Glassy dynamics of nanoparticles in semiflexible ring polymer nanocomposite melts

By employing molecular dynamics simulations, we explore the dynamics of NPs in semiflexible ring polymer nanocomposite melts. A novel glass transition is observed for NPs in semiflexible ring polymer melts as the bending energy (Kb) of ring polymers increases. For NPs in flexible ring polymer melts (Kb = 0), NPs move in the classic diffusive behavior. However, for NPs in semiflexible ring polymer melts with large bending energy, NPs diffuse very slowly and exhibit the glassy state in which the NPs are all irreversibly caged be the neighbouring semiflexible ring polymers. This glass transition occurs well above the classical glass transition temperature at which microscopic mobility is lost, and the topological interactions of semiflexible ring polymers play an important role in this non-classical glass transition. This investigation can help us understand the nature of the glass transition in polymer systems.

for the concentrated melts of long flexible ring polymers 24 . The effective interactions exerted on semiflexible ring polymers may lead to interpenetration with increasing concentration 25 , whereas flexible ring polymers adopt crumpled globular conformations in the melt state 26 . Moreover, a novel stack formation with a tube-like structure of quasi-parallel ring is found in semiflexible ring polymer melts 27 . 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 chains 28 , 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 = k B T. All neighbouring monomers are connected by the well-known finitely extensible nonlinear elastic (FENE) potential 29 : where r is the distance between two neighbouring monomers. The parameters K and r 0 are chosen as K = 30k B T/σ 2 and r 0 = 1.5σ 29 . 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 K b .
The NP/NP and NP/polymer are represented by truncated and shifted Lennard-Jones (LJ) potentials of the form [30][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 melts 30,[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 = 1.0k B T and r c = σ + r 2 EV 1/6 with r EV = 4σ for NP/NP interactions and r EV = 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 melts [30][31][32][33][34]37,38 .
In our simulation, the total number of NPs and polymer monomers are fixed to be N N = 6 and N P = 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 chains 30,31,38 . Then the systems were equilibrated in the NVT ensemble for very long time. Reduced units (ε = 1,σ = 1, m = 1, ρ −3 = 1, and τ σ ε = m / 0 2 = 1 are chosen to be the units of energy, length, mass, monomer density, and time, respectively) are used and the time step is τ = 0.001τ 0 . The reduced temperature is T* = 1.0 in units of ε /k B 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 package 39 . here L is the box size after the systems have equilibrated in the NVT ensemble for very long time 24 . However, NPs can diffuse quickly in flexible ring polymer melts (K b = 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, g 3 (t), which is defined as 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 (K b = 0) with a monomer density of C = 0.4, see Fig. 2(a), the motion at short times is ballistic, and g 3 (t) ~ t x where x is close to 2. At later times, the NPs move diffusively and the slope of g 3 (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 K b = 20, the motion at early times is superdiffusive with x ≈ 2. For NPs in ring melts with K b ≤ 20 the motion becomes diffusive at long times with g 3 (t) ~ t. However, for NPs in semiflexible ring melts with large bending energy such as K b ≥ 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 K b = 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 fluid 40 , or in a quasi-2D system of bidispersed particles fluidized by a uniform upflow 41 . A Similar dependence of g 3 (t) has been also observed in trapping motion of NPs in polymer naocomposites due to emerging entanglement constraints 7 . 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 K b = 20, see Fig. 2

Results and Discussion
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 polymers 24 . Diffusion coefficients depend on the stiffness of ring polymers seriously, and Fig. 3 shows that the ratio of D(K b )/D 0 decreases dramatically with the increase of stiffness of ring polymers (K b ).
Here D(K b ) is the diffusion coefficient of NPs in polymer nanocomposite melts with a stiffness of ring polymers  , for which all NPs are permanently trapped by the semiflexible ring polymers, i.e., for semiflexible ring polymer chains with K b > K b * . 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 K b * as the value of K b at which D(K b ) = 0. In practice, we define as "caged" states that display a D(K b ) one hundred times smaller than the diffusion coefficient of the flexible ring polymer case, D 0 . In fact, this method is also applied to investigate the glass transition of long flexible ring polymer melts 24 . From Fig. 3 one can notice that the decay of the D(K b ) 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 K b . For example, the root-mean-square radius of gyration R g = < > R g 2 is 2.29 for 30-monomer flexible ring chains (K b = 0) and 4.52 for 30-monomer semiflexible ring chains (K b = 40) with a monomer number density of C = 0.4. Therefore, the ratio of the polymer size to the NP radius η = R g /(d/2) = 2R g /d = (2 * 2.29)/5 = 0.92 for K b = 0 and η = 1.81 for K b = 40, which is consistent with the experimental result 5 . 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 3.6 5 , which is close to our simulation results.
Incoherent intermediate scattering function is used to study the dynamical process of the system, which is defined as 39,42,43 where q is a wavenumber and the brackets < > are averaged over many conformations and all NPs. In fact, F(q, t) is the space-Fourier transforms of the van Hove correlation function 40 , and is widely used in computer simulation of colloid science since it is readily available through light scattering experiments [42][43][44][45] . Figure 4 shows F(q, t) for 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(q. t) to fall to a level of 1/e, i.e., F(q,τ ′ ) = 1/e F(q.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 K b increases. For flexible ring polymer melts, as the exponential β ′ is close to 1, F(q. t) can be expressed approximately as exp(− t/τ ′ ), and this is the result for classical diffusion of NPs in liquid phase 46,47 , For semiflexible ring polymer melts, the relaxation time τ ′ increases abruptly as K b 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 law 48,49 , which is always widely used in studying the glass phase of particles, where A = 719 and K b c = 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 To study the non-Gaussian behavior of the NP diffusion, we compute the following non-Gaussian parameter 50 , < … .> 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 K b . In fact, non-Gaussian dynamics due to cage-escape processes have also been observed in single-particle diffusion in cluster crystals 51,52 , glasses 53,54 , and in a uniformly heat granular fluid 40 .
Extending an idea originally proposed for spin glasses 55 , one can construct a time-dependent 'order parameter' that compares the liquid configuration at two different times 56,57 : Here, r i in the second equality refers to the position of NP i, and − w r r ( ) i j is an "overlap" function which is unity for | − | ≤ a r r i j , where a is associated with the typical vibrational amplitude of the NPs, and we take  56,57 . 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 K b 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.,T g ) 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 Turner 24 , 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 determined 24 , which is different from the classical glass transition.
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 K b 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 (K b ) 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 C min ≈ 0.33, and is estimated according to the ratio of D(K b )/D 0 in Fig. 3. For C = 0.35, the glass transition occurs at K b = 40, while for C = 0.32, the ratio of D(K b )/D 0 is 0.0129 at K b = 100, which is greater than 0.01 yet. Therefore, we estimate that the absolute minimum  C min is approximately equal to 0.33, which is an extrapolation of the data from K b = 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 polymers 22 and this is a novel phase transition because this occurs well above the classical glass transition temperature at which microscopic mobility is lost 24 . 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 NPs 27 , and this is the main reason to form "caging" motion for NPs in semiflexible ring polymer melts.

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 (K b ) 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 τ′ − A K K exp( /( )) b c b , 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 C min with C min ≈ 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.