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Solvent-Induced Negative Energetic Elasticity in a Lattice Polymer Chain

Nobu C. Shirai shirai@cc.mie-u.ac.jp Center for Information Technologies and Networks, Mie University, Tsu, Mie 514-8507, Japan    Naoyuki Sakumichi sakumichi@tetrapod.t.u-tokyo.ac.jp Graduate School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
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 nn-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 n=20n=20 and analytic expressions for arbitrary nn 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 (GSG_{S}) and energetic (GUG_{U}) contributions to the elastic modulus (G=GS+GUG=G_{S}+G_{U}). In the case of conventional natural and synthetic rubbers, |GU|\left|G_{U}\right| is significantly smaller than GSG_{S} [2, 3, 4]. A small |GU|\left|G_{U}\right| 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 GUG_{U} 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, |GU|\left|G_{U}\right| is significantly larger than that of the energetic elasticity originating from conformational changes, and |GU|\left|G_{U}\right| reaches the same order of magnitude as GG, as shown in Fig. 1(a).

Refer to caption
Figure 1: Experimental results for a polymer gel [11, 13] and theoretical results of the lattice polymer chain model. (a) Temperature dependence of shear modulus GG of poly(ethylene glycol) (PEG) hydrogel for T=278T=278308308 K\mathrm{K}, including the midpoint T=293T^{*}=293 K\mathrm{K} [seven points, including the orange point (GG^{*}) in the center]. These data are obtained from Refs. [11, 13]. The black solid curve is the fitted quadratic function of these points. The dotted line is the tangent of the solid curve at the reference temperature TT^{*}, which intersects with the horizontal axis at TUT_{U}^{*} (>0>0) and the vertical axis at GUG_{U}^{*} (<0<0). Temperature dependencies of GUG_{U} (pink dashed curve) and GSG_{S} (blue dot-dashed curve) are calculated from the fitted quadratic function. (b) Temperature dependence of stiffness (k^\hat{k}) of lattice polymer chain model for (n,r)=(20,10a)(n,r)=(20,10a) and its energetic (k^U\hat{k}_{U}) and entropic (k^S\hat{k}_{S}) contributions. The tangent (dotted line) of k^\hat{k} at T^\hat{T}^{*} intersects with the horizontal axis at T^U\hat{T}_{U}^{*} (>0>0) and vertical axis at k^Uk^U(r,T^)\hat{k}_{U}^{*}\equiv\hat{k}_{U}(r,\hat{T}^{*}) (<0<0).

Although two previous studies [11, 13] measured the shear modulus GG over a narrow temperature (TT) range where GG can be approximated as a linear function of TT, the combination of both results shows that GG is an increasing convex function of TT, as shown in Fig. 1(a). Here, TT dependence of GG is fitted by quadratic function (black solid curve), and the tangent of the fitted curve (dotted line) at the reference temperature T=293T^{*}=293 K\mathrm{K} intersects with the horizontal axis at TUT_{U}^{*} and vertical axis at GUG_{U}^{*}. These two experiments had the same conditions, excluding the temperature ranges; Ref. [11] used T=278T=278298298 K\mathrm{K}, whereas Ref. [13] used T=288T=288308308 K\mathrm{K}. By shifting TT^{*} from 288288 (Ref. [11]) to 298298 K\mathrm{K} (Ref. [13]), the values of TUT_{U}^{*} increased. These results imply that (i) GG is an increasing convex function of TT, (ii) TUT_{U}^{*} depends on the reference temperature TT^{*}, and (iii) GUG_{U} is a monotonically decreasing function of TT.

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 GG is described as G=a(TTU)G=a(T-T_{U}^{*}) in a narrow temperature range, where aa depends on the polymer network topology, but TUT_{U}^{*} 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 (GU/GS=TU/TG_{U}^{*}/G_{S}^{*}=-T_{U}^{*}/T^{*}), suggesting that a subchain is sufficient to investigate GU/GSG_{U}^{*}/G_{S}^{*}. As shown in Figs. 1(a) and 1(b), this polymer chain model explains GUG_{U} 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 nn-step self-avoiding walk (SAW) [15], which has nn bonds connecting n+1n+1 consecutive and distinct lattice sites (i.e., the polymer segments). The energy function of the model is given by

E(ω)=εppmpp(ω)+εpsmps(ω)+εssmss(ω),E(\omega)=\varepsilon_{\mathrm{pp}}m_{\mathrm{pp}}(\omega)+\varepsilon_{\mathrm{ps}}m_{\mathrm{ps}}(\omega)+\varepsilon_{\mathrm{ss}}m_{\mathrm{ss}}(\omega), (1)

where ω\omega denotes the configuration of SAWs, and mpp(ω)m_{\mathrm{pp}}(\omega), mps(ω)m_{\mathrm{ps}}(\omega), and mss(ω)m_{\mathrm{ss}}(\omega) are the numbers of the polymer–polymer, polymer–solvent, and solvent–solvent contact pairs, respectively. Here, the interaction energies acting between each pair are εpp\varepsilon_{\mathrm{pp}}, εps\varepsilon_{\mathrm{ps}}, and εss\varepsilon_{\mathrm{ss}}, 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.

Refer to caption
Figure 2: (a) Two-component model of single polymer chain (open and filled black circles connected with lines) and solvent molecules (light-blue filled circles) on square lattice. There are three types of nearest-neighbor interactions. (b) Reduced model mathematically equivalent to (a). (c) Example of interacting self-avoiding walk on cubic lattice with on-axis constraint on end-to-end vector.

Once the entire lattice size is given, the total number of contact pairs, mpp(ω)+mps(ω)+mss(ω)m_{\mathrm{pp}}(\omega)+m_{\mathrm{ps}}(\omega)+m_{\mathrm{ss}}(\omega), is constant. In addition, we derive 2mpp(ω)+mps(ω)=(z2)n+z2m_{\mathrm{pp}}(\omega)+m_{\mathrm{ps}}(\omega)=(z-2)n+z by counting solvent molecules surrounding ω\omega. Here, zz is the coordination number of a lattice, e.g., z=4z=4 and 66 for the square and cubic lattices, respectively. Thus, when nn is constant, Eq. (1) is rewritten [14, 9] as

E(ω)=εm(ω),E(\omega)=\varepsilon\,m(\omega), (2)

where a constant term has been omitted, and εεpp2εps+εss\varepsilon\equiv\varepsilon_{\mathrm{pp}}-2\varepsilon_{\mathrm{ps}}+\varepsilon_{\mathrm{ss}} and m(ω)mpp(ω)m(\omega)\equiv m_{\mathrm{pp}}(\omega). 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 ε=0\varepsilon=0. Notably, many studies on the interacting SAW have focused on the self-attractive condition (ε<0)(\varepsilon<0) to investigate the collapsing transition [18, 19, 20, 21, 22]. By contrast, this study focuses mainly on the self-repulsive condition (ε>0\varepsilon>0) 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 ω\omega. Here, the length of the vector is rr, and the direction of the vector is the same as the xx axis, as shown in Fig. 2(c). The partition function with the on-axis constraint using Eq. (2) is given by

Z(r,T)=m=0mubWn,m(r)eεm/(kBT),Z(r,T)=\sum_{m=0}^{m_{\mathrm{ub}}}W_{n,m}(r)\,e^{-\varepsilon m/(k_{B}T)}, (3)

where kBk_{B} is the Boltzmann constant, TT is the absolute temperature, and Wn,m(r)W_{n,m}(r) is the number of possible ω\omega for a given set of nn, rr, and mm. In Eq. (3), mubm_{\mathrm{ub}} is an upper bound of mm summation [i.e., Wn,m(r)=0W_{n,m}(r)=0 for mmub+1m\geq m_{\mathrm{ub}}+1], which is discussed in Supplemental Material, Sec. S1  [23]. The corresponding free energy is A(r,T)=kBTlnZ(r,T)A(r,T)=-k_{B}T\ln Z(r,T).

We define the stiffness of the lattice polymer chain model with the on-axis constraint. In the continuum limit (nn\to\infty and lattice spacing a0a\to 0), the stiffness is defined as the second derivative of the free energy:

k(r,T)2A(r,T)r2=kBT[(Z(r,T)rZ(r,T))22Z(r,T)r2Z(r,T)].k(r,T)\equiv\frac{\partial^{2}A(r,T)}{\partial r^{2}}=k_{B}T\left[\left(\frac{\frac{\partial Z(r,T)}{\partial r}}{Z(r,T)}\right)^{2}-\frac{\frac{\partial^{2}Z(r,T)}{\partial r^{2}}}{Z(r,T)}\right]. (4)

Thus, we define the finite difference form of stiffness as

k(r,T)\displaystyle k(r,T) \displaystyle\equiv kBT[(1Z(r,T)m=0mubΔWn,m(r)Δreεm/(kBT))2\displaystyle k_{B}T\left[\left(\frac{1}{Z(r,T)}\sum_{m=0}^{m_{\mathrm{ub}}}\frac{\varDelta W_{n,m}(r)}{\varDelta r}\,e^{-\varepsilon m/(k_{B}T)}\right)^{2}\right. (5)
1Z(r,T)m=0mubΔ2Wn,m(r)Δr2eεm/(kBT)],\displaystyle\left.\quad-\frac{1}{Z(r,T)}\sum_{m=0}^{m_{\mathrm{ub}}}\frac{\varDelta^{2}W_{n,m}(r)}{\varDelta r^{2}}\,e^{-\varepsilon m/(k_{B}T)}\right],

where the first- and second-order differences of Wn,m(r)W_{n,m}(r) are given by ΔWn,m(r)[Wn,m(r+Δr)Wn,m(rΔr)]/2\varDelta W_{n,m}(r)\equiv\left[W_{n,m}(r+\varDelta r)-W_{n,m}(r-\varDelta r)\right]/2, and Δ2Wn,m(r)Wn,m(r+Δr)2Wn,m(r)+Wn,m(rΔr)\varDelta^{2}W_{n,m}(r)\equiv W_{n,m}(r+\varDelta r)-2W_{n,m}(r)+W_{n,m}(r-\varDelta r), respectively. Here, Δr2a\varDelta r\equiv 2a because ω\omega exists only for odd r^r/a\hat{r}\equiv r/a for odd nn and only for even r^\hat{r} for even nn (see Supplemental Material, Sec. S11 [23]).

We decompose the elasticity into its energetic and entropic contributions as k=kU+kSk=k_{U}+k_{S} in the same way as in Refs. [11, 12]. According to thermodynamics, A=UTSA=U-TS, where UU is the internal energy, and SS is the entropy. Thus, in the continuum limit, the energetic and entropic contributions are kU(r,T)2U(r,T)/r2k_{U}(r,T)\equiv\partial^{2}U(r,T)/\partial r^{2} and kS(r,T)T2S(r)/r2k_{S}(r,T)\equiv-T\partial^{2}S(r)/\partial r^{2}, respectively. From Maxwell’s relation, we have kS(r,T)=Tk(r,T)/Tk_{S}(r,T)=T\partial k(r,T)/\partial T. Thus, we calculate kS(r,T)=Tk(r,T)/Tk_{S}(r,T)=T\partial k(r,T)/\partial T and kU=kkSk_{U}=k-k_{S} in the lattice polymer chain model using Eq. (5).

Exact enumeration and derivation of polynomial functions.—We exactly enumerate Wn,m(r)W_{n,m}(r) for 1n201\leq n\leq 20 using the simplest recursive algorithm [27] with two pruning algorithms, considering the octahedral symmetry of the simple cubic lattice and the reachability of ω\omega to a specific endpoint (see Supplemental Material, Sec. S2 [23]). The complete lists of Wn,m(r)W_{n,m}(r) 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 Wn,m(r)W_{n,m}(r) with additional results up to n=26n=26 for r=(n8)ar=(n-8)a, we exactly derive the polynomial functions Wn,m((n2)a)W_{n,m}\big{(}(n-2)a\big{)}, Wn,m((n4)a)W_{n,m}\big{(}(n-4)a\big{)}, Wn,m((n6)a)W_{n,m}\big{(}(n-6)a\big{)}, and Wn,m((n8)a)W_{n,m}\big{(}(n-8)a\big{)} for arbitrary integers nn and mm, which are provided in Sec. S12 of SM [23].

Emergence of negative energetic elasticity.—From the lists of Wn,m(r)W_{n,m}(r), we can analytically calculate kk, kUk_{U}, and kSk_{S}. Figure 1(b) shows a representative result for ε>0\varepsilon>0 and (n,r)=(20,10a)(n,r)=(20,10a). Here, we introduce the dimensionless quantities k^a2k/ε\hat{k}\equiv a^{2}k/\varepsilon, k^Ua2kU/ε\hat{k}_{U}\equiv a^{2}k_{U}/\varepsilon, k^Sa2kS/ε\hat{k}_{S}\equiv a^{2}k_{S}/\varepsilon, and T^kBT/ε\hat{T}\equiv k_{B}T/\varepsilon. Figure 1(b) demonstrates the emergence of the solvent-induced negative energetic elasticity (kU<0k_{U}<0) in the model. Figure 2(c) displays an example of ω\omega for r=10ar=10a. In this Letter, we use r=10ar=10a for the illustration [e.g., Fig. 1(b)] because the maximum value of |kU|/k|k_{U}|/k is larger than r10ar\neq 10a. Although the extent of kU/kk_{U}/k depends on rr, the negative kUk_{U} can be observed for different n6n\geq 6 and rr. Notably, we find that kU<0k_{U}<0 for arbitrary n13n\geq 13, ε>0\varepsilon>0, and positive finite TT using the polynomial functions Wn,m((n2)a)W_{n,m}\big{(}(n-2)a\big{)}, Wn,m((n4)a)W_{n,m}\big{(}(n-4)a\big{)}, Wn,m((n6)a)W_{n,m}\big{(}(n-6)a\big{)}, and Wn,m((n8)a)W_{n,m}\big{(}(n-8)a\big{)} (see Supplemental Material, Sec. S13 [23]).

Figures 1(a) and (b) demonstrate the qualitative consistency between the previous experimental results for the shear modulus GG of the poly(ethylene glycol) (PEG) hydrogel [11, 13] and our results. These results suggest that negative energetic elasticity (GU<0G_{U}<0) in the polymer gel originates from a single chain (kU<0k_{U}<0).

To examine the effect of the sign of ε\varepsilon on the energetic elasticity, we show the dependence of the stiffness k^/T^=a2k/(kBT)\hat{k}/\hat{T}=a^{2}k/(k_{B}T) on 1/T^ε/(kBT)1/\hat{T}\equiv\varepsilon/(k_{B}T) in Fig. 3. [Note that k^\hat{k}\to\infty and k^S\hat{k}_{S}\to\infty at ε=0\varepsilon=0 (noninteracting SAW), whereas 0<k^/T^<0<\hat{k}/\hat{T}<\infty and 0<k^S/T^<0<\hat{k}_{S}/\hat{T}<\infty.] Figure 3 depicts that the energetic contributions are negative, zero, and positive for ε>0\varepsilon>0, ε=0\varepsilon=0, and ε>0\varepsilon>0, respectively, for (n,r)=(20,10a)(n,r)=(20,10a).

Refer to caption
Figure 3: Analytic curves of k^/T^\hat{k}/\hat{T}, k^U/T^\hat{k}_{U}/\hat{T}, and k^S/T^\hat{k}_{S}/\hat{T} as functions of 1/T^1/\hat{T} for (n,r)=(20,10a)(n,r)=(20,10a). The tangent (dotted line) of k^/T^\hat{k}/\hat{T} at 1/T^=01/\hat{T}^{*}=0 intersects with the horizontal axis at 1/T^U1/\hat{T}_{U}^{\infty}. The condition for negative energetic elasticity (k^U/T^<0\hat{k}_{U}/\hat{T}<0) is ε>0\varepsilon>0.

As depicted in Fig. 1(b), T^UT^U(T^)\hat{T}_{U}^{*}\equiv\hat{T}_{U}(\hat{T}^{*}) denotes the T^\hat{T} intercept of the tangent of k^=k^(T^)\hat{k}=\hat{k}(\hat{T}) at the reference temperature T^=T^\hat{T}=\hat{T}^{*}. For polymer gels, TUT_{U}^{*} [Fig. 1(a)] is a key factor in the analysis of negative energetic elasticity because TUT_{U}^{*} does not depend on the polymer network topology [11, 12]. In addition, in the lattice polymer chain model, TUT_{U}^{*} is a better measure of the negative energetic elasticity than kUk_{U}, because TU=εT^U/kBT_{U}^{*}=\varepsilon\hat{T}_{U}^{*}/k_{B} is independent of the lattice spacing aa, unlike kU=εk^U/a2k_{U}=\varepsilon\hat{k}_{U}/a^{2}, which depends on aa.

We define T^UlimT^T^U(T^)\hat{T}_{U}^{\infty}\equiv\lim_{\hat{T}^{*}\to\infty}\hat{T}_{U}(\hat{T}^{*}), which is a good indicator of the negative energetic elasticity in the sense that T^U>0\hat{T}_{U}^{\infty}>0 is identical to k^U<0\hat{k}_{U}<0 in the case of ε>0\varepsilon>0. As shown in Fig. 3, the 1/T^1/\hat{T} intercept of the tangent of k^/T^\hat{k}/\hat{T} at 1/T^=01/\hat{T}=0 corresponds to 1/T^U1/\hat{T}_{U}^{\infty}. Notably, T^U\hat{T}_{U}^{\infty} is a functional of Wn,m(r)W_{n,m}(r) and is a rational number for a given set of nn, rr, and mm (see Supplemental Material, Secs. S5, S6, and S7 [23]). In Fig. 4(a), we plot T^U\hat{T}_{U}^{\infty} that is calculated from Wn,m(r)W_{n,m}(r) (the exact rational numbers of T^U\hat{T}_{U}^{\infty} are listed in Sec. S14 of Supplemental Material [23]).

We successfully determine the analytic expressions of T^U\hat{T}_{U}^{\infty} as the rational functions of nn for r=(n2)ar=(n-2)a, (n4)a(n-4)a, and (n6)a(n-6)a, using the polynomial functions of Wn,m(r)W_{n,m}(r). For example,

T^U(n,(n2)a)=4(15n8356n7+3766n623016n5+88019n4213804n3+317784n2256008n+81484)(n1)(9n8204n7+2026n611648n5+42733n4102444n3+156272n2137656n+53028).\hat{T}_{U}^{\infty}\big{(}n,(n-2)a\big{)}=\frac{4(15n^{8}-356n^{7}+3766n^{6}-23016n^{5}+88019n^{4}-213804n^{3}+317784n^{2}-256008n+81484)}{(n-1)(9n^{8}-204n^{7}+2026n^{6}-11648n^{5}+42733n^{4}-102444n^{3}+156272n^{2}-137656n+53028)}. (6)

The other rational functions, T^U(n,(n4)a)\hat{T}_{U}^{\infty}\big{(}n,(n-4)a\big{)} and T^U(n,(n6)a)\hat{T}_{U}^{\infty}\big{(}n,(n-6)a\big{)}, 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 T^U\hat{T}_{U}^{\infty}.

Figure 4(b) shows T^U\hat{T}_{U}^{\infty} as a function of (nr^)α/n(n-\hat{r})^{\alpha}/n for α=3/4\alpha=3/4. Here, the three curves of the analytic expressions and the points collapse onto a single master curve, except for the small values of nn. A possible origin of the exponent α=3/4\alpha=3/4 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 ν=3/4\nu=3/4 of the two-dimensional SAW [38, 39] (see Supplemental Material, Sec. S8 [23]).

Refer to caption
Figure 4: (a) Exact values of T^U(n,r)\hat{T}_{U}^{\infty}(n,r) for n=5,6,,20n=5,6,\dots,20 and three analytic curves for r=(n2)ar=(n-2)a, (n4)a(n-4)a, and (n6)a(n-6)a, shown as gray dashed, light-green dot-dashed, and green solid curves, respectively. (b) Same points and curves are plotted as functions of (nr^)α/n(n-\hat{r})^{\alpha}/n, where α=3/4\alpha=3/4. These collapse onto a single master curve.

Microscopic mechanism of negative energetic elasticity.—To characterize the microscopic properties of the lattice polymer chain model with a negative kUk_{U}, we evaluate the thermal average of the mean free path of the interacting SAW as

(r,T)=1Z(r,T)m=0mubb=0n1nab+1Wn,m,b(r)eεm/(kBT),\ell(r,T)=\frac{1}{Z(r,T)}\sum_{m=0}^{m_{\mathrm{ub}}}\sum_{b=0}^{n-1}\frac{na}{b+1}W_{n,m,b}(r)\,e^{-\varepsilon m/(k_{B}T)}, (7)

where bb is the number of bending points of ω\omega, and Wn,m,b(r)W_{n,m,b}(r) is the number of ω\omega for a given set of nn, rr, mm, and bb. In Eq. (7), na/(b+1)na/(b+1) is the mean free path for each ω\omega (see Supplemental Material, Sec. S9 [23]). For example, b=4b=4 and 1313 for ω\omega in Figs. 2(b) and  2(c), respectively.

Refer to caption
Figure 5: Polymer–solvent interaction [ε/(kBT)\varepsilon/(k_{B}T)] dependences of k^/T^\hat{k}/\hat{T}, k^U/T^\hat{k}_{U}/\hat{T}, k^S/T^\hat{k}_{S}/\hat{T} (top panel), kU/kk_{U}/k (middle panel), and /a\ell/a (bottom panel) for (n,r)=(20,10a)(n,r)=(20,10a) and ε>0\varepsilon>0. As ε\varepsilon increases, the lattice polymer chain model (i.e., interacting SAW) becomes globally softer and locally stiffer.

Figure 5 indicates that k^/T^=a2k/(kBT)\hat{k}/\hat{T}=a^{2}k/(k_{B}T) increases with ε/(kBT)\varepsilon/(k_{B}T), whereas \ell decreases with ε/(kBT)\varepsilon/(k_{B}T). Here, k^/T^\hat{k}/\hat{T} characterizes the “global” stiffness of the whole polymer chain, whereas \ell characterizes the “local” stiffness of the chain. Thus, the global and local stiffnesses are negatively correlated, which is also observed in various sets of (n,r)(n,r) (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 ε>0\varepsilon>0 and k^U<0\hat{k}_{U}<0. This is the microscopic mechanism of negative energetic elasticity in the lattice polymer chain model.

Figure 5 also shows values corresponding to the SAW (ε0\varepsilon\to 0) and neighbor-avoiding walk (NAW; ε\varepsilon\to\infty[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 kU/kk_{U}/k is 0.554-0.554 at ε/(kBT)0.712\varepsilon/(k_{B}T)\simeq 0.712, where the interaction strength ε\varepsilon is the same order of magnitude as the thermal energy kBTk_{B}T.

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 TUT_{U}^{\infty} with respect to nn are derived from the enumeration results (Fig. 4), revealing that negative energetic elasticity exists for all finite n6n\geq 6. 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 TUT_{U}^{*}. This study was supported by JSPS KAKENHI Grant No. 22K13973 to N.C.S., and No. 19K14672 and No. 22H01187 to N.S.

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