Letter pubs.acs.org/NanoLett
Harnessing Interfacially-Active Nanorods to Regenerate Severed Polymer Gels Xin Yong,† Olga Kuksenok,† Krzysztof Matyjaszewski,‡ and Anna C. Balazs*,† †
Chemical Engineering Department, University of Pittsburgh, Pittsburgh, Pennsylvania 15261, United States Department of Chemistry, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, United States
‡
S Supporting Information *
ABSTRACT: With newly developed computational approaches, we design a nanocomposite that enables self-regeneration of the gel matrix when a significant portion of the material is severed. The cut instigates the dynamic cascade of cooperative events leading to the regrowth. Specifically, functionalized nanorods localize at the new interface and initiate atom transfer radical polymerization with monomers and cross-linkers in the outer solution. The reaction propagates to form a new cross-linked gel, which can be tuned to resemble the uncut material. KEYWORDS: Self-regeneration, polymer nanocomposites, atom transfer radical polymerization, dissipative particle dynamics
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regrowth. Second, the system must involve a means of continuing or propagating the desired growth. Finally, when the material reaches a certain size, there must be a means of terminating the ongoing reactions. By developing new computational models, we design a self-regenerating material that undergoes these processes to produce a remarkable form of regrowth when a significant portion of the system is removed. Figure 1a reveals the components of the uncut material: nanorods functionalized at one end and dispersed in a polymer gel, which is swollen in a good solvent. The solvent is weakly incompatible with the rods, but compatible with the endgrafted chains (in yellow), making the functionalized rods essentially amphiphilic. The rods contain initiator sites (in magenta) for the atom transfer radical polymerization (ATRP),7−10 which is a form of controlled/living radical polymerization.11,12 The synergy between these specific components enables a dynamic cascade of events that leads to gel regeneration when the sample is cut with the rods serving to trigger the ensuing, vital reactions. Finally, the surrounding simulation box is periodic in x and y but bounded by top and bottom hard walls (brown beads).
n elusive goal in materials science is designing systems that mimic the remarkable ability of amphibians to regrow limbs. While self-healing materials1−4 can mend local defects, there are virtually no examples of materials that can regenerate themselves. The advent of such regenerative materials could dramatically extend the useful lifetime of manufactured products. Through new computational models, we design a nanorod-filled gel that effectively regenerates the gel matrix when a layer of the material is sliced-off. With this layer removed, the nanorods diffuse to the newly formed interface and extend into the outer solution, which contains monomers and a small fraction of cross-linkers. Polymerization initiated from the rods’ surfaces leads to chains that become cross-linked to form a new gel that resembles the severed layer. After the initial cut, the regeneration requires no external intervention; synergistic interactions among all components in this system enable the vital processes leading to regrowth, which could be repeated with subsequent cuts. In biology, tissue regeneration5 is guided by an internalized “instruction set” that involves signaling molecules that mediate the vital processes in the regrowth: initiation, propagation, and termination.6 Analogous to biological systems, fully synthetic self-regenerating materials should incorporate the following elements. First, the system must encompass a component that not only senses the removal of material but also initiates the © XXXX American Chemical Society
Received: October 15, 2013 Revised: November 12, 2013
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dx.doi.org/10.1021/nl403855k | Nano Lett. XXXX, XXX, XXX−XXX
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Figure 1. System components and model validations. (a) Equilibrium state of a nanorod-filled polymer gel in a good solvent (not displayed). Green chains represent polymer strands, and brown beads form amorphous solid walls. Top left: enlarged view of initiator (magenta beads)-modified nanorod (white beads) with one end functionalized with polymer chains (yellow beads) and a cross-section of the hexagonal nanorod. Bottom left: initial configuration of the tetra-functional polymer gel network. (b) Polymer volume fraction as a function of temperature for a spherical gel, showing a continuous volume phase transition around 32 °C. Insets show gel at swollen and collapsed states at 12 and 56 °C, respectively. Hence, the simulation reproduces the LCST behavior of PNIPAM. (c) Schematic of the major reactions in living copolymerization of monomer and crosslinker. Asterisk indicates active radical. Bead colors represent following species: initiator (magenta), monomer (blue), cross-linker (orange). Open beads show unreacted or partially reacted species and filled beads show fully reacted species. Probabilities of initiation (Pir), propagation with p,X monomer (Pp,M r ), propagation with unreacted bifunctional cross-linker (Pr ), and cross-linking with partially reacted cross-linker with pendent p,X p,M functional group (Pp,P r ) can be varied independently. Values of Pr are twice greater than those of Pr , unless stated otherwise. (d) Comparison of DPD simulated gel points and ATRP experimental values for systematic variation of the initial concentration ratios of cross-linker to initiator, [X]0/ [Ini]0.
and r̂ij = rij/|rij|; all the relevant aij values are listed in Supporting Information Table SI1. The drag force is FDij = −γωD(rij)(r̂ij · vij)r̂ij, where γ is a simulation parameter related to the viscosity arising from the interactions between the beads (polymer− solvent, polymer−polymer, and solvent−solvent), ωD is a weight function that goes to zero at rc, and vij = vi − vj. The random force is FRij = σωR(rij)ξijr̂ij, where ξij is a zero-mean Gaussian random variable of unit variance and σ2 = 2kBTγ. The value of γ is chosen to ensure relatively rapid equilibration of the temperature in the system and the numerical stability of the simulations for the specified time-step.15 Finally, we use ωD(rij) = ωR(rij)2 = (1 − rij)2 for rij < 1.15 Because all three of these forces conserve momentum locally, hydrodynamic behavior emerges even in systems containing only a few hundred particles.13−15 The equations of motion are integrated in time with a modified velocity-Verlet algorithm. The factor kBT is taken as the characteristic energy scale, where kB is the Boltzmann constant. Because our simulations involve temperature variations, we chose room temperature as the reference, setting kBT0 = 1, where T0 = 298.15 K. The reduced temperature is therefore defined as T* = T/T0. The characteristic time scale is then defined as τ = (mr2c /kBT0)1/2 = 1. The remaining simulation parameters are γ = 4.5 and Δt = 0.02 τ with a total bead number density of ρ = 3.15 Here, we expand the DPD approach in two distinct ways. First, we modify the model to capture the response of chemically cross-linked gels to variations in temperature (as detailed in Supporting Information SI1) and second, we develop a scheme to simulate the living copolymerization of
This system is modeled using dissipative particle dynamics (DPD),13−15 which can be viewed as a coarse-grained molecular dynamics (MD) method. Relative to MD, one can examine larger systems for longer times in computationally realistic timeframes, making DPD an effective mesoscale method for simulating our complex system. Currently, however, there are no DPD approaches that explicitly incorporate the temperature-dependence of the polymer−solvent interactions in chemically cross-linked gels, making it difficult to realistically model gels characterized by specific values of the polymer− solvent interaction parameter, χps(T). Furthermore, there are no DPD models for the ATRP process. Consequently, we built on the method described below to develop the new approaches detailed in Supporting Information SI1−SI3. Dissipative particle dynamics is a mesoscopic particle-based computational method that provides an effective means of simulating the dynamic behavior of complex fluids and multicomponent mixtures.13−15 Unlike more atomistic MD simulations, a DPD bead represents clusters of molecules (see Supporting Information SI1D). Similar to MD simulations, DPD captures the time evolution of a many-body system through the numerical integration of Newton’s equation of motion, mdvi/dt = fi. The force acting on a bead consists of three parts, each of which is pairwise additive: fi(t) = ∑(FCij + FDij + FRij ), where the sum runs over all beads j within a certain cutoff radius rc. We take rc as our characteristic length scale and set the dimensionless value as rc = 1. The conservative force is a soft, repulsive force given by FCij = aij(1 − rij)r̂ij, where aij is the maximum repulsion between beads i and j, rij = ri − rj, rij = |rij|, B
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Figure 2. Nanorods diffusion and extension into outer solution. (a) Initial configuration of the gel after the upper layer was cut off. (b−e) Snapshots of the system when the rods diffuse and extend into the outer solution, taken at the following times: (b) t = 2500, (c) t = 10 000, (d) t = 23 700, and (e) t = 34 300. Numbers indicate the rods that extend into the outer solution. (f) Time evolution of the center of the mass of the four rods in the zdirection. Letters b−e correspond to the respective snapshots on the left. (g) Number density profiles of the system after nanorods have diffused into the outer solution. The weakly incompatible outer and inner solutions form an interface located at z ≈ 33.
Figure 3. Process of regenerating the cutoff layer. (a−e) Regrowth of the top layer at the following monomer conversions: (a) 0, (b) 25, (c) 50, (d) 75, and (e) 96%. Initial monomer and cross-linker concentrations are [M]0 = 30% and [X]0 = 0.53%. Initiator density σi is 0.25, which corresponds to a concentration [Ini]0 = 0.27%. (f−j) Top-down views corresponding to frames (a−e). Blue beads are the newly formed polymer chains. (k) Dependence of conversions and ln([M]0/[M]) of monomer (solid lines) and ln([X]0/[X]) of cross-linker (dashed lines) on reaction times during copolymerization, where [M] and [X] are the respective current concentrations of unreacted monomer and cross-linker. (l) Number density profiles of the newly formed (blue) gel at different monomer conversions. Black dashed line represents the corresponding number density profile of the original (green) gel at t = 3 × 104.
an angle potential between two consecutive bonds. Taking χps values from experimental data for poly(N-isopropylacrylamide) (PNIPAM),17 our model accurately reproduces the continuous volume phase transition occurring between 30 and 35 °C (Figure 1b), capturing the fact that PNIPAM exhibits a lower critical solution temperature (LCST) phase transition.17,18 Notably, our results for the polymer concentrations at the swollen and collapsed states are consistent with the respective experimental values17,18 (see Supporting Information SI1). This
monomer and cross-linker (as detailed in Supporting Information SI3). We validate these modifications via comparison with prior experimental studies. With respect to capturing the temperature-dependent behavior of the gels, the bottom image in Figure 1a shows that at the onset of the simulations, the tetra-functional network is arranged in a diamond-lattice structure.16 The polymer strands between the cross-links are modeled by a sequence of N = 30 DPD beads connected by harmonic springlike bonds with C
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Figure 4. Controlling properties of regenerated gel layer. (a,b) Density profiles of the system (including original gel, new gel, rods, and grafted chains) at monomer conversion 95% with (a) different initial monomer concentrations [M]0 for initiator density on the rod surface σi = 0.25 ([Ini]0 = 0.27%) and for initial cross-linker concentrations [X]0 varying with [M]0 by fixing [M]0/[X]0 ≈ 56; (b) with different initial densities σi ([Ini]0 varies from 0.09 to 0.54%) for [M]0 = 30% and [X]0 = 0.53%. Insets are the corresponding snapshots of the gel system in (a) with [M]0 = 20% and [M]0 = 50%, and in (b) with σi = 0.0834 and σi = 0.5. (c) Top-down view of the spatial distribution of the cross-links at monomer conversion 96% for the regenerated gel system with σi = 0.25 and [M]0 = 30% (corresponding to simulation in Figure 3e,j). Red beads are the inter-rod cross-links connecting chains grown from different rods and black beads are the intrarod cross-links connecting chains from the same rod. Smaller, differently colored beads represent the chains emanating from different rods. (d) Fraction of the number of inter-rod cross-links (Xinter‑rod) with respect to the total number of cross-links formed (Xtotal) as a function of initiator density and initial monomer concentration. Data are taken at cross-linker conversion of at least 96%. Error bars indicate the variations among four independent runs.
is the first DPD simulation to capture temperature-induced volume phase transitions in gels. The ATRP is a controlled/living polymerization process that leads to uniform chain growth. In the presence of cross-linkers, this process leads to gel formation.19−22 The relevant steps in this living copolymerization are shown schematically in Figure 1c, indicating the initiation, propagation, and cross-linking of the growing chains. The DPD scheme corresponding to Figure 1c is detailed in Supporting Information SI3. By using the latter scheme, we reproduce the experimentally observed23 dependence of the gel point in a bulk sample on the ratio between cross-linker and initiator concentrations, [X]0/[Ini]0 (Figure 1d). (This dependence marks a difference between ATRP23−25 and conventional free radical polymerization.) The excellent agreement between the experimental and simulation data provides validation for our newly formulated DPD approach. Using the above approaches, we probe the cascade of events that occur when the upper half of the nanorod-filled gel is removed (see Figure 2 and Supporting Information Video SI1). With this dramatic cut, the system is now exposed to the outer fluid, which is both weakly incompatible with the solvent in the gel and compatible with the rods, and contains dispersed monomer and a small concentration of cross-linking molecules. (Figure 2g highlights the interfacial region between the outer and inner fluids.) The amphiphilic rods act as crucial sensors, indicating a change in the state of the system. Specifically, rods near the new interface encounter the more compatible outer solvent and hence, diffuse toward this boundary to maximize their interaction with this fluid. Figure 2b−e illustrate the rods’ progression to and final orientation at the cut, and Figure 2f
shows the rods’ center of mass as a function of time. Because the chains on the rods are attracted to the gel and incompatible with the outer solution, they prevent these particles from diffusing out of the gel matrix. Hence, the grafted chains anchor the nanorods at gel-outer solvent interface. Some rods do remain buried within the gel (Figure 2e,f). The localization of the rods at the interface depends on their initial proximity to this boundary (see Supporting Information SI4); furthermore, steric hindrance from rods already at the interface can inhibit particles deeper in the matrix from reaching the cut. With rods at the interface, the initiator sites now become exposed to the monomer-laden solution. Hence, the system encompasses one of the critical events for regeneration: the initiation process. Importantly, chain growth in ATRP is significantly slower than the diffusion rate of rods to the interface.26,27 Thus, the polymerization primarily occurs once the rods are at the cut (not within the gel’s bulk). Importantly, the high surface area provided by the rods facilitates changes in initiator concentration, which plays a significant role in the structure of the new layer (as discussed further below). Figure 3 shows snapshots of the critical propagation step, as monomers form chains (in blue) from the surface of the rods via the ATRP. As indicated in Figure 1c, the cross-linkers become part of these growing polymers and can bridge two separate neighboring chains, thereby binding the system into a network. For these concentrations of monomer and crosslinker, the rods remain localized at the interface as the blue gel continues to grow (see Supporting Information Video SI2). The final component for biomimetic regeneration is a mechanism for turning off the propagating reaction and thereby D
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buried nanorods to the interface and hence to the repeated regrowth. These subsequent events do, however, require addition of new monomer and cross-linker to the host solution. The regrowth of a severed gel layer outlined here helps pave the way for creating materials that are both self-regulating and self-replenishing,29 leading to systems with improved functionality over significantly extended lifetimes.
regulating the height of the regrowing layer. Here, it is the concentrations of monomer and cross-linker that are the critical variables in the termination process; once these species have reacted, no further growth can occur. Figure 3k shows that with initial concentrations of 30% and 0.53% for the monomer and cross-linker respectively, both species become almost fully converted into the gel within the time scale of the simulation. The red curves (Figure 3k) show that the natural logarithms of the ratios of the initial to unconverted quantities for both monomer and cross-linker display a linear dependence with time (after t = 5000), as would be expected for this first-order reaction. These curves indicate that the model captures the correct kinetics for the reaction. The density profile in Figure 3l reveals that for a fixed initiator concentration, the percent conversion of the monomer plays a vital role in the density of the growing gel; at 38% conversion, the densities of the two gels are quite similar and at 96% conversion, the new gel exhibits a higher density than the original material. Importantly, the properties of the new gel can be tailored to resemble those of the original material by tuning the initiator density on the rod, σi, and initial monomer concentration, [M]0. Increasing [M]0 leads to increases in the density and height of the new layer (Figure 4a). The densities of the two layers can be reasonably matched by the appropriate choice of the initiator concentration, σi (Figure 4b). The density of the new gel is reduced with decreases in σi due to decreases in the number of growing polymer chains per unit volume.28 Notably, the structure of the regenerated gel layer is relatively heterogeneous, exhibiting a high fraction of cross-links between chains on a given rod (Figure 4c,d). While the fraction of interrod cross-links is relatively insensitive to σi (increasing only slightly at [M]0 = 50% with decreasing σi), it increases dramatically with increases in [M]0 for any given value of σi (Figure 4d). To conclude, we emphasize that the nanorods serve two important roles in this system. First, the rods act as sensors that indicate the removal of material from the system. Namely, when a layer of gel is severed, the nanorods near the newly formed interface “sense” the more compatible outer solution and, consequently, diffuse to the cut. Second, the nanorods encompass the initiation sites for the reaction leading to regrowth. The high-aspect ratio of the rods permits control over the concentration of initiator on the rod’s surface, σi; this is important since the density of the regrown gel can be tailored by tuning σi. In summary, the rods sense the damage, migrate to the cut, and thereby deliver controllable quantities of the initiators to the critical location. In this manner, the rods play a crucial role in both the initiation and propagation steps of the regrowth. Notably, the chains grafted to the ends of the rods also serve an important function: they provide a means tailoring the effective adhesion between the newly regenerated and old gels. We also emphasize that analogous to biological processes, the information for the regeneration in our rod-filled gels is inherent to the system; the “popping up” of the rods at the interface provides the signal to initiate the polymerization that leads to the formation of the new gel. The living copolymerization is halted when monomer and cross-linker are consumed, yielding a layer with a specified height, crosslinks, and density of the gel. As some of the rods remain buried in the bulk of the gel, subsequent cuts below the newly regenerated region can lead to localization of the previously
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ASSOCIATED CONTENT
S Supporting Information *
Section S1: detailed description and validation of DPD model for thermo-responsive gel. Section S2: details of modeling nanorod-filled gels. Section S3: detailed DPD scheme and validation of modeling ATPR process. Section S4: detailed discussion of nanorods’ diffusion and extension into outer solution. A video that shows nanorods’ diffusion and extension into outer solution after the upper half of the nanorod-filled gel is removed. A video for the process of regenerating the cutoff layer. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Notes
The authors declare no competing financial interest.
ACKNOWLEDGMENTS The authors gratefully acknowledge the DOE (for partial support of X.Y. for the computational systems) and the ARO (for partial support of O.K. for the analysis). A.C.B. also acknowledges helpful conversations with Dr. David Stepp and Professor Jennifer Lewis.
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REFERENCES
(1) Yang, Y.; Urban, M. W. Chem. Soc. Rev. 2013, 42, 7446−7467. (2) Blaiszik, B. J.; Kramer, S. L. B.; Olugebefola, S. C.; Moore, J. S.; Sottos, N. R.; White, S. R. Annu. Rev. Mat. Res. 2010, 40, 179−211. (3) Hager, M. D.; Greil, P.; Leyens, C.; van der Zwaag, S.; Schubert, U. S. Adv. Mater. 2010, 22, 5424−5430. (4) Wu, D. Y.; Meure, S.; Solomon, D. Prog. Polym. Sci. 2008, 33, 479−522. (5) Webber, M. J.; Stupp, S. I. J. Intern. Med. 2010, 267, 71−88. (6) Bénazet, J.-D.; Zeller, R. Cold Spring Harbor Perspect. Biol. 2009, 1, a001339. (7) Wang, J. S.; Matyjaszewski, K. J. Am. Chem. Soc. 1995, 117, 5614−5615. (8) Matyjaszewski, K.; Xia, J. H. Chem. Rev. 2001, 101, 2921−2990. (9) Matyjaszewski, K. Macromolecules 2012, 45, 4015−4039. (10) Wu, W.; Tsarevsky, N. V.; Hudson, J. L.; Tour, J. M.; Matyjaszewski, K.; Kowalewski, T. Small 2007, 3, 1803−1810. (11) Braunecker, W. A.; Matyjaszewski, K. Prog. Polym. Sci. 2007, 32, 93−146. (12) Ma, H. M.; Davis, R. H.; Bowman, C. N. Macromolecules 2000, 33, 331−335. (13) Hoogerbrugge, P. J.; Koelman, J. Europhys. Lett. 1992, 19, 155− 160. (14) Espanol, P.; Warren, P. Europhys. Lett. 1995, 30, 191−196. (15) Groot, R. D.; Warren, P. B. J. Chem. Phys. 1997, 107, 4423− 4435. (16) Jha, P. K.; Zwanikken, J. W.; Detcheverry, F. A.; de Pablo, J. J.; de la Cruz, M. O. Soft Matter 2011, 7, 5965−5975. (17) Hirotsu, S. J. Chem. Phys. 1991, 94, 3949−3957. (18) Schild, H. G. Prog. Polym. Sci. 1992, 17, 163−249. (19) Ide, N.; Fukuda, T. Macromolecules 1998, 32, 95−99. (20) Bannister, I.; Billingham, N. C.; Armes, S. P.; Rannard, S. P.; Findlay, P. Macromolecules 2006, 39, 7483−7492.
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(21) Bouhier, M.-H.; Cormack, P. A. G.; Graham, S.; Sherrington, D. C. J. Polym. Sci. A: Polym. Chem. 2007, 45, 2375−2386. (22) Gao, H. F.; Matyjaszewski, K. Prog. Polym. Sci. 2009, 34, 317− 350. (23) Gao, H. F.; Polanowski, P.; Matyjaszewski, K. Macromolecules 2009, 42, 5925−5932. (24) Gao, H.; Li, W.; Matyjaszewski, K. Macromolecules 2008, 41, 2335−2340. (25) Polanowski, P.; Jeszka, J. K.; Matyjaszewski, K. Polymer 2010, 51, 6084−6092. (26) Zhong, M. J.; Matyjaszewski, K. Macromolecules 2011, 44, 2668−2677. (27) Shibayama, M.; Isaka, Y. Macromolecules 1999, 32, 7086−7092. (28) Liu, H.; Li, M.; Lu, Z. Y.; Zhang, Z. G.; Sun, C. C. Macromolecules 2009, 42, 2863−2872. (29) Esteves, A. C. C.; Luo, Y.; van de Put, M. W. P.; Carcouët, C. C. M.; de With, G. Adv. Funct. Mater. 2013, DOI: 10.1002/ adfm.201301909.
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Harnessing Interfacially-Active Nanorods to Regenerate Severed Polymer Gels Xin Yong,† Olga Kuksenok,† Krzysztof Matyjaszewski,‡ and Anna C. Balazs*,† †
Chemical Engineering Department, University of Pittsburgh, Pittsburgh, Pennsylvania 15261, United States Department of Chemistry, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, United States
‡
S Supporting Information *
ABSTRACT: With newly developed computational approaches, we design a nanocomposite that enables self-regeneration of the gel matrix when a significant portion of the material is severed. The cut instigates the dynamic cascade of cooperative events leading to the regrowth. Specifically, functionalized nanorods localize at the new interface and initiate atom transfer radical polymerization with monomers and cross-linkers in the outer solution. The reaction propagates to form a new cross-linked gel, which can be tuned to resemble the uncut material. KEYWORDS: Self-regeneration, polymer nanocomposites, atom transfer radical polymerization, dissipative particle dynamics
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regrowth. Second, the system must involve a means of continuing or propagating the desired growth. Finally, when the material reaches a certain size, there must be a means of terminating the ongoing reactions. By developing new computational models, we design a self-regenerating material that undergoes these processes to produce a remarkable form of regrowth when a significant portion of the system is removed. Figure 1a reveals the components of the uncut material: nanorods functionalized at one end and dispersed in a polymer gel, which is swollen in a good solvent. The solvent is weakly incompatible with the rods, but compatible with the endgrafted chains (in yellow), making the functionalized rods essentially amphiphilic. The rods contain initiator sites (in magenta) for the atom transfer radical polymerization (ATRP),7−10 which is a form of controlled/living radical polymerization.11,12 The synergy between these specific components enables a dynamic cascade of events that leads to gel regeneration when the sample is cut with the rods serving to trigger the ensuing, vital reactions. Finally, the surrounding simulation box is periodic in x and y but bounded by top and bottom hard walls (brown beads).
n elusive goal in materials science is designing systems that mimic the remarkable ability of amphibians to regrow limbs. While self-healing materials1−4 can mend local defects, there are virtually no examples of materials that can regenerate themselves. The advent of such regenerative materials could dramatically extend the useful lifetime of manufactured products. Through new computational models, we design a nanorod-filled gel that effectively regenerates the gel matrix when a layer of the material is sliced-off. With this layer removed, the nanorods diffuse to the newly formed interface and extend into the outer solution, which contains monomers and a small fraction of cross-linkers. Polymerization initiated from the rods’ surfaces leads to chains that become cross-linked to form a new gel that resembles the severed layer. After the initial cut, the regeneration requires no external intervention; synergistic interactions among all components in this system enable the vital processes leading to regrowth, which could be repeated with subsequent cuts. In biology, tissue regeneration5 is guided by an internalized “instruction set” that involves signaling molecules that mediate the vital processes in the regrowth: initiation, propagation, and termination.6 Analogous to biological systems, fully synthetic self-regenerating materials should incorporate the following elements. First, the system must encompass a component that not only senses the removal of material but also initiates the © XXXX American Chemical Society
Received: October 15, 2013 Revised: November 12, 2013
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Figure 1. System components and model validations. (a) Equilibrium state of a nanorod-filled polymer gel in a good solvent (not displayed). Green chains represent polymer strands, and brown beads form amorphous solid walls. Top left: enlarged view of initiator (magenta beads)-modified nanorod (white beads) with one end functionalized with polymer chains (yellow beads) and a cross-section of the hexagonal nanorod. Bottom left: initial configuration of the tetra-functional polymer gel network. (b) Polymer volume fraction as a function of temperature for a spherical gel, showing a continuous volume phase transition around 32 °C. Insets show gel at swollen and collapsed states at 12 and 56 °C, respectively. Hence, the simulation reproduces the LCST behavior of PNIPAM. (c) Schematic of the major reactions in living copolymerization of monomer and crosslinker. Asterisk indicates active radical. Bead colors represent following species: initiator (magenta), monomer (blue), cross-linker (orange). Open beads show unreacted or partially reacted species and filled beads show fully reacted species. Probabilities of initiation (Pir), propagation with p,X monomer (Pp,M r ), propagation with unreacted bifunctional cross-linker (Pr ), and cross-linking with partially reacted cross-linker with pendent p,X p,M functional group (Pp,P r ) can be varied independently. Values of Pr are twice greater than those of Pr , unless stated otherwise. (d) Comparison of DPD simulated gel points and ATRP experimental values for systematic variation of the initial concentration ratios of cross-linker to initiator, [X]0/ [Ini]0.
and r̂ij = rij/|rij|; all the relevant aij values are listed in Supporting Information Table SI1. The drag force is FDij = −γωD(rij)(r̂ij · vij)r̂ij, where γ is a simulation parameter related to the viscosity arising from the interactions between the beads (polymer− solvent, polymer−polymer, and solvent−solvent), ωD is a weight function that goes to zero at rc, and vij = vi − vj. The random force is FRij = σωR(rij)ξijr̂ij, where ξij is a zero-mean Gaussian random variable of unit variance and σ2 = 2kBTγ. The value of γ is chosen to ensure relatively rapid equilibration of the temperature in the system and the numerical stability of the simulations for the specified time-step.15 Finally, we use ωD(rij) = ωR(rij)2 = (1 − rij)2 for rij < 1.15 Because all three of these forces conserve momentum locally, hydrodynamic behavior emerges even in systems containing only a few hundred particles.13−15 The equations of motion are integrated in time with a modified velocity-Verlet algorithm. The factor kBT is taken as the characteristic energy scale, where kB is the Boltzmann constant. Because our simulations involve temperature variations, we chose room temperature as the reference, setting kBT0 = 1, where T0 = 298.15 K. The reduced temperature is therefore defined as T* = T/T0. The characteristic time scale is then defined as τ = (mr2c /kBT0)1/2 = 1. The remaining simulation parameters are γ = 4.5 and Δt = 0.02 τ with a total bead number density of ρ = 3.15 Here, we expand the DPD approach in two distinct ways. First, we modify the model to capture the response of chemically cross-linked gels to variations in temperature (as detailed in Supporting Information SI1) and second, we develop a scheme to simulate the living copolymerization of
This system is modeled using dissipative particle dynamics (DPD),13−15 which can be viewed as a coarse-grained molecular dynamics (MD) method. Relative to MD, one can examine larger systems for longer times in computationally realistic timeframes, making DPD an effective mesoscale method for simulating our complex system. Currently, however, there are no DPD approaches that explicitly incorporate the temperature-dependence of the polymer−solvent interactions in chemically cross-linked gels, making it difficult to realistically model gels characterized by specific values of the polymer− solvent interaction parameter, χps(T). Furthermore, there are no DPD models for the ATRP process. Consequently, we built on the method described below to develop the new approaches detailed in Supporting Information SI1−SI3. Dissipative particle dynamics is a mesoscopic particle-based computational method that provides an effective means of simulating the dynamic behavior of complex fluids and multicomponent mixtures.13−15 Unlike more atomistic MD simulations, a DPD bead represents clusters of molecules (see Supporting Information SI1D). Similar to MD simulations, DPD captures the time evolution of a many-body system through the numerical integration of Newton’s equation of motion, mdvi/dt = fi. The force acting on a bead consists of three parts, each of which is pairwise additive: fi(t) = ∑(FCij + FDij + FRij ), where the sum runs over all beads j within a certain cutoff radius rc. We take rc as our characteristic length scale and set the dimensionless value as rc = 1. The conservative force is a soft, repulsive force given by FCij = aij(1 − rij)r̂ij, where aij is the maximum repulsion between beads i and j, rij = ri − rj, rij = |rij|, B
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Figure 2. Nanorods diffusion and extension into outer solution. (a) Initial configuration of the gel after the upper layer was cut off. (b−e) Snapshots of the system when the rods diffuse and extend into the outer solution, taken at the following times: (b) t = 2500, (c) t = 10 000, (d) t = 23 700, and (e) t = 34 300. Numbers indicate the rods that extend into the outer solution. (f) Time evolution of the center of the mass of the four rods in the zdirection. Letters b−e correspond to the respective snapshots on the left. (g) Number density profiles of the system after nanorods have diffused into the outer solution. The weakly incompatible outer and inner solutions form an interface located at z ≈ 33.
Figure 3. Process of regenerating the cutoff layer. (a−e) Regrowth of the top layer at the following monomer conversions: (a) 0, (b) 25, (c) 50, (d) 75, and (e) 96%. Initial monomer and cross-linker concentrations are [M]0 = 30% and [X]0 = 0.53%. Initiator density σi is 0.25, which corresponds to a concentration [Ini]0 = 0.27%. (f−j) Top-down views corresponding to frames (a−e). Blue beads are the newly formed polymer chains. (k) Dependence of conversions and ln([M]0/[M]) of monomer (solid lines) and ln([X]0/[X]) of cross-linker (dashed lines) on reaction times during copolymerization, where [M] and [X] are the respective current concentrations of unreacted monomer and cross-linker. (l) Number density profiles of the newly formed (blue) gel at different monomer conversions. Black dashed line represents the corresponding number density profile of the original (green) gel at t = 3 × 104.
an angle potential between two consecutive bonds. Taking χps values from experimental data for poly(N-isopropylacrylamide) (PNIPAM),17 our model accurately reproduces the continuous volume phase transition occurring between 30 and 35 °C (Figure 1b), capturing the fact that PNIPAM exhibits a lower critical solution temperature (LCST) phase transition.17,18 Notably, our results for the polymer concentrations at the swollen and collapsed states are consistent with the respective experimental values17,18 (see Supporting Information SI1). This
monomer and cross-linker (as detailed in Supporting Information SI3). We validate these modifications via comparison with prior experimental studies. With respect to capturing the temperature-dependent behavior of the gels, the bottom image in Figure 1a shows that at the onset of the simulations, the tetra-functional network is arranged in a diamond-lattice structure.16 The polymer strands between the cross-links are modeled by a sequence of N = 30 DPD beads connected by harmonic springlike bonds with C
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Figure 4. Controlling properties of regenerated gel layer. (a,b) Density profiles of the system (including original gel, new gel, rods, and grafted chains) at monomer conversion 95% with (a) different initial monomer concentrations [M]0 for initiator density on the rod surface σi = 0.25 ([Ini]0 = 0.27%) and for initial cross-linker concentrations [X]0 varying with [M]0 by fixing [M]0/[X]0 ≈ 56; (b) with different initial densities σi ([Ini]0 varies from 0.09 to 0.54%) for [M]0 = 30% and [X]0 = 0.53%. Insets are the corresponding snapshots of the gel system in (a) with [M]0 = 20% and [M]0 = 50%, and in (b) with σi = 0.0834 and σi = 0.5. (c) Top-down view of the spatial distribution of the cross-links at monomer conversion 96% for the regenerated gel system with σi = 0.25 and [M]0 = 30% (corresponding to simulation in Figure 3e,j). Red beads are the inter-rod cross-links connecting chains grown from different rods and black beads are the intrarod cross-links connecting chains from the same rod. Smaller, differently colored beads represent the chains emanating from different rods. (d) Fraction of the number of inter-rod cross-links (Xinter‑rod) with respect to the total number of cross-links formed (Xtotal) as a function of initiator density and initial monomer concentration. Data are taken at cross-linker conversion of at least 96%. Error bars indicate the variations among four independent runs.
is the first DPD simulation to capture temperature-induced volume phase transitions in gels. The ATRP is a controlled/living polymerization process that leads to uniform chain growth. In the presence of cross-linkers, this process leads to gel formation.19−22 The relevant steps in this living copolymerization are shown schematically in Figure 1c, indicating the initiation, propagation, and cross-linking of the growing chains. The DPD scheme corresponding to Figure 1c is detailed in Supporting Information SI3. By using the latter scheme, we reproduce the experimentally observed23 dependence of the gel point in a bulk sample on the ratio between cross-linker and initiator concentrations, [X]0/[Ini]0 (Figure 1d). (This dependence marks a difference between ATRP23−25 and conventional free radical polymerization.) The excellent agreement between the experimental and simulation data provides validation for our newly formulated DPD approach. Using the above approaches, we probe the cascade of events that occur when the upper half of the nanorod-filled gel is removed (see Figure 2 and Supporting Information Video SI1). With this dramatic cut, the system is now exposed to the outer fluid, which is both weakly incompatible with the solvent in the gel and compatible with the rods, and contains dispersed monomer and a small concentration of cross-linking molecules. (Figure 2g highlights the interfacial region between the outer and inner fluids.) The amphiphilic rods act as crucial sensors, indicating a change in the state of the system. Specifically, rods near the new interface encounter the more compatible outer solvent and hence, diffuse toward this boundary to maximize their interaction with this fluid. Figure 2b−e illustrate the rods’ progression to and final orientation at the cut, and Figure 2f
shows the rods’ center of mass as a function of time. Because the chains on the rods are attracted to the gel and incompatible with the outer solution, they prevent these particles from diffusing out of the gel matrix. Hence, the grafted chains anchor the nanorods at gel-outer solvent interface. Some rods do remain buried within the gel (Figure 2e,f). The localization of the rods at the interface depends on their initial proximity to this boundary (see Supporting Information SI4); furthermore, steric hindrance from rods already at the interface can inhibit particles deeper in the matrix from reaching the cut. With rods at the interface, the initiator sites now become exposed to the monomer-laden solution. Hence, the system encompasses one of the critical events for regeneration: the initiation process. Importantly, chain growth in ATRP is significantly slower than the diffusion rate of rods to the interface.26,27 Thus, the polymerization primarily occurs once the rods are at the cut (not within the gel’s bulk). Importantly, the high surface area provided by the rods facilitates changes in initiator concentration, which plays a significant role in the structure of the new layer (as discussed further below). Figure 3 shows snapshots of the critical propagation step, as monomers form chains (in blue) from the surface of the rods via the ATRP. As indicated in Figure 1c, the cross-linkers become part of these growing polymers and can bridge two separate neighboring chains, thereby binding the system into a network. For these concentrations of monomer and crosslinker, the rods remain localized at the interface as the blue gel continues to grow (see Supporting Information Video SI2). The final component for biomimetic regeneration is a mechanism for turning off the propagating reaction and thereby D
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buried nanorods to the interface and hence to the repeated regrowth. These subsequent events do, however, require addition of new monomer and cross-linker to the host solution. The regrowth of a severed gel layer outlined here helps pave the way for creating materials that are both self-regulating and self-replenishing,29 leading to systems with improved functionality over significantly extended lifetimes.
regulating the height of the regrowing layer. Here, it is the concentrations of monomer and cross-linker that are the critical variables in the termination process; once these species have reacted, no further growth can occur. Figure 3k shows that with initial concentrations of 30% and 0.53% for the monomer and cross-linker respectively, both species become almost fully converted into the gel within the time scale of the simulation. The red curves (Figure 3k) show that the natural logarithms of the ratios of the initial to unconverted quantities for both monomer and cross-linker display a linear dependence with time (after t = 5000), as would be expected for this first-order reaction. These curves indicate that the model captures the correct kinetics for the reaction. The density profile in Figure 3l reveals that for a fixed initiator concentration, the percent conversion of the monomer plays a vital role in the density of the growing gel; at 38% conversion, the densities of the two gels are quite similar and at 96% conversion, the new gel exhibits a higher density than the original material. Importantly, the properties of the new gel can be tailored to resemble those of the original material by tuning the initiator density on the rod, σi, and initial monomer concentration, [M]0. Increasing [M]0 leads to increases in the density and height of the new layer (Figure 4a). The densities of the two layers can be reasonably matched by the appropriate choice of the initiator concentration, σi (Figure 4b). The density of the new gel is reduced with decreases in σi due to decreases in the number of growing polymer chains per unit volume.28 Notably, the structure of the regenerated gel layer is relatively heterogeneous, exhibiting a high fraction of cross-links between chains on a given rod (Figure 4c,d). While the fraction of interrod cross-links is relatively insensitive to σi (increasing only slightly at [M]0 = 50% with decreasing σi), it increases dramatically with increases in [M]0 for any given value of σi (Figure 4d). To conclude, we emphasize that the nanorods serve two important roles in this system. First, the rods act as sensors that indicate the removal of material from the system. Namely, when a layer of gel is severed, the nanorods near the newly formed interface “sense” the more compatible outer solution and, consequently, diffuse to the cut. Second, the nanorods encompass the initiation sites for the reaction leading to regrowth. The high-aspect ratio of the rods permits control over the concentration of initiator on the rod’s surface, σi; this is important since the density of the regrown gel can be tailored by tuning σi. In summary, the rods sense the damage, migrate to the cut, and thereby deliver controllable quantities of the initiators to the critical location. In this manner, the rods play a crucial role in both the initiation and propagation steps of the regrowth. Notably, the chains grafted to the ends of the rods also serve an important function: they provide a means tailoring the effective adhesion between the newly regenerated and old gels. We also emphasize that analogous to biological processes, the information for the regeneration in our rod-filled gels is inherent to the system; the “popping up” of the rods at the interface provides the signal to initiate the polymerization that leads to the formation of the new gel. The living copolymerization is halted when monomer and cross-linker are consumed, yielding a layer with a specified height, crosslinks, and density of the gel. As some of the rods remain buried in the bulk of the gel, subsequent cuts below the newly regenerated region can lead to localization of the previously
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ASSOCIATED CONTENT
S Supporting Information *
Section S1: detailed description and validation of DPD model for thermo-responsive gel. Section S2: details of modeling nanorod-filled gels. Section S3: detailed DPD scheme and validation of modeling ATPR process. Section S4: detailed discussion of nanorods’ diffusion and extension into outer solution. A video that shows nanorods’ diffusion and extension into outer solution after the upper half of the nanorod-filled gel is removed. A video for the process of regenerating the cutoff layer. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Notes
The authors declare no competing financial interest.
ACKNOWLEDGMENTS The authors gratefully acknowledge the DOE (for partial support of X.Y. for the computational systems) and the ARO (for partial support of O.K. for the analysis). A.C.B. also acknowledges helpful conversations with Dr. David Stepp and Professor Jennifer Lewis.
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REFERENCES
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