Influence of Structural Characteristics on Stretching-Driven Swelling of

Aug 10, 2012 - The λ-dependent μ∞ under moderate elongation is attributed to a function characteristic of movable cross-links (pulley effect) whic...
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Influence of Structural Characteristics on Stretching-Driven Swelling of Polyrotaxane Gels with Movable Cross Links Akihiro Konda,†,§ Koichi Mayumi,‡,§ Kenji Urayama,*,† Toshikazu Takigawa,† and Kohzo Ito*,‡ †

Department of Materials Chemistry, Kyoto University, Nishikyo-ku, Kyoto 615-8510, Japan Graduate School of Frontier Sciences, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa 277-8561, Japan

‡

ABSTRACT: Stretching-driven swelling of polyrotaxane (PR) gels whose cross-linked cyclic molecules can slide along the network strands in response to imposed strain is investigated as functions of cross-link concentration (cx), the length of network strands (Ms) and the content of cyclic molecules threaded onto the network strands (ϕCD). For the PR gels with high cx, equilibrium Poisson’s ratio (μ∞) which is a measure of stretching-driven swelling is independent of imposed stretch (λ) as in the case of classical gels with fixed cross-links. In contrast, μ∞ for the PR gels with sufficiently low cx depends on λ: μ∞ increases with λ in the regime of λ < λc whereas μ∞ levels off in the region of λ > λc. The λ-dependent μ∞ under moderate elongation is attributed to a function characteristic of movable cross-links (pulley effect) which can vary the network topology under imposed deformation so that the configurational entropy can be maximized. The level-off behavior of μ∞ at high elongation is explained by the suppression of pulley effect caused by high stacking and/or localization of cross-linked cyclic molecules at the chain ends. The content ϕCD influences the variation in the λ dependence of μ∞ driven by a change in cx as well as the values of μ∞. These results indicate that (1) the cross-links in the PR gels move nonaffinely and (2) ϕCD influences how the cross-links slide along the network strands in nonaffine manner in response to imposed deformation.

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INTRODUCTION Polyrotaxane (PR) gels with the cross-links movable along network strands have received much attention as a new type of soft solid materials.1,2 The PRs have a necklace-like structure consisting of a single polymer chain with many cyclic molecules threaded onto it and two bulky moieties bound to each end.3−5 The PR gels are formed by linking the cyclic molecules in different PR chains.6 The resultant figure-of-eight cross-links are slidable along network strands in contrast to the cross-links fixed at specific sites by covalent bonds or physical aggregations in classical gels and elastomers. The PR gels with movable cross-links can vary the network topology in response to imposed deformation so that the configurational entropy could be maximumized. Such function of the movable cross-links is called “pulley effect”. The pulley effect has great potential for new functions and physical properties. Several peculiar features of the physical properties of the PR gels have been reported. The scattering studies revealed that the frozen structural disorder was less important than the thermal fluctuation in opposite to the cases of classical gels.7,8 The dynamic mechanical analysis showed a remarkable degree of viscoelastic relaxation at a characteristic frequency.9 The biaxial stretching experiments revealed that the effect of the strain in one direction on the stress in the other direction was considerably smaller in the PR gels than in the classical chemical gels.10 These features were explained by the variable network topology in response to external deformation via © 2012 American Chemical Society

pulley effect. We also reported an anomaly in the stretchingdriven swelling behavior of the PR gels.11 The strain-driven volume change is a general property of gels being a semiopen system where the solvent can flow in and out of gels.12−17 Gels that are fully swollen in solvents show further swelling when a constant tensile strain is imposed externally.14−17 The further swelling is induced by a force to increase the configurational entropy (i.e., reduce the anisotropy in configuration) of the deformed networks. The PR gels exhibited a pronounced strain dependence of the equilibrium (osmotic) Poisson’s ratio (μ∞) which is a measure of strain-induced swelling, while the classical gels showed the strain-independent μ∞.11 The stretch (α) dependence of μ∞ for the PR gels showed a crossover at α = αc: μ∞ increased with α in moderate stretching of α < αc whereas μ∞ leveled off at high stretching of α > αc. The α-dependent μ∞ under moderate stretching was attributed to the structural homogenization by the pulley effect which suppressed the subsequent stretching-induced swelling. The α-independent μ∞ under high stretching was explained by the suppression of the pulley effect caused by a considerable localization and/or stacking of many slide-rings to the chain ends (Figure 1c). The α-dependent μ∞ is a unique feature of the PR gels with the pulley effects. The PR gels have several structural Received: May 29, 2012 Revised: July 24, 2012 Published: August 10, 2012 6733

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with the same length as the strand in the state in the same solution. The aff ine deformation on the length scales of a network strand is assumed, and λ is the macroscopic linear deformation measured from the preparation state. The front factor ϕ/N corresponds to the number density of network strands with average degree of polymerization N. The scaling theory for semidilute solutions gives the ϕ dependence of R for free chains as19 R(ϕ) ≈ bN1/2ϕ(−ν + 1/2)/(3ν − 1)

(3)

For the equilibrium of free swelling from the preparation state (Figure 2b), equating πmix at ϕ = ϕ0 and πel gives Figure 1. Schematics of polyrotaxane gels with slidable cross-links (a) in the relaxed state, (b) under moderate stretching, and (c) under high stretching.

ϕ0

parameters (Figure 1), i.e., cross-link concentration, the filling rate of cyclic molecules in PRs, and chain length of PRs each of which may significantly influence the mobility of cross-linked cyclic molecules. The previous study11 employed the PR gels swollen in various solvents but with a unique set of these structural parameters. In addition, the PR chains employed had a wide length distribution,11 which obstructed the molecular interpretation of the phenomena. In this work, we investigate the effects of these structural parameters on the stretching driven swelling using the gels made of the PR chains with narrow length distributions. We also examine the reduction in tensile force caused by the induced swelling which was not discussed in previous work. The degree of force reduction is closely related to that of the induced swelling.14−17 The results in present work will contribute to full understanding of the function of movable cross-links. Theory of Equilibrium Poisson’s Ratio for Classical Gels. We theoretically consider here the equilibrium Poisson’s ratio (μ∞) for classical gels with fixed cross-links. The volume change before and after imposing a constant strain in the fully swollen state is considered on the basis of the Flory−Rehner’s concept,18 i.e, the swelling equilibrium is achieved by a balance between the osmotic swelling pressure (πmix) originating from the isotropic mixing of network and solvent, and the elastic resistive stress (πel) resulting from rubber elasticity. In previous papers,11,15−17 we addressed this issue on the basis of the classical model where the Flory type formula was employed for the elastic term of the free energy of a network (Fel) and the mean field theory was used for the mixing (osmotic) term (Fmix). We employ here the scaling approach for Fel and Fmix to obtain more general results considering the concentration fluctuation in the semidilute regime. According to the scaling theory, Fmix for semidilute solutions is given by19 Fmix π b3 ≈ mix ≈ ϕ3ν /(3ν− 1) kBT kBT

3ν /3ν− 1

⎛ R 00 ⎞2 = λ0 ⎜ ⎟ N ⎝ R0 ⎠ ϕ0

2

(4)

Figure 2. Double logarithmic plots of initial Young’s modulus (E0) versus degree of swelling (Q0) in freely swollen states. The slopes of the solid and dashed lines are 2.3.

where the subscript 0 denotes the state after free swelling, and λ0 is the linear isotropic expansion caused by free swelling. When a constant uniaxial deformation of α|| is imposed in the freely swollen state, the new equilibrium state (ϕ = ϕ) is achieved by a balance of πmix and the lateral elastic stress πel⊥ in the directions normal to the imposed strain as ϕ3ν /(3ν− 1) =

2 ϕ0 λ 0 2 ⎛ R 00 ⎞2 ϕ 2⎛ R 00 ⎞ ⎟ = ⎜ ⎟ λ⊥ ⎜ N ⎝ R ⎠ N α ⎝ R ⎠

(5)

where the subscript ⊥ represents the axis normal to the imposed strain, and the relations λ⊥ = λ0α⊥ and α||α⊥2 = ϕ0/ϕ are used (Figure 2d). From eqs 3−5, we obtain a simple relation between α|| and α⊥ as α⊥ = α −(1 − ν)/(1 + ν)

(6)

The exponent in eq 6 corresponds to the Poisson’s ratio μ∞ which is defined by the ratio of the true strains (ε) in the longitudinal and transverse directions:

(1)

μ∞ = −

where b is the monomer size, ϕ is the volume fraction of network, and ν is the excluded volume exponent depending on the quality of solvents. The elastic term Fel is expressed by the Panyukov form as20

ε⊥ ln α⊥ =− ε ln α

(7)

(2)

Equation 7 is a generalized definition of Poisson’s ratio which is valid in finite deformation as well as in the small deformation satisfying the linear elasticity. Accordingly, μ∞ is given by 1−ν μ∞ = (8) 1+ν

where R00 is the mean-square end-to-end distance of network strands in their preparation state, and R is that of a free chain

The quantity μ∞ is a measure of the degree of strain-induced swelling, because the equilibrium volumes of gels before and

2 Fel π b3 ϕ ⎛R ⎞ ≈ el ≈ λ 2⎜ 00 ⎟ kBT kBT N ⎝ R ⎠

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under stretching (designated as V0 and V, respectively) is given by ϕ V = 0 = α 1 − 2μ∞ V0 ϕ

Table 1. Characteristics of Gel Samples PEG sample

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94k-L-1 94k-L-4 94k-L-6 94k-L-13 94k-L-23 95k-H-0.6 95k-H-5 95k-H-28 120k-H-0.7 230k-H-0.4 230k-H-4 230k-H-19 115k*-21-3a PAAmG-0.5 PAAmG-27

As is evident from eqs 6 and 8, μ∞ depends only on ν, i.e., the solubility of constituent polymers in solvents: μ∞ = 1/4 for typical good solvents with ν = 3/5, and μ∞ = 1/3 for theta solvents with ν = 1/2. Importantly, μ∞ is independent of the degree of deformation α|| as well as the structural parameters of gels (such as the length of network strands N and preparation concentration ϕ00). Unexpectedly, the value of μ∞ (=1/4) for good solvents is identical with the result obtained by the classical theory where the excluded volume effect is not considered.11,16 This accordance will be a result of the cancellations of the absences of excluded volume effect in both Fel and Fmix in the classical theory. As in the case of μ∞, the ϕ0 dependence of the elastic modulus (E0) in the freely swollen state (Figure 2b) is universal, i.e., valid for any preparation concentration and any cross-link density21,22 E0 ≈

kBT b3

ϕ0 3ν /(3ν − 1) =

kBT b3

Q 0−3ν /(3ν − 1)

Mw/ Mn

ϕCD (%)

9.4 × 104

1.2

14

9.5 × 104

1.4

25

1.2 × 105

1.6

24

2.3 × 105

1.3

26

1.2 × 105

3.3

21

CDI (wt %)

E0 (kPa)

Q0

0.2 0.5 1.0 2.0 3.0 0.25 0.5 1.0 0.25 0.13 0.25 1.0

1.2 4.0 6.3 13 23 0.57 4.8 28 0.69 0.39 4.2 19 2.7 0.50 27

51 31 27 21 17 66 25 13 62 80 34 17 29 92 17

a

Data from ref 11 where the corresponding sample code is PRG/ DMSO.

some conditions. Our recent work28 using the contrast variation SANS technique showed that the CDs are not localized but are distributed randomly along the entire PEG chain when ϕCD (27%) and PR concentration (10%) are sufficiently small. The highest values of ϕCD and PR concentration in the swollen gels employed here are 26% and 7.7%, respectively, which indicates that the intermolecular and intramolecular aggregations of CDs are considerably suppressed in all specimens. The poly(acrylamide) (PAAm) chemical gels with different elastic modulus (designated as PAAmG-0.5 and PAAmG-27) were prepared by radical copolymerization of an acrylamide monomer and methylenebis(acrylamide) (cross-linker) using ammonium persulfate as an initiator of polymerization. For PAAmG-0.5 and PAAmG-27, the total reactant concentration in the aqueous solution was 5 and 10 wt %, and the molar ratio [monomer]/[cross-linker] was 600 and 200, respectively. Gelation was conducted at 5 °C for 24 h in a glass mold. The resultant rectangular gels were left to swell in water until the swelling attained equilibrium. Measurements. The rectangular gel specimens were elongated in each solvent at 25 °C using Tensilon-RTM500 equipped with a solvent bath. The typical size of the specimens was 20 × 5 × 1.5 mm. The swelling was equilibrated in the solvent bath prior to stretching. The samples were stretched at a cross-head speed of 100 mm/min up to the destination stretch ratio (α||). The transverse dimension of the samples under a constant stretch of α|| was measured as a function of time using a CCD camera. The tensile force was measured by a load cell with a capacity of 2 N. The imposed elongation was released after the swelling under elongation attained equilibrium. After the swelling was equilibrated in the relaxed state, the specimen was stretched to the next destination of α||. This cycle was repeated for different values of α|| ranging from 1.025 to 2.0. The reproducibility for the value of μ∞ for each specimen was confirmed using different samples but with same composition, and the experimental error for μ∞ was estimated to be ±0.01. The degree of equilibrium swelling in the relaxed state (Q0), defined by Q0 = wd/ws, was obtained from the weights in the dried and swollen states (wd and ws, respectively).

(10)

where Q0 is the degree of free swelling defined by the volume ratio, Q0 = ϕ0−1. It should be noted that, in contrast to eq 6, eq 10 does not rely on the assumption of affine deformation of network strands, and eq 10 only reflects the simple fact that the equilibrium swollen state is governed by the balance πmix = πel ≈ E0.21,22 As a result, the exponent in eq 10 is identical with that for πmix (eq 1).

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Mn

EXPERIMENTAL SECTION

Sample Preparation. Polyrotaxane (PR) composed of poly(ethylene glycol) (PEG) and α-cyclo-dextrin (CD) capped with 1adamantanamine was synthesized by using a method described elsewhere.23−25 The four PEG chains having relatively narrow molecular weight distributions (1.2 < Mw/ Mn < 1.6) with various lengths (9.4 × 104 < Mn < 2.3 × 105) were employed. The inclusion ratios (i.e., the CD filling rate to a full filling rate) for the PRs (ϕCD) were varied from 14% to 26%. The values of ϕCD were estimated by 1 H NMR analysis. The PR gels were prepared by the intermolecular cross-linking of the PRs using N,N-carbonyl diimidazole (CDI) as cross-linker. The PRs were dissolved with the cross-linker in dimethyl sulfoxide (DMSO) to achieve a PR concentration of 10 wt %. The cross-linker concentration was varied from 0.13 wt % to 3.0 wt %. The gelation was carried out at 50 °C for 3 days in a mold. The resultant gels were allowed to swell in DMSO until the equilibrium swelling was achieved. DMSO was renewed several times in order to wash out the unreacted materials. The characteristics of each PR gel are listed in Table 1. In the sample codes, “L” and “H” represent the gels made of the PRs with low ϕCD (14%) and high ϕCD (24, 25, 26%), respectively. The figures in the left and right sides of the sample codes denote the values of Mn of PEG and initial Young’s modulus (E0) in the fully swollen state. We employ E0 as a measure of the concentration of cross-linked CDs rather than the cross-linker (CDI) concentration in feed. This is because the cross-linking reaction might be influenced by MPEG and ϕCD to some extent, while the FT-IR analysis in our recent work26 confirmed that the degree of the reaction between CD and CDI was linearly proportional to the CDI concentration in feed in the same range of CDI concentration as in present study for the PR with MPEG = 1.1 × 105 and ϕCD = 25%. Several studies7,27−30 reported the presence of intermolecular and intramolecular aggregation of CDs in the PR chains in DMSO under

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RESULTS Figure 3 shows the double logarithmic plots of initial Young’s modulus (E0) in the freely swollen state versus equilibrium degree of swelling (Q0). The data for all PR gels fall on a straight line with a slope of −2.3, i.e., E0 ∼ Q0a with a = 2.3. As described in previous section (eq 10), a corresponds to the 6735

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Figure 3. Gels (a) in preparation state, (b) in freely swollen state, (c) just after imposing a constant stretch (isovolumetric deformation), (d) in fully reswollen state under constant stretch, and (e) just after releasing the imposed stretch in the state showin in part d. A reduction in tensile force ( f 0 > f∞) occurs as a result of a decrease in the effective strain acting on tensile force (denoted by the arrows) caused by the stretching-induced swelling.

scaling exponent for the concentration dependence of osmotic pressure. The value of a obtained is close to a theoretical value of a = 2.25 in typical good solvent systems (eq 10 with ν = 3/5) and the reported values for several gels in good solvents (a = 2.0−2.3).21,31−33 This result indicates that all PR gels examined correspond to the gels swollen in good solvents. The data for the PAAm hydrogels also provide a = 2.3, although they are slightly shifted in the vertical direction as compared to the data of the PR gels. This slight vertical shift is due to a small difference in the numerical front factor of the scaling relation depending on the chemical structures of the gels and solvents. Figure 4a shows the time (t) dependence of the transverse dimension (w) after the imposition of a stretch of α|| = 2 for 94k-L-23. In the figure, t = 0 is the point in time when the stretching reached the destination of α|| = 2. No appreciable strain-induced swelling occurred during elongation (t < 0), because the time required for the elongation of α|| = 2 (ca. 12 s) is much shorter than the characteristic time for strain-induced swelling (τ) which is described below. In fact, the Poisson’s ratio (0.490) at t = 0 estimated from eq 7 is close to 1/2 corresponding to no volume change during deformation. Under a constant stretch of α|| = 2, w slowly increases toward an equilibrium value in the long time limit. This type of straininduced swelling is observed at every value of α|| for each gel. The equilibrium Poisson’s ratio (μ∞) is obtained from eq 7 using the transverse true strain in the long time limit (ε⊥∞): μ∞ = −

ε⊥∞ ln α⊥∞ =− ε ln α

Figure 4. (a) Time (t) dependence of lateral dimension (w) and tensile force ( f) for 94k-L-23 after the imposition of a constant stretch of α|| = 2 at t = 0. (b) Semilogarithmic plots of [Δf(t)/Δf(0)] and [Δw(t)/Δw(0)] versus time. The characteristic time (τ) is estimated to be 8.8 × 103 s from the line slope.

intercept at t = 0) is theoretically expected, because the total swelling process obeys a multiexponential function.34 The diffusion constant (D) is estimated to be 1.53 × 10−10 m2/s using D ≈ d2/τ where d is the equilibrium thickness. This value of D is of the same order of the reported values for several gels.34,35 Figure 4 also includes the t dependence of tensile force ( f) simultaneously obtained with that of w. The tensile force gradually decreases and reaches the equilibrium value (Figure 4a). The total reduction in force is ca. 10% in this case. A finite scattering of the data is due to the small magnitude of the force. As is evident from Figure 4b where Δf(t) = f(t) − f(∞), the kinetics of the force reduction is similar to that of the induced swelling. This indicates that the force reduction is caused by the induced swelling. In fact, the PR gels exhibited no appreciable force reduction after the imposition of a step strain without surrounding solvent (i.e., in air),10 and thus the force reduction observed here is almost entirely attributed to the induced swelling. Figure 5 shows the α|| dependence of μ∞ for the PAAm hydrogels with fixed cross-links. The Poisson’s ratio (ca. 0.25) is independent of α|| as well as elastic modulus (i.e., cross-link concentration), in accordance with the theoretical expectation (eq 6). In contrast, μ∞ for the PR gels is α||-dependent, and the type of the dependence varies with cross-link concentration, which is shown in Figures 6−8. As can be seen in Figure 6, the gels

(11)

The value of μ∞ for the data in Figure 4a is 0.261, and the corresponding volume increase (eq 9) is ca. 40%. Figure 4b shows the semilogarithmic plot of Δw(t)/Δw(0) versus t for the data in Figure 4a. The quantity Δw(t) is defined as Δw(t) = w(∞) − w(t). The data points fall on a straight line. The process is well approximated by a single exponential function, and the characteristic time (τ) is evaluated to be 8.80 × 103 s from the inverse of the slope. A finite deviation from the single exponential function at the short times (i.e., nonzero 6736

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Figure 5. Equilibrium Poisson’s ratio (μ∞) as a function of imposed stretch (α||) for polyacrylamide hydrogels. The value (μ∞ ≈ 0.255) is independent of α|| as well as cross-link concentration.

Figure 7. Equilibrium Poisson’s ratio (μ∞) as a function of imposed stretch (α||) for 230k-H based PR gels with various moduli. The lines are guides for eyes.

The content of cyclic molecule in PR strands (ϕCD) also influences the α|| dependence of μ∞. Figure 8 shows μ∞ as a

Figure 6. Equilibrium Poisson’s ratio (μ∞) as a function of imposed stretch (α||) for 95k-H based PR gels with various moduli. The lines are guides for eyes.

composed of 95k-H exhibit the three types of α|| dependence of μ∞ depending on elastic modulus. For 95k-H-28 with the highest modulus among the three gels, μ∞ is independent of α|| as in the case of the classical gels. The lack of the data in the region of α|| > 1.6 is due to the poor extensibility. In the other two gels, μ∞ is dependent on α||, and the dependency exhibits a crossover at α|| ≈ 1.4. These two gels are different in the behavior in the region of α|| > 1.4: μ∞ is constant for 95k-H-0.6 with the lowest cross-link concentration whereas μ∞ gradually increases with α|| for 95k-H-5. The limited extensibility of 95kH-5 precluded the measurement in the region of α|| > 2. We designate the types of the dependency observed for 95k-H-0.6, 95k-H-5, 95k-H-28 as types A, A*, and B, respectively. These three types of the dependency are also observed for the PR gels made of 230k-H with various moduli, which is shown in Figure 7. From the data in Figures 6 and 7, the three types of the dependency is crudely classified on the basis of Young’s modulus (E0): The gel with E0 of the order of 102, 103, or 104 Pa, exhibits the behavior of types A, A*, and B, respectively. It should be emphasized that the α||-dependent μ∞ is not a general property of soft gels with low cross-link concentrations: The classical gel PAAmG-0.5 exhibits no dependence of μ∞ on α|| (Figure 5), although the modulus is comparable to those of 95k-0.6-H and 230k-0.4-H with α||-dependent μ∞.

Figure 8. Equilibrium Poisson’s ratio (μ∞) as a function of imposed stretch (α||) for 94k-L based PR gels with various moduli. The lines are guides for eyes.

function of α|| for the gels made of 94k-L with various moduli. For these gels, only types A and B were observed. The gel 94kL-4 exhibits the type A behavior with a definite plateau in the high α|| regime while similar in MPEG and E0 to 95k-H-5 showing the type A* behavior. For the 94k-L based gels, the transition between types A and B occurs in a narrow E0 range of 6 kPa < E0 < 13 kPa without showing type A*. Furthermore, μ∞ in type B is comparable to the plateau value of μ∞ in type A. This tendency for the gels made of 94k-L is different from those of 95k-H and 230k-H where μ∞ in type B is considerably smaller than the plateau value in type A (Figures 6 and 7). Figure 9 compares the type A behavior for the PR gels with various MPEG but with the similar ϕCD. The figure also includes the data of the PR gel 115k*-21-3 investigated earlier (designated as PRG/DMSO in ref 11) which was made of the PR with a wide length distribution. It appears that the 6737

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DISCUSSION

As is shown in Figures 6−8, E0 has a pronounced effect on the α|| dependence of μ∞ for the PR gels. For high cross-link concentrations, μ∞ is independent of α||, which is similar to the behavior of the classical gels (PAAm gels) with fixed cross-links (type B). In the case of sufficiently low cross-link concentrations, μ∞ becomes dependent on α||: At the moderate stretching of α|| < α||c, μ∞ steeply increases with α|| whereas at the high stretching of α|| > α||c, μ∞ reaches a plateau value (type A) or μ∞ gradually increases with α|| (type A*). The model assuming the affine displacement of cross-links predicts the α||independent μ∞ (eq 8), which well explains the behavior of the classical gels (Figure 5). The α||-dependent μ∞ of the PR gels under moderate stretching is attributed to the nonaffine displacement of cross-linked CDs via pulley effect. A major origin of stretching-driven swelling is an entropic force to decrease the configurational anisotropy of the networks caused by imposed stretching. The PR gels with highly mobile sliderings originally have a “pulley effect” to homogenize the network topology under imposed strain. Ideally, the pulley effect varies the network topology in response to applied deformation so that the configurational entropy can be maximized. As a result, the pulley effect suppresses the subsequent stretching-induced swelling, which results in an increase of μ∞ with increasing α||. If the slide-rings are immobile, μ∞ would remain the value in the small strain limit irrespectively of α||. At high elongation, many slide-rings including the free cyclic molecules are expected to be localized and/or stacked to the chain ends and lose the mobility. In this state, the pulley effect of the PR gels is considerably suppressed, and thus μ∞ becomes independent of α|| as in the case of classical gels with fixed cross-links. This scenario explains type A behavior. If the number of cross-linked slide-rings becomes too large, the mobility of the slide-rings is markedly suppressed due to the restriction of the slidable range along the strands. In such highly cross-linked PR gels, the pulley effect becomes much less active, which results in type B behavior. However, in contrast to the classical gels showing an E0-independent value of μ∞ ≈ 0.255, the values of μ∞ in type B behavior for the PR gels have a finite range of 0.22 < μ∞ < 0.29 which is beyond the scattering caused by the experimental error. The discussion about this point will be given later. The transition between types A and B behaviors by varying cross-link concentrations is observed independently of MPEG (Mn of PEG) as well as ϕCD (Figures 6−8). The type A* behavior is observed for the gels made of the PRs with relatively high ϕCD (95k-H and 230k-H), but it does not appear for the gels made of the PR with low ϕCD (94k-L) but with almost the same MPEG as 95k-H. This suggests that a crucial factor for the type A* behavior is ϕCD rather than MPEG. The type A* behavior appears at the moderate cross-link concentrations between the regimes for types A and B. In the case of the low ϕCD, the transition between types A and B directly occurs in a narrow E range of 6 kPa < E < 13 kPa without type A*. The high CD contents broaden the transition between types A and B, and type A* is regarded as an intermediate behavior between types A and B. If the free CDs between adjacent cross-linked CDs have broad number-distribution, the localization and/or stacking of CDs at the chain ends is expected to proceed gradually in a finite range of stretch (type A* behavior). The PR gels with high ϕCD may be larger in the number-distribution

Figure 9. Equilibrium Poisson’s ratio (μ∞) as a function of imposed stretch (α||) for the PR gels with various lengths of PR strands showing the type A behavior. The data of 115k*-21-3 was reproduced from ref 11.

stretch (α||c) where μ∞ starts to level off does not significantly depend on the chain length of PR. Figure 10 shows the magnitude of the total reduction of tensile force (Δf∞: Δf∞ = f 0 − f∞, where f 0 and f∞ are the

Figure 10. Degree of force reduction (Δf∞) as a function of imposed stretch (α||) for 94k-L-4, 115k*-21-3, 94k-L-23, and PAAmG-27. The force reduction is reduced by each initial force at t = 0 ( f 0). The solid and dashed lines are guides for eyes.

initial and equilibrium force, respectively) during the induced swelling. In the figure, Δf∞ is reduced by f 0. The figure includes the data of the PR gels 115k*-21-3 (investigated in previous study) and 94k-L-4 showing type A behavior, the PR gel 94k-L23 and the chemical gel PAAmG-27 showing type B behavior. The data of the gels with E0 of the order of 102 Pa are not displayed, because Δf∞ was too small to evaluate with sufficient accuracy. In the high α|| region of α|| > 1.4, all data collapsed into a single line, but the α|| dependence of Δf∞ in the low α|| region of α|| < 1.4 is classified into the two groups depending on the types of the α|| dependence of μ∞: The PR gels showing the type A behavior exhibit larger force reduction than the PR and PAAm gels showing the type B behavior. 6738

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SUMMARY The PR gels with sufficiently low cross-link concentrations exhibit a pronounced dependence of μ∞ on α||, while those with high cross-link concentrations show no α|| dependence of μ∞ as in the case of classical gels with fixed cross-links. This indicates that the high cross-link concentrations significantly suppress the slidability of the cross-linked CDs along the network strands. In the case of sufficiently low cross-link concentrations, μ∞ increases with α|| under the moderate stretching of α|| < α||c while it levels off in the high stretching of α|| > α||c. Correspondingly, the force reduction (Δf∞/f 0) caused by the induced swelling is considerably higher in the region of α|| < α||c than that in the regime of α|| > α||c. The constant μ∞ under high stretching is attributed to the localization and/or stacking of the CDs at the chain ends which makes the cross-linked CDs much less mobile. The scaling theory using the affine model well describes the results of the classical gels, i.e., μ∞ ≈ 0.25 independently of α||. The α||-dependent μ∞ for the PR gels is a result of nonaffine displacement of cross-linked CDs. The content of CDs in the PR strands has a finite effect on the values of μ∞: The values of μ∞ totally tend to become larger as ϕCD increases, when μ∞ depends on α||. For the PR gels showing the α||-independent μ∞, the values of μ∞ have a finite range mainly depending on ϕCD, and they tend to increase with a decrease in ϕCD as opposed to the case of α||-dependent μ∞. This indicates that the affine assumption will not be valid even for the PR gels with high cross-link concentrations although the slidability of the movable cross-links is considerably suppressed. The content ϕCD is expected to influence how the cross-linked CDs slide along the network strands in nonaffine manner in response to imposed deformation.

of free CDs between adjacent cross-linked CDs than those with low ϕCD. The values of μ∞ (≈ 0.255) for the PAAm gels are independent of α||, and they well agree with the value (0.26) estimated from the affine model (eq 8) with the ν value (0.59) which is obtained from the E0 dependence of Q0. Although all PR gels are similar in the value of ν to the PAAm gels (Figure 3), the values of μ∞ for the PR gels showing the type B behavior appreciably depend on ϕCD: μ∞ ≈ 0.22 for PR-H based gels and μ∞ ≈ 0.28 for PR-L based gels. This implies that the assumption of the affine displacement of cross-linked CDs will not be valid for the PR gels showing the type B behavior, although the pulley effect is considerably suppressed in these gels. It should be noted again that eq 8 relies on the affine assumption while this is not the case with eq 10. For the PR gels with type A behavior, each of the μ∞ values in the small strain limit and the plateau region also appears to be classified into the two groups depending on ϕCD: in the small strain limit, μ∞ ≈ 0.16 and μ∞ ≈ 0.21 for PR-L and PR-H based gels, respectively, while in the plateau region, μ∞ ≈ 0.25 and μ∞ ≈ 0.30 for PR-L and PR-H based gels, respectively. The values of μ∞ in the small and high elongation limits tend to become larger with an increase in ϕCD, while they are almost independent of MPEG and E0. As a consequence, the magnitude correlation of μ∞ in types A and B behaviors appears different between the gels made of 94k-H and 94k-L: μ∞ in type B behavior is larger than those at high elongation in type A behavior for 94k-L based gels, but the tendency is opposite for 95k-H based gels. Thus, ϕCD has a finite and complicated effect on the values of μ∞ for both types A and B behaviors, and ϕCD is expected to influence the nonaffine displacement of the crosslinked CDs in response to imposed deformation. The stretch α||c (≈1.4) is almost independent of both ϕCD and MPEG, in contrast to the simple expectation that each molecular parameter has a finite effect on α||c. Intuitively, a decrease in ϕCD or an increase in MPEG, each of which increases the slidable contour-length of cross-linked CDs, is expected to increase α||c. The reason for why none of ϕCD and MPEG has appreciable effect on α||c is not clear to us at present. The ranges of these parameters examined here may be too small to detect the influence on α||c. The stretching-driven swelling simultaneously causes a finite reduction in tensile force (Figures 4 and 10). This is because the volume increase reduces the effective strain for the tensile force (Figure 2).14−17 The PR gels with α||-dependent μ∞ (94kL-4 and 115k*-21−3) show a considerably larger degree of force reduction in the regime of α|| < 1.4 (i.e., α|| < α||c) than those with α||-independent μ∞, while no difference in stress reduction is observed in the region of α|| > α||c where μ∞ is constant. As is evident from the physical origin of the force reduction, Δf∞ increases as the degree of induced swelling increases (i.e., as μ∞ decreases). Thus, the large force reduction in the region of α|| < α||c is consistent with the α|| dependence of μ∞ observed in the corresponding regime. The modeling of the effect of slidable cross-links on μ∞ remains to be a challenging issue. In particular, we require an alternative form of eq 2 (affine model), i.e., Fel considering the nonaffine displacement of slidable cross-links which is considerably influenced by ϕCD. We need to characterize experimentally in detail the nonlinear elasticity of networks with movable cross-links and to deduce the form of Fel on the basis of the experimental data. This will be a subject of our future work.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (K.U.); [email protected] (K.I.). Author Contributions §

These authors contributed equally to this work.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was partly supported by a Grant-in-Aid for Scientific Research (S) (No. 20221005) from Japan Society of Promotion of Science (JSPS) and a Grant-in-Aid on Priority Area “Soft Matter Physics” (No. 21015014) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan.

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