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Modeling polymer grafted nanoparticle networks reinforced by high-strength chains† Matthew J. Hamer,a Balaji V. S. Iyer,a Victor V. Yashin,a Tomasz Kowalewski,b Krzysztof Matyjaszewskib and Anna C. Balazs*a Using a multi-scale computational approach, we determine the effect of introducing a small fraction of high-strength connections between cross-linked nanoparticles. The nanoparticles' rigid cores are decorated with a corona of grafted polymers, which contain reactive functional groups at the chain ends. With the overlap of neighboring coronas, these reactive groups can form weak labile bonds, which can reform after breakage, or stronger bonds, which rupture irreversibly and thus, the nanoparticles are interconnected by dual cross-links. We show that this network can be reinforced by the addition of high-strength connections, which model polymer arms bound together by bonds with energies on the order of 100 kBT. We demonstrate that in the course of these simulations, these high-strength connections can be treated as unbreakable chains. By subjecting networks with a random distribution of the unbreakable chains to tensile deformation at a constant strain-rate, we determine the distribution of strain at break and toughness. With even a small amount of unbreakable chains, the nanoparticle

Received 31st August 2013 Accepted 21st November 2013

networks can survive strains far greater than the networks without these connections. Furthermore, networks containing the high-strength connections tend to form long, thin threads, which enable a

DOI: 10.1039/c3sm52300d

larger strain at break. The findings provide guidelines for creating polymer grafted nanoparticles

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networks that could show remarkable strength and ductility.

I.

Introduction

Natural armor, such as the armadillo's shell, provides useful design concepts for the fabrication of high-strength, lightweight materials. A remarkable feature of the armadillo's shell is the structural hierarchy that enables this outer layer to be not only strong, but also exible.1 Notably, the armadillo's skin is composed of rigid elements that are interconnected by so biopolymers; consequently, this animal can withstand relatively high degrees of mechanical impact, with the outer composite layer providing a means of absorbing and dissipating energy. Based on the effectiveness of this and other natural dermal armor,1 it could be argued that structural hierarchy and exibility are two key components for creating synthetic materials that can emulate the desirable properties exhibited by various biological exemplars. In previous studies,2,3 we developed a computational approach that allowed us to design nanocomposite networks that encompassed these two important components and thus, could establish new design rules for fabricating materials that

a

Chemical Engineering Department, University of Pittsburgh, Pennsylvania 15261, USA. E-mail: [email protected]

b

Department of Chemistry, Carnegie Mellon University, Pennsylvania 15213, USA

† Electronic supplementary 10.1039/c3sm52300d

information

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(ESI)

available.

See

DOI:

are both strong and tough. Notably, researchers have fabricated a range of nanoparticle networks,4–7 with many of the systems displaying promising mechanical properties.4,8–10 Here, we formulate an approach for enhancing this useful mechanical behavior. Namely, we introduce a new structural element into our design scheme that enables the composite to resist significant mechanical deformation and thus, to emulate some of the desirable properties of natural armor. The schematic in Fig. 1a shows the salient features of our hierarchically structured nanocomposite networks, which encompass rigid particles, so interconnecting polymers and chemically reactive end groups. In particular, the fundamental unit in our system is a solid, spherical nanoparticle that is decorated with end-graed polymer chains. The ends of the graed polymers contain reactive groups that can interact with the chain ends on the neighboring polymer graed nanoparticles (PGNs). In our previous studies,2,3 these end groups could form either weak labile bonds, which can reform aer breakage, or strong, “permanent” bonds, which rupture irreversibly. Consequently, the nanoparticles were cross-linked by both labile and permanent bonds; we referred to this system as a “dual cross-linked” nanoparticle network. We previously showed that when this dual cross-linked composite is subjected to a tensile deformation, the permanent bonds form a strong “backbone” that provides structural integrity and strength. On the other hand, the labile bonds

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Fig. 1 (a) Schematic of a pair of cross-linked polymer grafted nanoparticle (PGNs). PGNs with overlapping coronas form bonded chains. Solid red lines with green circles denote the chains bonded through either labile or permanent bonds; dotted red lines with white circles represent free chains; the thick black curve represents an unbreakable chain connecting the two PGNs. (b) Example of the initial configuration of the cross-linked PGNs network generated with the average number of permanent bonds and unbreakable chains per neighboring pair at Pp ¼ 1 and Pn ¼ 0.15, respectively. The core is surrounded by a corona of grafted chains (blue and gray region). Labile bonds are represented by thin colored lines between cores, the unbreakable chains are indicated by thicker black lines, the cores with an unbreakable connection are colored red, and those without an unbreakable connection are colored blue. (c) Initial configuration of the network; only the cores connected by unbreakable chains are shown.

enable broken interconnections to be re-made and hence, permit re-shuffling of the bonds in the network. In particular, as the material is being strained, the labile bonds that are broken can cause the cross-linked nanoparticles to move relative to their original positions, allowing free, reactive end groups to bind to new neighbors. These newly formed bonds prevent the catastrophic failure of the material until the system is subjected to much higher strains. This ability of the material to recongure and form new bonds under deformation enhances the ductility of the system and leads to tough nanocomposites. By optimizing the energies of the labile bonds, we could in fact signicantly improve the toughness of the material. Furthermore, the strain at break (3b) was controlled by the fraction of


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permanent bonds and by varying this fraction, we could also tailor the strength of the material. In an attempt to achieve greater control of the mechanical behavior of these PGN networks and thus, further optimize the robustness of the material, we now introduce a fraction of highstrength interconnections between the nanoparticles (see Fig. 1a). We demonstrate that polymer arms that are bound together by bonds with energies on the order of 100 kBT (i.e., comparable to the energy of carbon–carbon bonds11), can be treated as unbreakable in the course of simulations. In essence, we are enhancing the structural hierarchy within the material, anticipating that these high-strength bonds will improve the ability of the material to resist deformation. By focusing on the structural rearrangements within the material, we show that the addition of just a small fraction of these unbreakable bonds can lead to a new mode of failure, which dramatically improves the mechanical performance of the sample. More generally, the inclusion of the unbreakable bonds provides a powerful means of mitigating the effects of mechanical damage. It is worth noting that our model encompasses salient features of double-network gels,12,13 since we combine strong and weak connections to create a tough and ductile network. The dual cross-linked PGN networks are, however, quite different from the double-network gels, which consist of the interpenetrating networks of stronger and weaker polymer chains that are crosslinked randomly and break irreversibly. Below, we rst describe our computational approach. We then describe the results of simulations that allow us to determine the properties of these cross-linked hybrid materials and uncover attributes that contribute to the superior mechanical response.

II.

Methodology

Our system consists of a swollen network of polymer graed nanoparticles (PGNs). The rigid core of each spherical nanoparticle has a radius r0. The graed polymers form a corona of stretchable chains around the nanoparticles (see Fig. 1a); the thickness of this corona is given by H ¼ qr0. The solvent within the system is assumed to be a good solvent for the polymer chains and the chains within the corona are assumed to be in the semi-dilute regime. In the ensuing studies, all the length scales in the system are given with respect to the core radius r0 and hence, we consider a PGN of core radius unity and a corona thickness q ¼ 0.75. As noted above, the end groups on the graed chains can form labile bonds that can reform aer breakage, or strong, “permanent” bonds that rupture irreversibly. Here, we x the bond energy of the labile bonds to be 37 kBT and that of the permanent bonds is 45 kBT (see Table 1). The bond energies lie in the range relevant to disulde bonds,14–19 which can undergo exchange and “shuffling” reactions that enable the material to exhibit the desired structural rearrangements.20,21 The black line in Fig. 1a indicates the new component in the system: a fraction of unbreakable interconnections between the nanoparticles. These unbreakable linkages represent polymer arms bound by bonds with energies of the order 100 kBT.11 The labile

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Soft Matter Table 1

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Parameters used in the simulations

5 Urep ðRÞ ¼ f 3=2 8 (

Dimensional units


Length: nanoparticle radius Time, t Velocity, v Force, F Toughness, W

r0 ¼ 50 nm T0 ¼ 1.41 102 s v0 ¼ r0T01 ¼ 3.55 mm s1 F0 ¼ 2.98 pN F0r0 ¼ 89.74 kJ mol1

Bond parameters

Labile bond

Permanent bond

Unbreakable chain

Bond energy, U0

37 kBT

45 kBT

Rupture rate at zero force, k0r Formation rate, k0f Bond sensitivity, g0

1.200 105T01 30T01 6F01

4.028 109T01 0 6F01

100 kBT (assumed innite) 5.235 1033T01 0 6F01

Characteristics of polymer graed nanoparticle (PGN) Corona thickness, H Kuhn length, lp Excluded volume parameter, y Number of graed arms, f Arm contour length, L Chain spring constant, k0 Repulsion parameter, s Cohesion parameter, A Cohesion parameter, B Cohesion parameter, C Mobility of PGN, m Pulling velocity, v

0.75r0 1 nm 0.25 156 8.89r0 7.81 102F0r01 3.02r0 1.15s 0.08s 60 kBT 0.57v0F01 0.001v0

and permanent bonds considered here break signicantly earlier and at much lower strains than these high-strength bonds and hence, relative to the 37 kBT and 45 kBT interconnections, the 100 kBT bonds can be considered as unbreakable (see below and the ESI†). As shown in Fig. 1b, these three-dimensional PGNs are arranged in a two-dimensional grid that contains 12 rows, with 15 particles in each row. Fig. 1c shows just the unbreakable chains within the sample (note that at the concentration shown in the gure, these bonds do not form a percolating network). Via this model, we can determine the effect that a small fraction of unbreakable chains has on the mechanical response of the cross-linked network. In the following discussion, we detail the nature of the interactions between the particles. In particular, the interaction between two PGNs is modeled through a sum of interaction potentials and is given by Uint ¼ Urep + Ucoh + Ulink. The rst term characterizes the repulsive interactions between the coated nanoparticles. We assume that the graed coronas are sufficiently thick that the rigid cores do not interact directly through the hard sphere repulsion; rather, the repulsion between the PGNs is due solely to the interaction between the corona chains when two PGNs come into close contact. We describe the repulsion at the center-to-center distance R through the effective pair potential proposed for the interactions between multiarm star polymers:22,23

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1 lnðR=sÞ þ 1 þ f 1=2 =2 ; R # s 1 1 þ f 1=2 =2 ðs=RÞexp½f 1=2 ðR sÞ=2s; R . s: (1) 1/2 1

Here, f is the number of arms, and s ¼ 2(1 + q)(1 + 2f ) is the range of the potential, which is related to the diameter of the last blob in the Daoud–Cotton (DC) model.22,24 The second term in the potential, Uint, describes the attractive cohesive interaction between the coated particles and chosen to have the following form:25 Ucoh(R) ¼ C{1 + exp[(R A)/B]}1

(2)

where C is an energy scale, and A and B are length scales that determine the respective location and width of the attractive well in the potential. The term Ucoh(R) is a pseudo-potential, which is constant for small values of R and balances the repulsion at the corona edges to allow for the overlap between neighboring coronas. The nal term in the potential, Ulink, describes the attractive interaction between the particles linked by the bonded polymer arms. When functional groups at the chain ends of neighboring PGNs form a bond, then the nanoparticles are connected by a chain formed by joining the two interacting polymer arms (see Fig. 1a). These interconnecting polymer arms introduce an attractive interaction between the particles due to the springlike entropic elasticity of the polymer chains. Consequently, the attractive force acting between the two bonded particles is given by the following equation: Flink(r) ¼ (Nb + Nu)k(r)r

(3)

where Nb is the number of bonds formed between the given pair of particles, Nu is the number of unbreakable chains between a given pair of particles, and k(r) is the spring constant, which increases progressively with the chain end-to-end distance r ¼ R 2. The stiffening of the chain is described by the following equation obtained for a worm-like chain:26 k(r) ¼ k0{1 + 2[1 r2(2L)2]2}

(4)

In the above equation, 2L is the contour length of the chain formed by bonding two polymer arms of length L. To calculate Nb, we examine the kinetics of bond formation and rupture at the individual bond level, using the Bell model27–31 to describe the rupture and re-formation of bonds due to thermal uctuations. In accordance with the Bell model, the rupture rate constant, k(p,l) , is an exponential function of the r force applied to the bond and is given by k(p,l) ¼ k(p,l) r 0r exp(g0F), (p,l) (p,l) (p,l) where k0r ¼ n exp(U0 /kBT) is the rupture rate in the zero force limit that depends on the energy barrier U(p,l) sepa0 rating the bound and the free states. Here, the superscripts p and l correspond to the permanent and labile bonds, respectively. For reversible bond formation in the zero force limit, the ratio of the formation to rupture rate constants is given by


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(l) (l) 28,29 k(l) In general, the rate constant of bond 0f /k0r ¼ exp(U0 /kBT). formation decreases under an applied force.30 The force due to chain extension, however, only affects the rate constants k(l) r and k(l) f at sufficiently large separations between a pair of PGNs. At the latter separations, the coronas do not overlap, so the total rate of labile bond formation is zero. Hence, for the sake of simplicity, we neglect the dependence of the rate constant of bond formation on force and consider k(l) 0f to be constant. We note that it is only the labile bonds that can reform aer breaking; the permanent bonds can only break, and the unbreakable chains by denition do not rupture. The number of graed chains available for linking a PGN to the neighboring particle located at the center-to-center distance R cannot exceed the number of chain ends that could be found in the overlap volume of the two coronas, Nmax(R). We estimate the value of Nmax by making the simplifying assumption that there is negligible distortion of the two coronas when they overlap, as described in ref. 2. While Nmax(R) corresponds to the maximum number of bonds that can be formed between two nanoparticles, the actual number of bonds formed, Nb, depends on the number of bonding pairs available in the corona overlap volume, and on the rates of formation and rupture of individual bonds. The evolution equation for the number of bonds can be written as:

dNb ¼ kr ðRÞNb þ Pc ðRÞ kf ðRÞ½Nmax ðRÞ Nb 2 ; dt

(5)

where Pc(R) is the probability of contact of two chain ends that, in turn, depends on the free-end distribution in the corona. For a single polymer arm, the probability p to have a bond with an arm belonging to the neighboring PGN is described by an equation similar to eqn (5): dp ¼ kr ðRÞp þ Pc ðRÞ kf ðRÞ½Nmax ðRÞ Nb ð1 pÞ: dt

(6)

The details of the probability of contact calculation can be found in ref. 2. We rst demonstrate the rupture behavior of the labile, permanent, and high-strength (100 kBT) bonds by focusing on a single pair of particles that are pulled apart at a constant velocity of v ¼ 0.001v0 (see Table 1 for the list of parameters used in the simulations). In particular, using eqn (6), we obtain the bond rupture rate,(dp/dt), as a function of strain, 3, for a single labile, permanent, and high-strength (100 kBT) bond, each with an initial equilibrium separation, of R0 ¼ 3.21. As seen in Fig. 2a, the labile bonds break at small strain (3 < 1) with a wide distribution of strains at break, while the permanent bonds survive longer and fail at a strain between 1.5 < 3 < 2. Notably, the high-strength bonds rupture only at large strains in the range of 3.75 < 3 < 3.8. Using our simulation approach, we determine the normalized total force as a function of strain for a pair of particles that are initially connected by all three bond types and whose equilibrium separation is R0 ¼ 3.15. As shown in Fig. 2b, small strains are sufficient to rupture multiple labile bonds, with the last labile bond failing at 3 ¼ 0.84. The permanent bonds survive up to a much higher strain of 3 ¼ 1.95. Finally, the high-strength


(a) Bond rupture rate, (dp/dt), as a function of strain, 3, for a single labile, permanent, and high-strength (100 kBT) bond, each with an initial equilibrium separation, R0 ¼ 3.21. Inset shows the plot for the high-strength bond in greater detail. (b) Force, F/F0, versus strain, 3, from a constant strain rate pulling simulation of a pair of particles connected by multiple labile (red) bonds along with a single permanent (gray) bond and a high-strength (black) bond, beginning at an equilibrium separation, R0 ¼ 3.15. The downward arrow indicates complete rupture of all bonds. Fig. 2

bonds rupture at a large strain of 3 ¼ 3.86. Fig. 2a and b demonstrate that the strains at break obtained in the simulations (Fig. 2b) are similar to the single bond predictions obtained by solving eqn (6). Given that (dp/dt) for the high-strength bonds is negligibly small at the strains leading to the rupture of labile and permanent bonds, it is a reasonable to treat the high-strength bonds as unbreakable in our simulations. The data provided in the ESI supports this assumption.† The dynamics of the system is assumed to be in the overdamped regime, with the motion of each particle governed by the equation dx/dt ¼ mFtot, where m is the mobility and Ftot is the total force on the polymer graed particle. The total force acting on a particle can be written as Ftot ¼ vUint/vx + Fext, where Fext is the external force acting on the edge particles of the particle array (see Fig. 1b). This equation is solved numerically by determining the number of bonds at any given time, Nb(t), with eqn (5), and using this value to calculate the spring force (as given in eqn (3)).2

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The initial state of the system is generated using the following six-step procedure. In the rst step, the nanoparticles are placed in an array such that their centers are separated by a horizontal spacing of 1.8(1 + q) and a vertical spacing of 1.62(1 + q), with a horizontal offset position of 0.855(1 + q) between adjacent rows. In the second step, we hold the sample in the initial conguration for 4 103T0 units of time to allow for the formation of the labile bonds. In the third step, we equilibrate the sample by allowing the particles to move for a period of time equal to 6 103T0. During equilibration, the stressed labile bonds are broken and new bonds are formed between the adjacent PGNs according to eqn (5). The equilibration process was monitored through the evolution of the average number of labile bonds and the sample length. We observed that the equilibration was almost complete aer the rst 103T0 of the equilibration time. During the remainder of equilibration step, the number of labile bonds and sample length changed by less than 1%. In the fourth step, we establish the permanent bonds within system. In this step, a certain number of already established labile bonds are converted randomly to permanent ones; as a result, each pair of neighboring particles has in average of Pp permanent bonds. Note that Pp is not an integer number and can be either Pp < 1 or Pp $ 1 depending on how many permanent bonds are introduced into the network. In the h stage, we introduce unbreakable chains between bonded neighbors in the network. For each neighboring bonded pair of particles, we assign an unbreakable chain with a probability Pn, only allowing at most one unbreakable chain per pair. Finally, in the sixth step, the resultant sample composed of a network of labile, permanent bonds, and unbreakable chains, is further equilibrated for 104T0 and then subject to tensile deformation by stretching at a constant velocity of 3.55 nm s1 (0.001v0). The latter value of velocity is similar to that used in single molecule pulling experiments.15 In the ensuing simulations, we consider a particle of core radius r0 ¼ 50 nm with f ¼ 156 polymer chains graed onto its surface. The value of f was chosen to satisfy the assumption that the corona chains are in the semi-dilute regime; this constrains 32,33 the value of f to f1/2 # yr0l1 where y is the polymer–solvent p , 32–34 interaction parameter. The value of f is used to determine the strength of repulsion through eqn (1) and the extent of bond formation between any two interacting particles in the calculation of Nmax(R).2 In simulating the bond kinetics, we x the bond energies of the labile bonds to be U(l) 0 ¼ 37 kBT, and the stronger permanent bonds to be U(p) 0 ¼ 45 kBT; these values were used in our previous studies of dual cross-linked nanocomposites2,3 and thus, provides a good basis of comparison. For convenience, all the parameters used in the simulations are collected in Table 1. We vary the average number of permanent bonds, Pp, in the range 1 # Pp # 1.3, and the probability of formation of unbreakable bonds, Pn, in the range 0 # Pn # 0.3. Each link between particles in the network, which contains multiple reformable bonds, may contain permanent bonds, and possibly an unbreakable chain. The rates of rupture and formation of labile bonds within the links varies as a function of corona

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overlap. The links stretch due to an applied deformation with the maximum extent of the stretching corresponding to the contour length of the graed chains. When the links are stretched, the force exerted on the bonds in the link increases with separation and facilitates rupture of only the labile and permanent bonds. On the other hand, as we detail below, the unbreakable chains stretch and act as transmitters of force.

III.

Results and discussion

Our aim is to determine how the introduction of just a small fraction of unbreakable bonds (i.e., high-strength connections) between the nanoparticles affects the response of the material to a tensile deformation. Specically, we focus on the strain at break, 3b (measured at the point of fracture), which characterizes the ductility of the sample. We also calculate the toughness of the sample, which can be dened as the work-to-break per one particle.2 Hence the toughness, W, is determined by integrating the force–extension curve, and dividing the resulting work by the number of nanoparticles in the sample, NPGN. To carry out these studies, we gradually increase the amount of unbreakable chains within the network. At the onset, the nanoparticles are cross-linked by both labile bonds and permanent bonds. The fraction of permanent bonds is set at Pp ¼ 1, which are distributed randomly throughout the system. This choice of Pp assures that there are a xed number of permanent bonds, i.e., 487, before the system is strained. For an innite 2D lattice with a triangular geometry (each node having six adjacent neighbors), the critical bond percolation probability is Pcrit ¼ 2 sin(p/18) z 0.35.35 We introduce unbreakable chains into our nite-sized samples in the range 0 # Pn # 0.3 < Pcrit. Our aim is to keep Pn sufficiently low that the randomly placed unbreakable bonds do not form a percolating network in the system (see below) and we can evaluate how a relatively few of these high-strength connections contribute to the mechanical properties. We dene Pt ¼ Pp + Pn to be the total number of permanent bonds and unbreakable chains per neighboring pair of particles, and hence, our range becomes 1 # Pt # 1.3. This notation facilitates the following discussion, but we emphasize that all the samples discussed below contain a large number of labile bonds between each particle (see Fig. 2b). To fully assess the effect of introducing the non-breakable bonds, we also carry out simulations with 1 # Pp # 1.3 and no unbreakable bonds. We performed 40 independent runs at each specied value of model parameters, and utilized Weibull statistics36 to determine the average values and standard deviations of strain at break and toughness. Note that for higher values of Pn, there is a possibility of the randomly placed unbreakable chains producing a percolating network. In our samples, for Pn ¼ 0.2, one out of 40 samples formed such a percolating network. With an increase of Pn to 0.3, seven out of 40 samples resulted in a percolating network. Since these samples cannot break, we excluded them from our analysis. To obtain a qualitative measure for the percolation threshold for the unbreakable bonds in our nite-sized system, we did consider cases involving Pn near Pcrit z 0.35; we found that at Pn ¼ 0.35, 42.5%


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of the samples exhibited percolating networks of unbreakable bonds and at Pn ¼ 0.4, this number increased to 67.5%. Fig. 3a and b show the respective plots of the strain at break and the toughness as a function of Pt; each plot shows data for systems with (red curve) and without (blue curve) the unbreakable bonds. For the strain at break, the data with just permanent bonds (Pn ¼ 0) remains approximately constant for increasing Pp, with all the data points showing relatively small standard deviations. In contrast, the system containing the unbreakable bonds (with Pp xed at 1) shows a systematic increase in 3b as Pn is increased. The standard deviations about the data points, however, become progressively larger for larger values of Pn. Hence, for this red curve, it is only the points Pt $ 1.25 that show a signicant difference from the Pp ¼ 1, Pn ¼ 0 data. From a comparison of the red and blue curves in Fig. 3a, it is evident that the systematic addition of unbreakable chains into the Pp ¼ 1 system does increase the average 3b relative to the system with no unbreakable bonds (i.e., at the same value of Pt but with Pn ¼ 0). Surprisingly, however, due to the large

(a) Strain at break and (b) toughness as a function of the average number of extra bonds per neighboring pair in the range 1 # Pt ¼ Pp + Pn # 1.3. Networks with solely permanent bonds, (1 # Pp # 1.3, Pn ¼ 0), are compared with networks that contain added unbreakable chains (Pp ¼ 1, 0 < Pn # 0.3). The average values and standard deviations were obtained according to the Weibull statistics. Fig. 3


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standard deviations, the difference between the two scenarios in Fig. 3a only becomes signicant for Pn $ 0.25. It is noteworthy that at Pt ¼ 1.30, the average value of the strain at break for the case encompassing the unbreakable bonds does show a two-fold increase relative to the system without these interconnections. The data for the toughness show trends that are similar to the ones described above. Fig. 3b does, however, show a slight increase in the average value of the toughness with greater Pp in the blue curve (without unbreakable bonds). This increase in toughness is more dramatic for networks with unbreakable chains. Again, however, the differences between points within the red curve and between points in the red and blue curves only become signicant for Pt $ 1.25 due to the large standard deviations. It is further worth noting that at Pt ¼ 1.30, the average value of the toughness shows a two-fold increase relative to the case with no unbreakable chains. We performed similar investigations on samples with 0.5 # Pt # 0.8 (data not shown). We obtained similar results as those depicted in Fig. 3, with the addition of a small amount of unbreakable chains leading to improvements in 3b and W. The potential for signicant improvements in 3b and W indicated in Fig. 3 with the addition of a sufficient number of unbreakable bonds led us to examine the origin of the large standard deviations shown in the plots and thereby obtain a deeper understanding of the behavior of the system. To this end, we determined the histograms shown in Fig. 4a and 5a for the strain at break and in Fig. 4b and 5b for the toughness. In Fig. 4a, the histograms indicate the number of samples that exhibit a specic value of 3b; to obtain the relative frequency, the number of occurrences in each bin is divided by the total number of samples. In Fig. 4, we show data sets for Pp ¼ 1, Pn ¼ 0, and Pp ¼ 1.15, Pn ¼ 0, along with an equivalent set with unbreakable chains: Pp ¼ 1, Pn ¼ 0.15. We picked this point to emphasize the fact that the average values plotted in Fig. 3 do not tell the whole story regarding the mechanical behavior of the samples. Notably, at Pt ¼ 1.15, the red and blue data points effectively overlap in Fig. 3, while the corresponding distributions in Fig. 4a are quite distinct. As seen in Fig. 4a for the strain at break, there is no noticeable change in distributions between Pp ¼ 1, Pn ¼ 0 (the green bars) and Pp ¼ 1.15, Pn ¼ 0 (the blue bars). On the other hand, for the network with unbreakable chains where Pp ¼ 1, Pn ¼ 0.15 (the red bars), there is a clear broadening of the distribution of data. Now, a signicant fraction of the samples yield considerably large values of 3b, well above the range for the data sets with the purely permanent bonds. Importantly, the samples with unbreakable bonds also display a clear reduction in samples that break at small strain. The histograms in Fig. 4b show the number of samples that exhibit a specic value of toughness, W; again, this number is divided by the total number of samples to obtain the relative frequency. For the toughness, there is a marginal shi in the distribution from Pp ¼ 1, Pn ¼ 0 to Pp ¼ 1.15, Pn ¼ 0, but the breadth of the distributions remains similar. On the other hand, for the network with unbreakable chains at Pp ¼ 1, Pn ¼ 0.15, the distribution of data points clearly becomes wider.

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different bond formation scenarios: the base case of permanent bonds with no unbreakable chains per pair (Pp ¼ 1, Pn ¼ 0), an increase in the average number of permanent bonds per pair (Pp ¼ 1.15, Pn ¼ 0), and an equivalent system with Pt ¼ 1.15, composed of permanent bonds at Pp ¼ 1 and the unbreakable chains introduced at Pn ¼ 0.15. The average value and standard deviation obtained according to the Weibull statistics is shown for each distribution.

Fig. 5 Histograms of the (a) strain at break and (b) toughness for three different bond formation scenarios: the base case of permanent bonds with no unbreakable chains per pair (Pp ¼ 1, Pn ¼ 0), an increase in the average number of permanent bonds per pair (Pp ¼ 1.3, Pn ¼ 0), and an equivalent system with Pt ¼ 1.3, composed of permanent bonds at Pp ¼ 1 and the unbreakable chains introduced at Pn ¼ 0.3. The average value and standard deviation obtained according to the Weibull statistics is shown for each distribution.

There are a signicant number of samples that are much tougher than the system involving just permanent bonds and there are far fewer weak (less tough) samples. To further demonstrate the effect and utility of adding a fraction of unbreakable bonds to the system, in Fig. 5 we show the representative histograms for samples with Pt ¼ 1.3, which corresponds to the case where the red and blue curves in Fig. 3 are signicantly different. In the absence of unbreakable bonds, the distribution of strain at break (Fig. 5a), does not change noticeably between the cases corresponding to Pp ¼ 1, Pn ¼ 0 (green bars) and the Pp ¼ 1.3, Pn ¼ 0 (blue bars). The visual difference between the histograms for the same Pp ¼ 1, Pn ¼ 0 data in Fig. 4 and 5 is due to the difference in bin sizes. A dramatic change in the distribution is observed, however, with the introduction of unbreakable bonds (Pp ¼ 1, Pn ¼ 0.3, the red bars). The distributions not only broaden signicantly toward higher strains, but they also clearly show a distinct separation from the histograms for the systems without unbreakable bonds. Only a small proportion of the networks encompassing the Pn ¼ 0.3 bonds break at strains that are sufficient to rupture the networks with Pn ¼ 0.

A similar trend is observed for the distributions of toughness shown in Fig. 5b. In the absence of unbreakable bonds (Pn ¼ 0), there is only a marginal increase of toughness with the increase of Pp from 1.0 to 1.3. On the other hand, the introduction of the unbreakable chains leads to a dramatic increase in the toughness of the material. Note that only a small fraction of the Pn ¼ 0.3 cases displayed toughness in the same region as networks with Pn ¼ 0. To better understand the observed broadening of the strain at break and toughness distributions, in Fig. 6 we examine force versus strain curves for two samples from each data set: Pp ¼ 1.15, Pn ¼ 0, and Pp ¼ 1, Pn ¼ 0.15. All four curves feature similar initial behavior: the samples reach a yield point at 3 z 0.1, then the force decreases until reaching a plateau at approximately F ¼ 10F0. The downward pointing arrows indicate the breakage of bonds that leads to the ultimate failure of the material (resulting in F ¼ 0). For Pp ¼ 1.15, Pn ¼ 0, both samples 1 and 2 fail in two stages, as indicated by successive drops in the force. In sample 1, there is a large initial drop in the force followed by a small fall from the plateau force of F ¼ 10F0. Sample 2 sustains

Fig. 4 Histograms of the (a) strain at break and (b) toughness for three

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Fig. 6 Force, F/F0, as a function of strain, 3, for four different simulations: (1) and (2) for networks containing solely permanent bonds (Pp ¼ 1.15, Pn ¼ 0), and (3) and (4) for an equivalent system with Pt ¼ 1.15, composed of permanent bonds and unbreakable chains at Pp ¼ 1 and Pn ¼ 0.15, respectively.

a large initial drop to F ¼ 5F0, then remains intact for a small period before failure. On the other hand, the samples containing unbreakable chains with Pp ¼ 1, Pn ¼ 0.15 display two distinct failure modes. While sample 3 suffers a sharp decline from F ¼ 10F0 at a small strain (3 ¼ 0.85), sample 4 survives an initial drop in the force to F z 4F0 at a relatively large strain (3 ¼ 1.20), and remains connected until a dramatically large strain (3 ¼ 1.80) before nally failing. We assess the behavior of these samples by investigating how force is transmitted through the networks and specically, through the unbreakable chains. The images in Fig. 7 show the elastic tension of each bond (calculated with eqn (3)) for the networks in sample 3 (Fig. 7a) and sample 4 (Fig. 7b) at strain 3 ¼ 0.85. Sample 3 is at the point of rupture, as indicated by several highly stressed bonds in the narrow region marked by the arrow labeled “A”. Note that the early failure of this sample is at least partly due to the lack of load being exerted on the unbreakable chains (shown in red). Alternatively, in sample 4, the tension is distributed rather uniformly throughout the network, loading both breakable and unbreakable bonds. In particular, the bond marked “B” is loaded with a high tension and is critically holding two large regions together. The random conguration of unbreakable bonds in this sample promotes this uniform distribution of bond tension, which signicantly delays overall fracture. To gain greater insight into the failure behavior of the samples with unbreakable bonds, we examined snapshots of networks 3 and 4, which are shown in Fig. 8 and 9, respectively. Sample 3 fails at a relatively small strain; the rst snapshot of this system is at an early time with strain 3 ¼ 0.42, and the second image shows the system at the point of rupture at 3 ¼ 0.85. In the top image, the arrow labeled “C” pinpoints the region of incipient failure in the material and in the lower image, the arrow marked “D” indicates the region where the material ultimately fractures. In particular, at 3 ¼ 0.85, multiple links are about to break within a short time period and the


Fig. 7 Distribution of force F/F0 on breakable and unbreakable links within the networks from Fig. 6. We display (a) sample 3 and (b) sample 4, both at strain, 3 ¼ 0.85. Blue: breakable links. Red: unbreakable links. Thin lines: 0 # F/F0 < 8. Lines of medium thickness: 8 # F/F0 < 16. Thick lines: 16 # F/F0 < 24.

material breaks into two unconnected sections. Note that even at low strain, a void has formed in the sample and the region in which the sample ultimately fails is completely devoid of unbreakable chains (lack of red lines, also seen in Fig. 7a). In Fig. 9, we display a series of snapshots at 3 ¼ 0.45, 3 ¼ 0.90, 3 ¼ 1.35, and 3 ¼ 1.80 for sample 4, which survives to a relatively large strain. The early picture at 3 ¼ 0.45 shows the network to

Fig. 8 Snapshots of the network in sample 3 with Pp ¼ 1, Pn ¼ 0.15 at strains 3 ¼ 0.42 and 3 ¼ 0.85 (at the point of rupture). The thick gray and black lines indicate the permanent bonds and unbreakable chains, respectively. The cores with an unbreakable connection are colored red, and those without an unbreakable connection are colored blue.

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dramatically extend the value of 3b relative to materials that do not contain the unbreakable bonds.


IV. Conclusions

Fig. 9 Snapshots of the network in sample 4 with Pp ¼ 1, Pn ¼ 0.15 at strains 3 ¼ 0.45, 3 ¼ 0.90, 3 ¼ 1.35, and 3 ¼ 1.80 (at the point of rupture). The thick gray and black lines indicate the permanent bonds and unbreakable chains, respectively. The cores with an unbreakable connection are colored red, and those without an unbreakable connection are colored blue.

be relatively intact, with only a small void beginning to form (at label “E”). At the next level of strain, 3 ¼ 0.90, the network is stretched considerably, and consequently, multiple voids have formed, most notably a large one shown by label “F”. At further strain of 3 ¼ 1.35, the region indicated by label “G” has been severed, although we nd that a single thread has formed that holds the entire network together. Finally, the sample does rupture at 3 ¼ 1.80 with “H” showing the bond that fails. At this point, the thread has grown signicantly in length and encompasses more nanoparticles than at the strain of 3 ¼ 1.35; namely, the thread contained ve crucial bonds at 3 ¼ 1.35, and this has grown to nine at 3 ¼ 1.80. This process occurs through a slow unraveling of particles from either side of the thread. Moreover, as bonds break under strain, particles are released and the thread can grow. Also, if the thread contains predominantly unbreakable chains that can span large distances under high tension, then this potentially enables the network to bridge huge separations. The threads containing mostly unbreakable chains enable the network to withstand high strain and lead to a distinctive mode of failure. It is important to note that for larger sample sizes, the system will contain not just one long-lived interconnecting strand (as shown in Fig. 9), but many such threads. Hence, the combined effect of these multiple strands is to

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We developed a computational, multi-scale model for rigid nanoparticles that are decorated with end-graed polymers. The free ends of these graed chains contain reactive groups that can bind to functional groups on adjacent polymer graed nanoparticles, and thus, the system can form a network of cross-linked PGNs. In our previous studies on these networks,2,3 the reactive end groups could form either reformable labile bonds or permanent bonds that rupture irreversibly. Here, we introduced unbreakable chains between nanoparticles and thus, modeled the effect of incorporating high-strength bonds (for example a carbon–carbon bond) into the networks. We found that the inclusion of a small amount of unbreakable chains has a dramatic effect on the response of the network to mechanical deformation. In particular, histograms for the strain at break and toughness exhibited a signicant broadening relative to comparable networks without unbreakable connections. Notably, a distinct fraction of samples with unbreakable links could survive strains far greater than networks containing only labile and permanent bonds. Under strain, links within the PGN network stretch and break. In order for a sample to fracture into two disconnected sections, a sequence of bonds must rupture across the width of the sample (normal to the direction of stretching). The presence of unbreakable links hinders this mode of failure and can lead to the formation of long threads, which are composed predominantly of unbreakable chains and hold the network intact. As more weak links break, particles are released from some neighbors and the thread can unravel, thus delaying sample rupture. This behavior is due to the ability of unbreakable chains to extend over large gaps and hold tremendous loads without rupturing. The formation of long thin threads that bridge distant groups and thus, enable networks to survive high strain is a new feature that arises due to the introduction of unbreakable chains. This distinctive mode of failure is not observed in networks without these interconnections. It is important to note that the size of our samples is relatively small (12 15 particles) and hence, the system subtended only a few or a single thread. On the other hand, many such long threads would be present in larger samples. This collection of threads would signicantly improve the ability of sample to withstand mechanical deformation. Given these ndings, our results provide valuable guidelines for creating materials that display remarkable ductility and strength. Notably, recent experimental studies have yielded self-healing networks formed from polymer graed nanoparticles that are interconnected by hydrogen bonds;37,38 the latter materials are examples of useful systems for testing our ndings.

Acknowledgements ACB and K.M. gratefully acknowledge funding from the DOE for partial support of M.J.H. ACB also gratefully acknowledges


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funding from the ARO for partial support of V.V.Y. and the AFOSR for partial support of B.V.S.I.


References 1 W. Yang, I. H. Chen, B. Gludovatz, E. A. Zimmermann, R. O. Ritchie and M. A. Meyers, Adv. Mater., 2013, 25, 31–48. 2 B. V. S. Iyer, I. G. Salib, V. V. Yashin, T. Kowalewski, K. Matyjaszewski and A. C. Balazs, So Matter, 2013, 9, 109–121. 3 B. V. S. Iyer, V. V. Yashin, T. Kowalewski, K. Matyjaszewski and A. C. Balazs, Polym. Chem., 2013, 4, 4927–4939. 4 N. J. Fernandes, T. J. Wallin, R. A. Vaia, H. Koerner and E. P. Giannelis, Chem. Mater., 2013, DOI: 10.1021/ cm402372q. 5 K. E. Mueggenburg, X. M. Lin, R. H. Goldsmith and H. M. Jaeger, Nat. Mater., 2007, 6, 656–660. 6 P. Voudouris, J. Choi, N. Gomopoulos, R. Sainidou, H. Dong, K. Matyjaszewski, M. R. Bockstaller and G. Fytas, ACS Nano, 2011, 5(7), 5746–5754. 7 D. Y. Lee, J. T. Pham, J. Lawrence, C. H. Lee, C. Parkos, T. Emrick and A. J. Crosby, Adv. Mater., 2012, 25(6), 1–6. 8 P. Akcora, H. Liu, S. K. Kumar, J. Moll, Y. Li, B. C. Benicewicz, L. S. Schadler, D. Acehan, A. Z. Panagiotopoulos, V. Pryamitsyn, V. Ganesan, J. Ilavsky, P. Thiyagarajan, R. H. Colby and J. F. Douglas, Nat. Mater., 2009, 8, 354–359. 9 M. McEwan and D. Green, So Matter, 2009, 5, 1705–1716. 10 J. Choi, C. M. Hui, J. Pietrasik, H. Dong, K. Matyjaszewski and M. R. Bockstaller, So Matter, 2012, 8, 4072–4082. 11 K. Kircher, Chemical reactions in plastics processing, Oxford University Press, New York, 1987. 12 J. P. Gong, Y. Katsuyama, T. Kurokawa and Y. Osada, Adv. Mater., 2003, 15(14), 1155–1158. 13 J. P. Gong, So Matter, 2010, 6, 2583–2590. 14 P. Dopieralski, J. Ribas-Arino, P. Anjukandi, M. Krupicka, J. Kiss and D. Marx, Nat. Chem., 2013, 5, 685–691. 15 A. P. Wiita, S. R. K. Ainavarapu, H. H. Huang and J. M. Fernandez, Proc. Natl. Acad. Sci. U. S. A., 2006, 103, 7222–7227. 16 S. Garcia-Manyes, J. Liang, R. Szoskiewicz, T. L. Kuo and J. M. Fernandez, Nat. Chem., 2009, 1, 236–242. 17 Y. Li, A. Nese, N. V. Lebedeva, T. Davis, K. Matyjaszewski and S. S. Sheiko, J. Am. Chem. Soc., 2011, 133, 17479–17484.


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18 N. V. Lebedeva, A. Nese, F. C. Sun, K. Matyjaszewski and S. S. Sheiko, Proc. Natl. Acad. Sci. U. S. A., 2012, 109, 9276– 9280. 19 Y. Li, A. Nese, K. Matyjaszewski and S. S. Sheiko, Macromolecules, 2013, 46, 7196–7201. 20 J. A. Yoon, J. Kamada, K. Koynov, J. Mohin, R. Nicolay, Y. Z. Zhang, A. C. Balazs, T. Kowalewski and K. Matyjaszewski, Macromolecules, 2012, 45, 142–149. 21 Y. Amamoto, H. Otsuka, A. Takahara and K. Matyjaszewski, Adv. Mater., 2012, 24, 3975–3980. 22 C. N. Likos, H. L¨ owen, M. Watzlawek, B. Abbas, O. Jucknischke, J. Allgaier and D. Richter, Phys. Rev. Lett., 1998, 80, 4450–4453. 23 A. Jusu, M. Watzlawek and H. L¨ owen, Macromolecules, 1999, 32, 4470–4473. 24 M. Daoud and J. P. Cotton, J. Phys., 1982, 43, 531–538. 25 F. Lo Verso, C. N. Likos and L. Reatto, Prog. Colloid Polym. Sci., 2006, 133, 78–87. 26 A. V. Dobrynin and Y. J. Carrillo, Macromolecules, 2011, 44, 140–146. 27 G. I. Bell, Science, 1978, 200, 618–627. 28 G. I. Bell, M. Dembo and P. Bongrad, Biophys. J., 1984, 45, 1051–1064. 29 S. K. Bhatia, M. R. King and D. A. Hammer, Biophys. J., 2003, 84, 2671–2690. 30 G. Diezemann and A. Janshoff, J. Chem. Phys., 2008, 129, 0849041–849110. 31 J. Ribas-Arino and D. Marx, Chem. Rev., 2012, 112, 5412– 5487. 32 D. Dukes, Y. Li, S. Lewis, B. Benicewicz, L. Schandler and S. K. Kumar, Macromolecules, 2010, 43, 1564–1570. 33 K. Ohno, T. Morinaga, S. Takeno, Y. Tsujii and T. Fakula, Macromolecules, 2007, 40, 9143–9150. 34 C. M. Wijmans and E. B. Zhulina, Macromolecules, 1993, 26, 7214–7224. 35 J. C. Wierman, Advances in Applied Probability, 1981, 13, 298– 313. 36 B. R. Lawn, Fracture of brittle solids, Cambridge University Press, New York, 2nd edn, 1993. 37 Y. Chen and Z. Guan, Polym. Chem., 2013, 4, 4885– 4889. 38 Y. Chen, A. M. Kushner, G. A. Williams and Z. Guan, Nat. Chem., 2012, 4, 467–472.

Soft Matter, 2014, 10, 1374–1383 | 1383

Electronic Supplementary Material (ESI) for Soft Matter This journal is © The Royal Society of Chemistry 2014

Modeling Polymer Grafted Nanoparticle Networks Reinforced by High-strength Chains Matthew J. Hamer,a Balaji V. S. Iyer,a Victor V. Yashin,a Tomasz Kowalewski,b Krzysztof Matyjaszewski b and Anna C. Balazs*a a

Chemical Engineering Department, University of Pittsburgh, Pennsylvania 15261, USA b

Department of Chemistry, Carnegie Mellon University, Pennsylvania 15213, USA

*Corresponding author; e-mail address: [email protected] Supplementary Information In this study, grafted polymers on neighboring nanoparticles can form a link between the particles through functional groups at the chain ends. Here, we validate our assumption that if the bond formed between the two polymer arms has an energy of 100 k BT , then the link can be represented by an unbreakable chain within the networks. Figure S1 shows histograms of the strain at break,  b , and toughness, W , from our simulations of stretching the chains at a fixed velocity ( v  0.001v0 ); the plots are for networks containing permanent bonds at Pp  1 and

Pn  0.15 of either unbreakable chains or polymers connected by high-strength bonds ( 100 k BT ). These histograms show that the distributions of the strain at break and toughness are very similar for these two different scenarios; even the average values and standard deviation ranges are comparable for the two cases. We emphasize this similarity through the use of Weibull statistics.S1 Namely, we calculate the Weibull probability, Pb , which is obtained using the Bernard and Bos-Levenbach approximation.S2 For example, for the strain at break, we determine the probability

Pb ( b   b )  j

j  0.3 , N test  0.4

(1)


where  b is the strain at break for the j th sample of the ordered set and N test is the total number j

of samples. For comparison, we added data for samples characterized by Pp  1.15 and Pn  0 (i.e., the system does not encompass either the unbreakable or the high-strength bonds). For each set of data, we also display the curve indicating the fit of the data to the Weibull probability in eq. (1). The plots in Fig. S2 clearly show that networks with unbreakable chains produce similar fitting curves to those with high-strength ( 100 k BT ) bonds, and both sets display a pronounced improvement over networks that do not encompass either of these connections. We repeated the analysis described above on samples with a greater probability of unbreakable chains or high-strength ( 100 k BT ) bonds so that the system was characterized by

Pn  0.3 . At this of value of Pn , the number of unbreakable chains are close to the percolation threshold; in our set of 40 runs, the chains formed percolating networks in seven of those cases. Figure S3 shows that for these Pn  0.3 cases, there is a difference between unbreakable chains and high-strength bonds for the strain at break and toughness. Figure S4 also shows that the networks with unbreakable chains produce more samples that survive at high strains than the samples with high-strength bonds. This behavior is due to the rare failures of high-strength bonds when subjected to extreme strain (as displayed in Fig. 2 within the main article). The highstrength bonds do, however, provide a significant improvement in ductility and strength relative to the networks characterized by Pp  1.3 and Pn  0 . References S1. B. R. Lawn, Fracture of brittle solids, 2nd ed., Cambridge University Press, New York, 1993. S2. A. Bernard, and E. C. Bos-Lovenbach, Statistica, 1953, 7, 163-173.


Fig. S1. Histograms of the (a) strain at break and (b) toughness for two equivalent systems with Pt 1.15 , composed of permanent bonds at Pp  1 and the unbreakable chains (red) or high-

strength connections with bond energy 100 k BT (black) both introduced at Pn  0.15 . The 

average value and standard deviation obtained according to the Weibull statistics is shown for each distribution.


Fig. S2. Plots of 40 data points in ascending order for three different cases displaying: (a) strain at break and (b) toughness versus the calculated Weibull probability, as well as the associated fitting curves. One set comprises networks (blue crosses) with permanent bonds at Pp  1.15 and no unbreakable chains Pn  0 . The other sets relate to networks with permanent bonds at Pp  1 and either unbreakable chains (red circles) or high-strength connections with bond energy

100 k BT (black squares) both at Pn  0.15 .


Fig. S3. Histograms of the (a) strain at break and (b) toughness for two equivalent systems with Pt  1.3 , composed of permanent bonds at Pp  1 and the unbreakable chains (red) or highstrength connections with bond energy 100 k BT (black) both introduced at Pn  0.3 . The average value and standard deviation obtained according to the Weibull statistics are shown for each distribution.


Fig. S4. Plot of 40 data points in ascending order (33 in the case of Pp  1 , Pn  0.3 (red circles) due to percolations) for three different sets displaying: (a) strain at break and (b) toughness versus the calculated Weibull probability, as well as including the associated fitting curves. One set comprises networks (blue crosses) with permanent bonds at Pp  1.3 and no unbreakable chains Pn  0 . The other sets relate to networks with permanent bonds at Pp  1 and either unbreakable chains (red circles) or high-strength connections with bond energy 100 k BT (black squares) both at Pn  0.3 .