Using Torsion for Controllable Reconfiguration of Binary Nanoparticle

Feb 28, 2017 - coronas, these reactive groups form labile bonds, which can reform after breakage. The labile bond energy between similar. PGNs (UAA, U...
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Using Torsion for Controllable Reconfiguration of Binary Nanoparticle Networks Tao Zhang, Badel L. Mbanga, Victor V. Yashin, and Anna C. Balazs* Chemical Engineering Department, University of Pittsburgh, Pittsburgh, Pennsylvania 15261, United States ABSTRACT: Mechanical deformation can potentially provide an effective means of controlling the nanoscale morphology in hybrid materials. The challenge, however, is establishing optimal couplings of the deformation and mechano-responsive components in the material to achieve nanoscopic structural reorganization without causing catastrophic damage. Through computational modeling, we investigate how torsion can be utilized to induce controllable structural changes in networks formed from binary mixtures (A and B) of polymer-grafted nanoparticles (PGNs). 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 the neighboring coronas, these reactive groups form labile bonds, which can reform after breakage. The labile bond energy between similar PGNs (UAA, UBB) is different than the energy between dissimilar species (UAB). By tailoring the relative values of these bond energies and the boundary conditions acting on the system, the application of a torsional deformation can result in a controllable reconfiguration of the network, leading to intertwining helical structures, or homogeneously mixed nanocomposites. In effect, our mechano-mutable system resembles a “Rubik’s cube” material, whose nanostructure, and hence global properties, can be tailored by mechanically twisting the sample. KEYWORDS: modeling polymer-grafted nanoparticle networks, torsion-induced structuring, labile chemical bonds, nanoparticle mixtures

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arms act as the elastic bands and the nanoparticles correspond to the individual blocks in the toy. Using computational modeling, herein we show that by applying torsion to this material, we can achieve significant control over the arrangement of a binary mixture of nanoparticles and, hence, tailor the nanostructure of the composite. In prior studies,2−5 we developed computational models to investigate the structural rearrangements of PGN networks under tensile deformation. We found that labile bonds between the flexible polymeric arms allowed the material to withstand large strains. The distinctive feature of labile bonds is that they can readily reform after they are broken, and thus, they enable the network to dynamically reorganize without undergoing catastrophic failure. We specifically isolated a scenario where the coupling between the tensile deformation and the labile bonds allowed the composite to undergo a smooth transition from an ordered hexagonal closed-packed (HCP) structure to a regular face-centered cubic (FCC) arrangement. 5 Our predictions about the self-healing behavior3 and mutability5 of these materials under tensile deformation have recently been experimentally confirmed.6−8

he ability to control the nanoscale structure of composite materials is key to tailoring the materials’ macroscopic mechanical, optical and electrical behavior, and thereby creating composites with superior properties. Precise control over the nanoscale morphology of hybrid materials that integrate polymers and nanoparticles remains, however, a considerable challenge. The presence of the polymers does provide the material with a degree of flexibility and mutability; when combined with the appropriate mechanical deformation, this mutability could provide a means of reorganizing the solid nanoparticles into useful structures. A fundamental problem is establishing how to combine the appropriate deformation with suitable mechanoresponsive elements within the material to achieve the desired structural reconfiguration. Here, we take inspiration from the early Rubik’s cube, where individual colored blocks were held together by elastic bands;1 as the structure was twisted, the blocks moved to form a different, stable arrangement of elements in the cube. To design an analogous mechanoresponsive system, we focus on polymer-grafted nanoparticles (PGNs), where each rigid nanoparticle core is decorated with a corona of polymer chains. The free ends of these chains encompass reactive functional groups that allow the polymers to form “arms” between neighboring particles and thus, interconnect the PGNs into a network. In effect, these polymer © 2017 American Chemical Society

Received: January 3, 2017 Accepted: February 28, 2017 Published: February 28, 2017 3059

DOI: 10.1021/acsnano.7b00037 ACS Nano 2017, 11, 3059−3066

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Figure 1. Morphologies obtained after four rotations of the top layer during torsional deformation of PGN networks formed from 1:1 mixtures of A (cyan) and B (pink) particles interlinked by labile bonds having the following bond energies: (a) UAA,BB = 39kBT, UAB = 33kBT, and (b) UAA,BB = UAB = 33kBT. In (c), the PGNs are interlinked by permanent bonds with bond energies of UAA,BB = 39kBT, UAB = 33kBT. The initial conditions correspond to random binary A/B mixtures within the bulk of the material; the bottom and top layers consist of solely A and B particles, respectively. (d)-(f) Evolution of the numbers of AA (red), BB (blue), and AB (black) links per particle as a function of the rotation angle θ for the systems shown in (a)−(c), respectively. The results are obtained by averaging over eight independent simulations and the error bars represent the standard deviations from those averages.

conditions, allowing us to design an intertwining helical morphology, which exhibits superior mechanical properties relative to the random binary structure. In contrast, if the AA and BB bonds are weaker than the AB bonds, the torsional deformation of an initially structured binary PGN network can result in the mixing of the A and B nanoparticles and forming more homogeneous hybrid materials. Hence, twisting our mechano-mutable “Rubik’s cube” material provides an effective route to tailoring the nanoscale morphology, and hence the global properties of the nanocomposite.

In these prior studies, we assumed that the networks were composed of a single type of PGN and all the labile bonds were characterized by the same bond energy. Herein, we introduce both A- and B-type nanoparticles, which encompass different reactive groups at the free ends of the grafted chains. The energies of the labile bonds formed between the same types of particles, UAA and UBB, are taken to be different than the bond energies between dissimilar types of particles, UAB. For simplicity, it is assumed that UAA = UBB ≡ UAA,BB. We focus on the behavior of the binary PGN networks under applied torsional deformations. In contrast to tensile deformations, torsion induces relatively small volumetric changes and hence, deformed PGN networks linked by labile bonds could exhibit useful restructuring without breaking. As discussed below, if the AA and BB bonds are stronger than the AB bonds, the torsion-induced restructuring of initially random, binary PGN networks results in the spatial separation of the A and B nanoparticles. The morphology of such phaseseparated PGN networks can be controlled by boundary

RESULTS AND DISCUSSION Our computational approach and the parameters used for modeling the PGN networks are detailed in the Methods section. Via this model, we examine the effects of applying torsion to cylinder-shaped samples composed of 986 PGNs. The PGNs are interconnected by chemically bonded polymer chains; a “link” refers to a group of chemically bonded polymers connecting two PGNs. The system is initially organized into a 3060

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Figure 2. Typical morphologies obtained after four rotations during the torsional deformation of initially random 1:1 mixtures of A and B particles within the bulk. In (a)−(c), both the bottom and top layers consist of 1:1 mixtures of A and B PGNs. In (d) and (e), the bottom layer consists of 1:1 mixtures of A and B PGNs and the top layer is formed solely from B particles. The particle coloring is the same as in Figure 1. The PGNs are interlinked by labile bonds with energies of UAA,BB = 39kBT and UAB = 33kBT. Note the formation of horizontally layered structures in (b)−(e).

PGN network exhibits an 8-fold decrease, whereas the numbers of stronger AA and BB links increase 2-fold, as seen in Figure 1d. The latter figure also reveals that the number of links remains constant after completion of the phase separation. Both a disparity in the energies of the chemical bonds, UAB < UAA,BB, and the ability of the chemical bonds to reform after breakage are crucial for the torsion-induced phase separation. If all the labile bond energies are set to the same value of UAA,BB = UAB = 33 kBT, then there is no observable change in the structure from the initial random configuration of the binary PGN network after four revolutions of the top boundary (see Figure 1b). The numbers of AA, BB, and AB links in the latter system exhibit a small initial decrease after the torsion is applied, and then remain constant, thus indicating the absence of structural changes in the course of deformation (Figure 1e). When the bond energies are reset to UAA,BB = 39kBT and UAB = 33kBT but the bonds cannot reform after breakage (“permanent” bonds), then again there is no observable change from the initial random morphology after four revolutions (Figure 1c). In the absence of bond reformation, the interparticle links decay irreversibly, as seen in Figure 1f. The weaker AB links break rapidly, with 99% of these links disappearing before the torsion angle reaches θ = 5°. The numbers of stronger AA and BB links decrease gradually (Figure 1f), but all these links are broken by θ/2π = 4. Hence, the sample shown in Figure 1c loses mechanical integrity. In the following studies, we focus on the cases where the labile bond energies are set at UAA,BB = 39kBT and UAB = 33kBT, and hence, the torsional deformations result in phase separation. We find that the phase separated morphology

FCC structure that is 11 layers in height. The middle nine layers (807 particles in total) are initialized as a 1:1 random mixture of A and B particles. The top and bottom layers contain 90 and 89 particles, respectively; the boundary conditions are altered by varying the A/B composition and morphology of these confining layers. The top layer of a sample is modeled as a rigid object that lies perpendicular to the central axis of the sample; this layer can move as a whole along the axis, and rotates around the axis under the applied torsion. The positions of particles in the bottom layer are fixed in space. After equilibration, the radius and height of the samples are approximately 16r0 and 25r0, respectively, where r0 = 50 nm is the radius of the nanoparticle core and is taken as the characteristic length scale of the simulations. After the equilibration of the sample (see Methods), the top layer is rotated in the counterclockwise direction at a constant angular velocity with period Trot = 1.2 × 104T0 ≈ 3 min, where T0 = 1.41 × 10−2 s is the time scale of the simulations. We first set the energies of the labile bonds to UAA,BB = 39kBT and UAB = 33kBT; these values lie within the range of energies characteristic of thiol/disulfide exchange reactions9,10 and supramolecular interactions11 that are used for fabricating self-healing polymers.12−14 The top layer of the sample is formed solely from B particles and the bottom layer contains only A particles. After four revolutions of the top boundary, the initially random mixture within the interior of the material phase-separates into distinct layers, with all the B particles clustered near the top boundary and all the A particles congregated at the bottom (see Figure 1a). In the torsioninduced restructuring, the number of weaker AB links in the 3061

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Figure 3. Structure formation in the course of torsional deformation of 1:1 mixtures of A and B PGNs at both the top and bottom layers consisting of the semicircular A and B patches. (a) Initial random configuration. (b) The columnar morphology formed after two revolutions of the top layer. (c) The helical morphology formed after three revolutions. (d) The evolution of the numbers of AA (red), BB (blue), and AB (black) links per particle as a function of the rotation angle θ; averaging is performed over eight independent simulations. (e) Torque variations in the course of torsion deformation obtained during two independent simulations. Particle coloring in (a)−(c) is the same as in Figure 1. The labile bond energies are the same as in Figure 2.

network, and thus promote the formation of a helical structure, for example. To achieve the latter morphology, the top and bottom layers are arranged into semicircular patches of A and B particles (see Figure 3a), while the bulk of the material initially has a random configuration. The A-type particles are preferentially attracted to the A semicircle and the B-type particles are preferentially attracted to the B semicircle on the confining layer so that distinct clusters are formed in the course of the torsional deformation. After two revolutions of rotation, the sample segregates into A and B columnar clusters that span the height of the sample and are bounded by the appropriate domains on the confining layers (Figure 3b). As the top layer is rotated further, the clusters become intertwined and thereby form the double helical structure shown in Figure 3c at the end of the third revolution. Further rotations of the top boundary lead to the consecutive breakage and reformation of the helices during each revolution. The formation and evolution of the helical structure are accompanied by notable variations in the numbers of AA, BB, and AB links in the PGN network. Figure 3d shows that the number of AB links decreases and the numbers of AA and BB links increase until the end of the second revolution, thus indicating the process of phase separation. After the helices have been formed (by the end of the third revolution), the numbers of links exhibit slight periodical variations as the helical structure breaks and reforms during each revolution (Figure 3d). The breakage and reformation of the helices result in the large-amplitude oscillations of torque measured on the top boundary. Figure 3e shows the variations of torque obtained from two independent simulations. Both curves in Figure 3e indicate that at θ/2π ≥ 3, the torque increases when the helices are formed and decreases when they are broken. (The blue

formed in the binary PGN networks in the course of torsion depends on the structure of the top and bottom boundary layers (boundary conditions). Notably, the layered structure in Figure 1a is formed when the top layer is formed solely from B particles and the bottom layer contains only A particles for any initial random configuration of the bulk of the system. On the other hand, if both the boundary layers are composed of random mixtures of A and B particles, then the structure formed under torsion depends on the initial arrangement of the two types of particles within the bulk. Figures 2a-c show the phase separated morphologies formed after four revolutions in the case where both the top and bottom boundaries are composed of 1:1 random mixtures of A and B particles. Depending on the initial configuration of the sample, the torsional deformations lead to one of three possible arrangements: the A and B clusters form either along the sample axis (Figure 2a), the B layers form on the top (Figure 2b) or at the bottom of the sample (Figures 2c). The number of possible particle arrangements within the sample is reduced when only one of the boundary layers is composed of a random mixture of A/B particles, but the other layer is composed solely of one type of particle. As seen in Figure 2d,e, the top boundary composed of B particles preferentially binds B particles, so that after four revolutions, the layer of B particles is always formed on the top. The layer of A particles forms directly beneath the B layer (Figures 2d,e). Depending on the initial random configuration, the Acontaining layer either extends to the bottom boundary (Figure 2d), or is formed in the middle of the sample and the A and B particles close to the bottom boundary are intermixed and not segregated (Figure 2e). The above findings indicate that the boundary conditions can be harnessed to control the morphology of the binary PGN 3062

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Figure 4. Details of the helical structure formed under applied torsion. (a) The clusters of A (cyan) and B (pinks) particles in Figure 3c shown separately. (b) Spatial distributions of the centers of mass of A (blue) and B (red) particles in the system shown in Figure 3c and sliced into nine layers, with the top and bottom particle layers excluded. (b) Radial distances from the sample axis, and (d) angular positions of the centers of mass of A and B particles as a function of layer number obtained by averaging over eight independent simulations.

curve in Figure 3e goes to zero after six revolutions because in that particular simulation, the two helices ultimately broke into irregular clusters and did not reform.) To prove that the structure in Figure 3c is indeed helical, we split the sample into two clusters, each consisting of one type of particles (see Figure 4a). We divide each cluster into nine layers along the sample axis (excluding the top and bottom boundaries), and determine the position of the center of mass for each layer. Figure 4b shows that the centers of mass of the two clusters form two spirals. As expected for a helical structure, the radial distances from the centers of mass to the sample axis are all equal, within the range of the error bars (Figure 4c). Additionally, the angular position of the layer center of mass increases linearly from layer to layer for both types of particles (Figure 4d), as is characteristic for helices. To demonstrate the advantage of forming such intertwining helical structures with respect to the mechanical properties of the material, we compare the tensile properties of PGN networks having a layered morphology (as in Figure 1a), random structure (as in Figure 3a), and the helical structure (as in Figure 3c). After equilibrating the samples for 2.0 × 104T0, we apply a strain-controlled deformation: the bottom layer of the sample is held fixed, and the top layer is pulled upward along the center axis of the cylinder at a constant velocity of v = 0.001v0 ≈ 3.55 nm/s. Figure 5 shows the force vs strain curves obtained from the tensile deformation of the samples with the three different structures. The samples having the layered and

Figure 5. Force vs strain curves obtained from the tensile deformation of the binary PGN networks. Here, the pink line is for systems where the applied torsion results in formation of random mixture of particles (as in Figure 3a); the blue link is for systems where the applied torsion leads to helical structures (as in Figure 3c); and the redline is for systems that form a layered structure (as in Figure 1a). Averaging is performed over eight independent simulations.

random structures break significantly earlier than the helical samples. The layer-structured samples (Figure 1a) are especially weak because the A and B layers are normal to the direction of pulling and the AB bonds linking the adjacent A and B layers are the weakest bonds in the system. The helical samples (Figure 3c) exhibit the superior tensile properties among the materials tested because the clusters, which are linked through the stronger AA and BB bonds, span the entire samples along the direction of pulling. Figure 5 shows that the strain at break, 3063

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course of mixing, whereas the number of stronger AB links increases 3.5 times.

tensile strength, and toughness of the helical samples (Figure 3c) are approximately 1.4, 2.4, and 4.3 times greater, respectively, than those for the random samples (Figure 3a). The above findings reveal that the applied torsion provides a facile route to fabricating binary nanocomposites that encompass distinct chiral structures. Here, the system was rotated in the counterclockwise direction, leading to the helices in Figure 4a. Notably, the handedness of each of these helices can be altered by rotating the sample in the clockwise direction. The ability to controllably form intertwining, double helices from achiral systems can have significant implications for controlling the optical properties of the composite.15,16 The torsional deformation can also be harnessed to transform an ordered binary structure into a mixed morphology if the AB bonds are stronger than AA and BB bonds. To illustrate this behavior, we set the energies of the labile bonds equal to UAA,BB = 33kBT and UAB = 39kBT and take the initial PGN network to consist of coaxial cylinders of A and B particles, as shown in Figure 6a. Specifically, the system

CONCLUSIONS The findings from our simulations reveal that the nanoscale morphology of binary PGN networks interlinked by labile bonds can be controllably tailored by applying a torsional deformation to the system and hence, point to an effective route for tuning the materials’ properties. The torsional deformation induces a reshuffling of the labile bonds without fracturing the sample. The resultant restructuring of the material depends on the energies of the labile bonds, and can be altered by switching the configuration of the bounding layers. For UAA,BB > UAB and boundary layers encompassing a design of A/B semicircles, the applied torsion led to the formation of helical structures within the binary PGN networks. These studies indicate a route for creating hybrid materials that encompass pronounced and controllable chirality; the handedness of the A and B helical clusters can be altered by twisting the material in the opposite direction (i.e., in the clockwise direction). In addition to improving the mechanical behavior, the double helical structure within this system could be exploited to tune the optical properties of the material. In the case where UAA,BB < UAB, torsional deformations applied to initially structured system resulted in the intermixing of the two types of particles and homogenization of the material. This intermixing is particularly important for improving the composite’s structural integrity when the bonds between the dissimilar species are stronger than the connections between identical units. Our results illustrate the utility of combining dynamic, reformable bonds and mechanical deformation to create nanocomposites that can be physically restructured into different configurations and thus, different useful materials. In effect, we designed a “Rubik’s cube” material, where a single sample could be reconfigured into multiple, stable configurations. We focused on bond energies that are representative of disulfide bonds, an ideal example of interconnections that can reshuffle under deformation. These finding can provide useful guidelines for creating mechano-mutable composites and thus, opening routes for materials’ processing.

Figure 6. Torsion-induced mixing of particles A (cyan) and B (pink) in a binary PGN network at UAA,BB = 33kBT and UAB = 39kBT. (a) The initial system consisting of the coaxial cylinders of A and B particles. (b) Structure of the sample after four revolutions of the top layer. (c) The evolution of the numbers of AA (red), BB (blue) and AB (black) links per particle as a function of the rotation angle θ; averaging is performed over eight independent simulations.

METHODS The PGNs consist of a rigid, spherical nanoparticle of radius r0, which is taken as the characteristic length scale, and f grafted polymer chains. The grafted chains form a corona of thickness q around the nanoparticles. The free end of each grafted chain contains a reactive functional group. The system is composed of the two types of PGNs, which differ in the type of reactive end groups, A or B. Upon overlap of the coronas belonging to two different nanoparticles, the endgroups form AA, AB, or BB chemical bonds, depending on types of the two PGNs. The rate of rupture of an interparticle bond depends on the bond type and the force acting on the bond. The interparticle chemical bonds can re-form after rupture (labile bonds). Table 1 contains the values of model parameters used in the simulations. The energy of interaction between two PGNs is given by the potential Uint, which is the sum of three pairwise interactions: Uint = Urep + Ucoh + Ulink. The repulsion interaction, Urep,17−19 is due to the steric repulsion between the corona chains:

contains 1008 PGNs, which are organized into 8 layers along the height dimension. The top and bottom layers contain 129 and 121 particles, respectively. The inner core, r < 6.37r0, and outer shell, r ≥ 12.74r0, of the sample consist of the A particles, which enclose the cylinder of B particles placed at 6.37r0 ≤ r < 12.74r0. After equilibration, the radius and height of a sample are approximately 19r0 and 18r0, respectively. After four revolutions of the top boundary, the initially separated A and B particles are mixed within the body of sample (see Figure 6b). As seen in Figure 6c, the numbers of weaker AA and BB links in the PGN network exhibit a 2-fold decrease in the 3064

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where Fext is the applied external force. The equations of motion for the PGNs are numerically integrated using the Euler scheme. The time step of the simulation is Δt = 10−2T0. The number of bonds between two PGNs, Nb, as a function of time t is described by the following rate equation:

Table 1. Simulations Parameters (a) Dimensional Units length: nanoparticle radius time, t velocity, v force, F toughness, W

r0 = 50 nm T0 = 1.41 × 10−2 s v0 = r0T−1 0 = 3.55 μm/s F0 = 2.98 pN F0r0 = 89.74 kJ/mol

dNb/dt = − k r(R )Nb + k f Pc(R )[Nmax(R ) − Nb]2

The first term on the right-hand-side (r.h.s.) of eq 4 is the rate of rupture of the bonds, where kr(R) = k(0) r exp[γ0F(R − 2)] is the rate constant, which depends on the interparticle distance R through the force F. The equation for kr describes an increase in the rate of rupture with the force according to the Bell model,22 where k(0) r is the rate of rupture at zero force, and γ0 is the sensitivity to force F(r) = κ(r)r, with r = R − 2 (cf. eq 3). The rate of rupture at zero force is calculated as = v0 exp(−U(l) k(0) r 0 /kBT), where ν0 is the characteristic frequency of molecular vibrations,22 and U(l) 0 is the bond energy taken equal to UAA, UBB, or UAB depending on the types of the two particles. The second term on the r.h.s. of eq 4 is the rate of bond formation, which depends on the distance R; here, kf is the rate constant of bond formation, Pc(R) is the probability of contact between two chain ends on two neighboring PGNs, and Nmax(R) is the maximum number of bonds that can be formed between the two particles.2 For each time step of the simulation Δt = 10−2T0, we first use the Euler method to solve eq 4, and then apply the rounding procedure to obtain the updated integer value for the number of bonds Nb.2 A sample is initialized into a FCC lattice. At 0 ≤ t < 4 × 103T0, the labile bonds are formed in the system of immobile particles. The particles in the middle layers are allowed to move at t = 4 × 103T0, whereas the particle positions in the bottom layer are kept fixed, and the top layer is allowed to move as a rigid object. The Z coordinates of all the particles in the top layer are kept equal. A constant downward force Fc = 100F0 is applied on the top layer at t = 4 × 103T0. The normal force Fc is divided equally between the particles in the layer. At 4 × 103T0 ≤ t < 2 × 104T0, the sample is equilibrated under the normal force. Finally, at t = 2 × 104T0, the torsional deformation is applied to the top boundary. The top boundary is rotated around the sample axis at a constant angular velocity under the applied constant normal force. To calculate the torque acting on the top boundary layer in the course of deformation, we first determine the horizontal component of force exerted on each particle in that layer by the particles in the body of the sample. For a given boundary particle, the vector cross product of the radius vector of the particle and the determined horizontal component of force gives the contribution of the particle to the torque. The value of torque is obtained by summing up the contributions from all particles in the top boundary. After equilibration, the samples maintain an essentially cylindrical shape. For each particle, the distance between the particle and central axis of sample is no greater than the cylinder radius rcyl ≈ 16r0. In the course of torsion, some particles could move across the side boundary of this initial cylindrical surface. The following special procedure is applied to confine the particles within the cylinder of radius rcyl. If, after a step of the simulation, the coordinates (x, y, z) of some particle are such that ρ = (x2 + y2)1/2 > rcyl, then the coordinates of this particle are updated as (xrcylρ−1, yrcylρ−1, z).

(b) PGN Characteristics corona thickness, q Kuhn length, lp no. of grafted arms, f arm contour length, L chain spring constant, κ0 repulsion parameter, σ cohesion parameters mobility of PGN, μ rotating period, Ttorsion pulling velocity, ν

0.75r0 1 nm 156 8.89r0 7.81 × 10−2F0r−1 0 3.02r0 U(0) coh = 60kBT; A = 1.15σ; B = 0.08σ 0.57v0F−1 0 1.2 × 104T0 10−3v0

(c) Bond Parameters bond parameters

labile bonds

bond energy, U0 rupture rate at F = 0 bond sensitivity, γ0 formation rate, kf

Urep kBT

=

33kBT 6.56 × 10−4T−1 0 6F−1 0 30T−1 0

39kBT 1.62 × 10−6T−1 0 6F−1 0 30T−1 0

5 3/2 f 18

⎧ ⎛R⎞ f 1/2 −1 ⎪− ln⎜ ⎟ + (1 + ) , R≤σ 2 ⎪ ⎝σ ⎠ ×⎨ −1 ⎡ f 1/2 (R − σ ) ⎤ ⎪⎛ f 1/2 ⎞ σ ⎥, R > σ ⎟⎟ ⎜⎜1 + exp⎢− ⎪ ⎪⎝ ⎢⎣ ⎥⎦ 2 ⎠ R 2σ ⎩

(1)

where R is the center-to-center distance between two particles, and σ = 2(1 + q)(1 + 2f−1/2)−1 is the spatial range of the interaction.17−19 The attraction due to cohesion, U coh , is described through the pseudopotential20 −1 ⎡ R − A ⎤⎫ (0) ⎧ ⎬ ⎨1 + exp⎢ Ucoh(R ) = − Ucoh ⎥ ⎣ B ⎦⎭ ⎩

(2)

U(0) coh

sets the energy scale of cohesion, and A and where the parameter B are respectively the position and width of the attractive well. The potential Ucoh is constant at small interparticle distances R, and balances the repulsion at the corona edges. The attraction potential, Ulink, describes the spring-like attraction between two PGNs upon formation of bonds. The force of attraction between the two interlinked nanoparticles is calculated as

Flink = Nbκ(r )r

(4)

(3)

Here, r = R − 2 is the end-to-end distance of polymer chains connecting the particles, Nb is the number of linking bonds, and κ(r) = 2 −2 −2 kBTR−2 0 {1 + 2[1 − r (2L) ] } is the stiffness of the spring-like force due to the formation of a bond according to the worm-like chain model.21 In the equation for κ(r), 2L is the contour length of the polymer composed of two linked grafted chains, R0 = 4lpL is the mean square end-to-end distance, and lp is the persistence length. The dynamic behavior of the system is assumed to occur at room temperature in the overdamped regime, where the motion of each PGN is described by dx/dt = μFtot. Here, x is the spatial position of the particle, t is time, μ is the particle mobility, and Ftot is the net force acting on the particle. The force is calculated as Ftot = −∂Uint/∂x + Fext,

AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. ORCID

Anna C. Balazs: 0000-0002-5555-2692 Notes

The authors declare no competing financial interest. 3065

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DOI: 10.1021/acsnano.7b00037 ACS Nano 2017, 11, 3059−3066