Insight into the Dispersion Mechanism of Polymer-Grafted Nanorods in

Article pubs.acs.org/Macromolecules

Insight into the Dispersion Mechanism of Polymer-Grafted Nanorods in Polymer Nanocomposites: A Molecular Dynamics Simulation Study Jianxiang Shen,*,† Xue Li,‡ Xiaojun Shen,† and Jun Liu*,§ †

College of Materials and Textile Engineering, Jiaxing University, Jiaxing 314001, P. R. China Department of Chemical and Textile Engineering, Jiaxing University Nanhu College, Jiaxing 314001, P. R. China § Key Laboratory of Beijing City on Preparation and Processing of Novel Polymer Materials, Beijing University of Chemical Technology, Beijing 100029, P. R. China ‡

ABSTRACT: Coarse-grained molecular dynamics simulations are performed to investigate the dispersion behavior and the underlying dispersion mechanism of polymer-grafted nanorods (NRs) in a polymer matrix. The influences of grafting density, grafted chain length, and the miscibility between grafted chains and matrix chains are systematically analyzed. The simulation results indicate that the dispersion state of grafted NRs is determined primarily by the excluded volume effect of grafted NRs and the interface between grafted chains and matrix chains. It is found that increasing grafting density and/or grafted chain length induces the conformational transition of grafted chains from mushroom to brush, enlarges the excluded volume of grafted NRs, and enhances the brush/matrix interface in the brush regime, resulting in the improvement of the NR dispersion state. By tuning the interaction strength between grafted chains and matrix chains in a wide range, three general categories of NR spatial organization are found: macroscopic phase separation of the NRs and polymer matrix, homogeneous dispersion of the NRs, and “tele-bridging” of the NRs via the matrix chains. The transition from a “wet” to “dry” brush is observed at the strong brush/matrix interaction. In addition, for the polymer brush grafted on the surface of NRs, the dependence of the brush thickness Tb on either grafting density Σ or grafted chain length Lg is always weaker than that of polymer brush grafted on flat surface or spherical surface, mainly due to the broader range of moving and the less stretching of polymer chains grafted to NRs. The Tb scales with Σ and Lg as Tb ∼ ΣαLgβ, where the ratio β/α is a constant equal to 3, independent of the grafted surface. In general, this work offers a deep insight into the dispersion mechanism of grafted NRs and thus is believed to provide some guidance on the design and preparation of high-performance polymer nanocomposites with tailored dispersion of NRs. separation at various length scales.17,18 Typically, the surface plasmon resonance is strongly dependent on the local orientation of NRs and their inter-rod separation,19,20 while the photovoltaic device prefers a uniform dispersion of NRs to take advantage of the large NR surface area.21 A promising strategy for desirable NR morphologies is to graft polymer chains onto the NRs for tuning the interaction between NRs and polymer matrix. 22−31 For example, Kumacheva et al. prepared Au NRs end-functionalized with polymer chains and observed the NRs were linked up with each other end-to-end to form chains via the association of grafted polymer chains.32,33 Besides, Wang et al. investigated the morphology of PNCs filled with polystyrene-grafted Au NRs and found that the Au NRs became more aggregated with the increase of the aspect ratio Ln/Dn and the ratio of matrix to

1. INTRODUCTION Polymer nanocomposites (PNCs), by incorporating nanoparticles of different shapes into a polymer matrix, offer a variety of substantially enhanced properties relative to pure polymer materials. In particular, PNCs filled with nanorods (NRs) have attracted considerable interest attributed to their unique inherent physical properties originating from the onedimensional shape of the NRs.1−5 For example, Au NRs have been considered as ideal candidates for applications involving optical and electronic devices,6,7 biosensing,8 and drug delivery,9 and CdSe NRs show potential applications in semiconducting devices10,11 and photovoltaic devices12,13 such as solar cells.14,15 Besides, carbon nanotube (CNT) with appropriate size can also be roughly regarded as a NR when considering the dispersion behavior of CNTs in a polymer matrix.16 However, a significant challenge to fully realize the potential of NR-based PNCs is to rationally control the spatial organization of NRs due to the complex interplay of enthalpic and entropic interactions leading to NR aggregation and phase © XXXX American Chemical Society

Received: October 20, 2016 Revised: December 21, 2016

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DOI: 10.1021/acs.macromol.6b02284 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules brush chain lengths Lm/Lg.6 Generally, the NR dispersion and aggregation can be controlled by varying the grafting density (Σ),34,35 the grafted chain length (Lg),36−38 the aspect ratio of NRs (Ln/Dn),6,39 the NR volume fraction (ϕNR),40 the confined spaces,41 and so on. Computer simulations offer a unique approach to investigate the dispersion behavior of grafted NRs by designing proper models and controlling interaction parameters. For example, Khani et al. performed dissipative particle dynamics (DPD) simulations to investigate the role of grafting density and grafted chain length in the phase behavior of NR-based PNCs and thus predicted a phase diagram with respect to the two parameters.34 They also observed three different morphologies, namely, the dispersion, partial aggregation, and aggregation. Besides, Horsch et al. employed Brownian dynamics simulations to study the self-assembly behavior of polymer-tethered NRs and found that the tethered NRs formed four structures such as the honeycomb phase and the tetragonally perforated lamellar phase.42 By using self-consistent field theory (SCFT), Ting et al. found that uniformly grafted NRs in the thermodynamic aggregation state preferred to align side-byside rather than end-to-end.30 Moreover, Monte Carlo (MC) simulations were conducted by Jiang et al. to investigate the assembly of polymer-grafted Au NRs as a function of NR volume fraction ϕNR.40 Their results showed that the NR arrangements evolved from well-dispersion to side-by-side alignment to end-to-end alignment with increasing ϕNR, and the NRs aligned end-to-end became connected to form a percolated network at high ϕNR. By the way, almost all the simulation work reported to date has been in situations where the grafted polymer and the matrix polymer are chemically identical. Although numerous theoretical and experimental researches have been carried out to investigate the dispersion behavior of polymer-grafted NRs, very few research efforts are concentrated on the mechanism behind the dispersion behaviors of grafted NRs. From the microscopic point of view, the dispersion behavior of polymer-grafted NRs depends critically on the extent to which the matrix chains penetrate into the grafted chains, i.e., the wettability of grafted chains by the matrix or the interface between them. Therefore, it is of significant importance to understand the microstructure of polymer chains tethered to the NRs as well as the matrix chains for insight into the dispersion mechanism of grafted NRs. To the best of our knowledge, the only available report on the microstructures of polymer chains tethered to NRs comes from Frischknecht et al.36 by employing density functional theory (DFT) and SCFT. As the polymer chains in their simulations were grafted on a single isolated NR, they could not simultaneously investigate the dispersion behavior of grafted NRs as well as the dispersion mechanism. Actually, considerable research works have been devoted to the microscopic structure of polymer chains tethered to spherical surface or flat surface, providing some information regarding the characteristics of grafted chains on NRs. For instance, the grafted chains on the spherical nanoparticles become more stretched with the increase of grafting density, forming the socalled brushes.43 When the grafted brushes are wetted by matrix chains (referred to as “wet brush”), the favorable brush−matrix interaction will promote the dispersion of spherical nanoparticles.44 However, if the matrix chains are too long or the grafting density is too high, the grafted brush will be unwetted by polymer matrix (referred to as “dry brush”) due to the

massive loss of the conformational entropy regardless of the gain in the mixing entropy, which has a detrimental impact on the dispersion of spherical nanoparticles.44−46 As described above, the exact mechanism behind the dispersion behavior of polymer-grafted NRs remains unclear due to the lack of detailed understanding of the microstructure of polymer chains tethered to the NRs and the matrix chains nearby. Here, by employing coarse-grained molecular dynamics (CGMD) simulations of PNCs filled with polymer-grafted NRs, we aim to elucidate the underlying dispersion mechanism of grafted NRs by systematically investigating the influences of grafting density and grafted chain length on the dispersion behavior of NRs and the chain characteristics. The brush thickness and the interface between grafted chains and matrix chains are also discussed. In addition, taking into account the practical situation that the grafted chains may be not chemically identical to the matrix chains, we examine the effect of the miscibility between grafted chains and matrix chains by adjusting the thermodynamic (Flory−Huggins) interaction parameter between them.

2. MODELS AND SIMULATION METHODS 2.1. Model and Interactions. In our simulation, a coarsegrained model of polymer-grafted nanorods (NRs) embedded in a homopolymer matrix was adopted. The polymer chains are represented by the standard bead−spring model, which is developed by Kremer and Grest.47 Each matrix polymer chain consists of 30 beads with the diameter σ and the mass m equal to unit. Although these chains are rather short, they display the typical Rouse behavior of polymer chains. Each bead would correspond to n = 3−6 covalent bonds along the backbone of a realistic chemical chain when mapping the coarse-grained model to a real polymer. The length of grafted polymer chains Lg is varied from 2 beads to 12 beads. At one end of each grafted chain is the grafting site bonding to the NRs. The total number of simulated polymer monomers is fixed to be 7200. The bonded interaction between adjacent polymer monomers is represented by the stiff finite extensible nonlinear elastic (FENE) potential: UFENE = −0.5kR 0

2

⎡ ⎛ r ⎞2 ⎤ ⎢ ln 1 − ⎜ ⎟ ⎥ ⎢⎣ ⎝ R 0 ⎠ ⎥⎦

(1)

where k = 30(ε/σ2) and R0 = 1.5σ, guaranteeing a certain stiffness of the bonds while avoiding high-frequency modes and chain crossing. The nonbonded interaction between all polymer monomers is described by the truncated and shifted Lennard-Jones (LJ) potential: ⎧ 12 6⎤ ⎡ ⎪ 4εpp⎢⎜⎛ σ ⎟⎞ − ⎜⎛ σ ⎟⎞ ⎥ + C r < 2.5σ ⎝r⎠ ⎦ Upp(r ) = ⎨ ⎣⎝ r ⎠ ⎪ ⎩0 r ≥ 2.5σ

(2)

where the LJ interaction is cut off at the distance r = 2.5σ to include the attractive part and C is a constant to guarantee that the potential energy is continuous everywhere. r is the distance between two interaction sites. The depth of the potential well εpp defines the pair interaction strength. In our simulation, εpp is set to be 1.0 except for the interaction between grafted chains and matrix chains εgm which is varied in section 3.2 to mimic B


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where Ng is the number of grafted chains per NR (see Table 2), Dn is the cross-section diameter of NR, and Ln is the length of NR. It is noted that the aspect ratio of NRs is 11 because of the reduced bond length of 2σ/3 for the NRs. By the way, although the grafted chains are free to rotate around the NR’s centerline and the dispersion state of NRs could be affected, the influence can be reduced due to the strong interaction between different grafted chains, and the obtained simulation results are also consistent with the experimental observation.34,37 If we map the coarse-grained bead−spring model employed in our simulations to real units by setting the diameter of the monomer σ = 1 nm, then a grafted density of Σ = 0.29 (Ng = 10) would correspond to 0.29 chains/nm2. By the way, as there is a oneto-one correspondence between Σ and Ng, we use the parameter Ng to denote the grafting density for clarity in most cases. 2.2. Molecular Dynamics Simulation. The initial configurations are generated by placing all the matrix chains and polymer-grafted NRs in a large simulation box. Then the NPT ensemble with T = 1.0 is adopted to compress the simulation box to make the number density of polymer beads around ρpolymer * = 0.85. The final equilibrium value of the overall number density is approximately ρoverall * = 0.90. Periodic boundary conditions are applied in all three directions to eliminate edge effects. Nosé−Hoover thermostat and barostat are used. The equations of motion are integrated using the velocity-Verlet algorithm with a time step δt = 0.001τ, where τ denotes the Lennard-Jones time unit τ = (mσ2/ε)1/2. The NVT ensemble with T = 1.0 is further performed to equilibrate the structures. After enough equilibrium of the system (at least 5 × 106 MD runs) to ensure that each chain has moved at least 2Rg, we collect the structure and dynamics data for ensemble average. All the MD simulations are executed in the large scale atomic/molecular massively parallel simulator (LAMMPS) molecular dynamics package.49 More detailed simulation information can be found in the literature.44,50,51

different miscibility between grafted polymer and matrix polymer. The rigid NR is represented by the bead−spring model as well. Each NR consists of 16 beads with σ = 1.0 and m = 1.0. A snapshot of a typical NR grafted with polymer chains is given in Figure 1. Each simulation system contains 50 NRs with the

Figure 1. Snapshot of an isolated nanorod (NR) grafted with four polymer chains, where the NR beads are in red, the grafted chains are in blue, and the grafting sites are in green. The grafting density is Ng = 4, and the grafted chain length is Lg = 6.

volume fraction equal to 4.9%. Followed by the literature,48 the bonded interaction between two consecutive beads is modeled by a stiff harmonic potential:

Ubond = k r(r − r0)2

(3) 3

2

where the spring constant is set to kr = 10 (ε/σ ). The equilibrium bond distance r0 is set to 2σ/3, thereby conferring to the NR a smoother surface. Meanwhile, the shape of the NR is governed by the bending potential: Ubend = kθ(θ − θ0)2

(4)

where kθ is equal to 200(ε/rad ) and θ0 is equal to 180° to ensure the rodlike character. The interaction between polymer and NRs, Upn, and between NRs and NRs, Unn, is governed by the same LJ potential function in eq 2 with the cutoff distance equal to rcutoff = 2.5σ. εpn and εnn are both set to be 1.0. For good comparison, the parameters of interaction potential energy between polymer− polymer, polymer−NRs, and nanrods−NRs are listed in Table 1. In our simulation, each NR is uniformly grafted with polymer chains (e.g., four grafted chains shown in Figure 1). The number of grafted chains is given by the grafting density (Σ): 2

Σ=

3. RESULTS AND DISCUSSION In this section, the effect of the grafting density Ng, the grafted chain length Lg, and the interaction between the grafted and matrix chains εgm on the dispersion behavior of NRs will be systematically examined. In order to probe the underlying dispersion mechanism, the chain characteristics of grafted chains and matrix chains, especially the brush thickness and the interface between grafted chains and matrix chains, will be analyzed in detail. 3.1. Effects of the Grafting Density Ng. 3.1.1. Dispersion Behavior of NRs. The grafting density Ng ranges from 2 to 20, as shown in Table 2. We choose to study a typical system where the grafted chain length is Lg = 6 and the matrix chain length is Lm = 30. In most experimental cases, the grafted chains are

Ng πDnLn

(5)

Table 1. Parameters of Interaction Potential Energy between Polymer−Polymer, Polymer−NRs, and NRs−NRs polymer−polymera nonbonded LJ potential εpp = 1.0 σ = 1.0 rcutoff = 2.5σ

bonded FENE potential k = 30(ε/σ2) R0 = 1.5σ

polymer−NRs nonbonded LJ potential εpn = 1.0 σ = 1.0 rcutoff = 2.5σ

NRs−NRs

bondedb FENE potential k = 30(ε/σ2) R0 = 1.5σ

nonbonded LJ potential εnn = 1.0 σ = 1.0 rcutoff = 2.5σ

bonded harmonic potential kr = 103(ε/σ2) r0 = 2/3σ

bending harmonic potential kθ = 200(ε/rad2) θ = 180°

a The interaction parameter between grafted polymer and matrix polymer εgm will be changed from 0.1 to 10.0 in section 3.2. bThe bonded polymer−NRs interaction refers the chemical bonding between the grafting site of the chains and the NR bead.

C


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Macromolecules Table 2. Grafting Density Parameters grafting density (Σ)

0.06

0.12

0.17

0.23

0.29

0.35

0.43

0.58

no. of grafted chains per NR (Ng) total no. of grafted chains in simulation box

2 100

4 200

6 300

8 400

10 500

12 600

15 750

20 1000

Figure 2. Snapshots corresponding to different grafting densities Ng at a constant grafted chain length Lg = 6. Note that the snapshots are shown from the perspective and orthographic aspect for better presentation. The NRs are in red, the grafted chains are in blue, and for clarity the matrix chains are not shown.

Figure 3. (a) Inter-rod radial distribution functions corresponding to different grafting densities. (b) The left axis denotes the average number of neighboring NR beads around each NR with respect to the grafting density, while the right axis denotes the total rod−rod interaction energy. (c) Percentages of isolated NRs and NRs in side-by-side alignment in the simulation system as a function of the grafting density.

D


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Figure 4. (a) Normalized distribution functions of the square radius of gyration Rg2 of grafted polymer chains in different systems with fixed grafted chain length Lg = 6 and varying grafting density Ng. (b) Mean-square radius of gyration ⟨Rg2⟩ of grafted chains vs grafting density.

always much shorter than the matrix chain. Therefore, it is reasonable and practical to consider the case of Lg < Lm. For an intuitive observation of the spatial distribution of NRs, the snapshots are given in Figure 2. At low grafting density, the depletion effect arising from the matrix polymer chains plays the dominant role, preferentially leading to the aggregation of NRs. With the increase of grafting density the gathering of NRs decreases, indicating that the dispersion state of NRs gets better. It should be noted that at low grafting density such as Ng = 2 or Ng = 4 the NRs are self-assembled to form local ordering structure with NRs aligned side-by-side, as shown clearly from the orthographic snapshots. For densely grafted NRs (Ng ≥ 15), the NRs are distributed and long distance disordered. This phenomenon is consistent with the observation from DPD simulations of Maia et al.34 Moreover, we measured the inter-rod radial distribution functions ginter‑rod(r), and the results are shown in Figure 3a. The peak located at approximately r = 1.0σ represents the directly contact aggregation of NRs, and the peak at approximately r = √2σ is equal to the length of the hypotenuse of the isosceles right triangle denotes the indirectly contact aggregation of NRs, as illustrated by the insets. Therefore, these two characteristic peaks together demonstrate the close packing of NRs in a side-by-side arrangement. According to Figure 3a, with the increase of grafting density, the aggregation of NRs assembled side-by-side gradually decreases, as evidenced by the decline in the strength of peaks at r = 1.0σ and r = √2σ. In particular, in the case of high grafting density (Ng = 20), the two characteristic peaks disappear, indicating no direct contact aggregation of NRs. It is noteworthy that the data shown in all the figures have been averaged over the simulation time to get statistically significant values. To quantitatively describe the dispersion state of NRs, the average number of neighboring NR beads around each NR is calculated. The neighboring beads are considered to be in contact with the NR within the distance of 1.25σ, the upper limit of the first peak in the ginter‑rod(r) graph. As shown by Figure 3b, when the grafting density Ng changes from 2 to 20, the number of neighboring NR beads decreases, demonstrating the improvement in the dispersion of NRs. The total rod−rod interaction energy (i.e., the sum of all pair interaction energies between NR beads) is also an agreeable indicator for the dispersion state of NRs. A lower interaction energy value denotes a more favorable aggregation of NRs. Figure 3b shows the interaction energy rises with the increase of grafting density, which means that the dispersion of NRs gets better. These results are consistent with the conclusions obtained from Figure 2.

Furthermore, the percentages of isolated NRs and NRs in side-by-side alignment are also calculated. As shown in Figure 3c, the majority of NRs exist in the side-by-side alignment manner at low grafting density of Ng = 2. With the increase of the grafting density, the NRs in side-by-side alignment are reduced, and meanwhile the isolated NRs account for an increasingly proportion in the polymer matrix. More specifically, the critical phenomenon is observed that the isolated NRs increases slowly below the threshold (designated as “the critical grafting density Nc”) and fast above Nc. We infer that it is the conformational transition of grafted chains from mushroom to brush that gives rise to this abrupt change in the dispersion state of NRs, and we will discuss it below. 3.1.2. Chain Characteristics. Next, we try to explore the underlying physical mechanism of grafted NRs in polymer matrix by investigating the grafted and matrix chain characteristics. Figure 4a shows the normalized probability distribution function of the square radius of gyration Rg2 of grafted chains with fixed chain length and varying grafting density. The probability distribution for Rg2 is the probability for Rg2 between Rg2 and Rg2 + d(Rg2). Interestingly, all the distribution curves look prominently similar, suggesting that the stretching behavior of grafted chains on NRs is statistically independent of the grafting density. Parenthetically, the Gaussian approximation is valid only for Rg2 much shorter than the maximum extension of the chains (for Rg2 ≪ (Rg2)max = 2.92 in our simulation), in accordance with the ideal linear chain model.52 Besides, the mean-square radii of gyration ⟨Rg2⟩ of grafted chains as a function of grafting density are calculated, as shown by Figure 4b. It is confirmed again that the grafting density has minimal effect on the stretching of grafted chains, with the values of ⟨Rg2⟩ oscillating about 1.17. Parenthetically, the ⟨Rg2⟩ of grafted chains is slightly higher than that of free chains, implying that the grafted chains are weakly stretched on account of the end-grafting effect. Actually, the stretching behavior of polymer chains grafted to NRs is distinct from those chains grafted to spherical nanoparticles or flat solid surface, which become more stretched with the increase of grafting density. Reasonably, less curved spherical or flat surface (compared with the surface of rod) constrains the range of grafted chains’ movement, and in the limited territory the grafted chains will be disturbed from their ideal random-walk state and stretch out to avoid each other as well as themselves. And the grafted chains in the more confined space such as in the cylindrical tube are even more stretched than the flat brush.53 However, the polymer chains grafted to NRs have a relatively greater freedom of movement and thus own greater ability for sustaining their original random-coil state with maximum conformational entropy. Also, it can be predicted E


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Figure 5. Schematic of dispersion states of grafted NRs when the grafting density is increased. The red spheres denote the NRs, and green points denote the grafting sites. Grafted and matrix chains are represented by blue lines and yellow lines, respectively.

that the grafted chains to NRs may become stretched at much higher grafting density.43 Despite the fact that the stretching behavior of grafted polymer chains basically maintains unchanged with the increase of grafting density, the overall arrangements of grafted chains distributed on the surface of NRs differ significantly from each other. From the fundamental point of view, bare NRs will aggregate in a side-by-side arrangement, which is intrinsically attributed to the entropic effects.30 When NRs are tethered with sparse polymer chains, these relatively shorter grafted chains prefer to twine around the rigid NRs rather than mix with longer matrix chains owing to the loss of entropy of mixing, as illustrated in Figure 5a. In this situation, the grafted chains are shaped like a mushroom, and the grafted NRs are unwetted by the polymer matrix. With the increase in the quantity of grafted chains, the surface of NRs gradually becomes saturated with these chains, and hence some of the grafted chains tend to adjust the spatial position so as to penetrate into the polymer matrix. Meanwhile, the matrix polymer chains penetrate into and interact with the grafted chains, giving rise to an improvement of the interface between the NRs and the polymer matrix. Consequently, the side-byside aligned structure of NR aggregates gradually disintegrates with the neighbored NRs being dissociated, a comprehensive effect due to the entropy and the enthalpy, as sketched in Figure 5b. Especially higher grafting density enables more favorable NR/matrix interface, resulting in the separation between NRs, as shown in Figure 5c. In this case (Ng > 8 in our simulation), the grafted chains penetrating into the matrix can be regarded as a brush as a whole though they almost still keep the random-coil state individually, and correspondingly the grafted NRs become wetted by the polymer matrix. Understandably, the excluded volume effect between the dense grafted brushes is another fundamental factor contributing to the improvement in the dispersion state of NRs. When the grafting density is high enough, the brush-decorated NRs become distributed uniformly in the polymer matrix, as illustrated in Figure 5d. On the basis of the above analysis, we shall next characterize the microstructure of the grafted chains. Figure 6 shows the average bond orientations of grafted chains along the NR’s axial direction as a function of grafting density. The bond orientations are measured via the second Legendre polynomial ⟨P2⟩: ⟨P2⟩ =

1 (3⟨cos2 θ ⟩ − 1) 2

Figure 6. Average bond orientations of grafted chains along the NR’s axial direction ⟨P2⟩ plotted vs the grafting density Ng.

orientation of the bonds. It can be observed from Figure 6 that the bonds show a preference to be oriented parallel to the NR’s axial direction at low grafting density, demonstrating that sparse grafted chains twine around the NRs, performing mushroom shape. However, with the increase of grafting density, the orientation of the bonds gradually approaches to zero, reflecting the randomly oriented tendency of the grafted chains far away from the grafting surface, performing brush shape. Moreover, Figure 7a shows the monomer density profile as a function of position from the NR diametrical axis of grafted chains that are grafted to the reference NR. As expected, with the increase of grafting density, more grafted chains penetrate into the polymer matrix away from the NR’ surface, and thus the grafted chains adopt a more extended conformation as a whole. By the way, the increase in the monomer density near the NR surface (r − RNR ≤ 1.0σ) at high grafting densities (Ng > 8) results from the fade away of grafted chains belonging to other NRs, an indication of the separation of NRs. In order to validate the conformational transition from mushroom to brush, we then calculate the deviation of simulated conformation from the hypothetical ideal brush using the Alexander−de Gennes approximation54,55 in the monomer density of grafted chains at NR surface ϕ(0), and the results are shown in Figure 7b. Obviously, the actual ϕ(0) is much higher than the ϕ(0) corresponding to the Alexander−de Gennes brush, which is attributed to the grafted chains twining around the NRs (the mushroom conformation). With the increase of grafting density, the deviation between the actual ϕ(0) and the ϕ(0) corresponding to the Alexander−de Gennes brush becomes larger, revealing that more grafted chains are collapsed on the NR surface. However, beyond the critical grafting density Nc, the deviation gradually decreases, mainly

(6)

where θ denotes the angle between the bond vector of grafted chains and the NR’s axial direction. Note that the maximum value ⟨P2⟩ = 1.0 indicates a perfect alignment parallel to the NR’s axial direction, while ⟨P2⟩ = 0.0 means a random F


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Figure 7. (a) Monomer density profiles as a function of position from the NR diametrical axis of grafted chains that are grafted to the reference NR. The curves are shifted by the NR radius. (b) Monomer density of grafted chains at NR surface with respect to the grafting density.

where z = r − RNR is the radial distance from the particle surface. According to the figure, the brush thickness Tb increases monotonically with increasing the grafting density Σ. More explicitly, Tb is approximately proportional to Σ0.12, which is much less dependent on the grafting density than the brush on a flat surface (∼Σ1/3) or the brush on a spherical surface (∼Σ0.2).44,59−61 The reason for this lies in the broader range of moving and the less stretching of the polymer chains grafted to the NRs. By the way, the brush thickness can be a measurement of the excluded volume effect of grafted NRs. Specifically, the excluded volume effect of grafted NRs plays an increasingly prominent role in the dispersion of the NRs. It is conventional to characterize the interface between the grafted NRs and the matrix based on the extent to which the matrix chains penetrate the grafted brush/mushroom. Accordingly, the monomer density profiles of matrix polymer chains around NRs corresponding to different grafting densities are computed. In Figure 9a, the grafting density ranges from Ng = 2 to 8. From the figure we can find that the peak at approximately r = 1.0σ, affected by the extent of matrix segment touching the surface of NRs, gradually falls off with the increase of grafting density, implying that the penetration of the matrix chains becomes increasingly difficult. And also the matrix chains are expelled from the outer layer of grafted mushroom when the grafting density increases, as manifested by the decrease in the local matrix density at r ≥ 1.0σ. Fundamentally, the loss of the interface between the grafted NRs and the matrix is attributed to the accumulation of the grafted chains collapsed on the surface of NRs in mushroom regime, intrinsically driven by entropy. Figure 9b shows the matrix monomer density profiles at higher grafting density with Ng ranging from 8 to 20. According to the figure, the matrix polymer density around the NRs has been raised by increasing grafting density, indicating an improved interface due to the interpenetration between the

because of some grafted chains penetrating into the polymer matrix away from the NR (the brush conformation). Accordingly, this yielding point can be served as the indicator of the conformational transition from mushroom to brush, as predicted. Furthermore, the brush thickness of grafted chains is also examined, as shown in Figure 8. Considering that the densely

Figure 8. Relationship between log(Tb) and log(Σ). The fitted line corresponds to the scaling relation, Tb ∝ Σ0.12.

grafted chains are uniformly distributed on the surface of NRs, the grafted chains thus form a quasi-two-dimensional bottle brush.56 Therefore, following the work of Binder’s group57 and Elliott et al.,58 the brush thickness can be defined as ∞

8 8 ∫0 zϕ(z) dz Tb = ⟨z⟩ = ∞ 3 3 ∫ ϕ(z) dz 0

(7)

Figure 9. Monomer density profiles of matrix polymer around NRs corresponding to different grafting densities ranging from (a) Ng = 2 to 8 and (b) Ng = 8 to 20. G


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Figure 10. Influence of the interaction between the grafted and the matrix chains on the dispersion of polymer grafted NRs: (a) the inter-rod radial distribution functions; (b) the changes of isolated NRs and NRs in side-by-side alignment. Note that the grafting density of the simulated system is fixed at Ng = 10 in the brush regime.

Figure 11. (a) Monomer density profile of grafted chains that are grafted to the reference NR. The inset shows the deviation of simulated conformation and the hypothetical ideal brush in the monomer density of grafted chains at the NR surface. (b) Change of brush thickness Tb as a function of εgm. (c) Mean-square radius of gyration ⟨Rg2⟩ of grafted chains as a function of εgm. (d) Monomer density profiles of matrix polymer around NRs with respect to εgm.

chains εgm, indicating an improvement of the dispersion state accompanied by the diminution of the directly contact aggregation of NRs. However, when εgm > 3.0, the NRs decorated by dense brushes show a tendency to form “telebridging” structure with longer range bridging, which is reflected by the slight increase in ginter‑rod(r) at larger distance (r ≥ 2.0σ). This observation implies that a good dispersion can be obtained at a moderate brush−matrix interaction. Besides, the changes of isolated NRs and NRs in side-by-side alignment as a function of the brush−matrix interaction strength εgm are also given by Figure 10b. According to the figure, when εgm ≤ 3.0, the NRs becomes more dispersed in the matrix with the increase of εgm. Yet as εgm further increases, the dispersion of NRs begins to deteriorate, albeit to a very small degree. In order to probe the underlying mechanism, we characterize the excluded volume effect of grafted NRs and the interface between the grafted brushes and the matrix. Figure 11a shows the effect of brush−matrix interaction on the brush monomer density profile. For εgm ≤ 3.0, the extension of the brushes get more pronounced with increasing εgm, while for εgm > 3.0, the

matrix chains and the grafted chains in brush regime. It is worthy to note that the peak at approximately r = 1.0σ had little change in brush regime (Ng ≥ 8), reflecting that some grafted chains still twine around the NRs, in good agreement with our theoretical analysis above. 3.2. Effects of the Interaction between the Grafted and Matrix Chains εgm. Considering that the grafted chains and the matrix chains are usually not chemically identical in practice, the influences of the miscibility between grafted chains and matrix chains on the dispersion state of NRs are then examined by adjusting the thermodynamic (Flory−Huggins) interaction parameter εgm. In our simulation, the grafting density is set as Ng = 10 in the brush regime, and εgm varies from 0.1 to 10.0, corresponding to the brush−matrix interaction strength from weakly repulsive to intensely attractive. To effectively characterize the dispersion state of the NRs, the inter-rod radial distribution functions ginter‑rod(r) are calculated, as shown in Figure 10a. The peaks located at approximately r = 1.0σ and r = √2σ gradually decrease with the increase of the interaction between the grafted and matrix H


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Figure 12. Average number of neighboring NR beads around each NR and the total rod−rod interaction energy as a function of the interaction strength between grafted chains and matrix chains εgm for three typical systems: (a) Ng = 2, (b) Ng = 10, and (c) Ng = 20.

Figure 13. Effective characterizations of the NR dispersion state influenced by the grafted chain length Lg: (a) the inter-rod radial distribution functions, (b) the average number of neighboring NR beads and the total rod−rod interaction energy, and (c) the percentages of isolated NRs and NRs in side-by-side alignment. Note that the grafting density of the simulated system is fixed at Ng = 10, and the matrix chain length is Lm = 30.

qualitative similarity between them is noticed. This observation reveals that the change of brush thickness with the increase of brush−matrix interaction is determined primarily by the stretching behavior of grafted chains which is induced by the enhanced attraction from the matrix. Furthermore, to understand the dependence of the interface between the grafted brushes and the matrix on the εgm, we then calculate the matrix monomer density around NRs, and the results are shown in Figure 11d. Under the condition of weak or moderate interaction (εgm ≤ 3.0), the brush/matrix interface substantially increases by raising εgm, indicating the wetting of brushes, which is conducive to the dispersion of NRs. However, as the εgm further increases (εgm > 3.0), the interface slightly decreases. Considering that the densely grafted chains adopt

brushes become more compact and less extended. Incidentally, from the inset of Figure 11a showing the deviation of simulated conformation from the hypothetical ideal brush in the monomer density at NR surface, it can be inferred that a few grafted chains still twine around the NRs regardless of the strong attraction from the matrix chains. More quantitatively, the brush thickness is calculated to describe the excluded volume effect of grafted NRs. As shown in Figure 11b, with the increase of εgm, the brush thickness Tb increases to reach a maximum value at εgm = 3.0 and then decreases slightly in the case of εgm > 3.0. Similar results are observed in the change of the mean-square radius of gyration ⟨Rg2⟩ of grafted chains as a function of εgm (shown in Figure 11c). By comparing the variation trend of Tb with that of ⟨Rg2⟩, I


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Figure 14. Influence of grafted chain length Lg on (a) the average bond orientation of the grafted chains along the NR’s axial direction ⟨P2⟩, (b) the monomer density profile of grafted chains that are grafted to the reference NR, (c) the brush thickness Tb, and (d) the matrix monomer density profiles.

the brush conformation, the exclusion of the matrix chains from the brush demonstrates a transition from “wet brush” to “dry brush” at high brush−matrix interaction strength. Actually, the gradual transition from a “wet” to a “dry” brush has been found as the grafting density, the NR radius, and/or the ratio of matrix to brush chain lengths is increased.36 On the basis of the above simulated results, it is concluded that at a low brush−matrix interaction macroscopic phase separation exists between the polymer-grafted NRs and the matrix polymer, and with the increase of the brush−matrix interaction, the NRs will be dispersed in the polymer matrix uniformly, while at high brush−matrix interaction, the grafted NRs tend to form “tele-bridging” structure via the matrix chains where one single matrix chain adsorbs to several brushes belonging to different NRs. Besides, the spatial organization of the NRs is essentially controlled by the excluded volume effect of grafted brushes and the brush/matrix interface. By the way, the three states of the NR organization, i.e., macroscopic phase separation of the NRs and the matrix polymer, homogeneous dispersion of the NRs, and “telebridging” of the NRs via the matrix chains, will not always appear in order by raising the interaction between the grafted chains and the matrix chains. As shown in Figure 12a−c, the average number of neighboring NR beads around each NR and the total rod−rod interaction energy are used to measure the dispersion state of the end-grafted NRs in three typical systems.

When the grafted chains are in the mushroom regime (Ng = 2), the phase separation of the NRs and the matrix polymer governs, and the dispersion state of NRs monotonically improves with the increase of εgm. If the grafted chains are in the “wet brush” regime (Ng = 10), the transition between the three states of the NR organization can be observed as the εgm increases. However, when the grafted chains are nearly in the “dry brush” regime (Ng = 20), the “tele-bridging” structure of the NRs via the matrix chains dominates, and the dispersion state of NRs will gradually deteriorate by increasing the εgm. 3.3. Effects of the Grafted Chain Length Lg. Since the grafted chain length is another deterministic factor that influences the dispersion behavior of the grafted NRs, we shall next change the grafted chain length Lg by fixing the grafting density Ng = 10 and the matrix chain length Lm = 30. For a description of the dispersion state of grafted NRs, we have computed the inter-rod radial distribution function with respect to the grafted chain length Lg, as shown in Figure 13a. Clearly, the dispersion state of the NRs is considerably improved by increasing the Lg, accompanied by the gradual disappearance of macroscopic phase separation of the NRs and the matrix polymer. The significant improvement in the NR dispersion is confirmed by the decrease of the average number of neighboring NR beads and the increase of the total rod−rod interaction energy (Figure 13b). Besides, the percentages of isolated NRs and NRs in side-by-side alignment are also J


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the interface between the grafted chains and the matrix chains in the mushroom regime, the excluded volume effect plays the dominant role in the dispersion state of grafted NRs at relatively high grafting density (Ng = 10), leading to the steady improvement in the dispersion of NRs.

calculated to indicate the dispersion state, and the results are shown in Figure 13c. According to the figure, with the increase of Lg, the parallel alignment of the NRs gradually breaks up, and simultaneously the isolated NRs continuously increase in the polymer matrix. Having known the dependence of the NRs’ spatial organization on the grafted chain length, we now embark on the effect of grafted chain length on the chain characteristics to explore the excluded volume effect of grafted NRs and the interface between grafted chains and matrix chains. Figure 14a shows the average bond orientation of the grafted chains along the NR’s axial direction ⟨P2⟩ with respect to the grafted chain length Lg. As is evident from the figure, the short grafted chains prefer to be oriented parallel to the NR’s axial direction, which is the characteristic of grafted chains in the mushroom regime on the basis of the above analysis. However, with the increase of grafted chain length, the bonds tend to be randomly oriented, corresponding to the characteristic of grafted chains in the brush regime. Hence, a conformational transition of grafted chains from mushroom to brush can be expected with increasing the grafted chain length. Furthermore, the monomer density profiles of grafted chains that are grafted to the reference NR are shown in Figure 14b. It is apparent that the monomer density profile of the longer grafted chains extends farther into the matrix, reflecting that the grafted chains adopt the brush conformation with greater exclude volume effect. Additionally, the excluded volume effect of grafted NRs is evaluated by the brush thickness Tb, as shown in Figure 14c. As the grafted chain length Lg increases, the increase in the Tb is observed, confirming the increase in the excluded volume effect. The quantitative relationship between Tb and Lg is further investigated, and the result shows that T b is approximately proportional to ∼Lg0.42, which is less dependent on the grafted chain length than the brush on a flat surface (∼Lg1.0) or a spherical surface (∼Lg0.6).61,62 Indeed, the difference in the range of moving for the grafted polymer chains makes the difference in the exponent of Lg. By the way, by combining the brush thickness Tb with both grafting density Σ and grafted chain length Lg, the general expression is given by Tb = A ΣαLg β

4. CONCLUSION In this work, we have systematically examined the effects of the grafting density, the grafted chain length, and the interaction strength between grafted chains and matrix chains on the dispersion behavior of NRs grafted with polymer chains, using coarse-grained molecular dynamics simulation. The underlying dispersion mechanism is also investigated through the comprehensive study of the chain characteristic. The results indicate that the dispersion state of grafted NRs is determined primarily by the excluded volume effect and the interface between the grafted chains and the matrix chains. Specifically, increasing grafting density and grafted chain length enlarges the excluded volume of grafted NRs and enhances the brush/matrix interface in the brush regime, resulting in the improvement of the NR dispersion state. By tuning the interaction between grafted chains and matrix chains in a wide range, we have observed three different states of NR spatial organization, i.e., macroscopic phase separation of the NRs and the matrix polymer, homogeneous dispersion of the NRs, and “telebridging” of the NRs via the matrix chains. Therefore, it is unnecessary to excessively increase the compatibility between the grafted chains with the matrix chains in experiments. Besides, the conformational transition of grafted chains from mushroom to brush is observed with the increase of grafting density and grafted chain length, and the transition from a “wet” to a “dry” brush is observed at strong brush/matrix interaction. For the polymer brush tethered to the surface of NRs, the dependence of the brush thickness Tb on either the grafting density Σ or the grafted chain length Lg is always weaker than that of polymer brush grafted on flat surface or spherical surface, which is mainly due to the broader range of moving and the less stretching of the polymer chains grafted to the NRs. The brush thickness generally obeys the mathematical expression of Tb ∼ ΣαLgβ, where the ratio β/α is a constant independent of the grafted surface. In general, this simulation study not only sheds light on the dispersion mechanism of polymer-grafted NRs in the polymer matrix but also provides guidelines for the design of polymer nanocomposites with tailored structures.

(8)

where A is a constant determined by multiple factors such as the curvature of grafted surface and the interaction between grafted chains and matrix chains. Interestingly, for the polymer chains tethered by one end to the rod surface, the spherical surface, or the flat surface, the relationship between α and β always follows the scaling relation of β/α = 3, and thus the ratio β/α can be considered as is a constant irrelevant to the grafted surface. Consequently, it is reasonable to speculate that the ratio β/α is probably decided only by the fractal dimension of the tethered polymer chains, which can be validated by theoretical calculations using the Alexander−de Gennes theory with the Flory approximation.61 Likewise, the monomer density profiles of the polymer matrix are examined to figure out the influence of grafted chain length Lg on the interface between the grafted chains and the matrix chains. The conformational transition of grafted chains from mushroom to brush is validated by Figure 14d. Specifically, the interface is reduced with the increase of Lg in the mushroom regime (Lg ≤ 6), while the interface improves in the brush regime (Lg > 6). Remarkably, despite the decrease in

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AUTHOR INFORMATION

Corresponding Authors

*E-mail [email protected]; Ph +086-753-8364 4878 (J.S.). *E-mail [email protected]; Ph +086-10-6445 5618 (J.L.). ORCID

Jianxiang Shen: 0000-0003-1991-8180 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported by National Natural Science Foundation of China (51503081, 11502096), Natural Science Foundation of Zhejiang Province, China (LQ15B040002), General Scientific Research Project of Department of Education of Zhejiang Province (Y201533251), the Science K


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and Technology Plan Projects of Jiaxing City, Zhejiang Province (2015AY11015), and Shanghai Supercomputer Center of China.

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