Single polymer growth dynamics

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Science  20 Oct 2017:
Vol. 358, Issue 6361, pp. 352-355
DOI: 10.1126/science.aan6837

Watching growth, step by step

Polymers can grow through the stepwise addition of monomers to an active end site. One might think that this would happen in a continuous linear process. With a focus on ring-opening metathesis polymerization of norbornene catalyzed by a Grubbs' catalyst, Liu et al. describe the growth of a single polymer chain. By attaching one end of a growing polymer to a bead exerting a constant force on it, they measured the extension occurring during the growth of the polymer. Oddly, the extension of the growing polymer did not increase continuously. Instead, it exhibited consecutive wait-and-jump steps, owing to conformational entanglements formed by newly incorporated monomers.

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In chain-growth polymerization, a chain grows continually to reach thousands of subunits. However, the real-time dynamics of chain growth remains unknown. Using magnetic tweezers, we visualized real-time polymer growth at the single-polymer level. Focusing on ring-opening metathesis polymerization, we found that the extension of a growing polymer under a pulling force does not increase continuously but exhibits wait-and-jump steps. These steps are attributable to the formation and unraveling of conformational entanglements from newly incorporated monomers, whose key features can be recapitulated with molecular dynamics simulations. The configurations of these entanglements appear to play a key role in determining the polymerization rates and the dispersion among individual polymers.

Catalytic polymerization is a key process in making synthetic polymers (16). In a typical chain-growth polymerization reaction, each catalyst molecule adds monomers to a growing polymer chain. These growing chains can adopt diverse conformations, which can affect the microenvironment of the catalyst, and can also interact with one another or themselves, leading to inter- or intrachain reactions. Consequently, the microscopic dynamics of polymer growth is expected to differ from one polymer to another, contributing to the differences in chain length, tacticity, and comonomer incorporation, which are crucial for their bulk properties. How a single polymer grows dynamically remains unknown, however, because polymerization dynamics has always been characterized at the ensemble level.

To visualize single-polymer growth in real time, we used magnetic tweezers (supplementary materials, materials and methods 1.1 to 1.4) (711) to monitor the lengthening of a polymer chain during ring-opening metathesis polymerization (ROMP) catalyzed by the second-generation Grubbs’ catalyst (G2) (Fig. 1, A and B) (12, 13). One end of the growing polymer is connected via a C=Ru bond to the G2 catalyst, whose N-heterocyclic carbene ligand is functionalized with silane groups (14) to attach to a ~1.5-μm-diameter magnetic particle; the other end is anchored to a coverslip via silane chemistry (figs. S7, S10, and S12). A constant magnetic force (up to ~17 pN) (supplementary materials, materials and methods 1.3) pulls the particle in the z direction and stretches the polymer but is not large enough to break covalent bonds (~4 nN) (15, 16). A constant force also keeps constant the ratio of the extension over contour-length of the polymer as described by the worm-like chain (WLC) model (supplementary materials, materials and methods 1.4) (17). During ROMP, the insertion of new monomers into the C=Ru bond (13) leads to a lengthening of the extension of the polymer tether (Fig. 1B). By tracking the z position of the magnetic particle, we can follow the growth of a single polymer in real time with ~6- to 10-nm precision (supplementary materials, materials and methods 1.2).

Fig. 1 Single-polymer growth measurements.

(A) Schematic of magnetic tweezers measurement of a growing polymer tethered between a coverslip and a magnetic particle. Monomers (blue spheres) are incorporated by the Ru-based catalyst. (Right) Linkage chemistry (scheme S3). (B) General mechanism of ROMP after initiation. Each catalytic cycle inserts a cyclic olefin into the C=Ru bond of the polymer tether. L, N-heterocyclic carbene ligand. (C) Real-time extension-versus-time trajectories for three growing polymers in 1 M norbornene (NB) or cis-cyclooctene (CO) under different magnetic forces. Solid black triangle marks monomer addition. (Inset) xy center positions of the tethered magnetic particle of the blue trajectory when rotating the magnets after polymerization. (D) Force-extension curve (black squares) of a single polymer [(C), green trajectory] after polymerization. Red line, WLC model fitting. (Inset) Distribution of polynorbornene’s persistence length p. Red line, Gaussian fit centered at 0.71 ± 0.12 nm. (E) Zoom-in of the boxed region in (C) (gray), with the smoothed (black) and the reconstituted trajectory after linear fitting of the waiting periods (red). Jump length j, waiting time τ, and slope s are defined here and in fig. S13. (Inset) Schematic of hairball formation and unraveling during single-polymer growth.

We first studied the growth of single polynorbornene molecules under ~17 pN at 1 M norbornene in toluene, a good solvent for polynorbornene; the monomer depletion during reaction is negligible owing to the high monomer-to-catalyst ratio (>107:1) (supplementary materials, materials and methods 1.12). The G2-loaded magnetic particles were initially tethered to surface norbornene or polynorbornene via metathesis, during which G2 had been initiated (supplementary materials, materials and methods 1.10) (13, 18, 19). Monomer addition led to the lengthening of the polymer extension (Fig. 1C).

The extension of a single growing polymer does not increase continuously but exhibits wait-and-jump steps (Fig. 1, C and E), even though in toluene and under the applied force, polynorbornene should follow WLC behavior, in which the extension scales linearly with chain length. The waiting periods are hundreds of seconds, during which the extension barely lengthens. The jumps are up to ~103 nm, which is equivalent to thousands of monomers (supplementary materials, materials and methods 1.4.2), and instantaneous within 0.05 s (fig. S14); they would correspond to apparent growth rates that are orders of magnitude faster than the average polymerization rate (~100 monomers s−1; discussed later). Therefore, the underlying polymer growth could not have occurred during the jump but must have occurred during the prior waiting period, when the newly incorporated monomers do not contribute to the extension and are “concealed” in a conformationally entangled “hairball” (Fig. 1E, inset) wrapping around the catalyst. With continuing growth and pulling force, this hairball would eventually unravel, releasing the newly grown chain length and giving an instantaneous extension jump.

To ensure that we observed single-polymer growth, we rotated the tethered magnetic particles at ~4.5-pN pulling force, under which the polymer conformation is in the aligned state (10), both before and after reaction. Single-polymer–tethered particles exhibit characteristic free rotation (movie S1) and circular precession movements in the xy plane (Fig. 1C, inset) (20), whereas multiple-tethered ones show irregular movements (fig. S15). The force-extension curves of individual polymers measured afterward follow the WLC model, giving an average persistence length p ~ 0.71 nm (Fig. 1D), which is consistent with those of related synthetic polymers (10), and the contour length L0 of the eventual polymer (for example, ~3618 nm in Fig. 1D, corresponding to ~5835 monomers) (supplementary materials, materials and methods 1.4.2).

Anchoring the catalyst on the coverslip also gives the stepwise extensions of single-polymer growth (fig. S16). After releasing the magnetic particle to fully relax the grown polymer, the subsequent force extension follows WLC behavior without discrete jumps (fig. S17). Both results rule out that the stepwise extension came from polymer interactions with coverslip or magnetic particle surfaces. Multiple observations rule out as possible reasons chain-transfer reactions or PCy3 ligand rebinding to the catalyst (supplementary text 2.6). Using cis-cyclooctene as monomer also gave stepwise extensions (Fig. 1C). Altogether, the stepwise extension, and thus the formation of the hairball, are a consequence of real-time growth under nonequilibrium, continual polymerization conditions.

We hypothesized that as new monomers are inserted, they do so with certain dihedral angles that introduce torsional strains in the newly grown chain, causing it to form entanglements, which are held together temporarily by intrachain van der Waals interactions (fig. S19). This torsional strain is likely dissipated quickly by backbone rotations that twist the neighboring segments into entangled conformations, forming a hairball. The nonequilibrium entanglements gradually relax, aided by Brownian chain-solvent collisions and by the applied force, until the hairball unravels and subsequently behaves as a typical WLC.

We used classical molecular dynamics simulations (21, 22) to test our hypothesis (supplementary text 4.1 to 4.3). To model a short chain bearing such strains, we created a polynorbornene chain with 146 monomers by connecting monomers randomly (2325), with the initial dihedral angles far from equilibrium (fig. S20). In its starting linear-chain configuration, the two chain ends and the initial dihedral angles were fixed while bending angles and bond lengths were allowed to relax. We then freed one chain end and let the chain relax; it spontaneously collapsed into an entangled hairball (Fig. 2A), concurrent with a shortening of its extension and fast relaxation of its dihedral strain (Fig. 2, B and C).

Fig. 2 Molecular dynamics simulation of hairball formation and unraveling.

(A) Structure of a hairball at 2 ns from a collapsed 146-oligomer polynorbornene chain with one chain end fixed and the other chain end pulled by a constant force. (B to D) Time evolutions of (B) extension, (C) dihedral potential, and (D) nonbonded pair potential of a 146-oligomer polynorbornene chain collapsing from a linear configuration into a hairball. (E to G) Subsequent evolutions upon applying a constant pulling force starting at 8 ns. (E) shows results from three applied forces. Red dashed lines, linear regression fits to the waiting and jumping portions of the 21 pN curve. (F) and (G), 21 pN only. (H) Topological structure of the entanglement in (A). The color changes from blue to red from one chain end to the other.

We then applied a constant pulling force on one end (2628) and monitored the extension over time. We observed a waiting period followed by a jump (Fig. 2E), analogous to those observed experimentally (Fig. 1C). The entanglements (Fig. 2H) slow down the extension until they unravel rapidly as an extension jump; no partial unraveling was observed in simulations.

A higher pulling force expectedly shortens the waiting time (Fig. 2E). During pulling, the torsional angle energy remains unchanged (Fig. 2F), whereas the nonbonded contacts in the hairball formed during collapse are broken as the hairball unravels (Fig. 2, D and G). After unraveling, the polymer extension follows the WLC model with a persistence length of ~0.67 nm (fig. S25B), which is consistent with experiments (Fig. 1D). Additional simulation tests are described in the supplementary text 4.5 to 4.8.

We then examined the experimental statistical properties of the wait-and-jumps in the extension-versus-time trajectories (Fig. 1E). The jump length j reflects the chain length newly grown into the hairball. Individual j follows a single exponential distribution (Fig. 3A). With 1 M norbornene and ~17 pN pulling force, the average jump length is 175 ± 11 nm, which is equivalent to 413 ± 26 monomers. Its prior waiting time τ is the duration of a hairball and reflects the hairball’s kinetic stability; it follows a multiexponential distribution, with at least two exponents (Fig. 3B), suggesting that either the kinetics of hairball unraveling involves a multiple rate-limiting step, or individual hairballs have different unraveling kinetics. The average waiting time is 333 ± 28 s, corresponding to an ~0.003 s−1 unraveling rate.

Fig. 3 Statistical properties of wait-and-jump behaviors of single-polymer growths.

(A to C) Histograms of jump length j, waiting time τ, and waiting-period-slope s from 43 polymers (1 M norbornene, 17 pN force). Red lines, single [(A) and (C)] and double (B) exponential fits; the fit in (C) excluded the first bin. (D to G) Average waiting time Embedded Image, average jump length Embedded Image, average slope S, and the average polymerization rate V of individual polymers under different conditions. [Olefin], monomer concentration in molar. Average values are defined in supplementary text 1.13). Error bars are SEM. (H to J) Correlation plots and Pearson’s correlation coefficients between τ, j, and s from polymers grown under 1 M norbornene and 17 pN force. Each solid black square indicates one wait-and-jump event. Open red squares indicate averages from bins of equal number of events. x and y error bars are SD.

For the individual wait-and-jump events, the jump length j has no significant correlation with its prior waiting time τ (Fig. 3H), indicating that the polymer length within each hairball and its kinetic stability are mostly decoupled (supplementary text 5.3). This decoupling could result from the polymerization occurring locally at the catalyst, whereas the stability of the hairball is controlled by its global configuration, and the global and local configurations of a hairball may not be strongly coupled structurally.

Moreover, ~75% of the waiting periods in the extension-versus-time trajectories exhibit resolvable positive slopes; the rest stays essentially flat (for example, Figs. 1E, event a and b, respectively, and 3C). Because the slope is a hairball’s expansion rate in physical dimension during polymerization, we postulated that it should relate to the hairball’s global structural “looseness.” Consistently, the slope s is anticorrelated with the waiting time τ (Fig. 3I) because looser hairballs (larger s) are expected to unravel faster (shorter τ). No significant correlation between the slope s and the jump length j was observed (Fig. 3J), which is consistent with the hairball’s global configuration being likely decoupled structurally from the local configuration around the catalyst.

Upon decreasing the pulling force to ~4.5 pN while maintaining norbornene monomer concentration (Fig. 1C), the average waiting time lengthens by ~53% (Fig. 3D); this is expected because hairball unraveling should be slower under smaller pulling force, as also reproduced in simulations (Fig. 2E). Average jump length also increases, by ~117% (Fig. 3E), likely because longer average waiting times would allow for more turnovers on average. The average slope of the waiting periods is essentially unchanged, ~0.12 nm/s (Fig. 3F), which should reflect a faster hairball expansion because the same extension at a lower pulling force corresponds to a larger contour length; the faster hairball expansion here is consistent with the global hairball configuration being looser under smaller pulling force. The average polymerization rate V increased by ~69% (Fig. 3G), indicating that pulling force impairs overall polymerization kinetics, perhaps via tightening the hairball to restrict access to, or distorting the structure at, the Ru center of the catalyst.

To actively manipulate ROMP activity, we decreased the norbornene concentration to 0.1 M while maintaining the ~17 pN pulling force. Expectedly, the average polymerization rate decreases by ~29% (Fig. 3G). The average jump length decreases by ~25% (Fig. 3E), reflecting smaller hairballs due to slower polymerization rates. The average waiting time increases by ~36% (Fig. 3D), reflecting slower unraveling kinetics and more stable hairballs, even though the hairballs are smaller. We speculate that the slower polymerization rate here might allow the newly incorporated monomers to form better nonbonded interactions within the hairball; consistently, the average slope decreases by ~38% (Fig. 3F), reflecting tighter hairballs.

We further manipulated the ROMP activity by using a less active monomer cis-cyclooctene (29). Under the same ~17-pN pulling force and 1 M monomer concentration, the average polymerization rate slows down expectedly (Fig. 3G). The average waiting time, jump length, and slope when using cis-cyclooctene monomer all show similar trends to those using a lower norbornene concentration (Fig. 3, D to F).

We then examined how the microscopic properties of hairballs relate to the polymerization rate of each polymer. Under any attempted reaction condition, the polymerization rates V of individual polymers differ greatly, up to a factor of ~50 (Fig. 4A and fig. S38, A, E, and I), even though the associated polydispersity index is 1.5 ± 0.4 (supplementary text 5.5). V of individual polymers shows strong correlations with their average jump length, waiting time, and waiting-period-slope, under all reaction conditions (Fig. 4, B to D, and fig. S38). Because the jump length, waiting time, and waiting-time-slope reflect, respectively, the size, kinetic stability, and structural looseness of the hairballs, these correlations indicate that faster polymerizations are associated with larger, less stable, and looser hairballs. Therefore, the microscopic configurations of the hairballs play key roles in each polymer’s growth, perhaps by controlling access to the catalyst, and the dispersion of the hairball’s microscopic configuration likely contributes substantially to the dispersion of polymerization rate among individual polymers under identical growth conditions.

Fig. 4 Origin of kinetic dispersion in polymerization among individual polymers.

(A) Histogram of the average polymerization rate V of individual polynorbornene molecules. Red line, lognormal fit. The polydispersity index of V is 1.5 ± 0.4. (B to D) Correlation plots between the average polymerization rate V of each polymer molecule and its average jump length, waiting time, and waiting-period-slope, and the respective Pearson’s correlation coefficients. Each solid black square indicates a single polymer. Open red squares indicate binned and averaged results. All data are from 1 M norbornene and 17 pN pulling force condition. x and y error bars are SD.

In typical solution synthesis of polymers, no stretching force exists, and the quality of solvent could be poorer. In such conditions, conformationally entangled hairballs could be more prevalent (supplementary text 4.8 and 5.7) (10, 30) and thus play a bigger role in altering polymerization kinetics.

Supplementary Materials

Materials and Methods

Supplementary Text

Figs. S1 to S40

Equations S1 to S15

Tables S1 to S4

References (3164)

Movie S1

References and Notes

  1. Materials and methods are available as supplementary materials.
Acknowledgments: The research is mainly supported by the Army Research Office (grant W911NF-14-1-0620), and in part by the Army Research Office (grants W911NF-13-1-0043 and W911NF-14-1-0377), U.S. Department of Energy (grant DE-SC0004911), Petroleum Research Foundation (grant 54289-ND7), and National Science Foundation (grant CMMI-1435852). The research uses Cornell Chemistry nuclear magnetic resonance facility, Extreme Science and Engineering Discovery Environment (XSEDE), and Cornell Center for Materials Research Shared Facilities supported by NSF (grants CHE-1531632, ACI-1053575, and DMR-1120296). We thank R. Khurana and A. DiCiccio for advice on synthesis; M. D. Wang, J. Ma, and J. Lin for advice on magnetic tweezers; and R. F. Loring for discussions. C.L. constructed instrument, performed measurements, coded software, and analyzed data; K.K. performed syntheses and measurements and contributed to instrument construction and data analyses; E.W. performed simulations; K.-S.H., F.Y., and G.C. contributed to experiments; F.A.E. supervised simulations; G.W.C. supervised syntheses; P.C. and G.W.C. designed experiments; P.C. conceived and oversaw research; C.L., K.K., E.W., F.A.E., G.W.C., and P.C. wrote the manuscript.
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