Award for Scientists and Engineers and Multidisciplinary UniversityResearch Initiative (MURI) on Exotic Quantum Phases, an ArmyResearch Office MURI on Atomtronics, and the David and LucilePackard Foundation. M.A.N. was supported by the U.S. Departmentof Defense through the National Defense Science andEngineering Graduate Fellowship Program. K.R.L. was supportedby the Fannie and John Hertz Foundation and the NSF GraduateResearch Fellowship Program. N.T. acknowledges funding from

NSF Division of Materials Research grant 1309461 and partialsupport by a grant from the Simons Foundation (343227) andthanks S. Todadri for hospitality at MIT during her sabbatical.T.P. acknowledges support from Brazilian National Council forScientific and Technological Development, Fundação de Amparo àPesquisa do Estado do Rio de Janeiro, and Instituto Nacional deCiência e Tecnologia on Quantum Information. M.R. was supportedby the U.S. Office of Naval Research.

SUPPLEMENTARY MATERIALS

www.sciencemag.org/content/353/6305/1260/suppl/DC1Supplementary TextFigs. S1 to S6References (41–47)

10 June 2016; accepted 18 August 201610.1126/science.aag3349

POLYMER SCIENCE

Quantifying the impact of moleculardefects on polymer network elasticityMingjiang Zhong,1,2* Rui Wang,2* Ken Kawamoto,1*Bradley D. Olsen,2† Jeremiah A. Johnson1†

Elasticity, one of the most important properties of a soft material, is difficult to quantify inpolymer networks because of the presence of topological molecular defects in thesematerials. Furthermore, the impact of these defects on bulk elasticity is unknown.We usedrheology, disassembly spectrometry, and simulations to measure the shear elasticmodulus and count the numbers of topological “loop” defects of various order in aseries of polymer hydrogels, and then used these data to evaluate the classical phantomand 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 loopfractions, RENTprovides predictions of the shear elastic modulus that are consistent withexperimental observations.

Molecular defects fundamentally governthe properties of all real materials (1–3).The language of crystallography has beensuccessfully used to describe defects andto model their impact in materials with

a degree of periodicity, such as silicon, steel, blockcopolymers, and liquid crystals. However, under-standing defects in amorphous materials presentsa continued challenge. In polymer networks, therelevant defects are largely of a topological nature:The properties of these amorphous materialsdepend primarily upon the way the moleculesin the material are connected. Understandingthe correlation between the network topologyand properties is one of the greatest outstandingchallenges in soft materials.Polymer networks can have a wide range of

shear elastic moduli (G′) from ~102 to ~107 Pa(4, 5), with different applications requiring moduliacross this entire range. Covalent polymer networksare generally formed via kinetically controlledprocesses; consequently, they possess cyclic topo-logical defects. The classical affine and phantomnetwork theories of network elasticity neglectthe presence of such defects (4, 5); they rely onidealized end-linked networks (Fig. 1A) that con-sider only acyclic tree-like structures, which leadsto overestimation of G′ (6, 7). In practice, G′ isfrequently calculated according to the equation

G′ = CneffkT, where kT is the thermal energy,neff is the density of elastically effective chains,and C is a constant that has a value of 1 for theaffine network model and 1 − 2/f for the phantomnetwork model (where f is the functionality of thenetwork junctions). Because polymer networksinclude elastically defective chains, neff is neverknown precisely, and thus neither theory is ableto accurately fit experimental data; a contro-versy continues over which theory, if either, iscorrect. Thus, despite decades of advances inpolymer network design, our inability to quan-titatively calculate the effects of defects on shearelastic modulus and to measure the correspond-ing defect densities in real polymer networksprecludes quantitative prediction of G′ and val-idation of the affine and phantom network mod-els (4, 8–12).To understand howmolecular structure affects

G′ and to use this knowledge to create a predictivetheory of elasticity, it is first necessary to quantifythe density of topological defects in a polymernetwork and to determine the impact of thesedefects on the mechanical properties of the net-work. Cyclic defects, created from intrajunctionreactions during network formation, are chem-ically and spectroscopically almost identical tononcyclic junctions, making them difficult to dis-tinguish and quantify (5, 13–16). We have devel-oped symmetric isotopic labeling disassemblyspectrometry (SILDaS) as a strategy to preciselycount the number of primary loops (Fig. 1B), thesimplest topological defects, in polymer networksformed fromA2+B3 andA2 +B4 reactions (17–20).Furthermore, we have developed Monte Carlo

simulations and kinetic rate theories that showthat cyclic defects in these polymer networksare kinetically linked, such that experimentalmeasurement of only the primary loops deter-mines the densities of all higher-order defectsincluding secondary (Fig. 1C) and ternary loops(Fig. 1D) (21). Here, we measured loop fractionsand G′ for a series of hydrogels, thus providingquantitative relationships between these param-eters. With this information, we examined theclassical affine and phantom network theoriesof elasticity, and we derived a modified phan-tom network theory—real elastic network theory(RENT)—that accounts for topological molec-ular defects.To rigorously determine how molecular topo-

logical defects affect elasticity, it is necessaryto measure the topological defect density andmodulus in the same gel. A class of stable yetchemically degradable gels was developed frombis-azido-terminated polyethylene glycol (PEG)(number-average molecular weight Mn = 4600,dispersity index Đ = 1.02) polymers with non-labeled or isotopically labeled segments neartheir chain ends, A2H and A2D, respectively (22)(structures are shown in Fig. 1A; for synthesisand characterization details, see figs. S1 to S3and figs. S17 to S34). Such labeling provides aconvenient method for precise measurement ofprimary loops by SILDaS (19). The PEG molec-ular weight ensures that the polymer solutionsused to form gels are well below the entangle-ment regime (5, 12). The labeled (A2D) and non-labeled (A2H) polymers (referred to herein as“A2 monomers”) were mixed in a 1:1 molar ratio,and this mixture was allowed to react with atris-alkyne (B3) or a tetra-alkyne (B4) (structuresare shown in Fig. 1A) in propylene carbonatesolvent to provide end-linked gels via copper-catalyzed azide-alkyne cycloaddition (23, 24).When the reactive group stoichiometry—azideand alkyne in this case—was carefully controlledto be 1:1, spectroscopic analysis demonstratedthat dangling functionalities (unreacted azidesor alkynes) could be minimized (19) such thattheir impact on elasticity is negligible. Gels withvaried fractions of topological defects were syn-thesized by varying the initial concentrations ofA2 and B3 or B4 monomers (22).For measurement of the shear elastic modulus

as a function of gel preparation conditions, gelsamples 1.59 mm thick were formed in situ inTeflon molds under an inert atmosphere (fig.S4). Gel disks (diameter 12 mm) were punched(Fig. 1A) and loaded onto an oscillatory shearrheometer equippedwith parallel-plate geometry.Propylene carbonate was chosen as the solvent

1264 16 SEPTEMBER 2016 • VOL 353 ISSUE 6305 sciencemag.org SCIENCE

1Department of Chemistry, Massachusetts Institute ofTechnology, Cambridge, MA 02139, USA. 2Department ofChemical Engineering, Massachusetts Institute ofTechnology, Cambridge, MA 02139, USA.*These authors contributed equally to this work. †Correspondingauthor. Email: bdolsen@mit.edu (B.D.O.); jaj2109@mit.edu (J.A.J.)

RESEARCH | REPORTSon June 13, 2020

http://science.sciencemag.org/

Dow

nloaded from

because of its high boiling point of 242°C, whichmitigates solvent evaporation during G′measure-ment. Representative rheology data for A2 + B3

( f = 3) and A2 + B4 ( f = 4) networks are provided

in Fig. 2, A and D, respectively (additional resultsare presented in figs. S5 and S6). Highly repro-ducible measurements of G′ were achieved forseveral samples of each network. As expected, G′

for both the trifunctional gels (Fig. 2B) and thetetrafunctional gels (Fig. 2E) increased with con-centration; increased concentration leads to fewertopological defects (17–19).

SCIENCE sciencemag.org 16 SEPTEMBER 2016 • VOL 353 ISSUE 6305 1265

0

Fig. 1. Cyclic defects in end-linked polymer networks. (A) Synthetic schemes for the preparation of A2 + B3 and A2 + B4 gels labeled for loop countinganalysis. (B to D) Topological structure of primary loops (the average number of primary loops per f-functional network junction is n1,f) (B), secondary loops(the average number of secondary loops per f-functional junction is n2,f) (C), and ternary loops (the average number of ternary loops per f-functional junction isn3,f) (D) formed in A2 + B3 (f = 3; top) and A2 + B4 (f = 4; bottom) gels, respectively.

Fig. 2. Shear rheology and loop measurements in PEG gels. (A to C) Tri-functional (A2 + B3) gels: (A) Representative rheological curves demonstratingreproducibility of this gelation system (A2 concentration for all samples, 14mM).(B) G′ and the average number of primary loops per junction in these f = 3networks (n1,3) versus A2 concentration. Error bars of G′ represent SD of threemeasurements; error bars of n1,3 represent SD of ninemeasurements. (C) Exper-imentally estimated coefficient C with and without first-order affine cor-

rection. (D to F) Tetrafunctional (A2 + B4) gels: (D) Representative rheologicalcurves demonstrating reproducibility of this gelation system (A2 concentrationfor all samples, 14 mM). (E) G′ and the average number of primary loops perjunction in these f = 4 networks (n1,4) versus A2 concentration. Error bars of G′represent SD of three measurements; error bars of n1,4 represent SD of ninemeasurements. (F) Experimentally estimated coefficient C with and withoutfirst-order affine correction.

RESEARCH | REPORTSon June 13, 2020

http://science.sciencemag.org/

Dow

nloaded from

After measurement of G′, we used SILDaS tomeasure the average number of primary loopsper junction (n1,f) in each gel. Basic hydrolysiscleaved the ester bonds near the chain ends ofthe polymer and converted the gel to soluble deg-radation products (22) (figs. S7 and S8). Theseparticular products have isotopic labeling pat-terns with mass/charge ratios (m/z) differingby +8 that depend exactly on the number orprimary loops in the parent gel. The relativeabundance of each labeled degradation pro-duct was quantified by mass spectrometry; theratios of the abundances were used to obtain n1, f[see (22) for details of the loop analysis pro-cedure for both the A2 + B3 and the A2 + B4

gels]. As shown in Fig. 2, B and E, n1, f decreaseswith increased concentration for both f = 3 andf = 4 networks because of a reduced probabilityof intrajunction reaction at higher networkconcentration.With this simultaneous knowledge of G′ and

n1, f , the elasticity of the gels could be comparedto theoretical models. An intuitive approach toobtain neff is to simply subtract the primary loopdensity from the total number density of polymerstrands in the gel, n0 [see (22) for detailed cal-culation], which is a first-order affine correction.Upon applying this correction, the data approachthe phantom network prediction as n1, f decreases(Fig. 2, C and F); however, they are still clearlybelow the predictions of both phantom and af-fine network theory.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 mea-sured G′ and classical elasticity theories. Higher-order loop defects (Fig. 1, C and D) should alsobe incorporated; because these defects are frac-tionally elastically effective, simply subtractingtheir density would not be appropriate. There-fore, to quantitatively understand the effect ofeach type of topological defect on network elas-ticity, we developed a real elastic network theory(RENT) [see (22) for detailed derivation] thatcaptures the effect of cyclic defects in the phan-tom network. For a loop-free system (fig. S9),the phantom network strands are ideal chainswith N monomers; the ends of the strands arejoined at junctions that are allowed to fluctuatearound their average position. Junctions areconnected to the nonfluctuating boundary ofthe network through repetitive branch con-nections, equivalent to a single effective phan-tom 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 theboundary (Fig. 3A and figs. S10 to S13). Thus,loop junctions are less constrained than idealnonloop junctions, which results in longer ef-fective phantom lengths of the loop strands(Fig. 3A, right) or, equivalently, fewer elasticallyeffective strands. The effective lengthm of strandsadjacent to loops also increases because of theirconnection to the boundary paths through loopstrands (Fig. 3B). To quantify these effects, RENTassumes that different cyclic defects are inde-pendent of each other; the gel can be envisioned

as an “ideal loop gas,” which enables an iso-lated treatment of each cyclic defect [the im-plications of this assumption are discussedbelow (21)]. The network is therefore a mix-ture of phantom strands with different effec-tive length in parallel (Fig. 3A). The impact ofeach strand on the elastic modulus is linearlyadditive, yielding

G0

kT¼Xi

N

Ni;phni ¼ f − 2

f

Xi

eini

¼ f −2f

nid þXl¼1

1 Xm¼0

1

nml; f eml; f

!ð1Þ

where N is the actual degree of polymerizationof each strand, and ni is the actual density of thetype i strand in the network with effective phan-tom length Ni,ph. The elastic effectiveness ei =Ni,ph/Nph is defined as the ratio of Ni,ph andthe ideal phantom strand Nph = fN/( f − 2).Polymer strands are divided into the ideal strands(with density nid and elastic effectiveness eid = 1)and nonideal strands (with density nml; f andelastic effectiveness eml; f , including both the loopstrands and the strands adjacent to loops). Theindices l and m denote the loop order and thetopological 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 dif-ferent topological distances from the loops must

1266 16 SEPTEMBER 2016 • VOL 353 ISSUE 6305 sciencemag.org SCIENCE

Fig. 3. Real elastic network theory (RENT). (A) Schematic depiction of the conversion of independent loops to a mixture of phantom strands withdifferent effective length. (B) Schematic that illustrates the calculation of effective phantom length of strands in lth-order loops at a distance m from the loop.(C) Elastic effectiveness of polymer strands as a function of topological distancem from a primary (l = 1) and secondary (l = 2) loops in f = 3 or f = 4 networks.(D) Elastic effectiveness of polymer strands in loops of different order l for f = 3 and f = 4 networks. (E) Fraction of secondary and ternary loops (n2,f and n3,f) asa function of the number of primary loops per junction (n1,f) in f = 3 and f = 4 networks.

RESEARCH | REPORTSon June 13, 2020

http://science.sciencemag.org/

Dow

nloaded from

be calculated. For primary and secondary loops(l ≤ 2), eml; f can be calculated exactly (figs. S10 toS14), which results in

em¼0l¼1; f ¼ 0 ð2Þ

em¼0l¼2; f ¼ ðf − 2Þ=f ð3Þ

and

em≥1l¼1; f ¼ 1 −

2

ð f − 2Þð f − 1Þ2m−1 ð4Þ

em≥1l¼2; f ¼ 1 −

2

f ð f − 2Þð f − 1Þ2m−1 ð5Þ

The key to obtaining Eqs. 4 and 5 is to developand solve the recursive relation that describesthe propagation of a topological defect in thephantom network (22).For larger loops (l ≥ 3), it is difficult to obtain

an exact solution of eml; f because of the complexcorrelation between loop junctions; an approx-imate treatment is provided. We consider thegeneral case, a strand (red strand in Fig. 3B) in anl th-order loop in the f-functional polymer net-work. The end junctions a of this loop strandare directly linked to two types of junctions.The first type is the ideal junction b, which isconnected to the network boundary throughrepetitive branches. The second type consists oftwo loop junctions g; these two junctions, aswell as the loop junctions between them (l − 2junctions in total), are simultaneously connectedto the boundary through repetitive branches.The length of the equivalent phantom chainconnected to junctions g can be derived fromthe convolutional features of the Gaussian chain.Neglecting long-range correlations between junc-tions a through these l − 2 loop junctions, theelastic effectiveness of the targeted polymerstrand (red strand in Fig. 3B) can be obtainedby combining the contributions from junctionsb and g, which yields

em¼0l≥3; f ¼ 1 −

2

ð f − 1Þ2ðl−1Þ þ 1ð6Þ

The elastic effectiveness of the strand adjacentto the larger loops is similarly obtained as

em≥1l≥3; f ¼ 1 −

2

ð f − 1Þ2ðlþm−1Þ þ ð f − 1Þ2m−1 ð7Þ

Equations 4 to 7 show that the effect of loops inreducing the elastic effectiveness of strands de-creases rapidly with increasing loop order ordistance from the loop (Fig. 3, C and D). Theelastic effectiveness of small loops is muchlower than 1, which implies that these strandswill have a non-negligible negative impact onthe gel modulus. In contrast, e(l ≥ 4, m = 0, f )

≈ 1, which indicates that strands in larger loops(l ≥ 4) behave nearly the same as ideal phantomchains (Fig. 3C). Furthermore, e increases mono-tonically with f; the loop effect is more pronouncedin networks with small junction functionality.Finally, the loop effect persists within only afew 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 nml; f interms of the average number of loops per junc-tion, nl, f, an analytical expression can be producedfor the shear modulus of a polymer network:

G0

n0kT¼ f − 2

f1 − Af n1; f −Bf n2; f −

Xl¼3

1

Cl; f nl; f

!

ð8Þwith

Af ¼5=2 for f ¼ 34=3 for f ¼ 42=ð f −2Þ for f ≥ 5

8<: ð9Þ

Bf ¼ 4ð f − 1Þf 2ð f − 2Þ ð10Þ

and

Cl; f ¼ 4

f

1

ð f − 1Þ2l−2 þ 1þ 1

ð f − 1Þ2l−1 − 1

" #ð11Þ

where n0 is the total number density of poly-mer 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 nec-essary to know the density of loops of order l inthe given network. As described above, SILDaScan precisely measure the primary loop fraction,n1, f. We have previously shown that that thereis a one-to-one correspondence between the den-sity of higher-order loops and the primary loop

density (21). Therefore, n2, f and n3, f can be calcu-lated once n1,f is measured [see (22) for MonteCarlo simulation algorithm] as shown in Fig. 3E.With knowledge of G′ and the densities of pri-

mary, secondary, and ternary loops, we appliedRENT to calculate the effect of topological defectson G′; excellent agreement between the experi-mental G′/n0kT (Fig. 4, black circles) and RENTtheory (Fig. 4, black lines) for low n1, f values(n1,3 < 0.20 and n1,4 < 0.28) was observed. Althoughprimary loops play the dominant role in reducingG′/n0kT because of their zero elastic effectiveness(see Fig. 3C), Fig. 4 shows that consideration ofprimary loops alone in RENT (blue lines, Fig. 4)does not give complete agreement with experi-ment; higher-order loops should be included to givethe best agreement. RENT also shows good agree-ment with simulation and experimental data fromother research groups (figs. S15 and S16) (25–27).Although RENT and experimental data show

excellent agreement at low n1,f values, there is adeviation between experiment and theory as n1, fbecomes large. This deviation is due to our as-sumption that loops can be treated as indepen-dent within an “ideal loop gas.” As the fraction ofloops in the networks becomes large, loops willbecome close to each other. In this case, the rulesfor elastic effectiveness as a function of m and l(as given in Fig. 3) break down; cooperative ef-fects between loops that are close in space mayhave a greater negative impact on elasticity thana linear combination of isolated loops in an “idealloop gas.” Notably, the deviation between RENTand experiment for both A2 + B3 and A2 + B4 net-works occurs at an A2 concentration of 12 to 13 mM,which is very close to the estimated polymer over-lap concentration c* for our PEG macromers (c* ≈13 mM) (12). We note that in most applicationsof gels, synthetic conditions are used (e.g., con-centration > c*; semidilute regime) that wouldavoid large numbers of loops; RENT as derivedhere should apply, given also that contributionsfrom entanglements are minimal. Nonetheless,

SCIENCE sciencemag.org 16 SEPTEMBER 2016 • VOL 353 ISSUE 6305 1267

‘

Fig. 4. Elastic moduli predicted by RENT. Comparison of experimentally determined G′/kTn0 (blackcircles) to that predicted by affine theory (green dashed line), phantom theory (green dashed line), RENTwith only a correction for primary loops (blue line), and RENTaccounting for primary, secondary, and tertiaryloops (black line). (A) f = 3 networks. (B) f = 4 networks. Error bars represent SD of nine measurements.

RESEARCH | REPORTSon June 13, 2020

http://science.sciencemag.org/

Dow

nloaded from

this work provides an important advance in quan-titative network theory and a foundation forfurther development of theories that account forinteracting network defects and entanglements.Combining RENT with primary and higher-

order loop measurements provides quantitativeagreement with measured elastic moduli for net-works with compositions and structures that arerelevant to common applications. In other words,RENT can predict the bulk mechanical propertiesof polymer networks on the basis of molecularinformation. We anticipate that RENT and loop-counting methods will be applicable to a widerange of polymer networks.

REFERENCES AND NOTES

1. E. J. Mittemeijer, in Fundamentals of Materials Science: TheMicrostructure-Property Relationship Using Metals as ModelSystems (Springer, 2011), pp. 201–244.

2. P. Capper, Ed., Bulk Crystal Growth of Electronic, Optical &Optoelectronic Materials (Wiley, 2010).

3. K. Müllen, Nat. Rev. Mater. 1, 15013 (2016).4. P. J. Flory, J. Chem. Phys. 66, 5720 (1977).5. M. Rubinstein, R. H. Colby, Polymer Physics (Oxford Univ.

Press, 2003).6. G. Hild, Prog. Polym. Sci. 23, 1019–1149 (1998).7. D. R. Miller, C. W. Macosko, Macromolecules 9, 206–211 (1976).8. M. Mooney, J. Appl. Phys. 11, 582 (1940).9. S. K. Patel, S. Malone, C. Cohen, J. R. Gillmor, R. H. Colby,

Macromolecules 25, 5241–5251 (1992).10. M. Rubinstein, S. Panyukov, Macromolecules 30, 8036–8044

(1997).11. Y. Akagi et al., Macromolecules 44, 5817–5821 (2011).12. Y. Akagi, J. P. Gong, U.-i. Chung, T. Sakai, Macromolecules 46,

1035–1040 (2013).13. R. F. T. Stepto, in Biological and Synthetic Polymer Networks,

O. Kramer, Ed. (Springer, 1988), pp. 153–183.14. S. Dutton, R. F. T. Stepto, D. J. R. Taylor, Angew. Makromol. Chem.

240, 39–57 (1996).15. K. Dušek, M. Dušková-Smrčková, J. Huybrechts, A. Ďuračková,

Macromolecules 46, 2767–2784 (2013).16. R. F. T. Stepto, J. I. Cail, D. J. R. Taylor, Mater. Res. Innovat. 7, 4

(2003).17. H. Zhou et al., Proc. Natl. Acad. Sci. U.S.A. 109, 19119–19124

(2012).18. H. Zhou et al., J. Am. Chem. Soc. 136, 9464–9470 (2014).19. K. Kawamoto, M. Zhong, R. Wang, B. D. Olsen, J. A. Johnson,

Macromolecules 48, 8980–8988 (2015).20. A. C. Balazs, Nature 493, 172–173 (2013).21. R. Wang, A. Alexander-Katz, J. A. Johnson, B. D. Olsen, Phys.

Rev. Lett. 116, 188302 (2016).22. See supplementary materials on Science Online.23. H. C. Kolb, M. G. Finn, K. B. Sharpless, Angew. Chem. Int. Ed.

40, 2004–2021 (2001).24. J. A. Johnson et al., J. Am. Chem. Soc. 128, 6564–6565 (2006).25. G. S. Grest, M. Pütz, R. Everaers, K. Kremer, J. Non-Cryst.

Solids 274, 139–146 (2000).26. G. S. Grest, K. Kremer, E. R. Duering, Eur. Phys. Lett. 19,

195–200 (1992).27. R. F. T. Stepto, Polymer 20, 1324–1326 (1979).

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.), theInstitute for Soldier Nanotechnologies via U.S. Army Research Officecontract W911NF-07-D-0004 (B.D.O.), and NSF Materials ResearchScience and Engineering Centers award DMR-14190807. All data areavailable in the supplementary materials. Author contributions:M.Z., R.W., K.K., B.D.O., and J.A.J. designed the research; M.Z. andK.K. performed all experimental work; R.W. developed the theory; andM.Z., R.W., K.K., B.D.O., and J.A.J. wrote the paper.

SUPPLEMENTARY MATERIALS

www.sciencemag.org/content/353/6305/1264/suppl/DC1Materials and MethodsSupplementary TextFigs. S1 to S32References (28, 29)

29 April 2016; accepted 18 August 201610.1126/science.aag0184

NANOMATERIALS

1D nanocrystals with preciselycontrolled dimensions, compositions,and architecturesXinchang Pang, Yanjie He, Jaehan Jung, Zhiqun Lin*

The ability to synthesize a diverse spectrum of one-dimensional (1D) nanocrystals presentsan enticing prospect for exploring nanoscale size- and shape-dependent properties. Here wereport a general strategy to craft a variety of plain nanorods, core-shell nanorods, and nanotubeswith precisely controlled dimensions and compositions by capitalizing on functionalbottlebrush-like block copolymers with well-defined structures and narrow molecular weightdistributions as nanoreactors.These cylindrical unimolecular nanoreactors enable a high degreeof control over the size, shape, architecture, surface chemistry, and properties of 1D nanocrystals.We demonstrate the synthesis of metallic, ferroelectric, upconversion, semiconducting, andthermoelectric 1D nanocrystals, among others, as well as combinations thereof.

The synthesis of isotropic nanomaterials hasprovided access to an array of nanoparticleswith controlled sizes, shapes, and function-alities. Going beyond zero-dimensionalnanoparticles (1, 2), one-dimensional (1D)

nanocrystals such as nanorods (3), nanotubes (4),nanowires (5), and shish kebab–like heterostruc-tures (6, 7) exhibit a range of properties [e.g.,optical (8), electrical (9), magnetic (10), and cat-alytic (11)] depending on their size and shape.Emerging approaches, including template-assistedsynthesis (12–14), chemical vapor deposition (15),and colloidal synthesis (16), enable the prepara-tion of intriguing 1D nanocrystals with controlleddimensions. However, some of these proceduresrequire tedious multistep reactions and purifica-tion processes and rigorous experimental condi-tions; in addition, they are difficult to generalize.In contrast to previously reported polymer

brushes for the synthesis of nanowires (12), ourbottlebrush-like block copolymers (BBCPs) areunimolecular cylindrical polymer brushes thathave a cellulose backbone with densely graftedfunctional block copolymers as side chains (arms),which consist of multiple compartments (innerand intermediate blocks for templating nano-crystal growth and outer blocks for solubility).Cellulose forms a rigid backbone because of intra-molecular hydrogen bridges between hydroxylgroups and oxygen atoms (17). In addition, thethree substitutable hydroxyl groups on celluloseallow dense polymer side chains to be graftedfrom the cellulose backbone. We used these ad-vantages inherent in cellulose to synthesize ourstraight cylindrical BBCPs (figs. S1 to S8 and sup-plementary text, sections I to III). Each fam-ily of BBCPs was synthesized by sequentialatom transfer radical polymerization (ATRP)or ATRP followed by a click reaction (Fig. 1 andfigs. S9 to S11).

Plain nanorods of varied diameters and com-positions were synthesized using amphiphiliccellulose-g-(PAA-b-PS) {cellulose-graft-[poly(acrylicacid)-block-polystyrene]} BBCP nanoreactors (Fig.1A; table S2; and supplementary text, sectionsIII and IV). Because each unit in the cellulosebackbone enables the growth of three PtBA-b-PSdiblocks (upper left panel in Fig. 1A; supple-mentary text, sections I and II), the denselygrafted PtBA-b-PS arms in conjunction with therigidity of the backbone force the cellulose-g-(PtBA-b-PS) BBCP to adopt a straight, rigid, cy-lindrical conformation (upper right panel in Fig.1A) [PtBA, poly(tert-butyl acrylate)]. Subsequenthydrolysis of the PtBA blocks yields amphiphiliccellulose-g-(PAA-b-PS) with inner hydrophilic PAAblocks and outer hydrophobic PS blocks (lowerright panel in Fig. 1A and fig. S47).When cellulose-g-(PAA-b-PS) BBCPs are dispersed in dimethyl-formamide (DMF) polar solvent (18), the resultingunimolecularmicelles canbeused as nanoreactors.The interaction betweenhighly polar DMFand theinner PAA blocks is stronger than that betweenDMF and the outer PS blocks. Thus, there is agreater repulsion between PAA chains in DMFthan between PS chains (18). As a result, the PAAchains are greatly stretched and form a large com-partment. On addition and dispersion of inor-ganic precursors into the solution, the cylindricalcompartment containing PAA blocks can accom-modate a large volume of precursors. In additionto the solvent polarity effect noted above, the pre-cursors are also preferentially partitioned in thecylindrical PAA block compartment as a result ofthe strong coordination interaction between themetal moieties of precursors and the carboxylicacid groups of PAA, which creates a high enoughconcentration of precursors to initiate the nucle-ation and growth of inorganic nanorods (19) (figs.S38 to S46 and S71; see supplementary text, sectionIV, for the proposed formation mechanisms). Thein situ capping of PS chains on the nanorods facil-itates dispersion and solubility in various organicsolvents (lower left panel in Fig. 1A).

1268 16 SEPTEMBER 2016 • VOL 353 ISSUE 6305 sciencemag.org SCIENCE

School of Materials Science and Engineering, GeorgiaInstitute of Technology, Atlanta, GA 30332, USA.*Corresponding author. Email: zhiqun.lin@mse.gatech.edu

RESEARCH | REPORTSon June 13, 2020

http://science.sciencemag.org/

Dow

nloaded from

Quantifying the impact of molecular defects on polymer network elasticityMingjiang Zhong, Rui Wang, Ken Kawamoto, Bradley D. Olsen and Jeremiah A. Johnson

DOI: 10.1126/science.aag0184 (6305), 1264-1268.353Science 

ARTICLE TOOLS http://science.sciencemag.org/content/353/6305/1264

MATERIALSSUPPLEMENTARY http://science.sciencemag.org/content/suppl/2016/09/15/353.6305.1264.DC1

REFERENCES

http://science.sciencemag.org/content/353/6305/1264#BIBLThis article cites 23 articles, 1 of which you can access for free

PERMISSIONS http://www.sciencemag.org/help/reprints-and-permissions

Terms of ServiceUse of this article is subject to the

is a registered trademark of AAAS.ScienceScience, 1200 New York Avenue NW, Washington, DC 20005. The title (print ISSN 0036-8075; online ISSN 1095-9203) is published by the American Association for the Advancement ofScience

Copyright © 2016, American Association for the Advancement of Science

on June 13, 2020

http://science.sciencemag.org/

Dow

nloaded from