Structure and Entanglement Network of Model Polymer-Grafted

Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

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Structure and Entanglement Network of Model Polymer-Grafted Nanoparticle Monolayers Jeffrey G. Ethier and Lisa M. Hall* William G. Lowrie Department of Chemical and Biomolecular Engineering, The Ohio State University, Columbus, Ohio 43210, United States

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S Supporting Information *

ABSTRACT: Ultrathin films containing polymer-grafted nanoparticles (PGNs) show promise for use in hybrid electronics and high energy density materials. In this work, we use a coarse-grained model to simulate a hexagonally packed monolayer of PGNs adsorbed on a smooth surface that is attractive to both nanoparticles and polymer chains. We find that decreasing graft density at the same graft length increases interpenetration of the polymer-grafted layers, as expected. We quantify both overall and interparticle entanglements (between polymers grafted to different PGNs). While the higher graft density particles have a higher overall number of entanglements per chain, the lower graft density particles have higher interparticle entanglements per chain due to their increased chain interpenetration. Finally, we apply uniaxial tensile deformation to the monolayers; the peak stress occurs at lower strain values for higher graft density particles, which is attributed to the relatively lower number of interparticle entanglements. Our analysis provides a molecular picture of how decreasing graft density leads to better interpenetration, increased interparticle entanglements, and increased toughness of PGN monolayers, though it can lead to slightly less uniformly spaced monolayers, in agreement with previous experimental observations. These tradeoffs are crucial to understand for the design of robust, well-ordered inorganic−organic hybrid films.

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INTRODUCTION Inorganic nanoparticles with polymer chains grafted to the surface, or polymer-grafted nanoparticles (PGNs), can form a variety of anisotropic or ordered structures in the bulk or in thin films.1−4 By using PGNs neat (without free matrix polymer or solvent), one can avoid issues of nanoparticle phase separation common in typical nanocomposites, as all polymer chains are chemically bonded to a particle. The organization of the nanoparticles can thus be tuned by changing the structure of the polymer layer surrounding the nanoparticle, for example, by changing graft length and graft density. While we consider neat systems here, previous work on PGNs in solvents provides important insights. In particular, models for planar brush surfaces and star polymers have been extended to spherical polymer brushes (PGNs) in solution, and experiments have observed the expected structural transitions.5−8 There are several regimes of graft conformational behavior that are determined by the graft chain length (molecular weight), graft density, and size of the nanoparticle (curvature). For example, at low graft densities and in poor solvent, the polymer chains are collapsed similar to low graft density planar brushes. Increasing graft density leads to a semidilute polymer brush regime, where chains are confined by neighboring graft chains and configurational entropy is reduced. At high graft densities, polymer chains are highly stretched due to increased confinement, creating a concentrated polymer brush layer. At © XXXX American Chemical Society

a critical radius away from the nanoparticle surface, the free volume becomes large enough for the chains to transition back to a semidilute regime. In neat PGN systems, neighboring chains rather than solvent act as the suspending medium,9−11 and the grafted polymer chains of neighboring particles can interpenetrate substantially and allow for semidilute conformational behavior.12 Overall, the graft length, graft density, and nanoparticle size are key parameters in understanding the selfassembly behavior and mechanical response of these materials and in optimizing their properties for applications such as hybrid electronics, sensors, dielectrics, and flexible energy storage devices.13−17 Numerous studies have analyzed the effect of the grafted polymer structure on the self-assembly,13,18−24 dynamics,12,25−30 glass transition,31−33 and mechanical properties28,34−36 of neat PGNs in bulk and thin films. While our interest here is on PGNs deposited on surfaces, we are guided by the work establishing the bulk properties of PGN systems. At low graft densities, the nanoparticles can orient such that their surfaces can come into contact, and neat PGNs selfassemble into anisotropic structures similar to PGNs in a polymer matrix (e.g., stringlike or spherical aggregate Received: June 27, 2018 Revised: October 22, 2018

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DOI: 10.1021/acs.macromol.8b01373 Macromolecules XXXX, XXX, XXX−XXX

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fillers showed increased dielectric breakdown strength for the PGN system compared to the analogous polymer nanocomposite.17 Neat PGN films containing polystyrene-grafted silica particles were imaged before and after breakdown using transmission electron microscopy (TEM), showing the significant morphology changes in the film after breakdown.42 The response of similar PGN thin films to high strain was also investigated; that work found that the failure mode depended on thickness, with thick films (above 200 nm) exhibiting crazing behavior.43 Very recent work showed how the molecular weight impacts the crack to craze transition and deformation properties of low graft density particles.44 It is well-known that the failure mechanism for entangled polymers involves craze formation. Thus, as expected based on bulk PGN literature, the thin-film literature points toward entanglements and interpenetration as quantities of interest in optimizing thin-film PGN performance. While the effects of entanglements on mechanical properties are easily measured, probing details of polymer conformations to understand polymer entanglements requires significant effort experimentally.45,46 Meanwhile, coarse-grained molecular dynamics (MD) simulations can easily track polymer conformations and allow for a clear description of the entanglement network. This has been demonstrated in prior works on model polymer melts,47−51 glasses,52,53 and nanocomposites.54−57 Several specific methods have been developed to determine the entanglement network from simulation data. One popular method is the geometrical primitive path analysis developed by Kröger, known as the Z1 algorithm.58 This analysis fixes the chain ends and iteratively attempts to locally straighten the chains’ paths, while disallowing chain crossing, to create “primitive paths”. Kinks in the primitive paths occur where topological constraints are imposed on a chain by other chains and are representative of entanglements. This method has been shown to be a computationally efficient method to analyze entanglement networks in homopolymer melts and to give a fast converging estimate of the entanglement length.48,59 An alternative “annealing” method employs a simulation technique in which intrachain excluded volume interactions are turned off and the polymer system is cooled to obtain the network of primitive paths.47,51 A comparison between geometrical and annealing methods showed the results are generally similar.48,59 The entanglements found in model systems and the resulting dynamical behavior also map well to the dynamic behavior of experimental systems; for example, Pütz et al. showed that the scaled polymer diffusion constants versus N/Ne for model and experimental polymer systems fell on a universal curve.60 More advanced methods to analyze entanglements from simulation continue to be developed; recently, Likhtman and coauthors suggested that entanglements can be understood based on the time-averaged coordinates of the polymer chains.61,62 Specifically, they consider entanglements to be long-lasting, tight contacts between the polymer chains. Any of these methods to analyze entanglements can be easily applied to systems that include surfaces or nanoparticles. For example, the Z1 algorithm has been applied to understand how the presence of bare nanoparticles impacts entanglements in polymer nanocomposites.55,57 For the adsorbed monolayer PGN systems of interest here, steric constraints imposed by the nanoparticles, the fact that chains are grafted to the particle surface, and the vacuum and attractive surface interfaces are all expected to impact the

morphologies).13,18,19,24 Alternatively, at high graft densities and relatively short graft lengths, the nanoparticles pack more like spheres, forming crystal-like lattices such as hexagonal and face-centered cubic arrays, making these materials of potential interest for multiple applications.18,20−22 However, processing highly ordered neat PGN assemblies is a challenge, as crack formation can occur during solvent evaporation due to the fragility of these systems.34 Thus, there is a need to design highly ordered but mechanically robust PGN assemblies. Interestingly, nanoparticle order at high graft densities can persist even with the addition of free matrix chains or with increased molecular weight of the graft chains.20,37,38 The addition of free polymer has been shown to enhance viscosity28 and reduce crack formation38 compared to pure PGNs. In the absence of matrix polymer, mechanical strength can be enhanced by using longer graft chains, presumably because this allows for better interdigitation and the formation of entanglements between grafted polymers. Specifically, toughening between polymer-grafted nanoparticles has been shown to increase when the grafted chain length allows it to be well into the semidilute regime.34,35 Recently, Agrawal et al. showed that increasing temperature can enhance the interdigitation of PEGME corona chains on silica particles, which induces nanoparticle coupling and increases elastic modulus.26 Enhanced interdigitation also promotes craze formation in glassy PGN materials during deformation.35,36 In addition to graft density and chain length, particle size also has an effect on the mechanical properties of neat PGNs. In particular, Schmitt et al. suggested that increasing particle size increases volume of interstitial areas, requiring chains to stretch more into the interstitial areas, which lowers the entanglement density and reduces fracture toughness.34 Overall, these studies established that interpenetration and entanglements are key properties that determine PGN mechanical behavior. Depositing PGNs on a substrate changes the grafted layer structure and can allow them to form highly ordered arrays. The spacing of the particles is dependent on the polymergrafted properties, surface chemistry, and quality of solvent or deposition process used.39,40 Recently, a study using gold nanoparticles with grafted poly(ethylene oxide) chains in good solvent showed that particle spacing agreed with scaling predictions (based on polymer chains in dilute solution), whereas using poor solvent resulted in disordered arrays.40 Furthermore, Che et al. analyzed polystyrene-grafted gold nanoparticles adsorbed on surfaces with different chemical treatments; monolayers and lower particle density systems were created. The authors showed that, with increasing polymer-surface interaction energy, the polymer “canopy”, or grafted layer, of an individual PGN spreads out to increase its interaction with the surface. Their sub-monolayer films contained strings of particle agglomerates, whereas the monolayer consisted of well-ordered hexagonally packed particles. Another study by Che and co-workers reported that PGN thin films were more stable as a function of surface energy and temperature compared to the analogous polymer thin film.41 Well-dispersed particle structures such as those of the PGN monolayers can also provide improved dielectric breakdown strength.14,17 In particular, Grabowski and coworkers recently reported little change in the low-frequency dielectric loss of polystyrene and poly(methyl methacrylate) grafted on silica with increasing particle volume fraction, though adding bare nanoparticles would increase dielectric loss.14 Additionally, similar films containing high permittivity B


ÄÅ l ÅÅ o o ÅÅi σ y12 i σ y6 o o o o 4ϵÅÅÅÅjjj zzz − jjj zzz − ULJ = m kr{ ÅÅk r { o Ç o o o o o 0 o n

Macromolecules entanglement network. While we are not aware of prior simulation studies analyzing entanglements in thin polymer nanocomposite films, several groups have analyzed entanglements of adsorbed polymer films without nanoparticles. For example, Li et al. recently used Monte Carlo simulations to show that adsorption on an attractive surface decreases entanglement formation.63 Additionally, Carrillo et al. used MD simulations to analyze the polymer conformations and dynamic scaling behavior of strongly adsorbed polymer chains, showing dynamics slow due to polymer segments forming dense layers in the interfacial layer.64 Our prior simulation study considered adsorbed PGN materials, but only isolated individual or pairs of PGNs on the surface were simulated (rather than a monolayer). We analyzed the effect of surface adsorption strength on the overall structure of the polymer canopy. Increasing wall interaction strength was seen to decrease interparticle entanglements, as the polymer chains spread out on the surface and the interparticle distance increases.65 Here, we simulate a thin film of PGNs placed in a hexagonally packed monolayer on an attractive surface, using periodic boundaries and constant pressure in the plane of the film to represent an infinite area. We compare PGN interparticle spacing and interdigitation in the monolayer. We then relate the interdigitation of PGNs to the entanglement formation and to the mechanical properties of the monolayer. For this initial monolayer study, we consider polymers in the melt state well above the glass transition temperature. Thus, our systems are representative of an experimental system during annealing before it would be cooled (setting in the overall structure present in the annealing step) and used as a structural material. We will consider glassy systems in future work. The paper is structured as follows. We begin by describing our coarse-grained model and simulation details. We compare the structure of PGN monolayers, considering PGNs with long versus short chains at moderate versus high graft density. We discuss the extent of interdigitation between PGNs based on radial distribution functions and relate the structure to the number of interparticle entanglements. Finally, we subject the monolayer systems with longer chains (at both moderate and high graft density) to uniaxial strain and show that the lower graft density system with increased interparticle entanglements also shows increased toughness.

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É 12 6Ñ ij σ yz ij σ yz ÑÑÑÑ jj zz + jj zz ÑÑ r ≤ r c jj r zz jj rc zz ÑÑÑ k c{ k { ÑÖ r > rc

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(2)

where we set the cutoff rc = 2 σ for interactions between bonded monomers and rc = 2.5σ for nonbonded monomers. We set ϵ = 1kBT for all monomer−monomer interactions. Nanoparticles (NPs) are modeled as a spherical beads of diameter dNP = 10σ and interact with monomers via the radially shifted 12−6 LJ potential ÅÄÅ l o ÅÅi σ y12 i σ y6 o o ÅÅjj o zz − jj zz o 4 r ≤ rc + Δ ϵ ÅÅj o z j z o Å o r r − Δ − Δ k { k { Å o Å o Ç o É o o 12 6Ñ shifted Ñ ULJ =m i y i y σ σ jj zz jj zz ÑÑÑÑ o o − + j z j z o jj r − Δ zz jj r − Δ zz ÑÑÑ o o o k c { k c { ÑÑÖ o o o o o o o 0 r > rc + Δ n 1/6

(3)

where Δ is set to 4.5σ, and the diameter of 10σ reflects the size due to the radial shift plus the strongly repulsive part of the potential. Therefore, monomer−monomer and monomer− nanoparticle interactions are identical other than the radial shift. NP−NP interactions would be similarly shifted, but they do not occur, as nanoparticles remain further than the cutoff distance apart from each other. The first bead in each polymer chain, the “graft bead”, is rigidly attached to the nanoparticle surface. Specifically, to create each PGN, the requisite number of graft beads for a given graft density are placed on the nanoparticle surface, and the rest of each polymer chain is grown from its graft bead. Grafts are placed randomly on the NP surface, except that placing a graft bead less than 0.9 σ from another graft bead is disallowed. Graft beads interact identically as other monomers, except that they are rigidly attached to the NP and are excluded from interactions with the NP or other graft beads. A flat, smooth wall is placed at z = 0 and interacts with both monomers and nanoparticles via the 9−3 LJ potential of the form ÄÅ l 9 ÅÅ o o ÅÅ 2 ij σ yz9 ij σ yz3 2 ijj σ yzz o o Å o j z j z z ≤ zc ϵ − − j z Å j z o j z w − pÅ o ÅÅ 15 k z { o 15 jjk zc zz{ z{ k o Å o Ç o É o 3Ñ ÑÑ Uw − p(z) = o m i y σ j z o jj zz ÑÑÑÑ o + o jj z zz ÑÑ o o o k c { ÑÑÖ o o o o o o o 0 z > zc n

MODEL AND SIMULATION METHODOLOGY

Coarse-Grained Model. We use a coarse-grained model for nanoparticles and monomers; polymers are represented as bead−spring chains similar to those of Kremer and Grest.66 Specifically, monomers are connected along the chain using the finitely extensible nonlinear elastic (FENE) potential É ÅÄÅ 2Ñ ij r yz ÑÑÑÑ k 2 ÅÅÅ j z UFENE = − R 0 lnÅÅÅ1 − jj zz ÑÑÑ j R 0 z ÑÑ ÅÅ 2 k { ÑÖ ÅÇ (1)

(4)

where ϵw‑p is the monomer−wall interaction strength, we set the cutoff zc = 2.5σ, and we shift the NP−surface interactions by the same amount as the NP−monomer interactions. The 9−3 LJ interaction is derived by integrating over the full (not cut off) 12−6 LJ interactions of one particle with an infinitely thick wall of uniform density of ρs of the same particles, which 2 results in a prefactor of ϵw − p = 3 πρs ϵ. For context, to consider a neutral surface (one that interacts with the same strength as a wall of particles identical to monomers at bulk density) and using the bulk monomer density ρs = 0.89 of our model, one

where k = 30ϵ/σ2 is the spring constant, R0 = 1.5σ is the maximum extent of the bond, σ is the monomer diameter and our unit of length, ϵ is the monomer interaction strength and our unit of energy, and r is the distance between two bonded monomers. These values of k and R0 are chosen to prevent chains from crossing or breaking apart.66,67 All monomers interact via the 12−6 Lennard-Jones (LJ) potential C


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ymer’s entanglement length of ∼48).65 Note that these particle volume fractions are lower and graft densities are higher than those of systems predicted by Ginzburg to microphase separate into nanoparticle-rich and polymer-rich phases.18 This set of two choices of two graft densities and chain lengths leads to four possible particle types. We name systems as in prior work according to their graft density and graft length; for example, PGN6−160 refers to a graft density of 0.6 chains/σ2 and N = 160 beads/chain. Snapshots of the four types of PGN systems analyzed herein are found in Figure 1.

would use ϵw‑p = 1.86. Note that this is the ideal value for a uniform density of 12−6 particles smeared out to make a perfectly smooth wall, and this analysis does not consider packing of particles in the wall, nor does it consider the cutoff applied to each pairwise interaction in the simulation. In our prior work, we varied monomer−wall interaction strength ϵw‑p from 1 to 4 in even increments of 1kBT to show a range of possible adsorption behaviors of individual and pairs of particles.65 At the highest ϵw‑p = 4 and with long chains, the chains were highly stretched on the surface, and large parts of the adsorbed polymer layer were only one or two monomers thick even in the region between two nearby PGNs, a situation that is presumably not of interest in creating monolayers of approximately uniform thickness. In contrast, at ϵw‑p = 1−3, the thickness of the polymer between two nearby adsorbed PGNs (whose polymer canopies are significantly interpenetrated) is more similar to the height of the PGNs. Because of their approximately circular height profiles and the moderate to high graft densities used (such that bare regions of NPs are small and do not come into contact with those on other PGNs), we expect all of these systems to pack hexagonally in the monolayer if the particles are able to form a stable monolayer. However, even though we initialize the particles in a hexagonally packed monolayer, at values of ϵw‑p = 1−3, we found in initial testing that most systems began to form thicker layers before the end of the simulation time (see Supporting Information). We then increased ϵw‑p to 3.5 and found hexagonally packed monolayers were stable during the time scale of the simulation for the four PGN systems analyzed herein (described further below). We do not attempt to determine the range of interaction strengths over which the monolayer is stable or metastable in the current work. However, a simple enthalpic accounting, without considering changes in local monomer density or entropic concerns, leads one to expect the monolayer to be preferred versus the bulk above ϵw‑p ≈ 2 × 1.86 ≈ 3.7 (the value of 1.86 for a neutral wall was discussed above). This is because PGNs that exist in a monolayer interact with the surface below and nothing above (in our systems particles only interact with themselves, each other, or the wall; there is no air). Thus, the surface must provide twice as much favorable enthalpic interactions as a neutral (representative of bulk density) surface would, to compete with the bulk system in which PGNs instead interact with polymers on nearby PGNs at approximately bulk density both above and below themselves. The value ϵw‑p = 3.5 used herein is similar to but somewhat below the naive approximation of the value at which the monolayer is preferred versus bulk, and thus the monolayer structure may be kinetically trapped (the bulk may be thermodynamically favored). Experimentally deposited polymer films may also be found in a kinetically trapped state. Simulation Details. To create the initial state for the simulation, monomers are placed along a chain one bond length away from the previously placed monomer in a random direction from the first graft bead, except they are disallowed from entering the wall or the nanoparticle. Polymer lengths and graft densities are chosen to match our prior work, and they allow us to consider either short and unentangled or longer and lightly entangled chains at moderate or high graft density, with similar nanoparticle volume fractions as those of Che et al.39,65 Briefly, we use graft densities of Γdens = 0.6 chains/σ2 or Γdens = 0.3 chains/σ2 and chain lengths of N = 35 (unentangled) or N = 160 (above the analogous homopol-

Figure 1. Four particle systems used for analysis: PGN6−35, PGN6− 160, PGN3−35, and PGN3−160. Polymer chains are smoothed by averaging particle positions over 31 snapshots, each spaced 100τ apart, at the beginning of the data collection period described in the text. This and all other snapshots were created using the VMD software.68

In all cases, 12 PGNs are initially placed in a hexagonally packed arrangement near the wall; the box has the appropriate dimensions for hexagonal packing and is periodic in the x and y directions (parallel to the wall). The average box dimensions are (57 × 66 × 17), (65 × 75 × 23), (80 × 92 × 32), and (103 × 119 × 37), where the values represent x × y × z dimensions in units of σ for the PGN3−35, PGN6−35, PGN3−160, and PGN6−160 systems, respectively. Other details are generally the same as in our prior work, including that the system can expand or contract freely above the wall (representing a vacuum above the simulated system). A short simulation is run with an increasingly repulsive soft potential between nonbonded monomers to remove any monomer overlaps; nanoparticle−monomer interactions are unchanged. We then turn on the LJ potentials described above and equilibrate the system in the NPT ensemble, using a Nosé−Hoover thermostat and barostat (the barostat is applied in the x and y directions, which are coupled to keep a constant side length ratio). We use a time step of δt = 0.01τ (where our time unit τ is the standard LJ reduced unit of time), a reduced pressure P* = 0 with damping parameter 100τ, and temperature of T* = 1.0 with damping parameter of 1τ. The monolayers are equilibrated for 2 × 106τ (PGN6−160), 2.75 × 106τ (PGN3− 160), 8.0 × 105τ (PGN6−35), and 1.4 × 106τ (PGN3−35), D


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against an already locally straight region of the primitive path of another chain (or another region of the same chain), leading to a “single kink” in that location. Both single and double kinks, as shown in Figure 2, are included in the calculation of the average entanglements per chain, ⟨Zkink⟩.58 We quantify the number of entanglements between PGNs by analyzing interparticle double kinks, in which the two chains involved in the double kink originate from different PGNs, as shown in Figure 2a. Both single kinks and intra-PGN double kinks are excluded. For the homopolymer film and all monolayer simulations, we average the results over 500 evenly spaced snapshots taken during the simulation. We report all average values with error bars representing one standard deviation from the mean. Uniaxial Deformation. Uniaxial tensile deformation is applied along the axial x or y direction at constant engineering strain rate, ε̇. Pressure is held constant during deformation in the other direction x or y using a Nosé−Hoover barostat with a damping parameter of 10τ (to allow for quicker changes to the box size than during the equilibrium simulations); therefore, the box length perpendicular to the strain direction decreases. The same thermostat is applied as above. During elongation, we record the average stress, the component of the pressure tensor in the direction of deformation, Pxx or Pyy. Note that the calculation of pressure depends on volume, and we define volume as the area times the height of the layer, where the height is the distance from the wall to the maximum z coordinate of any particle. We perform the deformation three times from three different starting configurations (the end of the data collection run described above, as well as 5 × 104τ and 1 × 105τ before the end of the run) and average over the results. We also take a moving average over a window of 0.1 in strain to smooth the stress−strain curves. We apply engineering strain rates of ε̇ = 1 × 10−3 and 5 × 10−4 τ−1 until a strain of 300% for all of our monolayers. Separately, we strain the simulation box up to 1400% at a strain rate of 5 × 103 τ−1 for the PGN6−160 and PGN3−160 monolayer systems to observe disentanglement. This strain is relatively high; however, these strains have been used in understanding crazing behavior for entangled polymer-grafted nanoparticles below the glass transition temperature.70 Our engineering strain rates are within the range of deformation rates in prior simulation studies of pure homopolymer melts as well as other bare/grafted nanoparticle−polymer composites.28,71−74 These strain rates are understood to show mechanical behaviors that are experimentally relevant, though the simulated rates are much faster than typical experimentally used tensile deformation rates. In all of the elongation simulations, we keep monomer−wall and nanoparticle−wall interactions unchanged from the equilibrium simulations. Thus, the system is still adsorbed to the wall, and the wall deforms uniformly and without changing density; this test allows us to consider a single type of controlled perturbation to our equilibrated system but is not representative of a particular experimental test. In designing experimental thin-film PGN systems of interest for use in flexible electronics, one may wish to optimize PGN parameters to provide greater robustness of the film and of the particle arrangement on the substrate as the substrate is deformed or bent in various ways. We hope these simpler uniaxial deformation tests on the substrate can provide insight toward such a design goal.

which is more than sufficient to allow chain conformations to relax from their initial state (see Supporting Information). The moderate graft density or longer chain systems take longer to equilibrate, as their chains can interpenetrate more. For all systems, the box dimensions and particle spacing fluctuated about a constant value during data collection. Data are collected for analysis during a further simulation of 2.5 × 105τ for the N = 160 systems and 2 × 105τ for the N = 35 systems. We also compare deformation results to those of homopolymer films prepared to be the same height as the PGN films. Specifically, we calculated the height of the high graft density PGN films as that at which the average monomer density reached 0.3σ−3. We then set the box length in the x and y directions to be 3 times the film height and add the number of polymers required to reach the proper height at the bulk density of 0.89. Simulations for the homopolymer films were run with constant area (no barostat) with other conditions the same as those used for the PGN systems; they were equilibrated for 4 × 105τ and run for data analysis for 2.0 × 105τ (N = 35) and 2.5 × 105τ (N = 160). All simulations were run using the large-scale atomic/molecular massively parallel simulator (LAMMPS) developed at Sandia National Laboratories.69 Primitive Path Analysis. Entanglements are quantified using the Z1 algorithm as implemented by Krö ger.58 Nanoparticles are replaced by a mesh containing 72 edges from 26 bonded dimers (see Figure 2) before the Z1 code is

Figure 2. Primitive paths of polymer chains from the Z1 algorithm showing (a) a double kink between two polymer chains belonging to different NPs (an interparticle double kink) and (b) a single kink involving chains on the same NP.

applied. The algorithm reduces each chain to a primitive path, where any kinks represent constraints on the paths (entanglements). When two chains (or two regions of the same chain) impose constraints on each other leading to them each kinking at the same point (within some small cutoff value), this is called a “double kink”. We use the cutoff distance of δr = 0.05 as in prior work.65 Alternatively, a chain can kink E


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Figure 3. Snapshots of nanoparticles (red) in the monolayer for (a) PGN6−35, (b) PGN6−160, (c) PGN3−35, and (d) PGN3−160 systems; top image is the top view (shows the xy plane) and bottom shows the xz plane with the wall at the bottom; in all cases the x axis is horizontal. Simulation box dimensions are shown in blue, and part of the periodic images in the x and y directions are shown. Graft beads are shown in black in the primary image.

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RESULTS AND DISCUSSION We simulated the four PGN systems as described above, representing all combinations of the two chain lengths N = 35 or 160 and graft densities 0.3 or 0.6 chains/σ2, at ϵw‑p = 3.5. We found that the moderate graft density, short chain (PGN3−35) system showed an uneven film height due to nanoparticles presenting their bare spots toward the top of the film (see Figure 1). This is likely because we set the nanoparticle−monomer interaction strength to be equal to the monomer−monomer interaction strength, which means that nanoparticle−monomer interactions are slightly unfavorable relative to the bulk monomer interactions. This is because a monomer at the nanoparticle’s surface is sterically prevented from multiple interactions of strength ϵ with other monomers it would otherwise have in the bulk, though it gains only one ϵ in its interaction with the nanoparticle.65,75 Thus, it is favorable for particles with large enough bare spots (gaps between graft beads) to present them to the vacuum interface and allow for a greater number of monomer−monomer interactions within the monolayer instead. This is not necessarily unrealistic, as nanoparticle−monomer interactions may be unfavorable depending on system chemistry; however, in future work, we plan to allow for improved stability at lower graft densities by using a more neutral “colloid” potential rather than the radially shifted LJ potential.75 At high graft density or high chain length, particle surfaces remain almost entirely covered, and this effect is not as important. Figure 3 shows the equilibrated spacing of the nanoparticles for the four PGN monolayer systems. All four particle systems show very uniform monolayers, and particles equilibrate to a stable hexagonally

packed structure. At the moderate graft density, however, the hexagonal spacing is somewhat perturbed, as individual particles arrange themselves slightly closer to or further from each other and the surface, likely due to bare regions moving toward the interfaces as described above. From the bottom images, it is noticeable that the distribution of particles in the z direction is wider for the longer chain systems and especially for the long chain, moderate graft density PGNs (the distribution in z is shown in the Supporting Information). To describe packing within the film, we calculate pair correlation functions (radial distribution functions) gij(r) in three dimensions (3D), even though our monolayers are relatively thin in one of the dimensions. We expect the 3D g(r) is more easily understandable and relatable to other systems versus a quantity based on cylindrical shells. To consider nanoparticle packing, we define gNP‑NP(r) in the standard way, as the average number of times an NP would observe another NP a distance between r and r + dr from itself, normalized by the expected number at uniform density (i.e., the number of other NPs in the system divided by the system volume). We define system volume as above (considering the height of the system as the height of the highest monomer). Because the spherical shell can extend into the vacuum and below the wall, this implies that g(r) will be below 1 for a uniform density film, decreasing significantly at distances greater than half of the film thickness. To consider packing of monomers around NPs, we calculate separate pair correlation functions gNP‑mon for attached (monomers of chains grafted to the current nanoparticle) and nonattached (other) monomers, in both cases excluding graft beads. For a clear comparison between F


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Macromolecules these two monomer types, we normalize both of these gNP‑mon(r) by the expected value of monomers in the given spherical shell if the shell were at a density of 0.89 (the density of the bulk polymer). In Figure 4, we show the nanoparticle−nanoparticle radial distribution function, focusing on the first (nearest neighbor)

Figure 5. Nanoparticle−monomer radial distribution functions g(r)NP‑mon for attached (solid line) and nonattached (dashed lines) monomers.

average over all 12 particles. Figure 5 shows identical monomer packing in the first two coordination shells for the high graft density particles; the moderate graft density systems show a higher first peak, as more monomers are able to reach the surface (graft beads are not included in gNP‑mon(r)). The attached gNP‑mon(r) for the short-chain particles tapers off quickly, becoming small after ∼15σ, whereas the longer chains extend to longer distances as expected. Comparing the two N = 160 particles, it is noticeable that the lower graft density particles have increased chain interdigitation from surrounding PGNs, indicated by the large increase in nonattached monomers close to the nanoparticle surface and decreased particle spacing versus the high graft density system. We quantify this by determining the number of neighboring monomers whose chains are attached to other particles and are within a distance r90, defined as the distance at which 90% of the monomers in a particle’s own grafted layer are found (see Supporting Information). The grafted layer surrounding a PGN3−160 particle contains ∼0.97 particles’ worth of unattached monomers, whereas the grafted layer surrounding a PGN6−160 particle contains ∼0.55 particles’ worth. Entanglements. The average number of kinks per chain, ⟨Zkink⟩, including both single and double kinks, is shown in Figure 6a. Compared to the homopolymer film, the moderate graft density particles have a similar number of entanglements per chain, but the higher graft density system has a greater number of entanglements per chain. One may expect that this would mean the PGN6−160 monolayer will be mechanically more robust than the moderate graft density system. However, ⟨Zkink⟩ includes a significant number of entanglements between chains attached to the same nanoparticle, which we expect to have a different impact on the overall mechanical properties of the system than entanglements between different nanoparticles. We also plot the number of interparticle double kinks per chain in Figure 6b and compare to the homopolymer double kinks (occurring between any two different homopolymers). In contrast to the ⟨Zkink⟩ results, we find that the higher graft density system has significantly fewer interparticle entanglements per chain. We can attribute this to the large decrease in interpenetration of the grafted polymer chains at the higher graft density. We show where the interparticle entanglements are located in the monolayer systems with areal density color plots in Figures 7 and 8. Entanglements are most often found in the interstitial areas between three particles,

Figure 4. First peak in the NP−NP radial distribution function gNP‑NP(r) for PGN3−35 (green), PGN6−35 (red), PGN3−160 (black), and PGN6−160 (purple) monolayers.

peak. In all cases, integrating the nearest neighbor peaks shown weighted by the volume of the shell and particle density in the system yields a value of six nearest neighbors. The peak is sharpest for the PGN6−35 monolayer, indicating this system has the highest degree of hexagonal ordering. The long chain, moderate graft density system shows a broader peak due to the somewhat different heights of the particles in the monolayer. Experimentally, at high graft densities, hexagonal packing was observed for long entangled chains, though at lower graft densities the particles are no longer well-ordered.21,22 Here, we considered a surface interaction strength favorable enough to yield a stable hexagonal monolayer over the time scale of the simulation. However, at lower surface interaction strengths, our moderate graft density particles also become more disordered and eventually become a bilayer, while the high graft density particles remain well-ordered (see Supporting Information). The location of the peak in g(r) shows the average particle spacing. We note that the average NP spacing is larger than that of the pairs of particles from our previous work at ϵw‑p = 3.0, though the spacing is similar for the PGN3−160 system.65 The larger spacing is likely due to both the monolayer configuration and to the more favorable wall− monomer interaction in this current study. The PGN3−160 system has a similar spacing to the pairs, as these PGNs are entropically driven to be highly interdigitated, and in the monolayer the driving force to pack more closely is larger, as multiple particles are participating in interdigitation on all sides. Comparing the two N = 160 PGNs, we find that decreasing graft density drastically reduces particle spacing, which indicates a large increase in interdigitation of the grafted polymer chains. The monomer packing and interpenetration of particles is further analyzed using the nanoparticle−monomer radial distribution functions for both attached and nonattached monomers (see Figure 5). Specifically, we calculate gNP‑mon(r) as described above between a particle and its attached monomers or from a particle to unattached monomers, then G


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Figure 6. (a) Average number of kinks per chain ⟨Zkink⟩ for the PGN3−160 (black), PGN6−160 (purple), and the N = 160 homopolymer film (gray). (b) Average number of interparticle double kinks per chain for the same thin film systems, where all double kinks between two distinct chains are included in the homopolymer film. Error bars show one standard deviation from the average over 500 snapshots spaced 500τ apart.

Figure 7. Average areal density of interparticle double kinks for (a) PGN3−160 and (b) PGN6−160 monolayers, viewed from the top of the film (view shows the xy plane as in Figure 3). Pure white represents 0 and solid black represents 0.20 interparticle double kinks per σ2. Areal binning begins at the bottom left corner, and strips of apparently low-density regions along the far box edges are visible where the bin does not include the full σ2 of area.

Figure 8. Average areal density of interparticle double kinks for (a) PGN3−160 and (b) PGN6−160 monolayers, viewed from the side; view shows the yz plane (see the Supporting Information for the view in the xz plane) with the wall at the bottom, looking through the entire film in the x direction (one periodic image, containing three repeat units in the x direction). Pure white represents 0 and solid black represents 0.45 interparticle double kinks per σ2. Areal binning begins at the bottom left corner, and strips of apparently low-density regions along the far box edges are visible where the bin does not include the full σ2 of area.

where the chains are furthest from the steric constraints of the particle surfaces, and this effect is most clearly seen in the top view of the higher graft density system (Figure 7b). The increased ability to entangle in the interstitial regions was discussed by Schmitt et al., who also showed fracture toughness relates to how far chains extend past the concentrated polymer brush regime into the semidilute brush regime (which allows for increased entanglements).34 This suggests that entanglements are found closer to the chain ends more often than they would be in a reference homopolymer, which may lead to weaker mechanical properties than expected for a given amount of entanglements. Interparticle entanglements are primarily located in the interstitial areas at both graft densities. Additionally, for the lower graft density system, entanglements are relatively more spread out across the monolayer and a significant number of entanglements are found above and below the locations of the nanoparticles. We expect this may lead to improved robustness in interparticle spacing under strain for the lower graft density system, as the particles will experience more uniform effects of interparticle entanglements that constrain their positions relative to each other. Dynamic and Mechanical Properties. We first analyze the grafts’ end-to-end vector relaxation. Specifically, we

calculate the end-to-end vector autocorrelation function (ACFee) ACFee =

⟨R ee(t ) ·R ee(0)⟩ ⟨R ee 2⟩

(5)

where Ree is the vector from the graft bead to the end monomer of a graft chain. Figure 9 shows the relaxation behavior of the four monolayer systems and homopolymer film reference systems. For the PGN systems, the end-to-end vectors do not completely decorrelate by the end of the time window shown. By observing the trajectories, we confirmed that most particles do not rotate significantly during the simulation; grafts are placed randomly rather than symmetrically, and therefore certain areas on particles may prefer to face the surface. Thus, we expect some amount of correlation in the grafts’ end-to-end vectors, as they generally point away from a certain position on the surface that does not rotationally decorrelate during the simulation. This is also observed in the chains’ segmental dynamics; in the Supporting Information, we H


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Figure 9. End-to-end vector autocorrelation function ACFee for the PGN3−35, PGN6−35, PGN3−160, and PGN6−160 monolayers. Homopolymer thin film systems N35 (chain length N = 35) and N160 (chain length N = 160) are included for comparison.

Figure 10. Stress−strain curves at a strain rate of ε̇ = 0.001 τ−1 for uniaxial deformation of monolayers in x (top) and y (bottom) directions; the hexagonally packed arrangement in the xy plane is shown in Figure 3.

show the bond orientational relaxation for bonds near the surface versus those near chain ends. Comparing the PGN systems to the homopolymers, we find that tethering chains slows their relaxation, as would be intuitively expected. Also as expected, the shorter homopolymers relax significantly faster than the longer homopolymers, and the N = 35 systems exhibit the fastest relaxation of the PGN systems. This effect is expected due to the polymer entanglements in the long polymer systems. In comparison to these major effects, the effect of graft density is small but noticeable. Specifically, the moderate graft density particle chains, having fewer constraints on their conformations, relax more quickly than the higher graft density particle chains. Overall, the dynamic relaxation behavior agrees with the average total entanglement results of Figure 6, in that the higher graft density system with more entanglements per chain (considering all single and double kinks) exhibits slower chain end-to-end vector relaxation. Stress−strain curves showing the response of the monolayers and homopolymer film reference systems to uniaxial deformation at a strain rate of 0.001 τ−1 are presented in Figure 10. Results at 0.0005 τ−1 are qualitatively similar and found in the Supporting Information. At this strain rate, we do not have a large linear response regime and do not attempt to calculate elastic modulus; however, it is apparent that the modulus (initial slope of the stress−strain curve) is higher for N = 160 homopolymers versus N = 35 homopolymers, as expected. Also, the PGN6−35 and PGN3−35 systems, which contain a relatively high fraction of hard particles, have a higher modulus than their reference homopolymer and than the lower nanoparticle volume fraction, longer-chain PGN systems. This trend is in qualitative agreement with previous simulations of bulk polymer-grafted nanoparticles, although that prior work used smaller nanoparticle sizes, shorter chain lengths, and more attractive polymer−nanoparticle interactions.76 The response of the longer-chain systems is relatively similar at low strains, and without more graft density and volume fractions to compare to, it is not clear to what extent differences in mechanical response arise from the differences in entanglement network due to graft density versus the different volume fractions. Looking to higher strains, the response in the

x and y directions is different due to the alignment of the hexagonally packed crystal planes. The particles are at their closest packing distance in the x direction but not in the y direction. When pulled upon in the y direction, close-packed rows of particles in the x direction are stretched apart, and particles become even closer together in x. In contrast, when pulled upon in the x direction, particles move apart in their originally close-packed direction and together in the y direction, until particles are nearest in the y direction. For the moderate graft density (PGN3−160) system, the particle spacing is less regular both initially and during strain; some particles become significantly closer together than others during strain. See the Supporting Information for snapshots of particles during strain. For the higher particle volume fraction systems considered (PGN3−35 and PGN6−35), when the simulation box is strained in the x direction, we see a plateau in the stress with increasing strain as this rearrangement occurs. As we increase strain further, we see the stress values begin to increase as the amount of polymer between the particles decreases to form valleys (see the Supporting Information). However, for the systems with longer chains and lower particle volume fractions, the particle arrangement is apparently less important, and the response is relatively similar to strain in either x or y. For these entangled PGN systems, failure occurs at much larger strains than for the N = 35 systems; we therefore performed simulations at a higher strain rates to observe the behavior at larger strains in a reasonable simulation time. Stress−strain curves for long-chain PGN systems and the reference homopolymer film at a strain rate of 0.005 τ−1 are plotted in Figure 11. It is easily noticeable that the two graft densities have very different mechanical properties. We notice a strain-stiffening effect for both PGN systems in both directions, where stress is increasing significantly more than linearly after a strain of ∼1.5; this effect is somewhat more pronounced for the higher graft density system. As we increase strain, a maximum stress is eventually reached, after which the I


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ment density, showing entanglements tend to form in the interstitial regions further from particle surfaces, especially for high graft density systems in which the chain conformations are more stretched. Finally, we applied uniaxial deformation to show the impact of entanglements on mechanical behavior. The short graft length PGN monolayers fail at relatively low strain, and the behavior is dependent on the strain direction. Considering the long-chain (entangled) systems, the lower graft density system is tougher, failing at higher stress and higher strain than the high graft density system. The high graft density system had a higher number of entanglements but a lower number of interparticle entanglements per chain. Thus, we attribute the increased toughness at moderate graft density to the increased interdigitation and interparticle entanglement formation, rather than the overall number of entanglements. We therefore suggest that a detailed understanding and analysis of the entanglement network and interparticle entanglements is necessary in understanding the spacing and mechanical properties of PGN materials. Such an analysis may also be applied to simulations of star polymers or microphaseseparated diblock copolymers. In future work, we plan to study glassy systems and more directly map to experimental materials.

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Figure 11. Stress−strain curves of entangled monolayer systems (N = 160) at a strain rate of ε̇ = 0.005 τ−1. Data are shown until complete failure was observed (until a gap was observed across the entire box).

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.8b01373. Simulation box height during equilibration of the PGN6−160 and PGN3−160 monolayers at ϵw‑p = 3.0; simulation box height as a function of time and equilibration details for all four PGN monolayers at ϵw‑p = 3.5; monomer density and nanoparticle number density profile as a function of distance from the surface (z = 0); average number of interpenetrating monomers within a polymer-grafted layer; monomer and double kink two-dimensional areal density profiles; segmental dynamics; stress−strain behavior; and snapshots of particles during deformation (PDF)

particles begin to break apart and eventually completely separate. We do not show the data past the point of complete separation (failure). Interestingly, we find that the moderate graft density (PGN3−160) system reaches a maximum stress at significantly higher strain than the high graft density (PGN6−160) system. The toughness, as measured by integrating the stress−strain curve, is larger for the PGN3− 160 monolayer. On the basis of this, we suggest that the initial response and strain hardening of these systems is more dependent on the overall entanglements (which are higher for the higher graft density system), while the higher strain and failure behavior depends more on interparticle entanglements, which hold particles together (and which are higher on a perchain basis for the lower graft density system). Thus, we attribute the increased toughness of the lower graft density system to the increased interpenetration and interparticle entanglements of this system. However, further analysis of the entanglement network and results for more graft densities and particle volume fractions would be needed to confirm and further understand these effects.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Lisa M. Hall: 0000-0002-3430-3494

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Notes

The authors declare no competing financial interest.

CONCLUSIONS Monolayers of hexagonally packed PGNs adsorbed on a wall were simulated in the melt state, considering particles with long grafted chains at moderate and high graft density, and short grafted chains at moderate and high graft density. The stability of the nanoparticle arrangement as well as PGN interdigitation was analyzed from pair distribution functions. Decreasing graft density increases interdigitation between PGNs and slightly decreases the degree of order of the nanoparticles. For long-chain systems, we quantified entanglements and found that high graft density PGNs have a higher average number of entanglements, but they have fewer interparticle entanglements per chain. This is due to the lesser degree of PGN interpenetration at high graft density. Additionally, we showed two-dimensional maps of entangle-

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ACKNOWLEDGMENTS We thank R. Vaia for useful discussions and experimental insights, the Ohio Supercomputing Center for computation time, and M. Kröger for code implementing the Z1 algorithm. We also acknowledge the support and high performance computing (HPC) resources of the DOD HPC Modernization Program. This work was partially supported by the AFRL/ DAGSI Ohio Student-Faculty Fellowship Program and by the Air Force Research Lab Summer Faculty Fellowship Program.

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Structure and Entanglement Network of Model Polymer-Grafted

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