(b), denotes the intercept of the tangent of at the reference temperature . For polymer gels, [Fig. 1(a)] is a key factor in the analysis of negative energetic elasticity because does not depend on the polymer network topology [11, 12]. In addition, in the lattice polymer chain model, is a better measure of the negative energetic elasticity than , because is independent of the lattice spacing , unlike , which depends on .
We define , which is a good indicator of the negative energetic elasticity in the sense that is identical to in the case of . As shown in Fig. 3, the intercept of the tangent of at corresponds to . Notably, is a functional of and is a rational number for a given set of , , and (see Supplemental Material, Secs. S5, S6, and S7 [23]). In Fig. 4(a), we plot that is calculated from (the exact rational numbers of are listed in Sec. S14 of Supplemental Material [23]).
We successfully determine the analytic expressions of as the rational functions of for , , and , using the polynomial functions of . For example,
(6) |
The other rational functions, and , are provided in Sec. S15 of Supplemental Material [23]. In Fig. 4(a), we overlay the three curves of these functions, which pass through all the corresponding points of .
Figure 4(b) shows as a function of for . Here, the three curves of the analytic expressions and the points collapse onto a single master curve, except for the small values of . A possible origin of the exponent is the reduction of the dimensionality from three to two by the on-axis constraint on the end-to-end vector, resulting in the universal critical exponent of the two-dimensional SAW [38, 39] (see Supplemental Material, Sec. S8 [23]).

Microscopic mechanism of negative energetic elasticity.—To characterize the microscopic properties of the lattice polymer chain model with a negative , we evaluate the thermal average of the mean free path of the interacting SAW as
(7) |
where is the number of bending points of , and is the number of for a given set of , , , and . In Eq. (7), is the mean free path for each (see Supplemental Material, Sec. S9 [23]). For example, and for in Figs. 2(b) and 2(c), respectively.

Figure 5 indicates that increases with , whereas decreases with . Here, characterizes the “global” stiffness of the whole polymer chain, whereas characterizes the “local” stiffness of the chain. Thus, the global and local stiffnesses are negatively correlated, which is also observed in various sets of (see Supplemental Material, Sec. S4 [23]). These results confirm that polymer chains become locally stiffer because of the attractive interaction with solvent molecules, and globally softer because of the smaller curvature of free energy in the case of and . This is the microscopic mechanism of negative energetic elasticity in the lattice polymer chain model.
Figure 5 also shows values corresponding to the SAW () and neighbor-avoiding walk (NAW; ) [40, 41, 32] (see Supplemental Material, Sec. S10 [23]). The emergence of negative energetic elasticity is characterized by the crossover between the SAW and NAW, which only possess entropic elasticity. The minimum of is at , where the interaction strength is the same order of magnitude as the thermal energy .
Concluding remarks.—We used the simplest lattice polymer chain model to explain both the energetic and entropic elasticities (Fig. 2). By exactly enumerating the configurations of this model, we obtained the stiffness of the chain and its energetic and entropic contributions (Fig. 3). This result demonstrates that the negative energetic elasticity originates from the polymer–solvent interaction. The three rational functions of with respect to are derived from the enumeration results (Fig. 4), revealing that negative energetic elasticity exists for all finite . We revealed a negative correlation between the stiffness of the whole polymer chain and the mean free path of the chain (Fig. 5). In short, locally stiffer chains are globally softer.
Although this simple model does not include chemical details, it qualitatively reproduces the temperature dependence of negative energetic elasticity observed in the experiments conducted on the PEG hydrogel [11, 13] (Fig. 1). This fact indicates that negative energetic elasticity would emerge in various single polymer chains [42, 43, 44] and various polymer gels other than the PEG hydrogel. Therefore, this model provides a starting point for the further understanding of negative energetic elasticity in polymer chains and networks in solvents.
Acknowledgements.
We thank Yuki Yoshikawa and Takamasa Sakai for allowing us to use the experimental data in Fig. 1(a) and for their useful comments, especially those regarding the temperature dependence of . This study was supported by JSPS KAKENHI Grant No. 22K13973 to N.C.S., and No. 19K14672 and No. 22H01187 to N.S.References
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