## Abstract

Elasticity, one of the most important properties of a soft material, is difficult to quantify in polymer networks because of the presence of topological molecular defects in these materials. Furthermore, the impact of these defects on bulk elasticity is unknown. We used rheology, disassembly spectrometry, and simulations to measure the shear elastic modulus and count the numbers of topological “loop” defects of various order in a series of polymer hydrogels, and then used these data to evaluate the classical phantom and affine network theories of elasticity. The results led to a real elastic network theory (RENT) that describes how loop defects affect bulk elasticity. Given knowledge of the loop fractions, RENT provides predictions of the shear elastic modulus that are consistent with experimental observations.

Molecular defects fundamentally govern the properties of all real materials (*1*–*3*). The language of crystallography has been successfully used to describe defects and to model their impact in materials with a degree of periodicity, such as silicon, steel, block copolymers, and liquid crystals. However, understanding defects in amorphous materials presents a continued challenge. In polymer networks, the relevant defects are largely of a topological nature: The properties of these amorphous materials depend primarily upon the way the molecules in the material are connected. Understanding the correlation between the network topology and properties is one of the greatest outstanding challenges in soft materials.

Polymer networks can have a wide range of shear elastic moduli (*G*′) from ~10^{2} to ~10^{7} Pa (*4*, *5*), with different applications requiring moduli across this entire range. Covalent polymer networks are generally formed via kinetically controlled processes; consequently, they possess cyclic topological defects. The classical affine and phantom network theories of network elasticity neglect the presence of such defects (*4*, *5*); they rely on idealized end-linked networks (Fig. 1A) that consider only acyclic tree-like structures, which leads to overestimation of *G*′* *(*6*,* **7*). In practice,* G*′ is frequently calculated according to the equation *G*′ = *C*ν_{eff}*kT*, where *kT* is the thermal energy, ν_{eff} is the density of elastically effective chains, and *C* is a constant that has a value of 1 for the affine network model and 1 − 2*/f* for the phantom network model (where *f* is the functionality of the network junctions). Because polymer networks include elastically defective chains, ν_{eff} is never known precisely, and thus neither theory is able to accurately fit experimental data; a controversy continues over which theory, if either, is correct. Thus, despite decades of advances in polymer network design, our inability to quantitatively calculate the effects of defects on shear elastic modulus and to measure the corresponding defect densities in real polymer networks precludes quantitative prediction of *G*′ and validation of the affine and phantom network models (*4*, *8*–*12*).

To understand how molecular structure affects *G*′ and to use this knowledge to create a predictive theory of elasticity, it is first necessary to quantify the density of topological defects in a polymer network and to determine the impact of these defects on the mechanical properties of the network. Cyclic defects, created from intrajunction reactions during network formation, are chemically and spectroscopically almost identical to noncyclic junctions, making them difficult to distinguish and quantify (*5*, *13*–*16*). We have developed symmetric isotopic labeling disassembly spectrometry (SILDaS) as a strategy to precisely count the number of primary loops (Fig. 1B), the simplest topological defects, in polymer networks formed from A_{2} + B_{3} and A_{2} + B_{4} reactions (*17*–*20*). Furthermore, we have developed Monte Carlo simulations and kinetic rate theories that show that cyclic defects in these polymer networks are kinetically linked, such that experimental measurement of only the primary loops determines the densities of all higher-order defects including secondary (Fig. 1C) and ternary loops (Fig. 1D) (*21*). Here, we measured loop fractions and *G*′ for a series of hydrogels, thus providing quantitative relationships between these parameters. With this information, we examined the classical affine and phantom network theories of elasticity, and we derived a modified phantom network theory—real elastic network theory (RENT)—that accounts for topological molecular defects.

To rigorously determine how molecular topological defects affect elasticity, it is necessary to measure the topological defect density and modulus in the same gel. A class of stable yet chemically degradable gels was developed from bis-azido-terminated polyethylene glycol (PEG) (number-average molecular weight *M*_{n} = 4600, dispersity index *Đ* = 1.02) polymers with nonlabeled or isotopically labeled segments near their chain ends, A_{2H} and A_{2D}, respectively (*22*) (structures are shown in Fig. 1A; for synthesis and characterization details, see figs. S1 to S3 and figs. S17 to S34). Such labeling provides a convenient method for precise measurement of primary loops by SILDaS (*19*). The PEG molecular weight ensures that the polymer solutions used to form gels are well below the entanglement regime (*5*, *12*). The labeled (A_{2D}) and nonlabeled (A_{2H}) polymers (referred to herein as “A_{2} monomers”) were mixed in a 1:1 molar ratio, and this mixture was allowed to react with a tris-alkyne (B_{3}) or a tetra-alkyne (B_{4}) (structures are shown in Fig. 1A) in propylene carbonate solvent to provide end-linked gels via copper-catalyzed azide-alkyne cycloaddition (*23*, *24*). When the reactive group stoichiometry—azide and alkyne in this case—was carefully controlled to be 1:1, spectroscopic analysis demonstrated that dangling functionalities (unreacted azides or alkynes) could be minimized (*19*) such that their impact on elasticity is negligible. Gels with varied fractions of topological defects were synthesized by varying the initial concentrations of A_{2} and B_{3} or B_{4} monomers (*22*).

For measurement of the shear elastic modulus as a function of gel preparation conditions, gel samples 1.59 mm thick were formed in situ in Teflon molds under an inert atmosphere (fig. S4). Gel disks (diameter 12 mm) were punched (Fig. 1A) and loaded onto an oscillatory shear rheometer equipped with parallel-plate geometry. Propylene carbonate was chosen as the solvent because of its high boiling point of 242°C, which mitigates solvent evaporation during *G*′ measurement. Representative rheology data for A_{2} + B_{3} (*f* = 3) and A_{2} + B_{4} (*f* = 4) networks are provided in Fig. 2, A and D, respectively (additional results are presented in figs. S5 and S6). Highly reproducible measurements of *G*′ were achieved for several samples of each network. As expected, *G*′ for both the trifunctional gels (Fig. 2B) and the tetrafunctional gels (Fig. 2E) increased with concentration; increased concentration leads to fewer topological defects (*17*–*19*).

After measurement of *G*′, we used SILDaS to measure the average number of primary loops per junction (*n*_{1,}* _{f}*) in each gel. Basic hydrolysis cleaved the ester bonds near the chain ends of the polymer and converted the gel to soluble degradation products (

*22*) (figs. S7 and S8). These particular products have isotopic labeling patterns with mass/charge ratios (

*m/z*) differing by +8 that depend exactly on the number or primary loops in the parent gel. The relative abundance of each labeled degradation product was quantified by mass spectrometry; the ratios of the abundances were used to obtain

*n*

_{1,}

*[see (*

_{f}*22*) for details of the loop analysis procedure for both the A

_{2}+ B

_{3}and the A

_{2}+ B

_{4}gels]. As shown in Fig. 2, B and E,

*n*

_{1,}

*decreases with increased concentration for both*

_{f}*f*= 3 and

*f*= 4 networks because of a reduced probability of intrajunction reaction at higher network concentration.

With this simultaneous knowledge of *G*′ and *n*_{1,}* _{f}*, the elasticity of the gels could be compared to theoretical models. An intuitive approach to obtain ν

_{eff}is to simply subtract the primary loop density from the total number density of polymer strands in the gel, ν

_{0}[see (

*22*) for detailed calculation], which is a first-order affine correction. Upon applying this correction, the data approach the phantom network prediction as

*n*

_{1,}

*decreases (Fig. 2, C and F); however, they are still clearly below the predictions of both phantom and affine network theory.*

_{f}The results shown in Fig. 2, C and F, suggest that the simple affine correction for primary loops is not sufficient to reach agreement between measured *G*′ and classical elasticity theories. Higher-order loop defects (Fig. 1, C and D) should also be incorporated; because these defects are fractionally elastically effective, simply subtracting their density would not be appropriate. Therefore, to quantitatively understand the effect of each type of topological defect on network elasticity, we developed a real elastic network theory (RENT) [see (*22*) for detailed derivation] that captures the effect of cyclic defects in the phantom network. For a loop-free system (fig. S9), the phantom network strands are ideal chains with *N* monomers; the ends of the strands are joined at junctions that are allowed to fluctuate around their average position. Junctions are connected to the nonfluctuating boundary of the network through repetitive branch connections, equivalent to a single effective phantom chain (Fig. 3A). However, in the real network, loops disturb the ideal repetitive tree structure: A loop junction is linked to other junctions, which reduces its effective connection to the boundary (Fig. 3A and figs. S10 to S13). Thus, loop junctions are less constrained than ideal nonloop junctions, which results in longer effective phantom lengths of the loop strands (Fig. 3A, right) or, equivalently, fewer elastically effective strands. The effective length *m* of strands adjacent to loops also increases because of their connection to the boundary paths through loop strands (Fig. 3B). To quantify these effects, RENT assumes that different cyclic defects are independent of each other; the gel can be envisioned as an “ideal loop gas,” which enables an isolated treatment of each cyclic defect [the implications of this assumption are discussed below (*21*)]. The network is therefore a mixture of phantom strands with different effective length in parallel (Fig. 3A). The impact of each strand on the elastic modulus is linearly additive, yielding
(1)where *N* is the actual degree of polymerization of each strand, and ν* _{i}* is the actual density of the type

*i*strand in the network with effective phantom length

*N*

_{i}_{,ph}. The elastic effectiveness ε

*=*

_{i}*N*

_{i}_{,ph}/

*N*

_{ph}is defined as the ratio of

*N*

_{i}_{,ph}and the ideal phantom strand

*N*

_{ph}=

*fN/*(

*f*− 2). Polymer strands are divided into the ideal strands (with density ν

_{id}and elastic effectiveness ε

_{id}= 1) and nonideal strands (with density and elastic effectiveness , including both the loop strands and the strands adjacent to loops). The indices

*l*and

*m*denote the loop order and the topological distance of the strand from the loop (see Fig. 3B;

*m*= 0 denotes the loop strand), respectively.

To predict *G*′, the elastic effectiveness of nonideal strands in different loops and of different topological distances from the loops must be calculated. For primary and secondary loops (*l ≤* 2), can be calculated exactly (figs. S10 to S14), which results in (2) (3)and (4) (5)The key to obtaining Eqs. 4 and 5 is to develop and solve the recursive relation that describes the propagation of a topological defect in the phantom network (*22*).

For larger loops (*l *≥ 3), it is difficult to obtain an exact solution of because of the complex correlation between loop junctions; an approximate treatment is provided. We consider the general case, a strand (red strand in Fig. 3B) in an *l*th-order loop in the *f*-functional polymer network. The end junctions α of this loop strand are directly linked to two types of junctions. The first type is the ideal junction β, which is connected to the network boundary through repetitive branches. The second type consists of two loop junctions γ; these two junctions, as well as the loop junctions between them (*l* − 2 junctions in total), are simultaneously connected to the boundary through repetitive branches. The length of the equivalent phantom chain connected to junctions γ can be derived from the convolutional features of the Gaussian chain. Neglecting long-range correlations between junctions α through these *l* − 2 loop junctions, the elastic effectiveness of the targeted polymer strand (red strand in Fig. 3B) can be obtained by combining the contributions from junctions β and γ, which yields (6)The elastic effectiveness of the strand adjacent to the larger loops is similarly obtained as (7)Equations 4 to 7 show that the effect of loops in reducing the elastic effectiveness of strands decreases rapidly with increasing loop order or distance from the loop (Fig. 3, C and D). The elastic effectiveness of small loops is much lower than 1, which implies that these strands will have a non-negligible negative impact on the gel modulus. In contrast, ε(*l* ≥ 4, *m* = 0, *f*) ≈ 1, which indicates that strands in larger loops (*l* ≥ 4) behave nearly the same as ideal phantom chains (Fig. 3C). Furthermore, ε increases monotonically with *f*; the loop effect is more pronounced in networks with small junction functionality. Finally, the loop effect persists within only a few nearest-neighbor strands, as shown in Fig. 3D; strands far away from the loop (*m* ≥ 4) behave as ideal phantom chains.

Substituting Eqs. 4 to 7 into Eq. 1 and rewriting the density of different nonideal strands in terms of the average number of loops per junction, *n _{l}*

_{,}

*, an analytical expression can be produced for the shear modulus of a polymer network: (8)with (9) (10)and (11)where ν*

_{f}_{0}is the total number density of polymer strands in the gel (loop or nonloop).

From Eqs. 8 to 11, to fully describe the effects of topological defects on the modulus, it is necessary to know the density of loops of order *l* in the given network. As described above, SILDaS can precisely measure the primary loop fraction, *n*_{1,}* _{f}*. We have previously shown that that there is a one-to-one correspondence between the density of higher-order loops and the primary loop density (

*21*). Therefore,

*n*

_{2,}

*and*

_{f}*n*

_{3,}

*can be calculated once*

_{f}*n*

_{1,}

*is measured [see (*

_{f}*22*) for Monte Carlo simulation algorithm] as shown in Fig. 3E.

With knowledge of *G*′ and the densities of primary, secondary, and ternary loops, we applied RENT to calculate the effect of topological defects on *G*′; excellent agreement between the experimental *G*′/ν_{0}*kT* (Fig. 4, black circles) and RENT theory (Fig. 4, black lines) for low *n*_{1,}* _{f}* values (

*n*

_{1,3}< 0.20 and

*n*

_{1,4}< 0.28) was observed. Although primary loops play the dominant role in reducing

*G*′/ν

_{0}

*kT*because of their zero elastic effectiveness (see Fig. 3C), Fig. 4 shows that consideration of primary loops alone in RENT (blue lines, Fig. 4) does not give complete agreement with experiment; higher-order loops should be included to give the best agreement. RENT also shows good agreement with simulation and experimental data from other research groups (figs. S15 and S16) (

*25*–

*27*).

Although RENT and experimental data show excellent agreement at low *n*_{1,}* _{f}* values, there is a deviation between experiment and theory as

*n*

_{1,}

*becomes large. This deviation is due to our assumption that loops can be treated as independent within an “ideal loop gas.” As the fraction of loops in the networks becomes large, loops will become close to each other. In this case, the rules for elastic effectiveness as a function of*

_{f}*m*and

*l*(as given in Fig. 3) break down; cooperative effects between loops that are close in space may have a greater negative impact on elasticity than a linear combination of isolated loops in an “ideal loop gas.” Notably, the deviation between RENT and experiment for both A

_{2}+ B

_{3}and A

_{2}+ B

_{4}networks occurs at an A

_{2}concentration of 12 to 13 mM, which is very close to the estimated polymer overlap concentration

*c** for our PEG macromers (

*c** ≈ 13 mM) (

*12*). We note that in most applications of gels, synthetic conditions are used (e.g., concentration >

*c**; semidilute regime) that would avoid large numbers of loops; RENT as derived here should apply, given also that contributions from entanglements are minimal. Nonetheless, this work provides an important advance in quantitative network theory and a foundation for further development of theories that account for interacting network defects and entanglements.

Combining RENT with primary and higher-order loop measurements provides quantitative agreement with measured elastic moduli for networks with compositions and structures that are relevant to common applications. In other words, RENT can predict the bulk mechanical properties of polymer networks on the basis of molecular information. We anticipate that RENT and loop-counting methods will be applicable to a wide range of polymer networks.

## SUPPLEMENTARY MATERIALS

www.sciencemag.org/content/353/6305/1264/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S32

## REFERENCES AND NOTES

**Acknowledgments:**We thank C. N. Lam and S. Tang for help with Teflon mold fabrication. Supported by NSF grant CHE-1334703 (J.A.J. and B.D.O.), the Institute for Soldier Nanotechnologies via U.S. Army Research Office contract W911NF-07-D-0004 (B.D.O.), and NSF Materials Research Science and Engineering Centers award DMR-14190807. All data are available in the supplementary materials. Author contributions: M.Z., R.W., K.K., B.D.O., and J.A.J. designed the research; M.Z. and K.K. performed all experimental work; R.W. developed the theory; and M.Z., R.W., K.K., B.D.O., and J.A.J. wrote the paper.