Aggregation–Dispersion Transition for Nanoparticles in Semiflexible

Article pubs.acs.org/JPCB

Aggregation−Dispersion Transition for Nanoparticles in Semiflexible Ring Polymer Nanocomposite Melts Zhenyu Deng,† Yangwei Jiang,† Linli He,‡ and Linxi Zhang*,† †

Department of Physics, Zhejiang University, Hangzhou 310027, China Department of Physics, Wenzhou University, Wenzhou 325027, China

‡

S Supporting Information *

ABSTRACT: By employing molecular dynamics simulations, we explored the conformation transition of nanoparticles (NPs) in semiflexible ring polymer nanocomposite melts. A novel aggregation−dispersion transition for NPs in ring polymer nanocomposites occurs when the bending energy of ring chains increases. The conformations of flexible ring chains near NPs are radial distribution, and the entropic depletion interactions between a pair of NPs in flexible ring polymer nanocomposite melts are attractive, however, the rod-like ring chains wrap around the NPs and the entropic depletion interactions between NPs in rod-like ring polymer melts are repulsive. The aggregation-dispersion transition for NPs induced by chain topology in polymer nanocomposites can provide a new access to achieve miscibility in producing high-performance polymer− nanoparticle composites by simply varying the topological structure of polymers. immersed in flexible linear polymer melts, such as depletion governed aggregation, miscible phase, and bridging state by varying the NP volume fraction and the polymer−NP interaction strength. Depletion and bridging phase separation occur at low and high attraction strengths of polymer−NP interaction, and many nanoparticle effects are found to always reduce miscibility.20 Meanwhile, a good filler dispersion state exists just at the intermediate polymer−NP interfacial interaction.10 In addition, Filippone et al. experimentally studied the relationship between structure and linear viscoelasticity of a model polymer nanocomposite system based on a mixture of fumed silica NPs and polystyrene, and alterations in the viscoelastic behavior are attributed to the structuring of primary silica aggregates.22 Meli et al. investigated the aggregation behaviors of ligand-stabilized gold NPs in poly(methyl methacrylate) (PMMA) thin films, and found that the growth process of NPs in PMMA thin films is described well by the classical coarsening mechanisms.23 Essentially, the dispersion/aggregation behavior of NPs in matrix is closely related to its size, shape, volume fraction, and polymer-NP interactions.6−10 Ring polymers are formed by the simple operation of joining together the free ends of a polymer chain, and topological properties manifest themselves on a variety of properties of ring because topological constraints of ring polymer chains

1. INTRODUCTION Materials that consist of mixtures of polymers and organic/ inorganic particles are used in a wide variety of applications, and the inherent macroscopic properties (such as mechanic, electronic, optical, and so forth) of polymer nanocomposites depend strongly on the microscopic morphology of constituent nanoparticles (NPs) in the polymer matrix.1−5 A good dispersion structure of NPs in nanocomposites is necessary in the development of high-performance materials because of the strong interparticle interactions and weak polymer−NP interfacial interactions.6 To address this issue, various experimental, theoretical, and simulation studies have been made.7−12 For example, Mackay et al. reported that a stable dispersion of NPs in polymer nanocomposites can be enhanced only when the radius of NPs is less than that of linear polymer chains.7 By functionalizing the surface of the NPs with compositions of polymer chains which shares identical chemistry with free polymers, miscibility (good dispersion) of NPs in nanocomposites can be achieved. In some cases, grafting density as well as the ratio between the length of grafted chains and free chains has a profound effect on NP dispersion in melts.13−18 For low grafting density and long grafting chains, free chains can easily penetrate into the grafted layer leading to a “wet” brush where NPs can be well dispersed into the matrix. In contrast, high grafting density and short grafting chains lead to the ejection of free polymers, where the brush becomes “dry” and nanoparticles tend to aggregate.16,17 Another way to tailor dispersion is to enhance polymer−NP interactions.10,19−21 Theories9,19,20 have predicted that there exist three main phases for NPs © 2016 American Chemical Society

Received: July 20, 2016 Revised: October 18, 2016 Published: October 18, 2016 11574

DOI: 10.1021/acs.jpcb.6b07292 J. Phys. Chem. B 2016, 120, 11574−11581

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The Journal of Physical Chemistry B

⎧ ∞ r ≤ rEV ⎪ 12 6 ⎪ ⎛ σ ⎞⎤ ⎪ ⎡⎛ σ ⎞ ULJ(r ) = ⎨ 4ε⎢⎜ ⎟ −⎜ ⎟ ⎥ + ULJ(rc) rEV < r < rc ⎢ ⎝ r − rEV ⎠ ⎥⎦ ⎪ ⎣⎝ r − rEV ⎠ ⎪ ⎪ 0 r > rc ⎩

decrease the conformational degrees of freedom. For example, ring polymer melts exhibit self-similar dynamics, yielding a power-law stress relaxation, instead of the entanglement plateau followed by exponential decay observed in entangled linear chains.24 The rings relax stress much faster than linear polymers and the zero-shear viscosity is found to vary as η0∼ N1.4, which is much weaker than the N3.4 behavior of linear chains.25 Moreover, the shapes of ring chains transit from prolate, crumpled structures to planar, rigid rings as the stiffness increases.26 Meanwhile, excluded volume of ring chains can induce an anisotropic increase of the main axes of the conformations,27 and ring polymer chains interact through non-Gaussian potentials, which is different from linear chains.28−31 Two ideal ring polymer chains experience a repulsive effective interaction between their centers of mass, while the same vanishes between two ideal chains or an ideal chain and a ring.32 The effective interactions exerted on semiflexible ring polymers may lead to interpenetration with increasing concentration,31 whereas flexible ring polymers adopt crumpled globular conformations in the melt state.30,33 Moreover, a novel stack formation with a tube-like structure of quasi-parallel ring is found in semiflexible ring polymer melts.34 Therefore, in this article, we report a new way to achieve the dispersion structure of NPs in nanocomposites, in which linear polymers are replaced by ring polymers in polymer nanocomposites. The dispersion structure of NPs is revealed in the presence of semiflexible nonconcatenated, unknotted ring polymer melts. And the article is organized as follows: in Section 2 the model and simulation details are provided. In Section 3 our results on the aggregation-dispersion transition of NPs in semiflexible ring polymer nanocomposites are given, and in Section 4 the conclusion will be presented.

(3)

The purely repulsive interactions of Weeks−Chandler− Anderson (WCA)39 are given by ε = 1.0kBT, and rc = rEV + 21/6σ with rEV = 4σ for NP/NP interaction and rEV = 2σ for NP/polymer interaction.40 The weak attractive polymer/ polymer interactions (nonbonded) are considered with ε = 0.1kBT, and rc = rEV + 2.5σ with rEV = σ. In our simulations, the total number of polymer monomers is fixed to be 28 800. To generate the initial configurations, we placed six NPs and the polymer chains of stretching conformation in a very large box. The NPT ensemble was used to compress the system of very low density to the desired equilibrium density of ρ0σ3 = 0.63. The systems were simulated in the NPT ensemble for at least 25 polymer chain relaxation times (Rouse times) at pressure P = 0.5−0.6 yielding the desired equilibrium density of ρ0σ3 = 0.63 for different bending energies of polymer chains.21,35,38,41 Then the systems were subsequently equilibrated in the NVT ensemble for 20−120 polymer relaxation times.21,35,38,41 Some discussions about the relaxation time τR for semiflxible ring polymers in nanocomposite melts are given in Supporting Information, see Figure S1. The reduced temperature is T* = 1.0 in units of ε/kB by using a Nose-Hoover thermostat where kB is the Boltzmann constant. Reduced units (ε = 1, σ = 1, m = 1, and τ0 = mσ 2/ε = 1 are chosen to be the units of energy, length, mass, and time, respectively) are used and the time step is τ = 0.001τ0. Periodic boundary conditions were employed during the whole runs, and all simulations were performed by the open source LAMMPS molecular dynamics package.42 Ring polymers are nonconcatenated and unknotted, and all data were sampled far from the polymer chain relaxation time. Meanwhile, all statistical properties are averaged over 50 independent simulation runs. In general, the polymer-mediated potential of mean force (PMF) U(r) between a pair of NPs can be calculated directly from simulations of polymer nanocomposites containing two NPs and given by43−47

2. MODEL In our simulation, a standard bead−spring model is used to model polymer chains,35 and each ring polymer chain consists of N monomers with a bead diameter of σ and a mass of m. All bonded monomers are connected by the well-known finitely extensible nonlinear elastic (FENE) potential:36 UFENE(r ) = −

⎡ ⎛ r ⎞2 ⎤ Kr02 ⎢ ln 1 − ⎜ ⎟ ⎥ , r < r0 2 ⎢⎣ ⎝ r0 ⎠ ⎥⎦

(1)

U (r ) = −kBT ln p0 (r )

Where r is the distance between two neighboring monomers, K = 30,kBT/σ2 is the spring constant, and r0 = 1.5σ.36 The stiffness of a polymer chain is described by the angle bending potential Ubending = Kb(1 + cos θ )

(4)

where p0(r) is the probability of finding NPs separated by distance r during simulations. Since p0(r) is quite small for separations of interest, we used the combining umbrella sampling43 with weighted histogram analysis method44 to calculate p0(r). In the umbrella sampling, the biasing potential is a simple harmonic spring

(2)

Where θ is the angle between two consecutive bonds and the rigidity of polymer chains is controlled by varying the value of Kb. In fact, the bending stiffness is related to the persistence length by lp = Kb/(kBT), which measures the distance over which the orientation of the tangent vectors are correlated.37 NPs are modeled as Lennard-Jones spheres of diameter D, and mass densities of NPs are the same as the polymers. The NP/NP and NP/polymer interactions as well as the nonbonded interactions between all polymer monomers are represented by truncated and shifted Lennard-Jones (LJ) potential of the form21,38−40

Uib(r ) =

1 k(r − R i)2 2

(5)

Here k = 4kBT/σ2. In our simulations, eight umbrella sampling simulations were run with Ri = 5.0σ, 6.0σ, ..., 12.0σ.

3. RESULTS AND DISCUSSION 3.1. Aggregation−Dispersion Transition for NPs in Ring Polymer Nanocomposite Melts. The conformations of ring polymers with different bending stiffness are first investigated and the statistical properties are characterized by the shape factor ⟨δ⟩, the characteristic ratio of mean square radius 11575

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The Journal of Physical Chemistry B of gyration ⟨S2⟩/Nb2 as well as the average curvature ⟨cosϕ⟩. Here the shape factor ⟨δ⟩ is defined as48,49 ⟨δ⟩ = 1 − 3

(L12L 22 + L12L32 + L 22L32) (L12 + L 22 + L32)2

(6)

The “equivalent ellipsoid” of a configuration can be obtained by evaluating the principal components L21>L22>L23 of the squares radius of gyration S2 = L21 + L22 + L23 of individual configurations taken along the principal axes of inertial, and the ⟨δ⟩ varies between 0 (sphere) and 1 (rod). The average curvature ⟨cosϕ⟩ is calculated by50 N

cos ϕ

=

⇀⇀

∑ u i ·u i + 1 i=1

(7)

⎯⇀ ⎯

Here u i is the unit tangent vector of i-th monomer, see the inset figure in Figure 1. For rod-like ring polymers (Kb = 200)

Figure 2. Simulation snapshots of multi-NP immersed in ring and linear polymer melts with different bending energies Kb = 0, 30, and 200. Here ρ0σ3 = 0.63 and D = 5.

aggregation−dispersion transition induced by chain topological constraints is shown clearly in Figure 2. The first column is for ring polymers with N = 30, and the middle one is for ring polymers with N = 60. Meanwhile, the typical conformation snapshots with 30-bond linear polymers are also shown in Figure 2 for comparison. Obviously, the NPs are aggregated in flexible ring polymer nanocomposite melts with two different chain lengths of N = 30 and 60, and there exists a single dominant peak at a contact interparticle separation of r = 5σ for both N = 30 and N = 60 in NP−NP pair correlation function g(r), see Figure 2I and II. Since the peak is located at r = 5σ, which is equal to the nanoparticle diameter, indicating that NPs exhibit direct contact aggregation in flexible ring polymer nanocomposite melts. However, when the stiffness (i.e., bending energy) of ring chains increases to Kb = 30, an aggregation− dispersion transition occurs and a good dispersion structure for NPs in semiflexible ring polymer nanocomposites is observed. Meanwhile, a similar dispersion structure of NPs in nanocomposites can maintain well for rod-like ring chains of Kb = 200. The detailed discussion about the dispersion structure of NPs in semiflexible ring polymer nanocomposites is given in Figure 2I and II because there is no peak in NP−NP pair correlation function g(r) for both Kb = 30 and 200. However, for linear polymer chains, the aggregation structures of NPs in nanocomposite melts remain well even for flexible linear chains (Kb = 0) as well as rod-like linear chains (Kb = 200). Although there are all contact aggregations for NPs in either linear flexible polymer or linear rod-like polymer nanocomposites, NPs are aggregated in a spherical structure for flexible polymer melts and are arranged in a linear-like structure for rod-like polymer melts, see the third column in Figure 2 and NP−NP pair correlation function g(r) in Figure 3III. A transition from isotropic “globular” aggregation to anisotropic “linear” aggregation is observed in linear polymer nanocomposte melts, which is consistent with the previous work.40 The aggregation− dispersion transition for NPs in nanocomposite melts can be determined by the maximum value g(r)max of NP−NP pair correlation functions g(r) for NPs in melts, and the transition occurs almost at the same bending energy of Kb = 6−8 for both N = 30 and 60, see the inset figure in Figure 3II. Meanwhile, some new results about the aggregation−dispersion transition for NPs in melts with different NP diameters of D = 3, 5, and 7,

Figure 1. (I) The shape factor ⟨δ⟩, the characteristic ratio of mean square radius of gyration ⟨S2⟩/Nb2 and (II) the average curvature ⟨cosϕ⟩ ⎯⇀ ⎯ of semiflexible ring chains as a function of Kb for N = 30 and 60. Here u i is the unit tangent vector of i-th monomer, and b is the bond length.

with N = 60, the shape factor ⟨δ⟩ is equal to 0.267, and for flexible ring polymers (Kb = 0) with N = 60, ⟨δ⟩ is given by 0.243, which is in agreement with previous theoretical results of ⟨δ⟩ 0.25 for rod-like ring polymers and 0.2464 for flexible ring polymers.26 Meanwhile, Figure 1II shows that the average curvature ⟨cosϕ⟩ keeps constant for Kb > 50, therefore, we can regard the ring polymers with Kb > 50 as the rod-like polymers and the statistical properties of the ring polymers are unchanged with Kb > 50, see Figures 1I and II. The dispersion structure of NPs in nanocomposite melts is very important in a wide variety of applications, and the 11576

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Figure 4. Entropic depletion potentials U(r) between a pair of NPs in polymer nano-composites with ring polymer chains (I), and linear polymer chains (II). Here N = 30 and D = 5.

of entropic interaction U(r) for NPs in nanocomposites with different ring polymer stiffness can explain clearly the aggregation−dispersion transition for NPs in polymer nanocomposite melts, see Figure 4I. Meanwhile, a comparison of entropic interaction U(r) for NPs in linear polymer nanocomposites is also made, and the results are shown in Figure 4II. For NPs in linear polymer nanocomposites, there exists an attractive interaction between NPs whether in flexible linear polymer melts (Kb = 0) or in rod-like linear polymer melts (Kb = 200), which are in good agreement with theoretic predictions44 as well as previous simulation.40,46 Meanwhile, the contact attraction and the range of entropic interactions between a pair of NPs in linear polymer nanocomposites increase monotonically with the increase of polymer stiffness, which is different from that in ring polymer nanocomposites, see Figure 4. In order to explore the origin of attractive-repulsive transition for entropic interactions U(r) in ring polymer melts, we studied the effective depletion zone around NPs in nanocomposite melts with different chain stiffness using molecular dynamics (MD) simulations. Figure 5 shows the density distribution of polymer chains around one single NP with a reduced density fluctuation map of polymer monomers ρ* (ρ* = ρlocal/ρ0 with ρlocal denoting the local monomer density) in the rxy−z plane for three typical bending rigidities of Kb = 0, 30, and 200. Here the first column is for ring polymer chains, and the second one is for linear polymer chain. Meanwhile, the z-axis is parallel to the nematic direction of the semiflexible linear chains,

Figure 3. NP−NP pair correlation functions g(r) in polymer nanocomposites for ring chains with N = 30 (I) and N = 60 (II), and for linear chains with N = 30 (III). The inset figure shows the maximum value g(r)max of NP−NP pair correlation functions for NPs in melts and D = 5.

and with different ring chain lengths of N = 5 and 120 are obtained and the typical simulation snapshots are given in Supporting Information. The same trend with the bending stiffness is also observed for different NP diameters of D = 3, 5, and 7. To investigate the reason why there exists an aggregation− dispersion transition for NPs in ring polymer nanocomposite melts, the entropic interactions U(r) between a pair of NPs in semiflexible ring polymer nanocomposites with different polymer stiffness at ρ0σ3 = 0.63 are calculated, and the results are shown in Figure 4. For flexible ring polymer nanocomposites (Kb = 0), the entropic interaction between NPs is short-range and attractive, and the contact attraction of entropic interaction reaches the minimum at r = D because the loss of the configurational freedom near NP reaches the maximum. However, for semiflexible ring polymer (Kb = 30) or rod-like ring polymer (Kb = 200) nanocomposites, the entropic interaction U(r) between NPs is repulsive, and the repulsive interactions increase monotonically with the ring polymer stiffness increasing. Therefore, the attractive-repulsive transition

rxy = ± x 2 + y 2 , and the sign of rxy is just used to distinguish between one-half (rxy < 0) and the other (rxy > 0). For both flexible ring and flexible linear polymers (Kb = 0), density fluctuations are isotropic. The area filled with cyan and blue 11577

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Figure 6. NP-polymer pair correlation functions gnp(r) in polymer nanocomposite melts for ring chains with different bending energies Kb = 0, 30, and 200. Here D = 5.

of NPs, the supposed attractive interactions between NPs are anomalously reversed to be repulsive when the bending energy increases. What causes the enrichment of polymer monomers near the NP-polymer interface for large bending energy? The problem is to be discussed in the following section. 3.2. Conformation Transitions of Semiflexible Ring Polymers in the Vicinity of NPs. The ring polymer conformations in the vicinity of NPs are studied and typical snapshots are shown in Figure 7 for flexible ring polymers with Kb = 0 and rod-like ring polymers with Kb = 200. Here back ground polymers are painted in transparent violet, while NPs and their surrounding polymers are strengthened with glossy. Meanwhile, one NP and its surrounding polymers are emphasized in the right columns. For flexible ring chains (Kb = 0), ring polymers are collapsed into random coils and stretched away from NPs in the nanocomposite melts, and NPs form an isotropic aggregation cluster. For semiflexible (or rodlike) ring polymers, on the contrary, NPs are wrapped by ring polymers, which act as a perfect barrier to isolate NP from each other and further give an access to explain the enrichment of polymer monomers near the NP−polymer interface. To demonstrate quantitatively the effects of chain stiffness on polymer enrichment in the vicinity of NPs, we have performed extensive simulations on one single NP immersed in ring polymer melts with various bending energies, and the results are shown in Figures 8 and 9. Here we use two parameters ⟨Np⟩ and ⟨Nm⟩ to describe ring polymer conformations in the vicinity of NPs. We first define an interfacial region between NP and polymers within the range of 0 < r < σ, where r denotes the spatial distance to the surface of NPs, see the inset figure in Figure 8. ⟨Np⟩ is the average number of polymers, and ⟨Nm⟩ is the average number of monomers per polymer whose monomers are located within the interfacial region of 0 < r < σ. In Figure 8, ⟨Np⟩ drops monotonically with the increase of bending energy Kb, while ⟨Nm⟩ increases abruptly for small Kb and then remains unchanged for Kb > 100. This trend is in accordance with typical snapshots observed from Figure 7, which implies that flexible ring polymer chains are distributed along radial direction with a large value of ⟨Np⟩ and semiflexible ring polymer chains are wrapped around the NPs because the value of ⟨Np⟩ decreases slowly with the increase of Kb. Meantime, the distribution of individual monomers of ring polymer chains in the vicinity of NPs is studied in Figure 9. Since there is no end monomer for ring polymer chains, we first define the nearest monomer of ring polymer chains within the interfacial

Figure 5. Reduced density fluctuation map for polymer beads ρ* around one NP in the rxy−z plane for ring chains (left column) and linear chains (right column) with different bending energies Kb = 0, 30, and 200. Here N = 30 and D = 5.

corresponds to the reduced density of 0.4 < ρ* < 0.9, which can be considered as the range of depletion zone due to its excluded volume effect. However, the difference between ring and linear polymers arises along with the increase of polymer stiffness. For linear polymers, an isotropic−nematic transition47 is observed when the chain stiffness increases from Kb = 0 to 200, and the reduced density increases monotonically in the longitudinal direction and oscillates in the lateral direction. For semiflexible ring polymers with Kb = 30, some ring-shaped stripes appear in the density fluctuation map. The yellow filled area of 1.1 < ρ* < 1.3, which reflects the enrichment of ring polymer monomers, alternates with the green marked area of 0.9 < ρ* < 1.1. The polymer-enriched area around NPs, which covers the depletion zone, is then naturally considered as the origin of repulsive interactions between NPs in semiflexible ring polymer melts because other NPs are expelled from the “crowded” environment. Moreover, the “repulsive area” extends for larger bending energy. The enrichment of polymer monomers in the vicinity of NPs for semiflexible ring polymers is also manifested in the NP-polymer pair distribution function gnp(r) shown in Figure 6. Unlike the smooth and steady trend for flexible ring polymer chains, two unexpected peaks are located at r = 3.4 and 4.4 for semiflexible ring polymers with Kb = 30, and r = 4.4 and 7.2 for rod-like ring polymers with Kb = 200, respectively, which fits well with the density fluctuation map in Figure 5. Clearly, by varying chain rigidity of linear polymer chains, one can successfully control the aggregation structure of NPs in linear polymer nanocomposite melts, but the ability to achieve miscibility is hindered. However, for ring polymer chains, it turns out to be an effective way to tailor dispersion for NPs in polymer nanocomposite melts by varying chain rigidity. Owing to the enrichment of semiflexible ring polymers in the vicinity 11578

DOI: 10.1021/acs.jpcb.6b07292 J. Phys. Chem. B 2016, 120, 11574−11581

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Figure 7. Typical snapshots of polymer nanocomposites for flexible ring chains (Kb = 0) and rod-like ring chains (Kb = 200). In the latter snapshots, only one NP and a portion of ring chains are shown for clarity.

the NPs. Of course, the average distance ⟨di⟩ of the middle monomers for flexible polymer chains is less than that for rodlike ring polymer chains because the average size for flexible ring polymers is less than that for rod-like ring polymers. Combining the intuitive conformations in Figure 7 and statistical properties of Figures 8, and 9, one can know clearly that flexible ring polymers in the vicinity of NPs exhibit a radial distribution while semiflexible ring polymers exhibit a wrapped distribution, see the inset figure in Figure 9. The radial distribution for flexible ring polymers is a consequence of the interfacial restrictions on polymer configuration entropy, and contacting NPs will result in great loss in configuration entropy for flexible ring polymers so that flexible ring polymers prefer to be expelled. However, semiflexible ring polymers are in favor of maintaining its original shape of an open ring such that free space in the central part allows NP to interpenetrate, and finally a wrapped distribution of ring polymers around NP is formed and functions as an idealized barrier to hamper NP aggregation. Therefore, the special topological structure of semiflexible ring polymers promotes NP dispersion in the melt states. In order to investigate the effects of the conformations of ring polymers in the vicinity of NPs on the aggregationdispersion transition for NPs in melts in more detail, we calculated the average number of rod-like polymers (Kb = 200) wrapping one single nanoparticle Nwrap for different chain lengths and different NP diameters, and the results are shown in Figure 10I. Meanwhile, the critical chain length for stiff polymers can also be determined by the maximum value g(r)max of NP−NP pair correlation function g(r) for NPs in melts, see Figure 10II. In Figure 10II, one can know that the aggregation− dispersion transitions for NPs in melts occur almost for the same chain length with different NP diameters of D = 3, 5, and 7, and there may exist the intrinsic relationship between Nwrap and the aggregation−dispersion transition for NPs in melts. In fact, the aggregation−dispersion transitions for NPs in melts depend on the bending stiffness (see the inset figure in Figure 3II), ring chain length, and the NP diameter (see Supporting Information) simultaneously. Therefore, it is difficult to determine the critical size ratio of NP size to the average ring diameter for aggregation-dispersion transitions. We try to describe the aggregation-dispersion transitions by the average curvature of ring polymer backbone, and the results are shown in Figure 10III. The aggregation-dispersion transition occurs for the average curvature of ring backbone of ⟨cosϕ⟩ = 0.65−0.70 with different chain lengths and different NP diameters.

Figure 8. Average number of ring chains ⟨Np⟩ and the corresponding average bead number per chain ⟨Nm⟩ within the determined interface as a function of bending energy Kb for ring chains with N = 30. Here D = 5.

Figure 9. Average distance ⟨di⟩ of the i-th monomer to the NP mass center for flexible ring chains (Kb = 0) and rod-like ring chains (Kb = 200). The definition of the first monomer and the schematic illustration of possible distributions are given in the inset figure. Here N = 30, D = 5, and ρ0σ3 = 0.63.

region mentioned above to the NP surface as the first monomer, and we then randomly select a bonded monomer as the second, see the inset figure in Figure 9. The average spatial distance between i-th monomer of ring polymer chains and NP mass center ⟨di⟩ is shown in Figure 9, and the good symmetry for both flexible and semiflexible ring polymers matches the expected topological structure because the second monomer is the same as the 30th monomer for 30-bond ring polymer chain. The average distance ⟨di⟩ from second to sixth monomers for flexible ring polymer chains (Kb = 0) is greater than those for rod-like ring polymer chains, which means that the flexible ring polymer chains are distributed in the radial direction, and the rod-like polymer chains are wrapped around

4. CONCLUSIONS In this article, detailed computational simulations on NPs immersed in ring polymer melts with various bending energies are presented. Unlike linear polymers increasing bending energy enhances NP aggregation, an unexpected transition from NP aggregation in flexible ring polymers to NP dispersion in semiflexible ring polymers is observed. Meanwhile, the effective 11579

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Article

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.6b07292. Mean square displacement of the rings center of mass g3(t) as a function of time t for flexible ring polymer with N = 30 and the relaxation time τR for different chain lengths and different bending energies; simulation snapshots of multi-NP immersed in ring polymer melts with different NP diameters for N = 60; with different chain lengths for D = 5; and a possible conformation for NPs in long semiflexible ring melts (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was financially supported by the National Natural Science Foundation of China (Grant Nos. 21374102, 21674082, 21674096). We are grateful to the reviewers of our manuscript for their detailed and insightful comments and suggestions.

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REFERENCES

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Figure 10. (I)The average number of rod-like polymers wrapping one single nanoparticle Nwrap, and (II) the maximum value of g(r)max for NP−NP pair correlation functions in melts as a function of chain length N for different NP diameters D. (III) The maximum value of g(r)max as a function of the average curvature ⟨cosϕ⟩ of ring backbone for different NP diameters D and different chain lengths N.

potential for NPs in ring polymer melts has transformed drastically from a short-range attractive interaction to a longrange repulsive interaction with increasing the bending energy of ring polymer chains. Moreover, the effective repulsive interactions between NPs in semiflexible ring polymer melts originate from the enrichment of ring polymers in the vicinity of NPs. Further investigation suggests that a conformational transition from a radial distribution for flexible ring polymer chains to a wrapping distribution for semiflexible ring polymer chains in the interfacial region near NPs occurs, and the wrapped semiflexible ring polymers near NPs act as a perfect barrier to avoid NP aggregation and promote NP dispersion. This investigation can provide a new access to tailor good dispersions of NPs in nanocomposites. 11580

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The Journal of Physical Chemistry B

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DOI: 10.1021/acs.jpcb.6b07292 J. Phys. Chem. B 2016, 120, 11574−11581