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Tuning the Mechanical Properties of Polymer-Grafted Nanoparticle Networks through the Use of Biomimetic Catch Bonds Badel L. Mbanga,† Balaji V. S. Iyer,‡ Victor V. Yashin,† and Anna C. Balazs*,† †

Chemical Engineering Department, University of Pittsburgh, Pittsburgh, Pennsylvania 15261, United States Department of Chemical Engineering, Indian Institute of Technology, Hyderabad, India

‡

ABSTRACT: The ability to precisely tune the mechanical properties of polymeric composites is vital for harnessing these materials in a range of diverse applications. Polymer-grafted nanoparticles (PGNs) that are cross-linked into a network offer distinct opportunities for tailoring the strength and toughness of the material. Within these materials, the free ends of the grafted chains form bonds with the neighboring chains, and tailoring the nature of these bonds could provide a route to tailoring the macroscopic behavior of the composite. Using computational modeling, we simulate the behavior of threedimensional PGN networks that encompass both high-strength “permanent” bonds and weaker, more reactive labile bonds. The labile connections are formed from slip bonds and biomimetic “catch” bonds. Unlike conventional slip bonds, the lifetime of the catch bonds can increase with an applied force, and hence, these bonds become stronger under deformation. With our 3D model, we examined the mechanical response of the composites to a tensile deformation, focusing on samples that encompass different numbers of permanent bonds, different bond energies between the labile bonds, and varying numbers of catch bonds. We found that at the higher energy of the labile bonds (Ul = 39kBT), the mechanical properties of the material could be tailored by varying both the number of permanent bonds and catch bonds. Notably, as much as a 2-fold increase in toughness could be achieved by increasing the number of permanent bonds or catch bonds in the sample (while the keeping other parameters fixed). In contrast, at the lower energy of the labile bonds considered here (Ul = 33kBT), the permanent bonds played the dominant role in regulating the mechanical behavior of the PGN network. The findings from the simulations provide valuable guidelines for optimizing the macroscopic behavior of the PGN networks and highlight the utility of introducing catch bonds to tune the mechanical properties of the system.

I. INTRODUCTION Polymer-grafted nanoparticle (PGN) networks integrate the desirable features of both polymers and nanoparticles. In particular, the polymer “arms” that interconnect the PGNs (see Figure 1) imbue the material with considerable flexibility, and the solid nanoparticles offer irreplaceable optoelectronic properties. Such PGN networks are a relatively new class of materials,1−7 and hence, there is a need for models to predict how to optimize the performance of these novel composites. A specific challenge is determining how to improve the strength and toughness of the material while maintaining its mutability. We recently developed computational models for “dual crosslinked” PGN networks that are interconnected by both high strength “permanent” bonds and more reactive labile bonds.8−14 Once the labile bonds are broken, they can readily re-form, in contrast to the less reactive permanent bonds, which break irreversibly. Hence, these labile connections allowed the bonds to rearrange and the material to reconfigure when the sample was mechanically deformed. Relative to PGN networks that were solely interconnected by permanent bonds, the introduction of the weaker, more reactive labile bonds improved the ductility and toughness of the material as well as enabled the composite to undergo self-healing.8−14 Notably, © XXXX American Chemical Society

our predictions on the self-healing behavior of PGN networks and the ability of these systems to form stable face-centered cubic (FCC) structures13 have recently been experimentally confirmed.15 In our previous studies,8−13 the permanent and labile bonds between the tethered polymer arms were both modeled as “slip” bonds, for which the bond lifetime decreases with an applied force. There is, however, another class of bonds referred to as “catch” bonds.16−31 A remarkable feature of a catch bond is that the bond becomes stronger or lives longer when subjected to a tensile deformation.17,30 Catch bonds are present in a number of biological systems, including the adhesive ligand−receptor interactions of biological cells (i.e., interactions involving selectin adhesion molecules).17 Certain nonbiological molecules can also exhibit catch bond behavior.32−34 Notably, however, such remarkable catch bond interactions have not been broadly exploited in the design of synthetic materials.32 The introduction of the mechanically resilient catch bonds into PGN networks might provide a unique means to “dial in” the Received: November 11, 2015 Revised: January 27, 2016

A

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material’s 3D mechanical behavior can be “tuned” through variations in C. To thoroughly explore the design space, we also vary the energy of the catch bonds, considering both relatively strong and weak catch bonds. Through these different sets of studies, we specifically examine how to combine the labile catch bonds and permanent bonds to optimize the strength and toughness of the composite. The studies described below can provide guidelines for designing novel mechano-responsive materials whose properties can be improved and controllably tailored through the application of force. These attributes can be enhanced by the presence of catch bonds, which are effectively strengthened by the applied force. As we show below, with the introduction of catch bonds, the applied force can yield PGN networks with significantly greater toughness. In effect, these composites mimic a salient feature of bone remodeling, which leads to the improvement of mechanical properties after deformation. Hence, by tuning both the energy and fraction of catch bonds, we can design mechano-mutable composites that mimic properties exhibited by biological materials systems. In the Methodology section, we describe our model for PGN networks that encompass three types of connections: labile slip, labile catch, and permanent slip bonds. We first set the bond energies of the labile (slip and catch) bonds to Ul0 = 39kBT(see Table 1); these reactive bonds can re-form when they have been broken. We then reduce the bond energy of the labile bonds to Ul0 = 33kBT to determine the role that the strength of the reactive, re-formable bonds has on the performance of PGN networks containing the mechano-responsive catch bonds. We chose these labile bond energies because they lie within the range that is relevant to disulfide bonds,35,36 which can undergo

Figure 1. (a) Schematic of a 3D PGN network under tensile deformation. (b) Close-up view of a pair of interacting PGNs. Upon overlap of their coronas, the reactive groups at the ends of the chains can form labile slip bonds (yellow), labile catch bonds (red), or permanent bonds (green).

desired mechanical properties by tuning the fraction of these bonds in the material. Herein, we test the above hypothesis by simulating threedimensional (3D) PGN networks with different fractions of catch bonds, C, and determining the material’s response to tensile deformation. In previous two-dimensional (2D) simulations,14 we specifically examined how the mechanical properties of the PGN network were affected by replacing all the labile slip bonds with catch bonds. By augmenting our computational framework, we can now investigate how the Table 1. Model Parameters Used in Simulations

dimensional units r0 = 50 nm T0 = 1.41 × 10−2 s v0 = r0T0−1 = 3.55 μm/s F0 = 2.98 pN F0r0 = 89.74 kJ/mol

length: nanoparticle radius time, t velocity, v force, F toughness, W PGN characteristics corona thickness Kuhn length, lp number of grafted arms, f arm contour length, L chain spring constant, κ0 repulsion parameter, σ cohesion parameters mobility of PGN, μ pulling velocity, v bond parameters rupture rate at F = 0, k0r Ul0

bond energy = 33kBT slip bond catch bond, state 1 catch bond, state 2 bond energy, Ul0 = 39kBT slip bond catch bond, state 1 catch bond, state 2 bond energy, Up0 = 45kBT permanent bond

H = 0.75r0 1 nm 156 8.89r0 7.81 × 10−2F0r0−1 3.02r0 U(0) coh = 60kBT; A = 1.15σ; B = 0.08σ 0.57v0F0−1 10−3v0 formation rate, k0f bond sensitivity, γ transition sensitivity, γI→F

transition rate at F = 0, k0I→F

6.55 × 10−4T0−1 6.55 × 10−4T0−1 6.55 × 10−4T0−1

0 6.55 × 10−4T0−1 1.965 × 10−3T0−1

30T0−1 [k21/(k12 + k21)]30T0−1 [k12/(k12 + k21)]30T0−1

6F0−1 6F0−1 2F0−1

6F0−1 6F0−1 1.6F0−1

1.62 × 10−6T0−1 1.62 × 10−6T0−1 1.62 × 10−6T0−1

0 1.62 × 10−6T0−1 4.86 × 10−6T0−1

30T0−1 [k21/(k12 + k21)]30T0−1 [k12/(k12 + k21)]30T0−1

6F0−1 6F0−1 2F0−1

6F0−1 6F0−1 1.6F0−1

4.03 × 10−9T0−1

0

0

6F0−1

6F0−1

B

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Macromolecules the exchange and “reshuffling” reactions that enable the material to exhibit structural rearrangements.37 While both these bond energies allow the material to undergo dynamic restructuring in response to tensile deformation, we previously found11 that these two values resulted in substantially different mechanical behavior and thus allowed us to characterize 39kBT and 33kBT as “strong” and “weak” labile bonds, respectively. The energy of the permanent bonds is held at 45kBT. By fixing these bond energies, we can focus on the effects of varying the fraction of catch bonds in the network.

mean-square end-to-end chain length, with lp being the persistence length. The number of bonds Nb between two PGNs as a function of time t is determined by the kinetics of the permanent and labile bond rupture and the labile bond re-formation. For “slip” bonds, the evolution of Nb is described by the equation dNb = −k r(R )Nb + k f Pc(R )[Nmax(R ) − Nb]2 dt

The first term on the right-hand-side (rhs) of eq 5 is the rate of bond rupture, where kr(R) is the rupture rate of an individual bond connecting two PGNs separated by the distance R. According to the Bell model43 for slip bonds, the rupture rate increases with force as kr = k0r exp(γ0F), where k0r is the bond rupture rate at F = 0 and γ0 characterizes the bond sensitivity to the applied force. The rupture rate at zero force depends on the bond energy U0 as k0r = v0 exp(−U0/kBT), where ν0 is the characteristic frequency of the molecular vibration. The rate of formation of slip bonds is described by the second term on the rhs of eq 5, where kf is the formation rate constant, which is assumed to be independent of the applied force F. The rate of the labile bonds formation depends on the interparticle separation R through the probability of contact between two chains’ ends on two neighboring PGNs, Pc(R), and the maximum number of bonds that can be formed between the two particles Nmax(R). Note that kf = 0 for the permanent bonds since they do not re-form after they have ruptured. Our model for characterizing the observed behavior of catch bonds was detailed in ref 14, and hence, here we provide just a brief description of our formulation. In particular, we employ a two-state model, which attributes the catch bond behavior to the existence of two conformational states. These states differ in their sensitivity to the applied force, F, and the rates of transition between the two states are controlled by F.18−20 Hence, among Nb catch bonds between two PGNs, N(1) b bonds are in conformational state 1, and N(2) b bonds are in state 2, so (2) that Nb = N(1) b + Nb . The catch bonds in states 1 and 2 rupture with the force-dependent rates k1 and k2, respectively. Transitions from state 1 to state 2 and the reverse transitions occur with the respective rates k12 and k21 that also depend on the force F. Upon breakage, the catch bonds can re-form in state 1 with the rate k(1) or in state 2 with the rate k(2) f f . The evolution of the number of catch bonds in the two states is described by

II. METHODOLOGY The PGN networks are composed of rigid nanoparticles that are interconnected by flexible polymer chains, which are endgrafted to the particles’ surfaces (Figure 1). The particles are monodisperse of radius r0 and contain f grafted polymer chains. The polymer chains are assumed to be in the semidilute regime and are swollen in a good solvent; they form a corona of thickness H = qr0 around the particle. Here, all length scales are expressed in units of r0, and thus, the corona thickness is simply H = q. The reactive groups at the free ends of the grafted chains can form chemical bonds with the ends of chains on neighboring particles. The chemical bonds are either “labile” or “permanent”. The reactive labile bonds can re-form after they have ruptured, whereas the stronger, permanent bonds break irreversibly. The interaction potential between two PGNs is written as Uint = Urep + Ucoh + Ulink. The first term describes the repulsion between the corona chains:38 Urep kBT

=

5 3/2 f 18

1/2 ⎧ /2)−1 , R ≤ σ ⎪− ln(R / σ ) + (1 + f ×⎨ ⎪ −1 1/2 1/2 ⎩(1 + f /2) (σ /R ) exp[− f (R − σ )/2σ ],

R>σ

(1)

where f is the number of polymer arms, R is the interparticle distance, and the range of the potential is39,40 σ = 2(1 + q)(1 + 2f−1/2)−1. The second tem, Ucoh, is a pseudopotential that captures the attractive cohesive interaction between the nanoparticles:41 (0) Ucoh(R ) = −Ucoh {1 + exp[(R − A)/B]}−1

(2)

The cohesive attraction is constant at small interparticle distances R and balances the repulsion at the corona edges; this interaction allows the neighboring coronas to overlap and form bonds. The energy scale of cohesion is set by the parameter U(0) coh, and A and B are the respective position and width of the attractive well. The term Ulink is the attractive interaction between two particles linked by Nb bonds, and hence, the attractive force between two linked particles is Flink(r ) = Nbκ(r )r

dN b(1) = −(k1 + k12)N b(1) + k 21N b(2) + k f(1)Pc(R ) dt × [Nmax(R ) − (N b(1) + N b(2))]2

(6a)

dN b(2) = −(k 2 + k 21)N b(2) + k12N b(1) + k f(2)Pc(R ) dt

(3)

where r = R − 2 is the end-to-end distance of polymer chains connecting the particles, and κ(r) is the stiffness of the springlike force due to the chain extension. The spring stiffness is obtained from the wormlike chain model42 and is calculated as κ(r ) = kBTR 0−2{1 + 2[1 − r 2(2L)−2 ]−2 }

(5)

× [Nmax(R ) − (N b(1) + N b(2))]2

(6b)

The rupture rates in states 1 and 2 are calculated according to the Bell model as k1 = k0r exp(γ1F) and k2 = k0r exp(γ2F), respectively. The force sensitivities γ1 and γ2 are chosen such that state 2 is less sensitive to the applied force than state 1. Specifically, we use γ2 = γ1/3 in the simulations. The transition rates between the two conformation states are calculated as k12 = k0r exp(γ12F) and k21 = 3k0r exp(γ21F) at γ12 > γ21. Hence, at

(4)

Here, 2L is the contour length of the chain resulting from the bonding of two polymer chains of length L, and R0 = 4lpL is the C

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permanent bonds. The resulting sample is further equilibrated for 104T0 during which labile bonds can either break or form. Following the setup and equilibration, the sample is subjected to a strain-controlled deformation, in the course of which one end of the sample (8 × 8 particles) is held fixed, and the other end is pulled along the long axis of the sample at a constant velocity of v = 3.55 nm/s. The tensile strain, i.e., the ratio of the sample elongation to its initial length, and the force on the sample’s moving edge is calculated and recorded. The sample is thus extended until it breaks into two separate parts, and the strain at break εb is recorded. The toughness W is calculated by integrating the force versus strain curve up to the break point and dividing this value by the number of particles in the sample, N. For each set of parameters, the data are obtained by averaging over eight independent simulations. Before discussing the results from these studies, we note that recent experiments15 have validated findings that have emerged from our computational studies involving this model. Namely, Williams et al. have shown that these PGN networks can undergo significant self-healing behavior, confirming our prior predictions.12 They have also confirmed our prediction that the PGN networks can form stable FCC superlattices.13

zero force, 75% of the catch bonds are in the more forcesensitive state 1. When the catch bonds are formed, these bonds are partitioned between the two conformation states according to the equilibrium partitions. In particular, the bond formation = kfk21(k12 + k21)−1 and k(2) = rates are calculated as k(1) f f −1 kfk12(k12 + k21) , where the rate constant of catch bond formation kf does not depend on the force. The rates of catch (2) bond formation in the specific conformation states, k(1) f and kf , depend on F through k12 and k21. We model the response of the PGN network to an applied external force Fext by solving the equations of motion in the overdamped regime. The dynamics of each particle is described by the equation dx/dt = μFtot, where Ftot = −∂Uint/∂x + Fext is the net force acting on the particle and μ is the particle mobility. A two-step approach is used to solve the equations of motion. First, we use an explicit Euler scheme to solve eq 5 for the permanent and labile slip bonds and eq 6 for the labile catch bonds to determine the number of bonds between the pairs of PGNs, Nb(t), at the moment of time t. Since Nb(t) is a real number, a rounding procedure8 is used to obtain integer values for the number of bonds. The determined numbers of bonds are used to calculate the springlike attraction forces between the particles according to eq 3. Then, the forces due to the repulsive and attractive cohesive interactions are calculated, and all forces are summed up to obtain Ftot for each particle. The subsequent step consists of numerically integrating the equation of motion using the Euler scheme. The time step of 10−2T0, where T0 is the unit of time of the simulation (see Table 1), is used for both steps of the simulation. The model parameters used in the simulations are collected in Table 1. In particular, the radius of a nanoparticle is r0 = 50 nm, the number of end-grafted chains on each particle is f = 156, and the thickness of the corona formed by the chains is q = 0.75. The parameters of the cohesive attractive potential are U(0) coh = 60kBT, A = 1.15σ, and B = 0.07σ, where σ is the range of the repulsive potential (eq 1). The mobility of a PGN is calculated as the inverse Stokes drag, μ = [6πηr0(1 + q)]−1, experienced by a particle of radius (1 + q)r0 in a fluid of viscosity η = 0.894 Pa·s (viscosity of glycerol). The energy U0 of the permanent bonds is 45kBT, and labile bonds are assigned the energy of 39kBT or 33kBT. The fraction of catch bonds is varied in the range 0 ≤ C ≤ 1, where C = 0 corresponds to a system in which all labile bonds are slip bonds and C = 1 to a system is which all labile bonds are catch bonds. Upon initialization, all bonds are labile slip bonds; we then randomly select a linked particle pair and reassign the bonds linking the particles to the labile catch type. This is done iteratively until the fraction of catch bonds in the sample reaches C. Further, we introduce the permanent bonds into the system and vary the average number of permanent bonds between a pair of linked particles at 0 ≤ P ≤ 1. The simulation setup consists of a rectangular parallelepiped composed of N = 896 PGNs. The system is initialized such that the PGNs form a face-centered cubic (FCC) structure of 14 × 8 × 8 particles along the respective length, height, and width dimensions. After holding the sample in this configuration for 4 × 103T0 to allow for the formation of labile bonds, we perform an equilibration of the sample for 6 × 103T0, during which overextended labile bonds break, and new bonds are re-formed. We subsequently add permanent bonds to the system by converting some randomly selected established labile bonds to

III. RESULTS AND DISCUSSION In the ensuing simulations, we systematically vary the number of labile catch bonds in the PGN network. We first consider the behavior of the stronger labile bonds Ul0 = 39kBT and then turn our attention to effects that emerge when this bond energy is reduced to Ul0 = 33kBT. For both these cases, we first examine the mechanical properties of the system in the absence of permanent bonds (P = 0) and then focus on changes that occur with the introduction of P = 0.5 and P = 1 permanent bonds between the linked PGNs. Through these different studies, we not only pinpoint the effect of varying Ul0 but also uncover the role of the permanent bonds at the different energies of the labile bonds. A. Behavior of the System at Ul0 = 39kBT. 1. No Permanent Bonds (P = 0). To illustrate the utility of introducing the catch bonds, we show snapshots of samples that are interlinked with purely slip bonds (C = 0) (Figure 2a,b) or with purely catch bonds (C = 1) (Figure 2c,d) and subjected to a tensile deformation. Both these samples do not encompass any permanent bonds (P = 0). The snapshots show

Figure 2. Snapshots of two PGN networks undergoing tensile deformation: (a, b) the PGNs are linked by labile slip bonds (C = 0); (c, d) the PGNs are linked by labile catch bonds (C = 1). Samples comprising catch bonds (d) can extend substantially further than their slip bonds counterparts (b). D

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Macromolecules the samples near the yield point (ε ≈ 0.5), and before the failure events, specifically, at 90% of the strain at break, εb. Comparison of the snapshots shown in Figures 2b and 2d clearly reveals that the catch bonds can dramatically improve the ductility of the composites. To quantify the response of these materials to deformation, in Figure 3a we plot the force versus strain for samples at C = 0,

Figure 4. Tensile properties of the PGN networks at the labile bond energy of Ul0 = 39kBT as functions of the fraction of the catch bonds C: (a) strain at break εb, (b) maximal force F*, and (c) toughness W. Line colors correspond to the permanent bond contents P = 0, 0.5, and 1 as indicated in the figures.

highlighted in Figure 4b, which shows the tensile strength, F*, as a function of C. The value of F* corresponds to the highest force that the material can sustain before the material fractures. As can be clearly seen, increasing C leads to greater values of F*. For C = 1, the shape of the curve in Figure 3a and the larger value of F* indicate that this composite has a greater tensile strength than the other samples. These changes in both εb and F* with variations in C are reflected in the fact that the toughness, W, depends on the value of C, as shown in Figure 4c. (Recall that W is determined by integrating the force−extension curve and dividing the resulting work by the number of nanoparticles in the sample, N.) Specifically, W shows a monotonic increase with increases in C, undergoing an essentially 3-fold increase in going from C = 0 to C = 1 for the P = 0 scenarios. Taken together, the plots in Figures 3 and 4 for P = 0 clearly reveal that the systematic increase in the fraction of catch bonds leads to directly proportional increases in εb, F*, and hence W and, thus, to significant improvements in the strength and toughness of the material. In other words, in the absence of the high-strength permanent bonds, the catch bonds provide significant structural reinforcement of the composite. Furthermore, the composite’s mechanical behavior can be tailored by tuning the value of C. To understand how the dynamics of the interparticle labile bonds affect the properties of the PGN network, we compare the network connectivity for the C = 0 and C = 1 samples (at P = 0) when the materials are strained to 90% of the strain at break. For this purpose, in Figure 5 we plot the histograms of the number of links in the samples shown in Figures 2b and 2d as functions of cos2 θ, where θ is the angle between the direction of a link and the x-axis, the direction of stretching. (A link between two nanoparticles consists of one or more pairs of

Figure 3. Force−strain curves obtained during tensile deformation of the PGN networks at the labile bond energy of Ul0 = 39kBT and the permanent bond contents of (a) P = 0, (b) P = 0.5, and (c) P = 1. The curve colors correspond to the fraction of catch bonds C = 0, 0.5, and 1 as indicated in the figures. The shading about the lines represents the error bars. Note that the figure shows the range of the force−strain response obtained for eight independents runs, before averaging (see Figure 4); the curves serve mainly as a guide to the eye. We chose this way of presenting the raw data in order to illustrate the range of deformations (strain) that these samples can sustain.

0.5, and 1.0 for P = 0. The material fractures at the highest value of strain in the curve. (Note that eight independent runs were performed for each value of C; the highest value of the strain in Figure 3a is the strain at which the most extended of the eight samples broke.) As anticipated from the images in Figure 2, the strain at break (εb) is essentially a factor of 2 greater at C = 1 than at C = 0. Notably, for C = 0.5, the value of εb lies between these two extreme cases, indicating that for P = 0 materials, the ultimate failure of the composite can be forestalled by increasing the fraction of catch bonds in the system. This observation is confirmed by the plot in Figure 4a, which shows a monotonic increase in εb with increases in C, the fraction of catch bonds. It is also apparent from Figure 3a that the C = 0 and 0.5 plots show a sharp decrease after the yield point, which marks the onset of plastic deformation. The C = 1 curve, however, displays a broad plateau and even a slight increase in force with an increase in strain after the yield point. These pronounced variations in force with increasing catch bonds content are E

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increases after the yield point, indicating an improvement in the tensile strength relative to the C < 1 cases in Figure 3. Figure 4a reveals, however, that the value of the strain at break, εb, is relatively insensitive to the value of P > 0. In particular, increasing the fraction of labile catch bonds has a more pronounced effect on the material’s strength (as measured by εb) than increasing the number of permanent bonds between the particles. Recall that the bond energy for all the labile bonds in the network (slip and catch) is fixed at Ul0 = 39kBT. At this bond energy, the behavior of εb does not show a significant dependence on the value of P because the primary reinforcement in this system is provided by the catch bonds, rather than the permanent bonds (which break at the high strains and cannot re-form). This behavior is evident from Figure 6, which shows the number of catch (Figure 6a) and

Figure 5. (a) Orientational distribution of the interparticle links for the samples shown in Figure 2b (red) and Figure 2d (blue). The sample strains correspond to 0.9εb. (b) Magnified region of the distributions in (a) around the direction of stretching cos2 θ = 1.

grafted polymer chains bound through the reactive end chains.) Note that the undeformed PGN networks exhibit FCC symmetry, and hence, the initial histograms show two distinct peaks: one at 0, corresponding to links that lie perpendicular to the stretching direction (along the y- and z-axis), and one at 0.5, corresponding to links oriented at 45° with respect to the stretching direction. Figure 5 indicates that at 0.9εb the most pronounced difference between the C = 0 and C = 1 cases is the relatively large number of links around cos2 θ = 1 for the sample that is solely interlinked by catch bonds (C = 1). Since the lifetime of the catch bonds increases with force (up to εb), there is both sufficient time and a sufficient number of unbroken bonds for these long-lived bonds to become aligned with the stretching direction. On the other hand, if the network contains only the slip bonds (C = 0), the majority of these bonds rupture before they have time to completely reorient as they are being pulled along the x-direction. The differences in Figure 5 help explain the underlying mechanisms that give rise to the greater ductility (and toughness) that can be achieved through the presence of the catch bonds. 2. Introducing Permanent Bonds (P ≠ 0). The mechanical properties of the material can be furthered tailored by integrating both permanent and catch bonds into the PGN network. The parameter P represents the average number of permanent bonds between any two linked nanoparticles. Hence, P = 1 indicates that on average, there is one permanent bond between any two nanoparticles in the network that are linked at t = 0. The force−strain curves in Figures 3b and 3c clearly show that setting P > 0 leads to an increase in the tensile strength, F*, or, in other words, produces an increase in the force needed to rupture the composite. This observation is summarized in Figure 4b, which shows F* versus C for P = 0, 0.5, and 1. Clearly, F*increases as P is increased. Notably, F* also increases with increasing C at all P considered here. It is particularly notable that at C = 1 for all these P values, the force

Figure 6. Average numbers of (a) catch and (b) permanent bonds per particle as functions of tensile strain ε for the PGN networks at the labile bond energy Ul0 = 39kBT and the catch bond fractions C = 0.5 (solid lines) and C = 1 (dashed lines). Line colors correspond to the permanent bond contents P = 0, 0.5, and 1 as indicated in the figures. The vertical lines about the curves represent the error bars.

permanent bonds (Figure 6b) per particle as a function of strain for three values of P. The number of catch bonds per particle is significantly higher than the number of permanent bonds per particle at both C = 0.5 and C = 1.0 at all levels of strain. In contrast to the strain at break, the toughness does depend on the value of P (Figure 4c). This dependence can be understood most clearly by comparing the force−strain plots in Figure 3 for the different values of P as well as examining Figure 4b. As P is increased, these plots show (1) an increase in F* and (2) a broadening of the force−strain curves. Both these factors contribute to increasing the area subtended by the curve and, hence, to the increase in toughness. Thus, for 0 < C ≤ 0.5, an increase in P can lead to an almost 2-fold increase in toughness. At C = 1, however, the number of catch bonds per particle is sufficiently high that the behavior of the system is relatively insensitive to increases in P (see Figure 6). It is worth noting the distinctive peak in the plots for the number of catch bonds as a function of strain in Figure 6. The initial increase is due to new bonds that are formed in the direction perpendicular to the tensile direction. In particular, as the material is stretched, the bonds can reshuffle, making new F

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Macromolecules connections as bonds in the lateral direction are broken. (Similar behavior is also seen for the labile slip bonds.) B. Behavior of the System at Ul0 = 33kBT. 1. No Permanent Bonds (P = 0). To highlight the role that the energy of labile bonds (slip and catch) plays on the behavior of the system, we reduce this value to Ul0 = 33kBT. The reduction in bond energy has a dramatic effect on the force as a function of strain, particularly at P = 0, as can be seen in Figure 7a. In

Figure 8. Tensile properties of the PGN networks at the labile bond energy of Ul0 = 33kBT as functions of the fraction of the catch bonds C: (a) strain at break εb, (b) maximal force F*, and (c) toughness W. Line colors correspond to the permanent bond contents P = 0, 0.5, and 1 as indicated in the figures.

bonds. In other words, at high strains, these stronger permanent bonds help maintain the structural integrity of the material, and a greater force is needed to break these permanent bonds. Again, F* is relatively insensitive to C (Figure 8a). Since the toughness W, i.e., the work-to-break per particle, integrates the features of both εb and F*, we see in Figure 8c that W also depends primarily on P and is relatively insensitive to the value of C. (Again, the large error bars are due to the fact that some fraction of the samples break at small strains.) Finally, Figure 9 reveals that the number of permanent bonds and catch bonds that survive at a high strain are more comparable in value than in the case where Ul0 = 39kBT (Figure 6). These plots further reveal that for P = 0 the catch bonds do not survive beyond even modest levels of strain. The addition of permanent bonds is crucial for postponing the breakage of PGN networks interlinked through weak labile bonds. Taken together, the data indicate that the permanent bonds play a significant role in reinforcing these samples.

Figure 7. Force−strain curves obtained during tensile deformation of the PGN networks at the labile bond energy of Ul0 = 33kBT and the permanent bond contents of (a) P = 0, (b) P = 0.5, and (c) P = 1. The curve colors correspond to the fraction of catch bonds C = 0, 0.5, and 1 as indicated in the figures. The shading about the lines represents the error bars.

essence, the material is so fragile that it breaks even at modest strains for all values of C considered here. This conclusion is reinforced by the data in Figure 8, which shows that the salient mechanical features of the system, εb, F*, and W, exhibit little or no dependence on the fraction of catch bonds in the P = 0 system. In this scenario, the addition of permanent bonds is vital to the mechanical stability of the material. Moreover, as we discuss below, the value of P has a more significant effect on the mechanical properties than the value of C. (Recall that the bond energy of the permanent bonds is set to Up0 = 45kBT.) 2. Introducing Permanent Bonds (P ≠ 0). Figures 7b and 7c reveal that these samples can withstand greater strain when permanent bonds are introduced into the sample. This can be seen clearly in Figures 8a, which reveals that increasing P to either 0.5 or 1 can lead to an almost 4-fold increase in εb. Both Figures 7 and 8a, however, show that increasing the value of C has essentially no effect on εb. (The large error bars in these measurements in Figure 8 reflect the fact that a number of the samples break at relatively small strains.) While P = 0.5 and P = 1 produce comparable values for εb, the value of F*increases with P (Figure 8b). Note that for P = 1 the force F increases with strain beyond the peak at the yield point (Figure 7c) and reflects the resilience of the permanent

IV. CONCLUSIONS A significant challenge for designing composites is establishing effective means of tuning the mechanical properties of the material and, thereby, tailoring the system for a range of applications. The PGN networks provide a particularly “tunable” composite because the strength and toughness of the material could be altered by varying the bonding interaction between the polymer arms that interconnect the material. Here, we introduced biomimetic catch bonds into the PGN networks, exploiting the ability of these bonds to become stronger under a tensile deformation. In particular, the bond lifetime can increase with increasing strain. Our challenge was to determine the G

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effectiveness of these connections within the network. Finally, the findings also highlight the advantage of these simulations in pinpointing optimal parameter ranges for tailoring the performance of these novel materials.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (A.C.B.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS A.C.B. gratefully acknowledges financial support from the DOE EFRC centered at Northwestern University for the development of the analytical model and the AFOSR for the development of the computational approach.

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Figure 9. Average numbers of (a) catch and (b) permanent bonds per particle as functions of tensile strain ε for the PGN networks at the labile bond energy Ul0 = 33kBT and the catch bond fractions C = 0.5 (solid lines) and C = 1 (dashed lines). Line colors correspond to the permanent bond contents P = 0, 0.5, and 1 as indicated in the figures. The vertical lines about the curves represent the error bars.

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optimal set of parameters for exploiting the mechanoresponsive, resilient behavior of these remarkable catch bonds. At the higher labile bond energy of Ul0 = 39kBT, we found that the properties of the material could be tailored by varying both the number of catch bonds and permanent bonds. In particular, the plots in Figure 4 provide guidelines for maximizing the mechanical properties of the system by increasing the number of catch bonds at a fixed value of P or increasing the number of permanent bonds at a fixed value of C. For example, as much as a 2-fold increase in toughness can be achieved by increasing C at fixed P. Comparable increases in toughness can be achieved increasing P at fixed C (e.g., by increasing P from 0 to 1 at C = 0.25). The tensile strength F* can also be tailored through the appropriate combination of C and P. Notably, the stress at break is relatively insensitive to increasing P beyond 0. Nonetheless, our findings show that increasing C can produce up to a 2-fold increase in εb for 0 ≤ P ≤ 1. At Ul0 = 39kBT, the number of catch bonds that survive high strain is sufficiently large (Figures 5b and 6a) that the applied deformation can be harnessed to drive the catch bonds from state 1 to the more force-resistant state 2. Hence, at these bond energies, we can take full advance of the catch bonds in tailoring the mechanical behavior of the system. The situation is quite different when the labile bond energy is reduced to Ul0 = 33kBT. In these systems, the materials rupture at significantly smaller strains than the composites that encompass the stronger labile bonds. The permanent bonds now serve a critical role in resisting the applied deformation and, thereby, contributing to the strength of the material. In particular, εb, F*, and W show at most a weak dependence on the value of C but show a strong dependence on the value of P. The results highlight the versatility of these PGN networks, which can be tailored to display both high strength (εb) and toughness (W). The findings particularly underscore the powerful utility of introducing catch bonds to tune the salient mechanical properties of system. Notably, however, the energy of the catch bonds is a critical parameter that influences the H

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DOI: 10.1021/acs.macromol.5b02455 Macromolecules XXXX, XXX, XXX−XXX