Introduction

Polymer nanocomposites, which involve nanoparticles (NPs) dispersed in polymer melts, have received a lot of attention recently because of their potential in fabricating materials with novel mechanical, flame resistance, thermal, electrical, and photonic properties1. In order to fully realize their practical potential, it is important to develop a detailed microscopic understanding of dynamical properties of polymer nanocomposites2,3. As a consequence there have been many investigations to study the dynamics of NPs in polymer melts to optimize properties and to facilitate their processing4,5,6,7,8,9,10. Owing to the fact that there are several length scales involved such as the radius of gyration of the polymer chain, the radius of the nanoparticle, and the correlation length of the polymer, it is rather complicated to investigate the dynamical phenomenon of NPs in polymer melts. The NP diffusion coefficient in the large limit follows the Stokes-Einstein (SE) relation, and the corresponding behavior of small NPs is also described by the SE relationship but with a length-scale-dependent viscosity that is smaller than the macroscopic value11,12. However, the motion of the intermediate sized NPs is faster than SE behavior owing to hoplike motions through the polymer’s entanglement mesh13, and faster diffusion for the intermediate sized NPs is also predicted by a microscopic force-level theory14. Meanwhile, the dynamical behaviors of NPs in polymer melts are also studied experimentally. Mukhopadhyay et al. studied the diffusion coefficient of gold NPs in poly(butyl methacrylate) melts by tracking their motion within a diffraction-limited focus of a laser and found that the gold NPs diffuse 250 times faster than predicted by the SE relation for longer chains9. Archer et al. reported interesting dynamical behaviors of NPs grafted with PEG in PMMA melts and found that NPs exhibit a transition from diffusive to hyperdiffusive when Mw becomes greater than the entanglement molecular weight15. Rubinstein et al. proposed a NP hopping mechanism for NP diffusion, in which NPs must overcome a hopping free energy barrier to move and the entanglement strands slip around the NPs resulting in localization of the NP in a neighbouring cage16. In a word, the dynamical behaviors of NPs in polymer melts appears to be complicated by hopping effects, length-scale-dependent entanglement forces, polymer-NP interactions, NP shape as well as polymer topological constraints.

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 polymers because topological constraints of ring polymer chains decrease the conformational degrees of freedom17,18,19,20,21,22,23,24,25,26,27. 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 chains22. 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 chains23. Moreover, topological interactions of threading can lead to a novel glass transition for the concentrated melts of long flexible ring polymers24. The effective interactions exerted on semiflexible ring polymers may lead to interpenetration with increasing concentration25, whereas flexible ring polymers adopt crumpled globular conformations in the melt state26. Moreover, a novel stack formation with a tube-like structure of quasi-parallel ring is found in semiflexible ring polymer melts27. Therefore, polymer chain topological constraints of ring polymers affects the statistical and dynamical properties of polymers seriously, and in this article, we investigate the dynamical behavior of NPs in semiflexible ring polymer nanocomposite melts. Our aim is to study the effects of polymer topological property and polymer stiffness on the dynamical properties of NPs in polymer nanocomposite melts. Fortunately, a novel glassy dynamics of NPs is revealed in the presence of semiflexible non-concatenated, unknotted ring polymer melts.

Model

In our simulation, a standard bead-spring model is used to model ring polymer chains28, and each ring polymer chain consists of N = 30 monomers with a monomer diameter of σ and a mass of m. To prevent overlap between monomers, a shifted and cut-off Lennard-Jones potential is used for all polymer monomers,

where r is the distance between two monomecrs, and ε = kBT.

All neighbouring monomers are connected by the well-known finitely extensible nonlinear elastic (FENE) potential29:

where r is the distance between two neighbouring monomers. The parameters K and r0 are chosen as K = 30kBT/σ2 and r0 = 1.5σ29.

Angle bending potential between adjacent bonds is used to describe the stiffness of ring polymer chains,

where θ is the angle between two neighbouring bonds and the stiffness of ring polymer chains is controlled by varying Kb.

The NP/NP and NP/polymer are represented by truncated and shifted Lennard-Jones (LJ) potentials of the form30,31,32,33

NPs are modeled as spheres of diameter d = 5σ, which is widely used in studying the statistical and dynamical properties of polymer nanocomposite melts30,34,35,36,37, and the density of NPs is same as the ring polymers. The purely repulsive interactions of Weeks-Chandler-Anderson (WCA)32 are given by ε = 1.0kBT and rc =  with rEV = 4σ for NP/NP interactions and rEV = 2σ for NP/polymer interactions. In order to avoid the competition between the NP/polymer interactions and the topological interactions of semiflexible ring polymers, the purely repulsive interactions for NP/polymer interactions are adopted here. Meanwhile, we only focus on the purely repulsive interactions for NP/polymer interactions here because our aim is to investigate the effects of polymer chain topological constraints of ring polymers on the dynamical properties of NPs in melts. In fact, the purely repulsive interactions for NP/polymer interactions are widely adopted in investigating the statistical properties of polymer nanocomposite melts30,31,32,33,34,37,38.

In our simulation, the total number of NPs and polymer monomers are fixed to be NN = 6 and NP = 28800. The NPs and ring polymers are placed randomly in a very large box, and the NPT ensemble was used to compress the system of very low density to the desired equilibrium density. The systems were simulated in the NPT ensemble for long time yielding the desired equilibrium density for different bending energies of polymer chains30,31,38.Then the systems were equilibrated in the NVT ensemble for very long time. Reduced units ( = 1, = 1, m = 1, ρ−3 = 1, and  = 1 are chosen to be the units of energy, length, mass, monomer density, and time, respectively) are used and the time step is τ = 0.001. The reduced temperature is T* = 1.0 in units of ε/kB by using a Nose-Hoover thermostat. Periodic boundary conditions were applied during the whole process, and NP coordinates were recorded to calculate some related parameters. All simulations were performed by the open source LAMMPS molecular dynamics package39.

Results and Discussion

Figure 1 shows different trajectories of one NP in the projective xy-plane with different bending energies. NPs are caged in the small region for Kb = 40, see Fig. 1(c). The monomer number density C is defined as C = (N * Np)/L3, here L is the box size after the systems have equilibrated in the NVT ensemble for very long time24. However, NPs can diffuse quickly in flexible ring polymer melts (Kb = 0), see Fig. 1(a). Therefore, the bending energy of ring polymers can seriously affect the diffusion of NPs in ring polymer nanocomposite melts. In fact, the motion of NPs in ring polymer nanocomposite melts can be characterized by the mean-square displacement of the center-of-mass, g3(t), which is defined as

Figure 1
figure 1

The trajectories of one NP in the projective xy-plane with different bending energies of Kb = 0(a), 20 (b), and 40 (c) in the interval of 105τ. Here the monomer number density is C = 0.4, and the total number of trajectories is 200.

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where r(t) is the position of a NP at time t, r(0) is its initial position, and the brackets < > represent ensemble averaging over many conformations and all NPs. For the case of NPs in flexible ring melts (Kb = 0) with a monomer density of C = 0.4, see Fig. 2(a), the motion at short times is ballistic, and g3(t) ~ where x is close to 2. At later times, the NPs move diffusively and the slope of g3(t) is close to 1. This shows that the NPs move randomly in flexible ring melts with C = 0.4. Meanwhile, the similar behavior is also observed for NPs in semiflexible ring melts with a low monomer density of C = 0.25, see Fig. 2(b). In fact, for NPs with various bending energies of ring polymer melts at C = 0.4 or with various densities of semiflexible ring melts at Kb = 20, the motion at early times is superdiffusive with x ≈ 2. For NPs in ring melts with Kb ≤ 20 the motion becomes diffusive at long times with g3(t) ~ t. However, for NPs in semiflexible ring melts with large bending energy such as Kb ≥ 30, see Fig. 2(a), the NPs are trapped by their neighbouring semiflexible ring polymers, and a plateau emerges at intermediate times where the motion is subdiffusive with 0.147 <x < 1. This plateau becomes obvious for Kb = 40 and represents the slowing down of NP motion due to the cage effect of semiflexible ring polymers. This behavior is reminiscent of transient caging motion of particles in a uniformly heated granular fluid40, or in a quasi-2D system of bidispersed particles fluidized by a uniform upflow41. A Similar dependence of g3(t) has been also observed in trapping motion of NPs in polymer naocomposites due to emerging entanglement constraints7. Meanwhile, trapping motion of NPs in nanocomposite melts is also observed for the higher monomer density of C = 0.55 with the lower bending energy of Kb = 20, see Fig. 2(b).

Figure 2
figure 2

g3(t) of NPs in ring polymer melts with different bending energies for C = 0.4 (a) and with different monomer densities of ring polymers for Kb = 20 (b).

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Diffusion coefficients are estimated from

Whose average is taken over all NPs and vanishes when all the NPs are caged by the topological interactions of semiflexible ring polymers24. Diffusion coefficients depend on the stiffness of ring polymers seriously, and Fig. 3 shows that the ratio of D(Kb)/D0 decreases dramatically with the increase of stiffness of ring polymers (Kb). Here D(Kb) is the diffusion coefficient of NPs in polymer nanocomposite melts with a stiffness of ring polymers Kb. For all systems with high monomer density there exists a critical frozen stiffness (Kb*), for which all NPs are permanently trapped by the semiflexible ring polymers, i.e., for semiflexible ring polymer chains with Kb > Kb*. NPs exhibit a transition from (at least partially) liquid, or diffusive, behavior to a glassy state in which the NPs are all irreversibly caged, although free to re-arrange their conformations to some extent. Therefore, it is natural to identify Kb* as the value of Kb at which D(Kb) = 0. In practice, we define as “caged” states that display a D(Kb) one hundred times smaller than the diffusion coefficient of the flexible ring polymer case, D0. In fact, this method is also applied to investigate the glass transition of long flexible ring polymer melts24. From Fig. 3 one can notice that the decay of the D(Kb) becomes increasingly steeper as the density increases. Meanwhile, the dynamical properties of NPs in polymer nanocomposite melts also rely on the chain length of semiflexible ring polymer chains, and some detailed discussions are given in Supplementary Information. In fact, the ratio of the polymer size to the NP radius plays an important role in the dynamical process of NPs in polymer nanocomposite melts. Of course, the size of semiflexible ring chains depends on the bending energy Kb. For example, the root-mean-square radius of gyration Rg =  is 2.29 for 30-monomer flexible ring chains (Kb = 0) and 4.52 for 30-monomer semiflexible ring chains (Kb = 40) with a monomer number density of C = 0.4. Therefore, the ratio of the polymer size to the NP radius η = Rg/(d/2) = 2Rg/d = (2 * 2.29)/5 = 0.92 for Kb = 0 and η = 1.81 for Kb = 40, which is consistent with the experimental result5. In the experimental study on dynamics of gold NPs in a polymer melt, the size of gold NP is about 5 nm, and the poly(n-butyl methacrylate) PBMA size is in the range of 2~18 nm, therefore, the ratio η ranges from 0.5 to 3.65, which is close to our simulation results.

Figure 3
figure 3

The ratio of D(Kb)/D0 as a function of Kb for NPs in ring polymer nanocomposite melts with five monomer densities of C = 0.25, 0.32, 0.35, 0.40 and 0.55.

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Incoherent intermediate scattering function is used to study the dynamical process of the system, which is defined as39,42,43,

where is a wavenumber and the brackets <> are averaged over many conformations and all NPs. In fact, F(, t) is the space-Fourier transforms of the van Hove correlation function40, and is widely used in computer simulation of colloid science since it is readily available through light scattering experiments42,43,44,45. Figure 4 shows F(, t) for different chain stiffness Kb with q = /5. For flexible ring polymers, F(, t) decays rapidly as NPs can move diffusively, and NPs are in the liquid phase. For NPs in the glassy phase (Kb  30), F(, t) decays slowly and it is little decorrelation since NPs are trapped by semiflexible ring polymers. In fact, Fig. 4 shows that F(,t) includes a two-step relaxation in the intermediate value of Kb: the fast (early time) β relaxation which is the diffusion inside the “cage” and the α relaxation which corresponds to the time it takes for the NP to diffuse out of the “cage”. When Kb increases, there exists a plateau and the α relaxation occurs at increasing longer timescales. The α relaxation at late times can be described by a Kohlrausch-Williams-Watts (KWW) function45

Figure 4: Time dependence of the intermediate scattering function F(q, t) for different bending energy of ring polymers with a monomer density of C = 0.40.
figure 4

The solid lines are fits to Eqn (8) with different value of β′.

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where τ′ is the relaxation time and β′ is the stretching parameter. The relaxation time τ′ is defined as the time it takes for the values of F(. t) to fall to a level of 1/e, i.e., F(,τ′) = 1/e F(.0)40. The exponential β′ and the fits of this stretched exponential form to the simulation data are shown as solid lines in Fig. 4. The exponential β′ decreases as the bending energy Kb increases. For flexible ring polymer melts, as the exponential β′ is close to 1, F(. t) can be expressed approximately as exp(−t/τ′), and this is the result for classical diffusion of NPs in liquid phase46,47, For semiflexible ring polymer melts, the relaxation time τ′ increases abruptly as Kb increases, and a significant slowing down of the dynamics is observed for large bending energy. Meanwhile, the relaxation time can also be described approximately by the Vogel-Fulcher law48,49, which is always widely used in studying the glass phase of particles,

where A = 719 and  = 74.57 are the fitting parameters. Figure 5 shows that the relaxation times of NPs in nanocomposite melts depend mainly on the stiffness of ring polymers. As NPs go through the transition from the fluid phase to the glassy phase gradually, relaxation time shows no particle feature. However, we make sure that NPs are in the glassy phase for semiflexible ring polymers with large bending energy owing to the caging motion of NPs in semiflexible ring polymer melts

Figure 5
figure 5

Relaxation time τ′ as a function of Kb for a monomer density of semiflexible ring polymers C = 0.4.

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To study the non-Gaussian behavior of the NP diffusion, we compute the following non-Gaussian parameter50,

<….> denotes an ensemble average over all NPs and initial time. This parameter quantifies the deviation from the Gaussian behavior of the probability density function for single-NP diffusion. For flexible ring polymers, the NPs are freely diffusing as the trapping cage isn’t formed by the surrounding ring polymer chains, and α(t) is very small with a typical normal liquid for NPs, see Fig. 6. For semiflexible ring polymers, the motion of each NP is hampered by its ring polymers and becomes subdiffusive, hence, α(t) is nonvanishing, indicating the dynamical heterogeneities. Additionally, we find that the peak height of α(t) increases upon increasing the bending energy Kb. In fact, non-Gaussian dynamics due to cage-escape processes have also been observed in single-particle diffusion in cluster crystals51,52, glasses53,54, and in a uniformly heat granular fluid40.

Figure 6
figure 6

Time-dependent non-Gaussian parameter α for a monomer density of ring polymers C = 0.4 with different bending energies.

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Extending an idea originally proposed for spin glasses55, one can construct a time-dependent ‘order parameter’ that compares the liquid configuration at two different times56,57:

Here, ri in the second equality refers to the position of NP i, and is an “overlap” function which is unity for , where a is associated with the typical vibrational amplitude of the NPs, and we take a = 2.5σ56,57. In fact, Q(t) is the number of “overlaping” NPs when configurations of the system at t = 0 and at a later time t are compared, and Q(t) counts the number of NPs that either remain within a distance a of their original position, or are replaced by another NP in an interval t. Figure 7 shows that for NPs in semiflexible ring polymer melts, Q(t) is characterized by a two-step relaxation, commonly observed in the intermediate scattering function F(q, t) in Fig. 4, as a result of the transient caging of NPs. At short times, NPs oscillate in a region smaller than the overlap radius a, i.e., < Q(t) > /M is close to 1, especially for large bending energy. Here M is the number of NPs. Meanwhile, there exists a short, initial relaxation of < Q(t) > and a longer, secondary relaxation. Therefore, we can conclude that the NPs in the semiflexible ring melts are caged and the liquid-glass phase transition occurs when the bending energy Kb increases. In general, the glass transition is associated with a reduction in the temperature of liquids or by an increase in density of granular materials. Therefore, there exist a critical temperature (i.e.,Tg) or a critical density (C*) of granular material for the classical glass transition. However, in this work, we obverse another type of glass transition for NPs in semiflxible ring melts and the glass transition occurs only with increasing the bending energy of semiflxible ring polymers. Because that this glass transition takes place well above the classical glass transition temperature at which microscopic mobility is lost, we can choose the system temperature to be fixed at T* = 1.0. In fact, a topologically driven glass in long flexible ring polymers has been observed by Michieletto and Turner24, and they found that a concentrated solution of long ring polymers can be driven to a kinetically arrested state by randomly pinning a small fraction of ring polymers. In their model, the system temperature is also fixed, and the glass transition can’t be determined24, which is different from the classical glass transition.

Figure 7
figure 7

Average time-dependent overlap “order parameter” < Q(t) > /M for a monomer density of ring polymers C = 0.4 with different bending energies.

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The behaviors of diffusion coefficient in Fig. 3 show an increasing steeper dependence on the monomer density of ring polymers as the chain stiffness of ring polymers Kb increases. This implies that systems made of semiflexible ring polymer nanocomposite with large bending energy become extremely sensitive to the monomer density of ring polymers. Figure 8 shows that the phase diagram of NPs in semiflexible ring polymer melts depends on the bending energy (Kb) as well as the monomer density (C) of ring polymers. There exists a critical density of polymer chains in nanocomposite melts: which is located at Cmin ≈ 0.33, and is estimated according to the ratio of D(Kb)/D0 in Fig. 3. For C = 0.35, the glass transition occurs at Kb = 40, while for C = 0.32, the ratio of D(Kb)/D0 is 0.0129 at Kb = 100, which is greater than 0.01 yet. Therefore, we estimate that the absolute minimum Cmin is approximately equal to 0.33, which is an extrapolation of the data from Kb = 40. This is the minimal monomer density of ring polymer chains to form the glass phase for NPs in ring polymer nanocomposite melts. In the glass phase, the NPs are caged by the topological interactions of semiflexible ring polymers22 and this is a novel phase transition because this occurs well above the classical glass transition temperature at which microscopic mobility is lost24. In fact, the topological interactions of semiflexible ring polymers play an important role in this non-classical glass transition. The stack structures for semiflexible ring polymers in nanocomposite melts hinder the motion of NPs27, and this is the main reason to form “caging” motion for NPs in semiflexible ring polymer melts.

Figure 8
figure 8

Phase diagram of NPs in semiflexible ring polymer nanocomposite melts relies on monomer density (C) and bending energy (Kb) of semiflexible ring polymers.

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Concluding Remarks

Dynamical behaviors of NPs immersed in semiflexible ring polymer melts with various bending energies are investigated by using molecular dynamics simulations. A novel glass transition for NPs in semiflexible ring polymer melts is observed as the bending energy (Kb) increases. For NPs in flexible ring polymers, the classical dynamics occurs and the motion of NPs is diffusive. However, for NPs in semiflexible ring polymer melts with larger bending energy, the NPs exhibit a transition from liquid behavior to a glassy state in which the NPs are all irreversibly caged. Moreover, the relaxation times of NPs in semiflexible ring polymer melts can be expressed approximately as , which is similar with the Vogel-Fulcher law. Meanwhile, the glass transition for NPs in semiflexible ring nanocomposite melts also relies on the monomer density of semiflexible ring polymers in melts. There exists a critical monomer density Cmin with Cmin ≈ 0.33, which is the minimal density of ring polymer chains to form the glass phase for NPs in ring polymer nanocomposite melts. As our glass transition occurs well above the classical glass transition temperature at which microscopic mobility is lost, our glass transition is different from classical glass transition, and there doesn’t exist the glass transition temperature for our simulation system. This investigation can help understand the nature of the glass transition in polymeric systems.

Additional Information

How to cite this article: Zhou, X. et al. Glassy dynamics of nanoparticles in semiflexible ring polymer nanocomposite melts. Sci. Rep. 7, 44325; doi: 10.1038/srep44325 (2017).

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