Open Access Article

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Chao-Yang Li*^{a},
Jian-Hua Huang^{b},
Hong Li^{c} and
Meng-Bo Luo^{d}
^{a}Department of Physics, Hangzhou Normal University, Hangzhou 311121, China. E-mail: cyli@hznu.edu.cn
^{b}Department of Chemistry, Zhejiang Sci-Tech University, Hangzhou 310018, China
^{c}Department of Physics, Wenzhou University, Wenzhou 325035, China
^{d}Department of Physics, Zhejiang University, Hangzhou 310027, China

Received
19th June 2020
, Accepted 20th July 2020

First published on 28th July 2020

The interfacial properties of polymer chains on spherical nanoparticles are investigated using off-lattice Monte Carlo simulations. Results show that the number of adsorbed monomers increases whereas the number of adsorbed polymers decreases with increasing the polymer–nanoparticle interaction strength. The interfacial layer thickness is independent of the nanoparticle size and chain length. The interfacial monomers exhibit layering behaviors with three distinct layers. The mobility of monomers in the innermost layer is strongly dependent on the polymer–nanoparticle interaction strength. The interfacial monomers always keep moving, and no glassy layer is present around the nanoparticle. Finally, our results show that the motion of nanoparticle can weaken the adsorption of polymers but does not change the conformational property of adsorbed polymers.

The influence of NPs on properties of polymers is interesting from the experimental and simulation investigations.^{20–24} However, there is still a lack of consistent conclusion on this specific issue. Small Angle Neutron Scattering (SANS) experiments on poly(dimethylsiloxane) (PDMS) containing trimethylsilyl-treated polysilicate NPs showed a decrease in the dimension of PDMS for σ_{n} ≈ R_{G0} and an expansion of PDMS for σ_{n} < R_{G0}, where σ_{n} is the diameter of NPs and R_{G0} is the radius of gyration of the polymer in dilute solution.^{20} Similarly, SANS experiment observed a 10–20% increase in the radius of gyration R_{G} of deuterated polystyrene (d-PS) when σ_{n} < R_{G0},^{21} and it was confirmed by small angle X-ray scattering (SAXS) experiments.^{3} SAXS found that NPs have no measurable effect on R_{G} of d-PS when σ_{n} > R_{G0}.^{3} These results imply that the polymer chain swelling is directly related to the ratio R_{G0}/σ_{n}. However, R_{G0}/σ_{n} is not the only or crucial parameter for the change of polymer size. Recent molecular dynamics (MD) simulation studies showed that, for σ_{n} < R_{G0} case, the polymers swell if σ_{n} is larger than the monomer size while they contract if σ_{n} is smaller than the monomer size.^{17} Monte Carlo (MC) simulation studies pointed out that, even for σ_{n} < R_{G0} case, polymer chains can expand, shrink, or be unaffected by NPs.^{22,23} Polymer dimensions are highly dependent on the polymer–NP interaction strength and NP–NP distance.^{22,23}

It is also essential to understand the dynamic behaviors of the interfacial polymer, including the adsorption/desorption process, the thermal motion, and the diffusion process. Mohammadreza et al. found that MCM-41 NPs in the reversible addition-fragmentation chain transfer polymerization (RAFT) can reduce the diffusivity of polymer chains and consequently slow down the propagation reactions.^{25} Using dynamic MC simulations, Hao et al. found that the diffusivity of polymers is controlled by NPs.^{26} Vacatello found that the attraction of NPs can slow the diffusion of polymers.^{27,28} And the diffusivity of polymers can be significantly reduced or even to zero if the attraction between polymer and NP is sufficiently strong.^{29–31} Moreover, the polymer dynamics is related to the concentration and distribution of NPs. The normal diffusion of polymers in dilute solution can be changed to the sub-diffusion in the media with stationary NPs.^{32} Although the diffusivity is slowed down by the attracting NPs, it was further pointed out that the polymer shows a normal diffusion in the system with orderly distributed NPs but a sub-diffusion in the system with randomly distributed NPs.^{22,23}

It is generally believed that the change in size and dynamics of polymers are induced by the NP's excluded volume effect and polymer–NP interaction. And the interfacial properties of polymers are important factors to understand the behavior of polymers in PNCs. In this paper, we utilize dynamic MC simulation to unravel the interfacial properties of polymers. The adsorption of polymers, the distribution of monomers, and the mobility of interfacial monomers are investigated. We find that the interfacial region of NP can be separated into three layers based on the distribution of monomers. The mean square displacement of monomers (MSD), the fraction of initial monomers f(t), and the mean probability of monomer movement P_{MM} in different interfacial layers are calculated to describe the dynamic properties of monomers. Our results can provide guidance for understanding the core–shell model and the large gradient of segmental mobility from the experiments.^{33,34}

Each polymer is modeled as a linear polymer chain by using a typical bead-spring model developed by Kremer and Grest.^{39} All the monomers in the polymer are identical. The mass and diameter of monomer are m and σ, respectively. The interaction between non-bonded monomers is defined as a Lennard–Jones (LJ) potential of the form

(1) |

(2) |

The interaction between polymer and NP is modeled by another expanded LJ potential of the form

(3) |

We adopted the off-lattice dynamic MC algorithm and the metropolis algorithm to simulate the random motion of polymer. Dynamic MC algorithm follows the evolution of one element of the statistical ensemble and simulates the time evolution of the system without dealing with the master equation directly.^{41} Thus, dynamical MC algorithm is suited to describe dynamic properties such as monomer mobility. At the beginning of the simulation, N_{p} polymers of length n are grown monomer by monomer according to the self-avoiding regulation in an amplified simulation system of size L_{a} × L_{a} × L_{a} (L_{a} = 24σ) because the polymers can be produced efficiently in an extensive system. Then, we randomly select one monomer and move it a small distance with dx, dy, and dz in x, y, and z directions. All dx, dy, and dz are random values within (−Δ, Δ). Here a small value Δ = 0.1σ is used. The attempted move will be accepted with a probability P = min[1, exp(−ΔE/k_{B}T)], where ΔE is the energy shift due to the move. The time unit used in this paper is the MC step (MCS), which is arbitrarily defined and can be rescaled to the real-time unit by experiment or MD simulation. In one MCS every monomer tries to move 100 steps on average. The NP in our simulation can move according to the same role as for monomers. The mobility μ of NP is defined as the ratio of NP's movement steps to monomer's. We reduce the simulation system size L gradually from the initial L_{a} = 24σ to the desired L = 20σ with a small length step ΔL = 0.05σ for every 10^{5} MCS. For every intermediate L, the coordinates of monomers and NP are multiplied by a factor (L − ΔL)/L accordingly. The small reduction step and long equilibrium time enable our system to reach equilibrium at every L. We find that the simulation results do not change if we further increase the equilibrium time or reduce the reduction step. When the system size reaches L = 20σ, the size of the system remains unchanged. The conformational and dynamic properties reported in this paper are obtained from the high density ρ = 0.85 at L = 20σ.

In the paper, the main variables are the polymer–NP interaction strength ε_{pn}, the polymer–NP distance, and the NP mobility μ. Our simulation results are averaged over 1000 independent runs. The statistical errors of our simulation results are found to be so small and can be negligible.

Fig. 1 presents the dependences of the mean number of adsorbed polymers 〈N_{p_ad}〉 and the mean number of total adsorbed monomers 〈n_{cn}〉 on the polymer–NP interaction strength ε_{pn}. Here 〈 〉 represents an ensemble average over all contact states. We can find that 〈N_{p_ad}〉 decreases whereas 〈n_{cn}〉 increases with the increase in ε_{pn}, which results in an increase in the mean adsorption degree 〈D_{ap}〉 with ε_{pn} as shown in the inset of Fig. 1. The variations of 〈N_{p_ad}〉 and 〈n_{cn}〉 with ε_{pn} indicate that the incremental adsorbed monomers mainly come from the already adsorbed polymers and at the same time some of the adsorbed polymers are pushed away because of the excluded volume of the newly adsorbed monomers. The adsorption of the polymer will reduce the conformational entropy. Therefore, the simultaneous adsorption of too many chains is bad for free energy.

The attractive NP may change the polymer conformation which is usually characterized by the mean square radius of gyration 〈R_{G}^{2}〉. We calculate 〈R_{G}^{2}〉 of all adsorbed polymers. Fig. 2 presents the dependence of 〈R_{G}^{2}〉 on the number of polymer–NP contacts n_{cp} for several ε_{pn}s. 〈R_{G}^{2}〉 increase slightly at first and then decrease quickly with the increase in n_{cp}. When a few monomers are adsorbed on the NP, the adsorbed polymers are stretched, resulting in the initial small increase in 〈R_{G}^{2}〉. The adsorption usually takes place at the end monomers of polymer because such a configuration has larger configuration entropy S than that adsorbed at the middle monomers. A snapshot of adsorbed polymers at ε_{pn} = 3 is presented in the inset of Fig. 2. When more and more monomers are adsorbed on the NP, the size of polymers will become smaller and smaller. These observations are in general agreement with our previous simulation results.^{43}

(4) |

The distribution of monomers can be explained by the attraction of NP and the interaction among monomers. The peak in the L_{1} layer is mainly attributed to the attraction of NP as the NP–monomer distance is very short in this region. Thus, the influence of NP on the L_{1} layer is the most important, and the peak of the L_{1} layer increases significantly with ε_{pn}. Although the adsorbed monomers will pull the polymer near the NP because of the FENE interaction, the excluded volume effect of polymers pushes them away, which results in the peaks in the L_{2} and L_{3} layers. The two peaks are however microscopic because of large volumes of the L_{2} and L_{3} layers relative to that of the L_{1} layer. Also, the two peaks are independent of ε_{pn} because of the weak attraction of NP at the relative long NP–monomer distance. The monomers near the NP are adsorbed compactly on the NP, whereas those far away from the NP are loosely distributed, as shown in the inset of Fig. 2. For the monomers in the L_{4} layer, the NP–monomer distance is larger than the cut-off distance r_{c}, and the influence of NP on monomers can be neglected. Also, at the junction of L_{1} and L_{2} layers, ρ/ρ_{0} exhibits a valley due to the repulsion of monomers in the L_{1} layer. As the number of monomers in the L_{1} layer increases with ε_{pn}, the valley becomes deeper.

The influence of NP size and chain length on the distribution of monomers is also studied. For chain length n = 64, we calculated the distribution of interfacial monomers for different NP sizes. Fig. 4A–C present the dependence of the relative number density of monomers ρ/ρ_{0} on r for σ_{n} = 1, 4, and 6, respectively. For NP size σ_{n} = 5, we calculated the distribution of interfacial monomers for different chain lengths. Fig. 4D presents the dependence of ρ/ρ_{0} on r for n = 8, 48, and 80, respectively. We can see that the interfacial area also can be divided into three layers and the layer width is about 0.8σ. That is to say, the distribution of monomers around NP is independent of the diameter of NP and the chain length of the polymer, which is different from the experiment results. Dielectric spectroscopy found that the interfacial layer thickness increases with the diameter of NP.^{33,34} We conjecture that the different results between simulation and experiment are induced by the difference in concentration of monomers. So we further study the distribution of monomers for ρ_{0} = 0.28, corresponding to a semi-dilute solution.^{35} Fig. 3B shows the dependence of ρ/ρ_{0} on the radial distance r from the NP center. We can also see three interfacial layers where ρ/ρ_{0} is uneven. Compared with the interfacial layers at ρ_{0} = 0.85, the three interfacial layers have different thickness. Interestingly, the L_{3} layer has a very wide range with ρ/ρ_{0} > 1, resulting in a larger size interfacial region. That is to say, the concentration of polymer plays an important role on the interfacial layer thickness, which is consistent with the experimental findings.^{33} The difference in interfacial layer thickness can be explained from the competition between the chain crowding and the NP size. The chain crowding imposes stronger steric hindrances at the NP surfaces and reduces the interfacial layer thickness, whereas the larger NP has the larger volume-to-surface ratio and can increase the interfacial layer thickness. At ρ_{0} = 0.85, the chain crowding dominates and the effect of NP size can be neglected. Therefore, we only present our simulation results for σ_{n} = 5σ and n = 64 in this paper.

〈Δr^{2}〉 = 〈[r→(t) − r→(0)]^{2}〉,
| (5) |

Fig. 5 Mean square displacement of monomers vs. the residence time t for monomers in (A) L_{1}, (B) L_{2}, and (C) L_{3} layers, respectively. NP mobility μ = 0. |

The decrease of MSD with increasing ε_{pn} for monomers in the L_{1} and L_{2} layers indicate that the attraction of NP retards the diffusion of monomers. The number density of monomers ρ in the L_{1} layer increases quickly with ε_{pn}, as shown in Fig. 3A. The dynamics of monomers is slowed down by the crowded environment as well as by the strong attractive effect of NP. The attraction of NP can reduce the dynamics of monomers along the radial direction. Thus the mobility in the L_{1} layer decreases with an increase in ε_{pn}. The monomers in the L_{2} layer may connect with that in the L_{1} layer through FENE interaction. Thus the mobility of monomers in the L_{2} layer is partly slowed down. But the influence of NP on the monomers dies away for small ε_{pn} or large NP–monomer distance d_{nm}.

It is well known that the monomers in the glass state are immobilized. In comparison with the glass state, the monomers in the interfacial regions can still move even at large ε_{pn}. To investigate the move of monomers in interfacial layers, we count the number of monomers, n_{im}, which are initially in the layer at time t = 0. Fig. 6 presents the evolution of the fraction of initial monomers

(6) |

Fig. 6 Dependence of the fraction of initial monomers f(t) on the simulation time t for monomers in the L_{1} (A) and L_{3} (B) layers at different ε_{pn}s. NP mobility μ = 0. |

In our previous works, we have simulated the diffusion of a linear polymer in the sparse environment with periodically distributed NPs.^{22} We found that one polymer can be firmly adsorbed on one or two NPs and stop diffusion if the attraction is strong enough. Compared with the monomers in the sparse environment, polymers in the dense environment can always keep in motion. The reason is that only several monomers of a polymer can be adsorbed on the NP because of the crowded environment, as shown in the inset of Fig. 2. The irregular thermal motion of desorption monomers can help the adsorbed monomers to depart from the NP. So there is no glassy state or glassy layer around the NP in the dense environment.

To quantify the effect of polymer–NP interaction on the mobility of monomers, we calculate the half-life period T_{1/2} of f(t) for the three interfacial layers. T_{1/2} means the time duration during which f(t) decreases from 1 to 0.5. Fig. 7 presents the dependence of T_{1/2} on the polymer–NP interaction strength ε_{pn}. We can see that T_{1/2} increases exponentially with ε_{pn} as T_{1/2} ∼ exp(αε_{pn}). Here α = 0.32, 0.093, and 0.025 for monomers in the L_{1}, L_{2}, and L_{3} layers, respectively. We know that there is a potential barrier between two neighboring layers. The monomer motion from one layer to another must overcome the potential barrier. The potential barrier height increases with the increase in ε_{pn} and the decrease in monomer–NP distance. Thus, the mobility of monomers is sensitive to the attraction of NP in the L_{1} layer, and the value of α in the L_{1} layer is the largest. In brief, the monomers in the interfacial region exhibit a gradient mobility along the radical direction of NP.

During the dynamical MC simulation, the attempted movements of monomers are accepted according to the Metropolis algorithm. The acceptance probability can also reflect the mobility of monomers. We define the mean probability of monomer movement P_{MM} as

(7) |

Here n_{m} is the number of monomers in one layer, n_{mm} is the number of monomers accepted to move in the same layer, and 〈 〉 represent an ensemble average over all MC steps, respectively. Fig. 8 shows the evolutions of P_{MM} in different layers. In the L_{1} layer, P_{MM} is strongly dependent on the polymer–NP interaction strength ε_{pn}. With the increase in ε_{pn}, the attraction of NP becomes more potent, and the attempted movement of monomer needs more energy, resulting in an obvious decrease in P_{MM}. However, P_{MM} roughly remains constant in the L_{2} and L_{3} layers. Moreover, P_{MM} ≈ 0.64 in the L_{3} layer is roughly the same as that in bulk solution. The results further prove that there is a gradient of monomer mobility in the vicinity of NP.

However, the NP of σ_{n} = 5σ is 125 times as heavy as the monomer if both NP and monomer have the same mass density. The approximate NP mobility is about μ = 0.008 based on the conservation of momentum. From Fig. 9, we can see that there is no remarkable difference in adsorption properties between the polymers at μ = 0.008 and μ = 0. Also, we study the distribution of interfacial monomers for different NP mobility μ at ε_{pn} = 3. The results show that the relative number density of monomers ρ/ρ_{0} has the same behaviors as that at μ = 0 shown in Fig. 3A. The conformational properties of adsorbed polymers are also calculated for different NP mobility μ at ε_{pn} = 3. Fig. 10 presents the dependence of 〈R_{G}^{2}〉 on μ. 〈R_{G}^{2}〉 keeps nearly a constant, indicating that the tiny change of 〈D_{ap}〉 caused by the NP mobility has no apparent effect on the conformation of adsorbed polymers and the distribution of interfacial monomers. The dependence of the mobility of interfacial monomers on the NP mobility μ will be investigated in the future.

Fig. 10 Mean square radius of gyration 〈R_{G}^{2}〉 of adsorbed polymers vs. the NP mobility μ. Here ε_{pn} = 3. |

By analyzing the number density of monomers ρ along the radial direction of NP, we find that the interfacial region can be divided into three layers marked as L_{1}, L_{2}, and L_{3}, respectively. The interfacial layer thickness is independent of the NP size and the chain length because of the chain crowding. By analyzing the mean square displacement (MSD), the fraction of initial monomers f(t), and the mean probability of monomer movement P_{MM}, we find that the mobility of monomers increases with the decrease in ε_{pn}. Also, we find that the monomers in interfacial layers always keep moving, and there is no glassy layer around the NP. Finally, we have checked the influence of the mobility of NP on the interfacial properties of polymer chains. Results show that the motion of NP can weaken the adsorption of polymers but does not change the conformational property of adsorbed polymers.

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