Impact of Hydrogen Bonding Interactions on Graft–Matrix Wetting and

Mar 4, 2019 - One can achieve equivalent grafted layer wetting in. PNCs with ..... We vary the number of matrix chains depending on the chain length N...
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Impact of Hydrogen Bonding Interactions on Graft−Matrix Wetting and Structure in Polymer Nanocomposites Arjita Kulshreshtha,† Kevin J. Modica,† and Arthi Jayaraman*,†,‡ †

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Department of Chemical and Biomolecular Engineering, 150 Academy Street, Colburn Laboratory, University of Delaware, Newark, Delaware 19716, United States ‡ Department of Materials Science and Engineering, 201 Dupont Hall, University of Delaware, Newark, Delaware 19716, United States S Supporting Information *

ABSTRACT: We present a new coarse-grained (CG) model that captures directional interactions between graft and matrix polymer chains in polymer nanocomposites (PNCs) comprising polymer grafted spherical nanoparticles in a matrix polymer. In this CG model we incorporate acceptor and donor CG beads along with graft and matrix monomer CG beads and optimize the bonded and nonbonded interactions to mimic directional and specific H-bonding between the acceptor and donor sites on graft and matrix chains, respectively. Using this CG model and molecular dynamics simulations we show that Hbonding interactions between graft and matrix polymer chains increase the grafted layer wetting by matrix chains compared to that at the purely entropic limit. One can achieve equivalent grafted layer wetting in PNCs with directional acceptor−donor interactions and PNCs with isotropic graft−matrix interactions, but the directional acceptor−donor interaction strength needs to be much stronger than the isotropic graft−matrix monomer attraction strength. Strikingly, despite equivalent grafted layer wetting and graft chain conformations, on average, each graft chain interacts with fewer matrix chains and has a lower free volume in PNCs with H-bonding interaction as compared to PNCs with isotropic graftmatrix attraction. These trends are seen both at high (brush-like) and low grafting densities, and in PNCs with equal graft and matrix chain lengths as well as PNCs with matrix chain length three times the graft chain length.

I. INTRODUCTION Macroscopic properties of polymer nanocomposites (PNCs) are linked to the PNC morphology and, in particular, the dispersed or aggregated state of nanoparticles in the polymer matrix. Past studies have shown that PNC morphology and the effective interactions governing the morphology can be tuned by grafting the nanoparticle surface with polymers and by tailoring the graft and matrix polymer chemistry, molecular weights, and grafting density.1−12 In particular, tuning the extent of interpenetration between graft and matrix chains, i.e., wetting of grafted layer by matrix chains (or grafted layer wetting, in short), is key to controlling the PNC morphology. In PNCs with chemically identical graft and matrix chains, grafted layer wetting depends on grafting density, ratio of graft to matrix molecular weight, and nanoparticle curvature. At high grafting density where the particle surface is covered with a densely grafted polymer layer, the PNC morphology is driven purely by entropic driving forces. The grafted layer wetting by matrix chains increases the mixing entropy gain and matrix conformational entropy loss. The net entropic driving forces can be made to favor wetting by reducing the matrix molecular weight relative to the graft molecular weight,13−18 reducing particle size relative to the size of graft chains,16,19−21 decreasing the flexibility of graft and matrix chains,22,23 or increasing the polydispersity of graft chains.10,24−26 In PNCs with chemically dissimilar graft and matrix chains, the interplay of entropic and enthalpic driving forces determine morphol© XXXX American Chemical Society

ogy; the enthalpic driving forces are tuned via interactions between the graft and matrix chains. Previous studies have shown that by choosing graft and matrix monomer chemistries with a negative Flory−Huggins interaction parameter, χ, one can offset the entropic losses and increase the grafted layer wetting, and improve nanoparticle dispersion within the PNC.27−31 Furthermore, Martin et al. have also shown that in chemically dissimilar, negative χ, graft−matrix PNCs, with increasing temperature the PNC undergoes a sharp first-order nanoparticle dispersion−aggregation transition and a simultaneous gradual wetting−dewetting transition.29 They also showed that the dispersion−aggregation transition occurred at the temperature where the grafted layer wetting becomes equivalent to an analogous chemically identical graft−matrix PNC. This showed that chemically dissimilar graft−matrix PNCs offer control over morphology by tuning of graft−matrix interactions to achieve desired grafted layer wetting and dewetting. The above studies have focused on graft and matrix chains that have isotropic interactions, driven by entropy alone or through a balance of entropy and (isotropic) graft−matrix interactions. Favorable graft−matrix interactions can also be achieved by the introduction of intermolecular hydrogen Received: December 14, 2018 Revised: March 4, 2019

A

DOI: 10.1021/acs.macromol.8b02666 Macromolecules XXXX, XXX, XXX−XXX

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Figure 1. Schematic of the CG model of the polymer grafted nanoparticle and matrix polymer in the absence of H-bonding sites (a) and with Hbonding sites (b) (not drawn to scale).

and matrix polymers (i.e., chain lengths, grafting density, particle size, monomer chemistry dictated H-bond strength, etc.) impacts the grafted layer wetting and, as a result, nanoparticle dispersion or aggregation in the PNC. Here we describe the details of this model and demonstrate its use for studying PNCs with H-bonding graft and matrix chemistry for a variety of PNC design parameters (e.g., grafting density, graft and matrix chain lengths, and graft and matrix interactions). We also elucidate new physics that emanates from the directional nature of the H-bonding graft−matrix interaction in contrast to the isotropic graft−matrix interactions. The paper is organized as follows. In Section II, we describe the model, simulation method and analyses. In Section III, we present the details of the simulation results and likely reasons underlying the simulation observations. In the Conclusion section we summarize the key results and outlook for future work based on the results in this paper.

bonds (or H-bonds) between graft and matrix polymers. Experimental studies on PNCs with poly(styrene-r-2-vinylpyridine) grafts in a poly(stryrene-r-4-vinylphenol) matrix have confirmed the existence of a thermally reversible dispersion− aggregation transition mediated by H-bonds between graft and matrix polymers.32 Previous studies on H-bonding interactions in carbon nanotube (CNT) nanocomposites have shown that improving the accessibility of H-bonding groups leads to enhanced dispersion of CNTs in polymer matrix and improved mechanical and electrical properties.33 Similarly, H-bonding interactions have been used to tune miscibility in polymer blends34−37 and to create supramolecular assemblies of nanoparticles in matrix dependent on temperature38 and pH.39 These studies support the fact that PNCs with Hbonding interactions have a promising potential for programmable assembly and for fabrication of materials with unique properties such as self-healing and stimuli responsiveness attributed to the dynamic properties of H-bonds. Even though experimental studies have established that Hbonds can serve as an effective handle to alter polymer morphology and properties, there is a dearth of theoretical and simulation studies exploring H-bonding interactions in PNCs. While simulations with atomistic models can describe Hbonding interactions in chemical detail, they can become computationally expensive and, thus, cannot capture the whole range of length and time scales associated with H-bonds and polymer chains. Therefore, atomistic simulations are not viable for predictive morphological studies of PNCs with H-bonding polymer chemistries. In this regard, coarse-grained (CG) models capable of capturing large length and time scales of polymers are a suitable alternative, especially if they can also capture the directionality and specificity of an H-bond as opposed to treating the H-bond as an isotropic interaction between graft and matrix chains. Such CG models exist for DNA, like in the model of Sciortino and co-workers which incorporates a “sticky bead” to represent an H-bonding donor/ acceptor site for DNA.40,41 Jayaraman and co-workers42,43 have extended this CG model to mimic directionality and specificity of the H-bonding interactions in DNA42 and in collagen-like peptides (CLP)43 and replicated experimentally observed DNA and CLP melting trends with varying DNA and CLP design. In this paper, we present a CG model based on these ideas above, optimized to capture directional interactions between graft and matrix polymer chains in PNCs. With this model we can computationally predict how the design of graft

II. MODEL AND SIMULATION II.A. Model. Figure 1 shows our CG model where the polymer grafted particle has a spherical nanoparticle core of diameter D, grafted with bead−spring44 polymer chains with each CG bead of diameter d representing a monomer. The graft chains are attached to the nanoparticle core through harmonic bond potentials. Matrix chains are also modeled like the graft polymer chains. The bonded interactions include a harmonic spring between bonded CG monomers in the graft and matrix chains, as well as a harmonic angle potential between three bonded CG monomers whose force constant allows us to vary the graft and matrix chain flexibility in a manner similar to Lin et al.22 (see also Figure S1). The nonbonded interactions between pairs of graft (G) and matrix (M) monomers are modeled using Lennard-Jones (LJ)45 potential with σij = 1.0d and rcut=2.0d, and εGG = εMM= 0.5kT and εGM varying from 0.2kT (to get χGM = +0.3) to 1.0kT (to get χGM = −0.5). For PNCs with H-bonding graft and matrix monomer chemistries, we place an acceptor bead (A) on the graft bead and a donor bead (D) on matrix bead, as shown in Figure 1b, to act as intermolecular H-bonding sites between graft and matrix monomers. These A and D beads of size 0.3d are maintained at specific positions with respect to the G and M beads’ centers via bonded harmonic interactions of force constant k bond = 1000kT /d 2 and equilibrium bond distance B

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package50 using a protocol similar to our previous work.29 In this study, we conduct the simulations for PNCs with a single polymer grafted nanoparticle with D = 5d, graft chain length NG = 20, grafting densities Σ = 0.65 chains/d2 and 0.32 chains/ d2 placed within a matrix with chain lengths of NM = 20 and 60. The grafting densities Σ = 0.65 chains/d2 and 0.32 chains/ d2 mimic the brush-like high grafting density and intermediate grafting density and are selected to show the effects of Hbonding in both regimes of grafted chain conformations. The matrix chain lengths of 20 and 60 are chosen to be the two cases where (a) the graft and matrix chain lengths are equala condition that is shown to be favorable for grafted layer wetting in the entropically driven limit at high grafting density, and (b) the matrix chain length being larger than the graft chain lengtha condition that is shown to be unfavorable for grafted layer wetting in the entropically driven limit at high grafting density. The total volume packing fraction, η, quantifies the fraction of the simulation box volume that is occupied by particle, graft, and matrix beads and is maintained in this study at η = 0.367 ; this value was chosen to achieve melt-like conditions in PNC as proven in the Supporting Information and Figure S3. Since we simulate a single polymer grafted particle within the polymer matrix, the grafted filler

ro = 0.37d . There are no A-G-G-A or D-M-M-D bonded dihedral potentials in this work; however, this can easily be introduced in our model to mimic torsional constraints imposed on the H-bonding donor and acceptor atoms in some graft and matrix chemistries.46−48 The nonbonded A-D interaction is defined using LJ potential with σAD = 0.3d , rcut = 2σAD, and εAD = 13kT to mimic the maximum strength of OH:N H-bond pair. We choose this specific pair of interactions to mimic polymer chemistries like the ones in the experimental studies by Hayward and co-workers where they used poly(styrene-r-2-vinylpyridine) grafts in a poly(stryrene-r4-vinylphenol) matrix and found improved particle dispersion mediated by H-bonds between graft and matrix polymers.32 Even though we choose 13kT as the Lennard-Jones well depth for the A-D interaction in this study, we can vary the strength of A-D attraction (i.e., value of εAD) to mimic varying Hbonding donor−acceptor pair chemistry. All other pairwise nonbonded interactions involving the nanoparticle, P, (i.e., AP, D-P, G-P, and M-P), and involving graft, matrix, acceptors, and donors (i.e., G-A, M-A, G-D, and M-D) are modeled as purely repulsive using the Weeks Chandler Andersen49 (WCA) potential. The directionality of A-D interactions is brought about by the relative size and placement of A and D beads with respect to their attached G and M beads’ center, as shown in Figure 2a.

fraction defined as ϕG =

Vgraft + Vparticle Vgraft + Vparticle + Vmatrix

is kept at 0.02 for

the high grafting density, and 0.01 for the low grafting density, where Vgraft , Vparticle , and Vmatrix represent the total volume occupied by graft, particle, and matrix beads, respectively. When selecting the system size we wish to simulate, we choose to have a minimum box size of 44d to eliminate any finite size effects. We vary the number of matrix chains depending on the chain length NM to achieve the η desired (keeping the number of matrix beads fixed for all PNCs) based on the volume occupied by the chosen polymer grafted nanoparticle design. To obtain an initial configuration for the simulations, we first create a polymer grafted particle of size D = 5d whose surface is tessellated by grafting site beads of size 1d. For D = 5d, we have 51 grafting site beads to achieve a grafting density of 0.65 chains/d2 and 25 grafting site beads for a grafting density of 0.32 chains/d2. We graft polymer chains to the grafting site beads through harmonic bonds. To ensure relaxation of graft chains from the chosen initial configuration, we simulate this single polymer grafted particle (without any matrix chains) keeping all interactions (graft-graft, acceptor-acceptor, acceptor-graft, acceptor-particle and graft-particle) as purely repulsive, specified by WCA potential, over 1 million time

Figure 2. H-bond interaction energy (UAD) experienced by a donor bead (not shown) as a function of its position (x, y) as it approaches one A site (part a) and as it approaches A and D sites at varying distances from each other (parts b-d).

steps where each time step = 0.001 σ

0.5

( mε )

. Then, the

equilibrated polymer grafted particle is placed at the center of the simulation box and the matrix chains of length NM are added into the simulation box volume that is much larger (ranging 2−200 times, depending on grafting density) than the actual simulation box we use for the production run. We start with a larger box size because it helps accommodate matrix chains around the polymer grafted particle without significant overlaps. We take this large simulation box configuration and run equilibration over another 1 million time steps to promote mixing of graft and matrix chains in the NVT ensemble at T* = 1. Temperature is controlled using the Nose−Hoover thermostat implemented in LAMMPS with a damping parameter of 1 τ (each time step being 0.001 τ ) for PNCs at high grafting density, and a Langevin thermostat is used with a damping parameter of 10 τ for PNCs at low grafting density

The specificity of the H-bonding interaction (i.e., preference for A-D over A-A and D-D and preventing possibility of A-D-A or A-D-D interactions) is brought about by A-A and D-D repulsive interactions modeled using WCA potential with σAA = σDD = 2.3 σAD and rcut = 1.1225 σAA and εAA = εDD = 0.5kT. This repulsive potential ensures that a D or A site sees no energetically favorable patch as it approaches another pair of A and D sites that are close to each other and interacting via A-D attraction (see Figure 2d). We direct the reader to the Supporting Information section I and Figures S1 and S2 where we present all details of the CG model to facilitate future studies using this model. II.B. Simulation Method. We use the above CG model in a molecular dynamics (MD) simulation within LAMMPS C

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Figure 3. (a) Schematic for monomer concentration profile and brush height calculation described mathematically in Section II.C. (b) Graft and matrix monomer concentration profiles for PNCs with no A-D sites (black curve) and with repulsive A-D sites (modeled with WCA potential) with attractive A-D interactions turned off (red dashed line) with the grafted layer (brush) heights shown as vertical dotted lines. Probability distribution of the end−end distances, P(Ree) versus Ree for (c) graft and (d) matrix chains. These results are for D = 5d, graft chain length NG = 20, grafting density Σ = 0.65 chains/d2 placed within a matrix of chain length NM = 20. Error bars have been calculated from the standard deviation in three independent simulation runs.

standard deviation in average values obtained from three independent simulation runs. The effective thickness of grafted layer is represented by brush height, which is calculated as the root-mean-square of the distance of the grafted beads from the particle surface.

with time integration performed using the velocity Verlet algorithm. After the initial configuration relaxation of graft and matrix chains, the simulation box is gradually compressed over at least 2 million time steps to the desired box size to achieve η = 0.376. This final system size is equilibrated in NVT ensemble at T* = 1 for at least another 5 million time steps. The equilibrated system is then used as a starting point to generate production run configurations which are saved every 105 time steps and used for calculating ensemble averaged concentration profiles and performing other structural analysis. There are minor variations in equilibration protocol (e.g., thermostats, number of time steps for the above stages, starting large box size, etc.) of high grafting density and low grafting density PNCs, yet as we see in the results, they do not alter the key results and the scientific trends. II.C. Analyses. To quantify wetting and interpenetration between graft chains (G) and matrix chains (M), we plot the monomer concentration profiles of graft (CG) and matrix (CM) chains22 as a function of distance from the particle surface. Cx (x = G or M) is given by Cx =

n

Hb =

nG

(2)

where Hbis the brush height, ri is the distance of grafted bead i from the particle surface, and nG is the total number of graft beads in the polymer grafted nanoparticle. We calculate the number of matrix chains that interact with each graft chain by tracking the matrix chains whose beads lie in the interaction region (defined by potential cutoffs) for both isotropic G-M interaction and directional A-D interaction. We also calculate the average number of matrix beads within the grafted layer thickness as an additional method to quantify the extent of grafted layer wetting. The conformations of the graft and matrix chains are described by plotting the probability distribution P(R ee) of end-end distance R ee . For each chain the R ee is calculated as follows:

nx(r ) 4πr 2Δr

∑1 G ri2

(1)

R ee =

where nx(r ) is the ensemble average number of monomers of type x within a shell of thickness Δr = 1d at a distance r from the particle surface. We report for all calculated quantities the

|ri , l − ri ,1|2

(3)

where ri , l and ri,1 are the position vectors of the last and first beads of the ith graft or matrix chain. D

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Figure 4. (a) Graft and matrix monomer concentration profiles and (b) probability distribution of the end−end distances, P(Ree) vs Ree for graft chains in PNCs with χGM = 0 and attractive H-bonding interaction εAD = 13kT (magenta curve) and PNCs at the purely entropic limit with χGM = 0 and repulsive interactions between A and D sites (black curve). All results are for D = 5d, NG = 20, Σ = 0.65 chains/d2, and NM = 20. Error bars (calculated as standard deviation from three independent simulation runs) when not visible are smaller than marker size. (c) Simulation snapshot obtained using Visual Molecular Dynamics (VMD)53 of the PNC showing H-bonding interaction between the acceptor beads (yellow) of one graft chain with donor beads (blue) of different matrix chains.

present structural effects that H-bonding brings about for varying interactions between graft and matrix monomers. After describing the H-bonding effects for the PNCs at high grafting density and equal matrix and graft chain lengths, we present results for two cases: the corresponding results at low grafting density 0.32 chains/d2 (NG = NM = 20) in Section III.B and the corresponding results for NM = 60 (NG = 20 at Σ = 0.65 chains/d2) in Section III.C. III.A. High Grafting Density and NG = NM = 20. First, we prove that any new physics for wetting of the grafted layer in the presence of attractive acceptor−donor interactions using the CG H-bonding model is because of the directional interactions between A-D sites and not as an artifact of the physical presence of A-D sites or their repulsive interactions. We do this by comparing the graft and matrix monomer concentration profiles which quantify the interpenetration between graft and matrix chains and grafted layer wetting, for a purely entropically driven PNCs with no A-D sites (Figure 1a) and for PNCs in the presence of these A-D sites interacting purely repulsively via WCA potential for all pairs of A and D (i.e., A-D, A-A, and D-D) with σAA = σAD = σDD = 0.3d, rcut = 1.1225(0.3d), and εAA = εAD= εDD= 0.5kT. The latter is also effectively a purely entropically driven PNC and, therefore, should behave identically to the former PNC if there are no artifacts of the chosen A-D based CG model. Figure 3 shows that there is indeed no difference in graft and matrix monomer concentration profiles and brush heights between these two PNCs. The small deviation in graft monomer concentration at the particle surface for the two PNCs (Figure 3b) can be attributed to the repulsion between A and D sites which likely prevents effective packing of graft chains at the particle surface. Furthermore, the chain conformations quantified by proba-

We also calculate the free volume to quantify the unoccupied space surrounding each graft bead. We use the method of Voronoi tessellation of space,51 whereby the vectors joining each graft bead to all the other beads are perpendicularly bisected to obtain polyhedrons around each graft bead. Since the acceptor and donor beads in our model lie within parent graft and matrix beads and act merely as sites to facilitate hydrogen bonding between graft and matrix chains, we exclude them from the calculation of free volume. Our interpretation of free volume is the unoccupied volume around each graft bead and is calculated by subtracting the occupied volume of the graft bead (i.e., the volume of a spherical bead of diameter 1d) from the volume of the smallest polyhedron surrounding it. This calculation is performed by utilizing the open source library VORO++.52 The sum of free volumes of NG graft beads that constitute a graft chain gives the effective free volume of a graft chain. We plot the probability distribution of free volume per graft chain from the ensemble of graft chains in all configurations in all three trials. For one case, we also report free volume of each graft chain averaged over three independent trials. Errors bars are calculated by standard deviation from the values of the average free volume in three trials.

III. RESULTS AND DISCUSSION The outline of this section is as follows. In Section III.A, we present results for PNCs with a polymer grafted nanoparticle of particle diameter D = 5d tethered with NG = 20 chains at high grafting density, Σ = 0.65 chains/d2, placed in NM = 20 matrix polymers. For this case, we first prove that the Hbonding CG model performs well without any unphysical effects due to the selected interactions in our model, and then E

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Macromolecules bility distribution of graft and matrix chain end-end distances in Figure 3c,d confirm that the chain conformations are not altered by the physical presence of the A and D sites in the CG model. Thus, either of these results in Figure 3 are representative of PNC with NG = 20, NM = 20, and Σ = 0.65 chains/d2 at the purely entropic limit, i.e., in the absence of attractive interactions, isotropic or directional, between G and M monomers. Next, we use the CG model described above with attractive interactions between A and D sites to show the effect that the graft and matrix H-bonding has on grafted layer wetting and chain conformations. In Figure 4a, the matrix monomer concentration profile demonstrates the increased penetration of the matrix into the grafted layer for attractive A-D interactions as compared to the repulsive A-D χGM = 0 PNC (i.e., PNC at the purely entropic limit). As one would expect, this increased matrix penetration implies greater wetting of the grafted layer by matrix chains when hydrogen bonding is present as compared to the entropic limit. The brush heights show that the grafted layer extends into the matrix in the presence of H-bonding interactions, as confirmed by the conformations of graft chains in Figure 4b. A simulation snapshot presented in Figure 4c provides insight into the grafted chain conformations in the presence of attractive A-D sites, showing that each graft chain extends into the matrix to allow for the formation of A-D contacts. Although each individual A-D interaction is indeed specific, every graft chain interacts with multiple matrix chains. This is contrary to what is often seen in DNA duplexes where a single “graft” DNA chain zips together with its complementary single “matrix” DNA chain. These differences are likely due to (a) our polymer model lacking the dihedral interactions that mimic stacking in the DNA CG model and (b) all donors interacting equivalently with all acceptors in the polymer model, in contrast to DNA where Watson−Crick type base pairing encourages each DNA strand to specifically bind to its complementary sequence DNA strand. Despite multiple matrix chains interacting with each graft chain, the matrix chain conformations (not shown) are not impacted by the A-D attraction, as they are dominated by matrix chains in the bulk that are not interacting with the polymer grafted particle; the volume fraction of the single polymer grafted particle in the system is small (ϕG ≈ 0.02), and as a result, the bulk matrix chains dominate the overall matrix chain conformations. Having established that attractive A-D interaction increases grafted layer wetting compared to the purely entropic limit, we consider PNCs with unfavorable graft-matrix interaction (i.e., positive χGM) to show if/how favorable H-bonding interactions increase grafted layer wetting when χGM > 0. Figure 5a shows that in the absence of attractive A-D interactions, both grafted layer wetting and brush height decrease when χGM increases from χGM = 0.0 to +0.3. Moreover, PNCs with χGM > 0 are marked by enrichment of grafted beads near the particle surface; this grafted layer collapse occurs to minimize unfavorable enthalpic interactions between graft and matrix chains. Figure 5b shows that, in the presence of attractive A-D interactions, we see an increase in the grafted layer wetting. In other words, the strongly favorable H-bonding interactions are able to overcome the energetically unfavorable graft bead− matrix bead interactions and cause more matrix chains to penetrate the grafted layer. The increase in brush heights proves that the graft polymers extend to enable the favorable acceptor−donor contacts even if the graft−matrix interactions are unfavorable (i.e., χGM > 0). The probability distribution of

Figure 5. Graft and matrix monomer concentration profiles (a,b) and probability distribution of graft end−end distance (c,d) for PNCs with repulsive A-D interaction (a,c) and attractive A-D interaction (b,d). For all plots in a−d, the legend is as follows: χGM = 0.1 with repulsive (a,c)/attractive (b,d) A-D interaction (blue diamonds and dashed line), χGM = 0.3 with repulsive(a,c)/attractive (b,d) A-D interaction (red squares and solid line), and the purely entropic case of χGM = 0 and repulsive A-D interaction (black circles). Simulation snapshots of the polymer grafted particle (matrix chains hidden) are shown for (e) χGM = 0.1 with repulsive A-D interaction and (f) χGM = 0.1 with attractive A-D interaction. All results are for D = 5d, NG = 20, Σ = 0.65 chains/d2, and NM = 20. Error bars (calculated as standard deviation from three independent simulation runs) when not visible are smaller than marker size.

end−end distance for graft chains show collapsed grafted layer conformations in the absence of attractive A-D interactions (Figure 5c) and an extended grafted layer in the presence of attractive A-D interactions (Figure 5d). These conformations are also depicted visually in simulation snapshots shown in Figure 5e,f. Based on the well-established link between increased wetting and improved particle dispersion as described in the many papers cited in the introduction, so far, our results suggest that by introducing directional interactions between graft and matrix chains (either through H-bonding or other chemistries with specific and directional interactions) we could increase the propensity for particle dispersion even in PNCs where dewetting and nanoparticle aggregation would otherwise be favored. Since past studies have shown that grafted layer wetting can also be increased purely through choice of graft and matrix chemistries with a negative χGM,29,31 we will now F

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Figure 6. (a) Graft and matrix monomer concentration profiles for PNCs with isotropic graft−matrix interaction and repulsive A-D interaction and for PNCs with attractive A-D interaction, εAD = 13kT and χGM = 0 . Probability distribution of the end−end distances, P(Ree) vs Ree for (b) graft chains and (c) matrix chains. Legend below part a is applicable to all plots in this figure. (d) Simulation snapshots showing a graft chain and the individual matrix beads interacting with it for PNCs with attractive A-D interaction and χGM = 0 and for PNCs with χGM = −0.5 and repulsive A-D interaction. (e) Probability distribution of free volume per graft chain for the two cases in part d. All results are for D = 5d, NG = 20, Σ = 0.65 chains/d2, and NM = 20. Error bars (calculated as standard deviation from three independent simulation runs) when not visible are smaller than marker size.

chemistry with a lower value of H-bond strength or εAD, it would result in reduced wetting (see results in Figure S4), and hence, a lower value of isotropic graft−matrix interaction εGM would be needed to achieve equivalent wetting. One could use more rigorous ways to confirm this hypothesis of equivalent ef fective G-M interactions by conducting computationally intensive potential of mean force calculations54 to quantify the WGM(r) for (a) χGM = −0.5 with repulsive A-D interaction and (b) χGM = 0 and attractive A-D interaction of 13kT. Given the extensive simulation work that goes toward calculating these potentials of mean force, we have ongoing work to calculate the potential of mean forces through a theoretical approach. In Figure 6b, the graft chains adopt extended conformations corresponding to the increasing grafted layer wetting shown in Figure 6a. Interestingly, the PNCs with identical wetting also have identical graft chain conformations, irrespective of whether the wetting was driven by isotropic G-M interactions or directional A-D interaction. We note here that this is likely true only for fully flexible graft chains considered here and may change as flexibility decreases. The matrix chain conformations (Figure 6c) are dominated by the matrix chains in the bulk (away from the grafted particle) and exhibit no differences at all with changing PNC interactions. Even though the graft and matrix conformations are the same between the two PNCs with equivalent wetting, there are some key differences in how the graft makes contacts with the matrix monomers for the PNC with isotropic G-M attraction versus PNC with directional A-D interactions. Visually, in Figure 6d we show the matrix beads (green) interacting with the graft chains for

compare the increased wetting achieved due to directional/ anisotropic A-D attraction to that induced by isotropic graft− matrix attraction. Figure 6a shows that in the absence of attractive A-D interaction as χGM becomes progressively more negative, the grafted layer wetting increases, as expected. This figure also shows that a PNC with attractive A-D interaction of εAD = 13kT and χGM = 0 has similar graft and matrix monomer concentration profiles as PNCs with isotropic χGM = −0.4 and χGM = −0.5, both in the absence of attractive A-D interaction. We calculate the number of matrix beads within the grafted layer thickness to prove that the PNC with isotropic χGM = −0.5 in the absence of attractive A-D interaction has equivalent grafted layer wetting as the PNC with attractive A-D interaction of εAD = 13kT and χGM = 0 (see Supporting Information Table S1). This means that isotropic and anisotropic PNC interactions are able to achieve equivalent wetting, but with an order of magnitude difference in strength; isotropic graft−matrix attraction εGM = 1.0kT that leads to χGM = −0.5 achieves equivalent wetting as directional A-D interaction of strength εAD = 13kT. We think that equivalent wetting is achieved because the ef fective G-M interactions in both cases are equivalent. We confirm this by calculating the second virial coefficient B2 from the gGM(r) for the two PNCs with equivalent wetting. The B2 for the PNC with isotropic graft−matrix attraction (χGM = −0.5 with repulsive A-D interaction) is 215.44 ± 0.55 and the B2 for the PNC with directional graft−matrix attraction (εAD = 13kT and χGM = 0) is 209.30 ± 1.42; these similar values confirm that the equivalent wetting is due to equivalent effective graft−matrix interaction. So, if one chose a different graft and matrix G

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Macromolecules the two cases. In the case of attraction created via directional A-D interactions, each graft chain interacts with 12.4 ± 0.63 matrix chains. In contrast, when the attraction is created via isotropic G-M interactions (χGM = −0.5) each graft interacts with 32.77 ± 0.13 matrix chains. Interestingly, despite the graft chains interacting with fewer matrix chains in the case of attractive A-D interactions, the free volume for each graft chain (Figure 6e) is lower for PNCs with directional A-D interaction as compared to the PNC with isotropic G-M attraction. This means that even though structurally the wetting (i.e., net interpenetration of the graft and matrix chains) is the same, and the graft and matrix conformations are also similar, the graft and matrix contact is “tighter” in the case of PNCs with H-bonding interaction as compared to PNCs with isotropic GM attraction or χGM < 0. We also confirm (Figure S5) that this observation of lower free volume per graft chain for PNCs with H-bonds is not due to the choice of smaller A-D bead sizes relative to G-M beads in our model leading to smaller cut-offs in the potentials and closer bead−bead contact for each A-D interaction. So far, this result of directional A-D interactions leading to fewer matrix chains directly interacting with each graft chain and a lower free volume per graft chain despite equivalent wetting as that of isotropic G-M interactions is exciting as it has implications on the thermomechanical properties of the PNCs. It suggests that the choice of H-bonding graft and matrix chemistries can lead to differences in glass transition temperature and mechanical and viscoelastic properties compared to graft and matrix chemistries that have a negative χGM. III.B. Low Grafting Density and NG = NM = 20. In Figure 7a,b, we present analogous results as Figure 6a,b but at low grafting density. Overall, the effects of attractive A-D interactions on the grafted layer wetting (Figure 7a) and graft chain conformations (Figure 7b) are qualitatively the same as that described at high grafting density (Figure 6). We do not show the matrix chain conformations in Figure 7 because they essentially mimic the trend in Figure 6c that at this low filler fraction the matrix chains in the grafted layer do not impact the matrix conformations in the bulk. One should note that for each interaction case (note the legends in Figures 6 and 7), the PNC with lower grafting density always exhibits significantly higher grafted layer wetting than its counterpart at the high grafting density. This behavior is well-known based on past literature cited in the Introduction (as well as by intuition) that decreasing grafting density should increase matrix penetration into the grafted layer. To quantify if the extent to which A-D attractive interactions improve grafted layer wetting is equal at low and high grafting density, we calculate the crossover point in the concentration profiles. This crossover point is defined as the value of r where C(r)graft = C(r)matrix. In Figure 7a (at low grafting density) the crossover point goes from r = 1.45d to no crossover as one compares a PNC at the purely entropic limit to a PNC with directional AD attraction (i.e., as we go from black line to navy blue line). For the corresponding high grafting density cases in Figure 6a, the crossover point shifts from r = 3.5d to r = 0.5d. The shift in crossover point shows an increase in grafted layer wetting due to A-D attraction for both grafting densities, but the shift is smaller for the PNCs at the low grafting density because the grafted layer is significantly more wet even at the purely entropic limit.

Figure 7. (a) Graft and matrix monomer concentration profiles for PNCs with isotropic graft−matrix interaction and repulsive A-D interaction and for PNCs with attractive A-D interaction. (b) Probability distribution of the end−end distances, P(Ree) vs Ree for graft chains. Legend next to part a is applicable to all plots in this figure. (c) Probability distribution of free volume per graft chain for the two cases that show equivalent wetting in part a and equivalent end−end distances in part b. These results are for D = 5d, NG = 20, Σ = 0.32 chains/d2, and NM = 20. Error bars (calculated as standard deviation from three independent simulation runs) when not visible are smaller than marker size.

The isotropic G-M interaction (the χGM value) that provides equivalent concentration profiles to PNCs with directionally attractive A-D interaction is χGM = −0.4. Table S1 also shows that the number of matrix beads in the grafted layer are similar for the two cases; in fact, one could expect that a PNC with −0.5 < χGM < −0.4 may provide a quantitative match in wetting to the PNC with directionally attractive A-D interaction of 13kT. Rather than exploring the value of χGM that provides quantitatively equivalent wetting, we note the similarity between the results at high and low grafting density which support the hypothesis that the ef fective graft−matrix interaction is likely equivalent for the isotropic G-M interaction of 0.9−1.0kT and directionally attractive A-D interaction of 13kT. Finally, the trends of the number of matrix chains interacting with each graft chain and the free volume per graft chain we highlighted at the end of Section III.A are also seen at low grafting density. The number of matrix chains interacting with each graft chain is 11.39 ± 0.16 for attractive A-D interactions and χGM = 0 , and it is 25.52 ± 0.16 for repulsive A-D interactions and χGM = −0.4 . Also, the free volume per graft chain (Figure 7c) is lower in the case of directional A-D attraction than isotropic G-M attraction, despite similar wetting. III.C. High Grafting Density and NG = 20 and NM = 60. In this section we consider PNCs with matrix chain lengths greater than the graft chain length. It is well understood from past work that as the relative length of matrix chains to graft H

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Macromolecules chains increases, the interpenetration of the longer matrix chains into the grafted layer leads to a higher loss in conformational entropy of matrix chains than for the lower matrix length case. By comparing PNCs with NG = 20 and NM = 20 described in Section III.A and PNCs with NG = 20 and NM = 60 cases we show if and how this difference in entropic driving force impacts the effects of directional A-D interactions we have described so far. As seen for PNCs with NM = 20 (Figure 5), for NM = 60 also the PNCs with directional H-bonds can increase the grafted layer wetting even if the graft−matrix chemistries exhibit a positive χGM. Similar to the wetting behavior seen in the absence of attractive A-D interactions for NM = 20 (Figure 5a), for NM = 60 also the monomer concentration profiles (Figure 8a) show that wetting and brush heights decrease as positive

attractions. In the case of attractive A-D interaction, each graft chain interacts with 11.6 ± 0.6 matrix chains and in the case of isotropic graft−matrix interaction, the number of interacting matrix chains per graft chain is 27.5 ± 1.5. As a reminder, the corresponding numbers for NM = 20 are 12.4 ± 0.63 and 32.77 ± 0.13 matrix chains. These results are in line with the expectation that at constant graft length as the matrix chain length increases, the matrix contacts with the graft chains and matrix chain penetration in the grafted layer should decrease. Finally, as seen for NM = 20 (Figure 6e), the free volume per graft chain (Figure 9e) is smaller for the A-D attraction compared to the isotropic graft−matrix interaction. The reduction in free volume per graft chain is identical for NG = 20 and NM = 20 and NG = 20 and NM = 60 (direct comparison not shown here).

Figure 8. Parts a-d are same as Figure 5a-d but for NM = 60.

IV. CONCLUSION AND FUTURE OUTLOOK In this paper we present a new coarse-grained (CG) polymer model that captures specific and directional hydrogen bonding interactions between the graft and matrix polymers for PNCs comprising polymer grafted nanoparticle placed in a polymer matrix. Using this CG model, we show the impact of introducing hydrogen bonding chemistries in the graft and matrix polymers on the grafted layer wetting, grafted chain conformations, and the free volume in the grafted layer. Our results show that directionally attractive interactions between graft and matrix chains improve the grafted layer wetting over both the purely entropically driven PNC as well as PNCs with unfavorable isotropic G-M interactions. The grafted chains extend toward the matrix chains to make acceptor−donor contacts thereby increasing the grafted layer thickness and increasing the interpenetration of matrix chains into the grafted layer. Comparison of PNCs with isotropic graft−matrix interactions and PNCs with directional hydrogen bonding type interactions between graft and matrix chains shows that one can achieve equivalent wetting (as seen by overlapping monomer concentration profiles, graft conformations and number of matrix beads within grafted layer) with the directional hydrogen bonding interaction strength that is an order of magnitude larger than the isotropic graft−matrix attraction strength. Interestingly, despite equivalent wetting and grafted chain conformations, we find that PNCs with directional hydrogen bonding type interactions between graft and matrix chains have a lower free volume per graft chain than PNCs with isotropic graft−matrix interaction. This suggests that directional acceptor−donor interactions induce a tighter graft−matrix contact in PNCs. We find the above trends to hold both at the high and low grafting density regimes as well as matrix chain length/graft chain length ratio of 3 and 1. These results suggest that incorporating hydrogen bonding between graft and matrix monomers can be an effective means for improving grafted layer wetting and as a result, increasing particle dispersion in PNCs. At the same time, the thermomechanical properties of PNCs with H-bonding interactions between graft and matrix monomers may be different from another PNC that has isotropic graft−matrix interactions and an identical grafted layer wetting. In terms of future directions, one can use this CG model for other hydrogen bonding polymeric systems beyond PNCs, like polymer blends, block copolymer mixtures, and so forth. Even though the focus of this paper is on structural features like grafted layer wetting, graft and matrix chain conformations and free volume, there are ongoing efforts in our group to evaluate

χGM increases. In the presence of attractive A-D interactions, there is a significant increase in wetting and brush heights (Figure 8b). Grafted chain conformations for NM = 60 show similar features as for NM = 20 (Figure 5c and d), with a collapsed grafted layer in the absence of attractive A-D sites (Figure 8c) and grafted chains extending into the matrix to allow for the formation of A-D contacts in the presence of attractive A-D interactions (Figure 8d). In summary, these results show that even if the entropic driving forces are less favorable for wetting (in the case of NM = 60 versus NM = 20), the H-bonding interactions can be used to improve grafted layer wetting. As done for the PNCs with NG = 20 and NM = 20 in Figure 6, in Figure 9, we find the conditions for NM = 60 where there is equivalent wetting for PNCs with directional A-D attraction and PNCs with isotropic G-M interaction. As seen for NM = 20, for NM = 60 in Figure 9a an isotropic graft−matrix strength of 1kT, corresponding to χGM = −0.5, is needed to achieve wetting equivalent to that due to attractive A-D interaction with εAD = 13kT and χGM = 0. As seen for NM = 20, the number of matrix chains interacting with each graft chain differs significantly for the directional A-D and isotropic G-M I

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Figure 9. (a) Graft and matrix monomer concentration profiles for PNCs with isotropic graft−matrix interaction and repulsive A-D interaction and for PNCs with attractive A-D interaction, εAD = 13kT and χGM = 0 . Probability distribution of the end−end distances, P(Ree) vs Ree for (b) graft chains and (c) matrix chains. The legend below part a is applicable to all plots in this figure. (d) Simulation snapshots showing a graft chain and the individual matrix beads interacting with it for PNCs with attractive A-D interaction and χGM = 0 and for PNCs with χGM = −0.5 and repulsive A-D interaction. (e) Average free volume per graft chain for the two cases in part d. All results are for D = 5d, NG = 20, Σ = 0.65 chains/d2, and NM = 60. Error bars (calculated as standard deviation from three independent simulation runs) when not visible are smaller than marker size.

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how well this CG model captures dynamic information at both the small time scale of H-bonding (e.g., hydrogen bond lifetimes) and long time scale of polymer relaxation.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.8b02666. Additional information about the coarse-grained model and simulation method (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Arthi Jayaraman: 0000-0002-5295-4581 Notes

The authors declare no competing financial interest.

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REFERENCES

ACKNOWLEDGMENTS

This work was supported by the U.S. Department of Energy, Office of Science grant number DE-SC0017753. This research was supported in part through the use of Information Technologies (IT) resources at the University of Delaware, specifically the high-performance computing resources of the Farber supercomputing cluster. J

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