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Effect of Polymer Architecture on the Structure and Interactions of Polymer Grafted Particles: Theory and Simulations Kevin J. Modica,† Tyler B. Martin,† and Arthi Jayaraman*,†,‡ †

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

ABSTRACT: We use Langevin dynamics simulations and Polymer Reference Interaction Site Model (PRISM) theory to study polymer grafted nanoparticles specifically to explain the impact of comb polymer architecture on the grafted layer structure and effective interparticle interactions in solvent and in matrix polymer. First, we use simulations to study a single particle grafted with comb polymers with varying comb polymer design (i.e., spacing and length of side chains along the comb polymer backbone), grafting density (i.e., polymer chains/particle surface area), and particle curvature in implicit solvent at the athermal limit. We find that increasing side chain length or decreasing side chain spacing along the comb polymer effectively swells and extends the polymer backbone due to the increasing side chain monomer crowding. For particles at finite curvature with increasing side chain monomer crowding, the monomer concentration profile of the comb polymer backbone at short distances from the surface resembles the concentration profile of a semiflexible linear polymer and at farther distances resembles that of flexible linear polymers grafted to a flat surface. As the particle curvature decreases to zero (i.e., flat surface), increasing side chain crowding has a simpler effect of expanding the grafted layer without changing the overall shape of the concentration profile. To understand how architecture affects the interactions of the comb polymer grafted particles, we use PRISM theory to calculate the potential of mean force (PMF) between comb polymer grafted particles in implicit solvent, explicit solvent, and explicit matrix of athermal linear polymers. On the basis of the PMFs calculated for a wide range of design parameters (grafting density, comb polymer design), we find that, compared to linear polymers, the comb polymers exhibit stronger effective attraction in the PMF between the grafted particles in both small molecule solvent and polymer matrix due to the increased crowding in the grafted layer from the comb polymer side chains. Interestingly, the PMF between the grafted particles in a small molecule solvent is more sensitive to the comb polymer design (i.e., side chain length and spacing) than the PMF between the grafted particles in a polymer matrix. emphasized the role of polymer grafted particles as fillers in nanocomposites linking the effects of grafted polymer chemistry, molecular weight, grafting density to the grafted layer structure, and filler dispersion/aggregation/ordering as a function of particle chemistry, particle size, medium (solvent/ matrix) chemistry, and molecular weight.1−4,14−21,23−33 In particular, these studies have established the fundamental rules emphasizing the role of the grafted layer in dictating the enthalpic and entropic driving forces behind wetting/mixing of the medium (solvent/matrix polymer) with the grafted chains. Most of the past work has been on linear grafted polymer

I. INTRODUCTION Polymer grafted nanoparticles are a class of hybrid materials where organic/inorganic nanoparticles are functionalized with synthetic polymers or biopolymers that are tethered/anchored at a desired grafting density. These hybrid materials have been the topic of many computational and experimental studies in the past decade that have focused on the synthesis of these polymer grafted nanoparticles, on their structural characterization in solution, melts, and in polymer matrix, and on their use in a myriad of applications.1−40 These applications include, but are not limited to, the use of polymer grafted particles as fillers in polymer nanocomposites, their assembly into optically active materials, their role as compatibilizers in fluid−fluid interfaces, and their use as carriers for drugs/other cargo in biomedical applications. The majority of these past studies have © 2017 American Chemical Society

Received: March 11, 2017 Revised: May 3, 2017 Published: June 9, 2017 4854

DOI: 10.1021/acs.macromol.7b00524 Macromolecules 2017, 50, 4854−4866

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Macromolecules

Figure 1. (a) Schematic of the coarse-grained (CG) model of the particle core and one of the grafted comb polymers. The black and gray CG beads on the particle make up the rigid body of the particle, with the black beads denoting the grafting sites. The yellow and blue CG beads in the grafted polymer depict the side chain and backbone monomers of the comb polymer, respectively. This color distinction between the side chain and backbone monomers is purely for visual appeal; we study the systems at the athermal limit where there is no chemical distinction between the backbone and side chain monomers. The definitions of the various design parameters of the comb polymer, Nbb, Nsc, and lsc, are also depicted. (b) Schematic of a full comb polymer grafted particle in its initial configuration at the start of a simulation with Nbb = 20, lsc = 2, Nsc = 3, diameter D = 5d, and grafting density Σ = 0.32 chains/d2; as the simulation proceeds, the polymer chains relax to more realistic chain configurations.

these comb polymers. Through the design of the side chains and side chain density, comb polymers have found success as stimuli-responsive polymers, as patterning agents for photonic devices, and in various biomedical applications.54,55,66 Despite their success in being used as both free polymers and grafted to flat surfaces, to our knowledge there has not been a comprehensive simulation and theory study looking at the effect of the comb polymer architecture on the grafted layer structure and interactions of polymer grafted nanoparticles. In this paper, we use a combination of coarse-grained Langevin molecular simulations and liquid state theory (PRISM) to elucidate the effect of comb polymer architecture on the grafted layer structure and on the effective interparticle interactions for polymer grafted nanoparticles in solvent and matrix, in the purely athermal limit. First, using coarse-grained molecular simulations, we study how varying design aspects of the comb architectures (e.g., side chain placement along comb polymer backbone, side chain length) at varying grafting density and surface curvature affect the grafted layer structure quantified by monomer concentration profile, grafted chain radii of gyration, and backbone scaling exponent. We find that increasing monomer crowding along the comb polymer backbone, by using either long side chains or closely placed side chains, causes the (flexible) polymer backbone to swell and extend farther than the corresponding linear flexible polymers with the same number of backbone segments. On curved particles at moderate to high grafting density, we see dual regimes in the monomer concentration profile from the particle surface. Near the surface, the comb polymer concentration profile mimics the shape of concentration profiles seen for grafted semiflexible linear polymers. Farther from the surface, the comb polymer concentration profile mimics that of linear flexible polymers grafted on flat surfaces. To understand how the comb polymer design impacts the effective interactions with other grafted particles in either solvent or polymer matrix, we use PRISM theory to calculate the potential of mean force between the particles in implicit and explicit solvent as well as in a polymer matrix. Compared to linear grafted polymers, the

architectures, and the impact of exotic new architectures of the graft polymer (e.g., comb, bottlebrush) on the grafted layer structure and/or effective interactions in a polymer matrix has largely been left unexplored. In addition to their use in nanocomposites, polymer grafted particles have also been used in biomedical applications as imaging or delivery agents.36−40 When used for imaging, the particle itself serves as the imaging agent with the grafted layer providing a stealth coverage. When used for biological delivery, the functional biomolecule/drug could either be loaded in the grafted layer or be the particle itself, with the grafted biocompatible polymers (e.g., poly(ethylene glycol)) protecting/shielding the drug(s) during circulation in both cases. The impact of varying the grafted polymer architecture on the efficacy of polymer grafted particles in these biological relevant environment/medium remains to be studied. One should be motivated by the success of biocompatible polymers in dendrimer or star architectures (without particles) as stealth carriers of therapeutic materials to explore beyond linear architectures. The recent advances in synthesis of particles grafted with sophisticated, new, and exotic polymer architectures26,38,41−46 and the lack of computational and experimental studies aimed at establishing the effects of complex nonlinear polymer architecture on grafted layer structure motivate our work in this paper. One particular nonlinear chain architecture that is the focus of this paper is the comb polymer architecture.47−61 Comb polymers constitute a family of polymers that have a linear polymer backbone to which side chains are connected at regular or random intervals and are distinct from polymers with dendritic architecture,45,62−65 which does not have a welldefined linear backbone. Comb polymers with high side chain density (e.g., one or two side chains for every backbone monomer) are called bottlebrush polymers, and comb polymers with low side chain density are often called comb/graf t polymers. Irrespective of the side chain density, the physical and chemical features of the side chains provide an expanded set of tuning parameters when designing materials systems with 4855


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Macromolecules comb polymer architecture increases the effective midrange attraction in the potential of mean force between grafted particles in both small molecule solvent and polymer matrix, with the small molecule solvent medium being more sensitive than the polymer matrix to the specific design of the comb architecture. This paper is organized as follows. In section II, we present the details of the molecular simulations as well as PRISM theory along with the analyses techniques. In section III, we present the simulation results of the grafted layer structure followed by the theoretical results of the potential of mean force. In the Conclusions section, we summarize the results and present some future directions for this work.

II. APPROACH II.A. Molecular Dynamics Simulations. Model. We study comb polymer grafted nanoparticles and flat surfaces using a coarse-grained model. We model polymer chains with linear or comb architecture as bead−spring polymers, where each bead of diameter d along the chain depicts a repeating unit/ monomer of the polymer. The comb polymers are composed of a backbone (blue beads in Figure 1) of length Nbb and side chains (yellow beads in Figure 1) of length Nsc at a specified side chain spacing (lsc) along the backbone starting with the second monomer away from the surface/particle. Both the flat surfaces and spherical particles are modeled as rigid bodies of hard sphere beads of diameter d. In the case of the spherical nanoparticles, we use overlapping surface beads to obtain the correct diameter (D) with a specific number of (roughly) equidistant surface beads serving as grafting sites (denoted by black sites in Figure 1) to achieve a target grafting density (Σ). In the case of flat surfaces (Figure 2a,b) we use several layers of hexagonally close-packed beads with a separate set of beads on embedded in the surface as grafting sites (Figure 2a). The linear or comb polymers are attached to the flat surface on both sides of the surface, and we choose the surface thickness to be large enough to prevent any interactions between the polymers on either side of the surface slab (Figure 2b). Grafting density is varied by changing the number of grafting sites and by changing the x and y dimensions of the flat surface. The star polymer case (Figure 2c) is similar to the spherical nanoparticle with core diameter D = 1d, modeled as a single bead, and comb polymers directly attached to the core bead instead of grafting sites. All bonded interactions between monomer beads are modeled using a harmonic potential Ubond(r ) = k bond(r − r0)2

Figure 2. (a) Top-down view of a representative flat surface with gray beads being the surface beads and blue beads depicting the first tethered bead of the grafted comb polymers. Simulation snapshots of (b) a flat surface grafted with comb polymers and (c) comb polymers in the star configuration after partial relaxation of the grafted comb chains.

athermal limit has been studied extensively for the linear architecture of the grafted polymer (summarized in review articles2−4,68,69), providing a baseline for comparison and for isolating the effect of varying chain architecture. Simulation Method. Using the above CG model and LAMMPS package,70 we simulate a single comb grafted particle/star/surface system in implicit solvent via Langevin dynamics at a reduced temperature of T* = 1.0 and time step of dt = 0.001. Initially, to remove any unphysical overlaps in the chosen initial configuration, we run for 1 × 104 timesteps with all nonbonded pairwise interactions set to a soft potential defined as ⎡ ⎛ π r ⎞⎤ Usoft(r ) = A⎢1 + cos⎜ ⎟⎥ , ⎢⎣ ⎝ rc ⎠⎥⎦

(1)

(3)

where the prefactor A is gradually increased from 0 to 100 in units of ϵ, and rc is the cutoff distance equal to diameter of monomer bead (1d). After this initialization, the nonbonded interactions are switched to WCA potential, as described in eq 2, and the simulation is run for 25 × 106 timesteps collecting snapshots every 1 × 105 timesteps for analyses. We verify that the relaxation time of the system is much faster than our sampling frequency by plotting the time series and autocorrelation of the graft chain radius of gyration as shown in Supporting Information Figure S1. Analyses. Using the snapshots from the simulations, we calculate the radial concentration profile of all monomers (referred to as “total concentration profile”) using

with kbond = 100kBT/d2 and r0 = 1d. The nonbonded interactions between all pairs of beads (backbone, side chain, particle) are modeled through a Weeks−Chandler−Andersen (WCA) potential67 ⎧ ⎡⎛ ⎞12 ⎛ ⎞6 ⎤ σ σ ⎪ ⎪ 4ε⎢⎜ ⎟ − ⎜ ⎟ ⎥ + ε r ≤ 21/6σ ⎝ ⎠ ⎝ r⎠ ⎦ UWCA(r ) = ⎨ ⎣ r ⎪ ⎪ ⎩0 r > 21/6σ

r < rc

(2)

where ε = 1.0kBT and σ is representative of the diameter of the bead in units of d. The choice of this interaction is motivated by the need to first understand how the architecture of the grafted polymer impacts grafted layer structure, wetting by solvent and matrix, and effective interactions in the athermal limit. The

C total(r ) = 4856

N (r ) 4πr 2Δr

(4) DOI: 10.1021/acs.macromol.7b00524 Macromolecules 2017, 50, 4854−4866

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Macromolecules

predicts the structural correlations in isotropic, molecular fluids and is an extension of Ornstein−Zernike liquid state theory. PRISM theory has no mean-field or incompressibility assumptions and captures density fluctuations over a range of length scales. It has been successfully applied to predict liquid structure in a broad range of soft-materials systems including melts,71,74,77 blends,74,78,79 nanocomposites,76,80−84 and polyelectrolytes.85,86 Besides the prediction of structure and thermodynamics, PRISM theory has also been useful as a coarse-graining engine87−91 and for providing models for experimental scattering data.92,93 We only provide a brief overview of this theory in this paper and direct the reader to previous papers cited above for additional details. The fundamental PRISM equation is written as

where N(r) is the average number of polymer beads in the spherical shell bounded by r and r + Δr, where Δr = 1d. Similarly, for the flat geometries, the total concentration profile is calculated as C total(z) =

1 N (z ) 2 lxlyΔz

(5)

where N(z) is the number of beads in the cuboid shell with dimensions equal to the simulation box lengths lx, ly, and Δz = 1d. The 1/2 in eq 5 is to average the contributions counted from both the grafted layers above and below the flat surface. To compare linear and comb architecture with different number of monomers in the grafted layer, but equal number of backbone beads, we also calculate the concentration profiles of only the backbone beads, Cbackbone(r), where N(r) and N(z) are restricted to the backbone monomers in the chain. We characterize the average height of the grafted layer H as the square root of the second moment of the normalized total concentration profile C(r) ⎡ ∫ r 2C (r ) dr ⎤1/2 total ⎥ H=⎢ ⎢⎣ ∫ C total(r ) dr ⎥⎦

Ĥ (k) = Ω̂(k)Ĉ(k)[Ω̂(k) + Ĥ (k)]

with Ĉ (k) representing the direct correlation function, Ω̂(k) the intramolecular correlation function, and Ĥ (k) the total intermolecular correlation function, all as Fourier space functions of wavenumber k. At each k, Ĉ (k), Ω̂(k), and Ĥ (k) are symmetric N × N matrices with N being the number of site types in the model. In this study, we use P, G, M, and S as labels for the particle, graft, matrix, and solvent sites, respectively. For example, the 3-site Ĉ (k), Ω̂(k), and Ĥ (k) matrices for a polymer (G)-grafted particle (P) in an explicit polymer matrix (M) would be written as

(6)

The calculation is handled in the same way for the linear concentration profiles, Ctotal(z). We also present the probability distribution of squared radius of gyration Rg2 of the grafted polymer where Rg2 for a single chain is calculated as Rg2 =

1 N

⎡ c ̂ (k) c ̂ (k) c ̂ (k) ⎤ GM GP ⎢ GG ⎥ ̂ ⎢ C(k) = cGM ̂ (k) cMM ̂ (k) cMP ̂ (k)⎥ ⎢ ⎥ ⎢⎣ cGP ̂ (k) cMP ̂ (k) cPP ̂ (k) ⎥⎦

N 2 ⃗ ) ∑ ( ri ⃗ − rcom

(7)

i=1

with ri⃗ denoting the location of bead i on the chain, N denoting the number of beads along the chain, and rc⃗ om denoting the location of the center of mass of the chain. We calculate the Rg2 probability distribution considering all monomers in the polymer as well as considering the backbone monomers only. We also calculate the fractal dimension, D, for the polymer backbones in both linear and comb architectures using the intramolecular pair correlation function. First, we calculate the intramolecular correlation function of the backbone monomers using the Debye scattering relation:71,72 ω̂ backbone(k) =

1 Nbb

Nbb

∑ i,j

(10a)

⎡ ρ ωGG ̂ (k) ̂ (k) (ρG + ρP )ωGP ̂ (k) ⎤ (ρG + ρM )ωGM ⎢ G ⎥ ̂ (k) ρM ω̂MM (k) (ρM + ρP )ω̂MP (k)⎥ Ω̂(k) = ⎢(ρG + ρM )ωGM ⎢ ⎥ ⎢(ρ + ρ )ω̂ (k) (ρ + ρ )ω̂ (k) ρ ω̂ (k) ⎥ ⎣ G ⎦ GP MP P M P P PP

(10b)

⎡ ρ ρ h ̂ (k) ρ ρ h ̂ (k) ρ ρ h ̂ (k) ⎤ G M GM G P GP ⎥ ⎢ G G GG ⎢ ̂ (k) ρ ρ h ̂ (k) ρ ρ h ̂ (k)⎥ Ĥ (k) = ⎢ ρG ρM hGM ⎥ M M MM M P MP ⎥ ⎢ ̂ ̂ ̂ ⎢⎣ ρG ρP hGP (k) ρM ρP hMP (k) ρP ρP hPP (k) ⎥⎦

sin(krij) krij

(9)

(10c)

where ρX is the number density of site X, ĉXY(k) is the X−Y direct correlation function, ω̂ XY(k) is the X−Y intramolecular correlation function, and ĥXY(k) is the X−Y total correlation function. In the current work, the explicit solvent and explicit matrix calculations are conducted using 3-site PRISM (G, P, M or G, P, S), while the implicit solvent case is considered using 2site PRISM (G, P). For numerically solving the PRISM equations, two pieces of information must be provided: the full Ω̂XY(k) at all k, and a set of closure relations linking the direct correlation functions, cXY ̂ (k), to the total correlation functions, ĥXY(k). To obtain the full Ω̂XY(k) for the polymer grafted particle and matrix polymer, i.e., ω̂ GG, ω̂ GP, or ω̂ MM, we first conduct a molecular dynamics (MD) simulation of either a single polymer grafted particle or a single matrix chain, using the simulation model and procedure outlined in the previous section. Then, using the Debye scattering relation,71,72 we directly calculate the ω̂ XY(k):

(8)

where Nbb is the backbone length, k is the wavenumber, rij is the distance between sites i and j, and the angle brackets represent ensemble averaging over all grafted chains and uncorrelated snapshots in the simulation trajectory. We then extract D from the slope of the linear fit to ω̂ backbone(k) in the fractal scaling regime of 1/⟨Ree2⟩0.5 < k/2π < 1/rb, where ⟨Ree2⟩ is the meansquared end-to-end distance of the grafted chain, and rb is the polymer Kuhn segment size.73 The fractal dimension, D, is related to the scaling exponent ν through D = −1/ν, as shown in Figure S2. II.B. Polymer Reference Interaction Site Model (PRISM) Theory. To predict the particle−particle potential of mean force, WPP(r), as a function of grafted polymer architecture and medium (e.g., implicit solvent, explicit solvent, explicit polymer matrix), we employ the Polymer Reference Interaction Site Model (PRISM) theory.74−76 PRISM theory 4857


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1 NMNtot

NM NX

NY

∑∑∑ m

i

j

For the PRISM calculations in explicit solvent or explicit matrix, we maintain the total system volume fraction to be η = 0.35 at the dilute filler limit (i.e., the filler fraction ϕ = 0.001). We consider matrix lengths of N = 10 or 60, shorter and longer than backbone length Nbb = 20. For the implicit solvent case, we set the total system volume fraction to η = 0.0035 to mimic the same grafted particle volume fraction as the explicit solvent and matrix cases. Both the solvent bead diameter and matrix monomer diameters are kept equal to the graft monomer diameter, d.

sin(krij) krij

(11)

where X and Y represent site types, NM is the total number of molecules containing type X and type Y sites, NX is the total number of sites of type X in each molecule, NY is the total number of sites of type Y in each molecule, Ntot = (NX + NY) if X ≠ Y, otherwise Ntot = NX, rij is the distance between sites i and j, and the angle brackets represent ensemble averaging over uncorrelated snapshots in the simulation trajectory. For ω̂ PP, there is only one P site in each polymer grafted particle, and for ω̂ SS, there is only one site S in each solvent; therefore, ω̂ PP(k) = ω̂ SS(k) = 1. All pairs of site types that do not exist within the same molecule have ω̂ XY(k) = 0; therefore, ω̂ MP(k) = ω̂ GM(k) = ω̂ SP(k) = ω̂ SP(k) = 0. For closure relations, we use a combination of the Percus− Yevick (PY) and hypernetted chain (HNC) closures PY:

HNC:

cXY (r ) = (1 − e βUXY (r))(hXY (r ) + 1)

III. RESULTS III.A. Effect of Polymer Architecture on Grafted Layer Structure. We present in Figures 3a and 3b the monomer

(12a)

cXY (r ) = hXY (r ) − ln(hXY (r ) + 1) − βUXY (r ) (12b)

where UXY(r) is the interaction potential between sites X and Y at distance r. In this study, we use the HNC closure for the PP pair and the PY closure for all other pairs. For all site pairs, we describe the interactions using a hard sphere potential where UXY(r) = 0 for r ≥ σXY and UXY(r) = ∞ for r < σXY. Finally, we numerically solve the PRISM equations using a Newton−Krylov solver built into the KINSOL package.94 The result of a successful calculation is the converged Ĥ (k) matrix which we invert to real space and then use to calculate the intermolecular potential of mean force: WXY (r ) = −kBT ln(hXY (r ) + 1)

(13) Figure 3. Total (a, c) and backbone (b, d) concentration profiles for polymer grafted particle of diameter D = 5d and grafting density Σ = 0.32 chains/d2 with linear and comb polymers with backbone length Nbb = 20. Parts a and b are for systems with side chain spacing lsc = 2 and varying side chain length Nsc while parts c and d are for systems with Nsc = 3 and varying lsc. Vertical dotted lines represent the rootmean-square brush height. The legend in part a is also applicable for part b, and the legend in part c also holds for part d. The error bars in all parts represent standard deviation.

II.C. Parameters Varied. In this paper, we focus on two architectures of the grafted polymers: comb and linear architectures. As shown in Figure 1, within the comb architecture we can tune three features: backbone length Nbb, side chain length Nsc, and side chain spacing lsc. We maintain the backbone length to be constant at 20 beads in this study and vary side chain length Nsc from 3 to 8 and side chain spacing lsc from 2 to 8. For example, a system with Nsc = 8 and lsc = 2 implies the comb polymer of 20 beads backbone has a side chain of length 8 emanating from every other backbone bead. We note that the first side chain is connected to the second backbone bead along the chain, starting from the surface. While we maintain the particle diameter in the (linear or comb) polymer grafted particle to be constant at D = 5d (where d is the diameter of the CG polymer bead), we study effects of varying curvature by also studying a star polymer with D = 1d core and polymers grafted on a flat surface (mimicking an infinite D surface). The grafting density of the polymers on these surfaces, Σ, is quantified as the number of chains per unit area, d2. On the polymer grafted particle, the grafting density is chosen to be Σ = 0.12, 0.32, 0.65, and 0.76 chains per d2. The grafting densities for the star polymer and flat surfaces are chosen so that one can fairly compare across these three systems of varying curvature. In all the systems considered in this paper, the pairwise interactions between all pairs of beads are maintained as athermal, to ensure all effects we capture are entropically driven.

concentration profiles for a polymer grafted particle of size D = 5d and grafting density Σ = 0.32 chains/d2 with comb polymers with Nbb = 20 and lsc = 2 for varying Nsc. As Nsc (length of the side chain) increases, the number of monomers in the grafted layer increases. Going from linear polymers to comb polymers with increasing Nsc, the Ctotal increases and shifts to larger r in Figure 3a. To fairly compare the linear (with no side chains) and comb architectures, in Figure 3b, we present the monomer concentration profiles of backbone beads only, as all architectures have the same number of backbone beads. We note that the linear case is the same in both total and backbone concentration plots (Figures 3a and 3b) as there are no side chains in the linear case. In Figure 3b, going from linear to comb polymers with increasing Nsc, the Cbackbone extends to larger r, indicating increasing backbone stretching with increasing Nsc. This demonstrates that the presence of side chains in the comb architecture increases the steric repulsion 4858


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Figure 4. Representative simulation snapshots of linear and comb grafted particles of diameter D = 5d at a grafting density Σ = 0.32 chains/d2 and backbone length Nbb = 20. The titles of the figures depict the side chain spacing lsc and side chain length Nsc of the grafts in those figures. The blue beads depict the backbone monomers and yellow beads the side chain monomers. In both parts, we also show the snapshots with the side chains hidden to clearly show the effective stiffening of the backbones due to the side chains crowding.

Figure 5. Probability of squared radii of gyration of the linear and comb polymers (Nbb = 20) grafted on a particle of diameter D = 5d at a grafting density Σ = 0.32 chains/d2. Parts a and b present the Rg2 (in d2) of all monomers along the graft chains while parts c and d present the Rg2 of the backbone monomers only. The side chain length Nsc and spacing lsc are denoted by the row labels and legend, respectively.

three grafting densities the backbone monomer concentration profile of the comb polymer grafted particle most closely resembles that of the linear semiflexible polymer grafted particle ε with angle potential of strength kab = 2 θ 2 . In previous work, we

along the polymer backbone, effectively stiffening the backbone. Interestingly, a comparison of the linear and comb-grafted cases in Figure 3b shows that the shape of the concentration profile is different for the two architectures. Near the particle surface, the comb polymer backbone concentration profile resembles visually that of semiflexible linear chains tethered on curved surfaces.16 One could expect this as near the surface the shell volume is small (4πrΔr is small as r is small), making the impact of side chain crowding on the backbone stiffening stronger. Thus, near the surface the comb polymer system approaches the behavior of semiflexible chains grafted on curved surfaces. In Figure S3, we directly compare the backbone concentration profiles of one of the comb polymer architectures, namely, Nsc = 3 and lsc = 2 to linear polymers with same Nbb, grafted on same particle diameter and grafting density but with varying semiflexibility. We find that for all

ε

had found the persistence length of linear chains with kab = 5 θ 2 grafted on spherical particles of similar size and Σ = 0.65 chains/d2 to be between 5.7d and 7.1d.16 We note that despite the overall correspondence between the semiflexible linear polymer grafted particles and comb polymer grafted particles (with flexible backbones), there is one key difference, with the linear semiflexible polymer lacking the “knee” at intermediate distances (shown clearly in Figure S3). At intermediate to larger distances, as shown by the inset in Figure 3b, the comb polymer backbone concentration profile adopts the shape seen with linear flexible chains grafted to flat surfaces at low grafting densities.29,95 This is also expected as at large r where the 4859


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completeness, in Figures S4−S6, we also show the total and backbone monomer concentration profiles along with the Rg2 probability distribution of the polymer grafted particles of size D = 5d at the three different grafting densities shown in Figure 6.

curvature effect is smaller, the comb polymers are spread farther apart, giving more volume for the side chains and reducing their stiffening effect on the grafted chain conformations. Next, to understand if the stiffening effects brought upon by increasing side chain length at constant spacing is similar to the effects brought upon by decreasing side chain spacing at constant side chain length, in Figures 3c and 3d, we present concentration profiles for polymer grafted particle with comb polymers of Nsc = 3 and varying lsc. As lsc (spacing of the side chains) decreases, the number of monomers in grafted layer increases, analogous to increasing Nsc. Not surprisingly, as we go from (flexible) linear to comb systems with decreasing lsc, the Ctotal increases and shifts to larger r as there are larger number of monomers in the grafted layer. The height of the grafted layer, denoted by the vertical dotted lines in Figure 3, also shows a shift to higher values as the grafted layer becomes more crowded and extends. Also, in Figure 3d we see that with decreasing lsc, Cbackbone extends to larger r, indicating increasing backbone stretching. The inset and the plot in Figure 3d show the same shape changing effect brought about by increasing Nsc as seen in Figure 3b. Thus, increasing Nsc at constant lsc or decreasing lsc at constant Nsc, both seem to bring about similar effects on the structure of the grafted layer. In both cases, increasing crowding of side chains effectively increases the stiffness of the polymer backbone and extends the concentration profile to larger distances, increasing the grafted layer thickness/height. Figure 4 shows simulation snapshots of these polymer grafted particles visually confirming our quantitative observations in Figure 3. While the concentration profiles give a cumulative picture of the grafted layer, the probability distribution of radii of gyration of the grafted polymers (Figure 5) elucidates the statistics of the individual chain conformations as a function of the grafted chain architecture. In Figures 5a and 5b, the Rg2 distribution for the grafts including both backbone and side chain monomers show that as the side chain spacing decreases, the chain dimensions become larger, in agreement with the grafted layer swelling and backbone extension shown in Figure 3. These data also show that the addition of a small number of side chains (comparing the linear and lsc = 8 cases) cause the graft Rg2 distribution to widen, but upon continuing to decrease lsc the distributions become narrow again. This trend is confirmed by calculating the standard deviation of these distributions, as shown in subplots i and j of Figure S4. As Nsc increases from 3 to 8, the effect of decreasing lsc on the graft Rg2 distribution is stronger because the steric crowding of the side chains along the backbone is higher for longer side chains. Effectively, the grafting density of side chains along the polymer backbone is increasing and causing the backbones to extend. Another way to quantify the grafted backbone extension with decreasing side chain spacing is through the backbone scaling exponent, ν, which we calculated from the fractal dimension obtained from fractal regime of the intramolecular correlation functions (see Figure S2). For all grafting densities, as the comb polymer side chain spacing (lsc) decreases, the backbone ν increases and deviates more from the value of the linear polymers (shown with the horizontal dashed line). These data also show that at constant lsc the backbones at the higher grafting densities have a higher ν than their counterparts at lower grafting densities. This clearly shows that as side chain crowding increases, the backbones and as a result the grafted layer effectively extend(s), and this impact of the side chains crowding is stronger at higher grafting density. For

Figure 6. Backbone scaling exponent, ν, as a function of side chain spacing, lsc, for particles of diameter D = 5d grafted with comb polymers with backbone length Nbb = 20 and side chain length Nsc = 3 at grafting densities shown in the legend. The data for the equivalent linear polymer grafted particles are shown as horizontal dashed lines.

So far, in all the results we compare polymer grafted particles with linear and comb architectures having the same backbone length (Nbb = 20) and at constant grafting density, but changing total number of monomers. If we choose to keep the graf ting density and total number of monomers in the grafted layer constant, we would have to decrease the graft backbone length to compensate for increasing side chain length or decreasing side chain spacing. Similarly, if we choose to keep the backbone length and total number of grafted layer monomers constant, the grafting density would have to be changed. Regardless of what we choose to keep constant, there must be variation in some other system parameters to satisfy the constraint of constant number of graft monomers. For comparison, in Figure S7, we show the concentration profile and radius of gyration distribution data for linear and several comb systems all with the same backbone length Nbb = 20 and total number of monomers in the grafted layer. We find it impossible to decouple the effect of varying grafting density from varying architecture, both of which impact the concentration profile. We therefore continue our discussion in the next sections while keeping the backbone length and grafting density constant. While grafting density can tune the crowding of the grafted layer, changing surface curvature can also bring about similar effects (see schematic in Figure S8). In the limits of curvature, i.e., flat or star-like, the monomer concentration profiles of linear polymer grafted surfaces take on piecewise power-law dependencies with exponents of 2 and −4/3, respectively, with finite curvature particles falling somewhere between these regimes.96−98 In Figure 7, we explore three curvaturesflat, spherical particle, and starin order to understand how varying curvature and crowding mediates the ef fect of graf ted chain architecture. Our goal is not to exhaustively understand star or flat polymer grafted systems but to use these systems to understand how curvature mediates the effect of the comb polymer architecture. We compare these systems with varying curvature both at constant grafting density and at constant number of grafted chains (but varying surface area). We note that for the constant grafting density case we are restricted to 4860


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Figure 7. Monomer concentration profiles (a, c, e) and Rg2 probability distributions considering backbone monomers only (b, d, f) for flat (a, b), spherical particle (c, d), and star (e, f) geometries. In all parts, the backbone length is Nbb = 20 and the grafting density (in units of chains/d2) is as shown in the legend. The comb polymer cases (triangles) have backbone length Nbb = 20, side chain spacing lsc = 2, and side chain length Nsc = 3. Part a is a linear concentration profile in the direction normal to the flat surface while parts c and e are radial concentration profiles. The error bars in parts a, c, and e represent standard deviation.

Figure 8. Potential of mean force between particle cores, WPP(r), in kT as a function of interparticle distance, r (minus the particle size D), in units of d for polymer grafted particles with diameter D = 5d, varying grafting density (0.65 to 0.13 chains/d2), comb architectures with backbone length Nbb = 20, side chain length Nsc = 3 for varying side chain spacing lsc in explicit one bead (1d) solvent (open symbols) and implicit solvent (closed symbols). The corresponding linear architecture is shown in all plots for comparison. We note that the range of y-axis in the main plots as well as in the inset are not the same as the grafting density changes, and the y-axis ticks in the inset are scaled by the value at the top of the y-axis.

considering a low value of grafting density case as the grafted layer of the flat configuration becomes too dense for simulation at much lower grafting densities compared to systems with finite curvature. First, considering the systems at constant number of grafts (filled or solid symbols in Figure 7), going from linear architecture (filled circles) to comb architecture (filled triangles), the flat geometry shows the largest extension in the backbone concentration profile (Figure 7a). Going from flat

to spherical particle to star (i.e., increasing curvature), the differences between the linear and comb polymer systems decrease. The reasoning for this diminishing effect of comb polymer architecture with increasing curvature is analogous to that for decreasing grafting density on the spherical particle. Interestingly, unlike the spherical particle results discussed in Figures 3−6, the flat geometry concentration profile does not exhibit a noticeable change in shape. This is easily explained because the shell volume does not change with increasing 4861


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Macromolecules distances from the flat surface (AΔz does not change with z, where z is distance from surface and A is surface area). Thus, the crowding of the side chains in the grafted layer remains the same near and far from the surface of zero curvature. To rule out any effect of the polymer grafting pattern on the flat surface, we compare the results in Figure 7a to those from an identical system where the chains are randomly grafted to the surface instead of a uniform grid. Figure S9 shows that there is no discernible effect of the grafting pattern for the systems shown in Figure 7a. The star case (D = 1d) exhibits little to no effect of the comb architecture even though 10 grafted chains amounts to a much higher grafting density of 3.20 chains/d2 on the D = 1d core than D = 5d particle. This emphasizes that by choosing a high curvature one can diminish the impact of increased side chain crowding in comb polymer architectures on the grafted layer structure. We can compare the three curvatures at a constant grafting density of 0.07 chains/d2 (open red symbols in Figure 7) and see that the differences between comb and linear architecture on the concentration profile and grafted chain conformations is the most drastic in the flat surfaces. Interestingly, these effects of the comb architecture on the flat surfaces at a small grafting density of 0.07 chains/d2 (open symbols in Figure 7a,b) are more drastic than the star geometry at a much higher grafting density of 3.20 chains/d2 (solid symbols in Figure 7e,f). These results in Figure 7 show that one can tune the impact of the comb polymer architecture through grafting density and curvature of the surface. So far, we have seen that increasing monomer crowding, brought about by either increasing side chain length or decreasing side chain spacing, extends the comb polymer backbone and swells the grafted layer. The shape of the backbone concentration profile is changed upon the introduction of graft side chains but this effect is only observed for systems where the polymers are grafted to surfaces with nonzero curvature (spherical particle but not flat surface). The next step is to explain how these changes affect the interactions of these particles in a given medium (solvent or matrix). III.B. Effect of Graft Polymer Architecture on Effective Interactions in Solvent/Matrix. In Figure 8, we show the potential of mean force (PMF) between two grafted particles at varying interparticle distances with the midrange PMFs shown in the inset. These PMF data describe the effective interactions between the two grafted particles in the implicit or explicit medium, including all enthalpic and entropic effects brought about by the medium itself. PMFs are useful not only to simulate more coarse-grained polymer grafted particles at larger scales but also to provide detailed information on the effective interactions at various interparticle distances unlike a single parameter quantifying the effective interactions, like the second virial coefficient or a χ-parameter.32 It has been previously shown that in the case of linear grafted particles the near surface (low-r) behavior comes from the steric repulsion between the grafted layers on the two particles.96,99−101 The attractive well at intermediate distances arises due to “dewetting” of the grafted layer by the medium (solvent or free polymers) similar to the classic Asakura−Oosawa depletion attraction.2,19,21,31,102 The “dewetting” manifests as an attraction at high grafting densities at the distance where the edges of the grafted layers on the two particles are in contact and/or at low grafting densities at the particle−particle contact distance. We show the short-range behavior of these PMF data in Figure S10 to highlight the presence/absence of a contact well. By analyzing

the depth and location of the attractive wells in these PMFs as a function of graft architecture, we can infer how the architecture increases the overall attraction or repulsion of the grafted particles and, in turn, their propensity for aggregation or dispersion in solution or polymer matrix. At the high grafting density of 0.65 chains/d2 and side chain length Nsc = 3 (Figure 8a), going from linear to the two comb architectures with decreasing lsc, we see increasing strength and range of repulsion at 0 < r − D < 15d, both in implicit solvent (i.e., devoid of medium-induced depletion-like attraction) and in explicit one bead solvent. This is because of increased grafted layer crowding and steric repulsion brought about by decreasing side chain spacing. The inset in the case of implicit solvent shows that the point where the PMF approaches zero shifts to larger distances as we go from linear to the two comb architectures with decreasing lsc, and there is no significant midrange attraction due to the lack of a depletant with configurational entropy. The shift in the zero-crossing is another manifestation of the extending backbones and grafted layer swelling that causes the particles to effectively sterically repel each other at larger interparticle distances. The inset also shows that in the case of explicit one bead solvent, at intermediate distances an attractive well develops for the comb architectures and deepens for the most crowded comb polymer. This is likely due to the solvent beads dewetting the grafted layer due to increasing graft monomer crowding as we go from linear to lsc = 8 to lsc = 2. Not surprisingly, these trends hold, and in some aspects are exaggerated for Nsc = 8 systems (Figure S11). At intermediate grafting density of 0.32 chains/d2 and Nsc = 3 (Figure 8b), we see all of the trends seen in Figure 8a but with two small differences: (1) the magnitude of the steric repulsion near the particle surface and attraction at intermediate interparticle distances is smaller, and (2) the midrange attractive well is weaker in strength and shifts to smaller interparticle distances in Figure 8b as compared to Figure 8a. To avoid any confusion, we note that y-axis is changing, both in the main plot and in the inset, as we go from one grafting density to the other. Overall, the trends with changing architecture for grafting density 0.32 chains/d2 are similar to those for 0.65 chains/d2. The decreasing magnitude of the short-range repulsion with decreasing grafting density (Figure 8a,b) at constant architectures is indicative of decreasing side chain induced crowding in the grafted layer with increasing spacing between the grafted chains on the surface. As grafting density decreases at the same lsc, the midrange attractive well becomes significantly shallower. This is because as the crowding in the grafted layer is lowered with decreasing grafting density, the solvent can “wet” or penetrate the grafted layer more easily, and as a result, the “dewetting” driven attraction is decreased. These trends are also seen for Nsc = 8 (Figure S11). The only point to note is that on the heavily crowded (Nsc = 8) comb architecture the effect of lowering the grafting density is not as drastic as seen for Nsc = 3 (going from Figure 8a to 8b). At the lowest grafting density of 0.13 chains/d2, for both Nsc = 3 (Figure 8c) and Nsc = 8 (Figure S11c), the differences between architecture (linear to comb with decreasing lsc) is smaller than those seen at intermediate and higher grafting density. Furthermore, the differences between the PMF in implicit solvent and explicit solvent is smaller at all distances except at contact. At contact, we see a strong depletion attraction (Figure S11c) develop in the presence of explicit 4862


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Figure 9. Potential of mean force between particle cores, WPP(r), in kT as a function of interparticle distance, r (minus the particle size D), in units of d for polymer grafted particles with D = 5d, varying grafting density (0.65 to 0.13 chains/d2), comb architectures with backbone length Nbb = 20, and side chain length Nsc = 3 for varying side chain spacing lsc in linear matrix polymer of length Nm = 10 (solid symbols) and Nm = 60 (open symbols). The corresponding linear architecture is shown in all plots for comparison. We note that the range of y-axis in the main plots as well as in the inset is not the same as the grafting density changes, and the y-axis ticks in the inset are scaled by the value at the top of the y-axis.

polymer architecture. We explain this behavior based on our understanding of linear graft−linear matrix polymer nanocomposite physics. In athermal systems with linear graft and linear matrix polymers, it has been well established in past studies that the presence of an attractive well and the “autophobic dewetting” that drives this attraction can be explained based on competing entropic driving forces.2,19,31,102 While the gain in mixing entropy drives the matrix polymer to wet/penetrate the grafted layer, the loss in conformational entropy by entering the dense and crowded grafted layer drives the matrix polymer to dewet. Past studies have shown that the mixing entropy is impacted by the volume of the grafted layer (i.e., grafted layer thickness), and the loss in configurational entropy of the matrix chain is primarily impacted by the grafted layer crowding. Going from linear to comb polymer with decreasing lsc, purely based on the increasing grafted layer crowding, we would expect increasing loss in configurational entropy. However, with decreasing lsc, we also see an increase in the grafted layer thickness (Figure 3), which should increase the grafted layer volume and, in turn, the mixing entropy gain. In our study, in the case of matrix polymer, these two competing entropic driving forces must be balancing each other out to leave the magnitude of the attractive well similar in all three architectures. In the case of the solvent, the loss in configurational entropy of solvent is likely dominating, causing the deepening of the attractive well going from linear to comb architectures.

solvent that is absent in the implicit case. All these trends are explained by decreasing crowding attributed to the low-grafting density, leading to the solvent effectively pushing the particles together due to depletion effects. Analogous to the explicit solvent results in Figure 8, in Figure 9 we show the PMFs in the presence of a linear flexible polymer matrix with chain lengths less than (Nm = 10) and greater (Nm = 60) than backbone length Nbb = 20 of the grafted chains. We show the short-range behavior of these PMF and the Nsc = 8 data in Figures S12 and S13. We consider these two chain lengths for the polymer matrix based on the well-established physics in composites of polymer grafted particles with linear graft and matrix chains, where the ratio of matrix to graft chain length dictates the extent of mixing of matrix and graft chains.2,19,21,31,102 As the matrix chain length to graft chain length ratio increases from below 1 to much greater than 1, the extent of mixing or “wetting” decreases. Thus, for a graft polymer with Nbb = 20, we choose to study linear matrix polymers of lengths Nm = 10, resulting in Nm/Nbb = 0.5, and Nm = 60, resulting in Nm/Nbb = 3. For brevity and to avoid repeating the trends seen in the case of explicit one bead solvent, we will only highlight the new effects brought about because of the matrix polymers of these two chain lengths (10 and 60). Going from linear to comb architecture with decreasing lsc, the repulsion in the PMF at low r − D increases in strength and range, and the position of the well in the PMF at intermediate distances (inset) shifts to larger r. In contrast to the one bead explicit solvent, we see that the magnitude of the well at intermediate distances (shown in the insets) is significantly deeper and does not change with the grafted chain architecture. For example, in Figure 9a going from linear polymers to comb polymers with decreasing ls, in the inset the attractive well shifts slightly in r but remains at a constant magnitude of ∼−0.3kT. In contrast, in Figure 8a inset, as we moved from linear (black) to comb polymers with increasing side chain monomer crowding (blue to red), the magnitude of the well deepens from ∼−0.01kT to −0.03kT, and the location of the well shifts in r. This suggests that the PMF of the polymer grafted particles in the presence of one-bead (or small molecule) solvent is more sensitive than in linear matrix polymers to the changes in graft

IV. CONCLUSIONS Using theory and simulations, we have elucidated the effect of comb polymer architecture on the grafted layer structure and effective interactions of polymer grafted particles. At moderate grafting densities, the presence of side chains in the comb polymer architecture has distinct effects on the grafted layer structure. By increasing side chain length or decreasing side chain spacing, we increase the number of side chain monomers, and the resulting monomer crowding, in the grafted layer. At short distances from the particle surface, the monomer concentration profiles are similar to that of linear, semiflexible chains; at farther distances from the particle, the concentration profiles resemble that of linear flexible chains grafted to flat 4863


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for supporting K.J.M.’s undergraduate summer research work on this paper.

surfaces. The former effect is due to the increased steric repulsion created by the side chains, especially at distances close to the particle surface, that effectively stiffens the comb polymer backbone. These effects of the comb polymer architecture are enhanced on surfaces with lower curvatures (e.g., flat surfaces) and/or higher grafting density (e.g., 0.65 chains/d2) and diminish with increasing curvature (e.g., star polymers with a core size commensurate with the monomer size) and/or lower grafting density (e.g., 0.12 chains/d2). These structural effects brought about by the comb polymer architecture affect the effective interactions of the polymer grafted particles in solvent or polymer matrix. Increasing number of side chain monomers causes stronger attraction between the grafted particles, either at midrange (for intermediate and high grafting density) or at contact distances (for low grafting density). This increased attraction is observed in both explicit solvent and polymer matrix, although the attraction in the explicit solvent case is more sensitive to the design of the comb polymer (e.g., side chain spacing). For both solvent and polymer matrix, the thermodynamic reasoning behind this increased attraction is due to the side chains causing increased crowding in the grafted layer, which in turn reduces the ability of the solvent or matrix chains to mix with or “wet” the grafted layer. This paper provides valuable guidelines for materials design by contrasting the effects of comb polymer architecture to that of the more commonly used linear architecture on the structure and thermodynamics of polymer grafted particles. While all effects demonstrated in this paper are for comb polymers and particles at the purely athermal limit capturing entropic driving forces only, the use of comb copolymers where the side chains and backbones are chemically distinct and/or charged, could be a direction of future study. The combination of the steric crowding and the enthalpic effects brought about by the chemical dissimilarity between side chain and backbone monomers and/or electrostatics could lead to striking nonintuitive architectural effects not seen so far in their linear counterparts.

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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.7b00524. Figures S1−S14 (PDF)

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (A.J.). ORCID

Arthi Jayaraman: 0000-0002-5295-4581 Author Contributions

K.J.M. and T.B.M. made equal contributions. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank National Science Foundation (NSF) Division of Materials Research (DMR-CMMT) grant number 1609543 for financially supporting this work. K.J.M. and A.J. also thank the University of Delaware summer fellows program 4864


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