polymer 
 nanocomposites that integrate nanoparticles ( nps ) into 
 polymer melts often possess superior mechanical , thermal , optical , 
 or electrical properties in comparison with pure polymeric materials . fabrication and processing of polymer nanocomposites 
 require a good understanding of their viscoelastic behavior . central to the viscoelasticity of polymer nanocomposites is the 
 coupling between the motion of nps and the relaxation dynamics of 
 matrix chains . the dynamical coupling between nps and linear 
 polymers depends 
 on the particle diameter d compared to the polymer 
 entanglement mesh size a , as shown in experiments , simulations , and 
 theoretical studies . for d < a , the motion of nps 
 is coupled to the unentangled dynamical modes of local chain segments 
 up to sizes d . in contrast , nps with d > a are confined by the entanglement 
 mesh for time scales greater than the relaxation time e of an entanglement strand , as illustrated in figure 1a . brochard - wyart and de gennes argued that nps with d > a have to wait for the complete relaxation of surrounding 
 entangled polymers to diffuse . however , cai et al . recently proposed a hopping diffusion mechanism for nps 
 moderately larger than a to overcome the confinement 
 by the entanglement network . a qualitatively different activated hopping 
 mechanism for nps to overcome confinement is described in a nonlinear 
 langevin equation theory for np motion in cross - linked networks and 
 entangled melts of polymers . microscopic 
 force - level theories without considering the mechanism 
 of np hopping also predict a partial coupling between nps and the 
 entanglement network for d moderately larger than a. schematic sketch of a np ( light orange sphere ) in ( a ) 
 entangled 
 linear polymers ( red lines ) and ( b ) entangled nonconcatenated ring 
 polymers ( green lines ) . the discrepancy between the dynamics of nps smaller and larger 
 than the entanglement mesh size in linear polymer melts reflects the 
 impact of entanglements on np motion . entanglements also play an important 
 role in the viscoelasticity of high molecular weight polymers with 
 nonlinear architectures , such as ring , star , and branched polymers . one would anticipate intriguing dynamical couplings between nps 
 and entangled polymers with nonlinear architectures . we study the 
 motion of nps in entangled nonconcatenated ring polymers as illustrated 
 in figure 1b . ring 
 polymers are particularly interesting as their unique topology gives 
 rise to distinctive conformational and dynamic properties . entangled nonconcatenated ring polymers adopt compact conformations rather than the almost gaussian random - walk conformations of linear 
 polymers . the entanglements in ring polymers do not create long - lived 
 tubes that confine polymer motion as in entangled linear polymers . the entangled dynamics of ring polymers is self - similar , and there 
 is a power - law time dependence of the stress relaxation function without 
 the rubbery plateau characteristic of entangled linear polymers . nonconcatenated entangled ring polymers are 
 good models of the chromatin packed in cell nucleus and of the strands of deswollen gel networks . molecular dynamics 
 ( md ) simulations of nps in melts of entangled 
 nonconcatenated ring polymers and entangled linear polymers are performed . the models of polymers and nps are similar to those in previous simulations . the bead all beads of size  and mass m interact via 
 the lennard - jones ( lj ) potential with interaction strength  , 
 cutoff distance rc = 2.5 , and characteristic 
 time . neighboring beads in a polymer are connected 
 additionally by the finitely extensible nonlinear elastic ( fene ) potential . a bond bending potential v = k(1 + cos  ) is applied with k = 1.5. nps are modeled as smooth 
 spheres that interact with each other through a purely repulsive potential . the mass density of a np is  = m/. interactions between nps and polymers are weakly attractive , 
 so nps are well dispersed in polymers for the linear polymers , the number of monomers per entanglement 
 strand ne  28 , the number 
 of monomers per chain n = 800  30ne , the average spacing between entanglements a  5 , and the 
 entanglement time e the root - mean - square end - to - end distance 
 of the linear polymers is re  
 41.2. the root - mean - square spanning distance of ring polymers 
 between pairs of beads separated by n/2 beads is r  15.3 and 7.0 for n = 800 and 100 , respectively . the samples of nps in linear 
 polymers were built using the same 
 method in previous simulations by kalathi et al . the 
 samples of nps in ring polymers were made by inserting nps into well - equilibrated 
 melts of nonconcatenated ring polymers from previous simulations by 
 halverson et al . the insertion of nps into ring 
 polymers started with creating well - separated small voids with volumes 
  in a ring polymer melt . then the volumes 
 of the voids were increased until each void was large enough to accommodate 
 a np . during the insertion , one ring of each concatenated pair was removed from the simulation 
 box to ensure nonconcatenation . it is sufficiently 
 low for np motion to be barely affected by the interactions between 
 nps , while the corresponding number of nps nnp is sufficiently large for good statistics of np motion . nnp ranges from 650 for d = 
 3 to 5 for d = 15. we also simulated 
 monomers with d = 1 dispersed in polymer melts 
 at volume fraction 10% . subsequent simulations were run at constant 
 volume v for up to 10 depending 
 on d. the mean - square 
 displacements ( msds ) of nps in entangled ring and linear polymer melts 
 are compared in figure 2a . for nps in linear chains , the smallest nps with d = 3 < a move subdiffusively with the 
 minimum log  log slope of msd min  
 0.7 . e  
 4000 ( see dashed black line in figure 2a ) . larger nps with d  a  5 also move subdiffusively after the ballistic 
 regime , but the crossover to fickian diffusion occurs at t > e ( see dashed color lines 
 in figure 2a ) . min decreases from 0.58 for d = 5 
 to 
 0.14 for d = 15 , as shown in the inset 
 of figure 2a . ( a ) mean - square 
 displacements r(t) of nps with sizes d = 315 
 in entangled melts of nonconcatenated 
 ring polymers ( solid lines ) and linear polymers ( dashed lines ) with n = 800 . thick red line indicates the log  log slope 
  = 1 for fickian diffusion . thick blue line indicates the expected 
   0.48 for complete dynamical coupling between nps 
 and rings . the inset shows min vs d for np - linear ( filled circles ) and np - ring ( filled squares ) systems 
 and for np - linear systems ( open circles ) in previous simulations with smaller n / ne  9 and larger entanglement mesh size a  7. ( b ) ratio of the msds of nps in nonconcatenated 
 rings r(t)np - ring to the corresponding msds in linear 
 chains r(t)np - linear . 4000. the insets illustrate the trajectories of a 
 np with d = 10 in entangled rings ( dark yellow 
 line ) and linear chains ( violet line ) for a time period of 10  25e . for nps in linear chains , the subdiffusion of nps with d < a and the subdiffusion of nps with d > a for t < e arise from the coupling between np motion and the unentangled 
 dynamics of chain segments . the theoretical prediction is  
 = 1/2 based on the rouse model of unentangled polymer dynamics . this predicted asymptotic scaling regime is not well developed in 
 the present simulations . for d < a ,  in the subdiffusive regime is affected by the crossover 
 to the fickian regime . for d > a ,  in the subdiffusive regime for t < 
 e is affected by the crossover to the regime for t > e . the motion of nps with d > a in 
 linear chains is further suppressed for t > e because of confinement by the entanglement network . while 
 the theory by brochard - wyart and de gennes predicts that min decreases to 0 sharply at d  a , min in 
 the simulations progressively decreases with d over 
 a broad range up to d  3a ( see the inset of figure 2a ) . for d moderately larger than a , nps can overcome the confinement of the entanglement 
 network through the hopping diffusion mechanism . in hopping diffusion theory , the entanglement network is 
 described as many overlapping yet independent elementary networks , 
 the centers of which are separated by an average distance . 
 each hop of a np confined by elementary networks corresponds to the 
 displacement from one network cage to a neighboring cage over a distance 
 . a successful hop needs to overcome a free energy 
 barrier associated with the fluctuation of one network strand from 
 the average size a to d. the barrier 
 height is proportional to d / a , and 
 therefore as d increases the probability of hopping 
 decreases as exp(d / a ) . a 
 qualitatively different picture of activated np hopping is presented 
 in a microscopic force - level theory . it 
 predicts that hopping distance of a large np in an entangled polymer 
 melt increases with d , and the hopping barrier height 
 is asymptotically proportional to the volume of a np . as hopping becomes less likely with increasing d , the confinement of the entanglement network is enhanced , 
 and min is reduced . for d a , hopping diffusion is highly unlikely , and nps have to 
 wait for 
 the relaxation of the entanglement network to diffuse . in this case , 
 min would be independent of d. 
 however , min is expected to be nonvanishing for n / ne  30 because the 
 entanglement network can partially relax through longitudinal rouse 
 modes in a confining tube , tube length fluctuation , and constraint 
 release . hopping events are not 
 directly observable in the trajectories 
 of nps for a < d < a/ , as they 
 are masked by the thermal fluctuations of nps confined by individual 
 cages . the msd due to thermal fluctuation of a np confined in an entanglement 
 cage is rfluct a/(d ) . for a < d < a/ , rfluct is larger than  resulting from one hopping event . for d a/ , rfluct <  , but 
 the propability of hopping exp(d / a ) is lower . the contribution of hopping diffusion 
 to the msd is manifested 
 in the dependence of min on d. 
 kalathi et al . have simulated nps in 
 linear polymers with k = 0.75 , ne  45 , n = 400  
 9ne , and a  7. 
 the dependence of min on d for 
 their simulations is shown in the inset of figure 2a by the open circles . compared to the previous 
 simulations , the present simulations with n / ne  30 and a  
 5 enable the exploration of hopping diffusion over a broader 
 range of d / a , as the largest d / a is increased from 2 to 3 . higher n / ne is favorable 
 because the reptation time (n / ne ) is longer , and thus the contribution 
 to np mobility from hopping diffusion is more dominant over the competing 
 contribution from chain reptation . min at the largest d / a  
 0.14 in the present simulations is reduced with respect to 0.3 
 in the previous simulations . to our knowledge , this is the first time 
 such a strongly suppressed motion of nps in entangled linear polymer 
 melts is observed in simulations . nps in entangled 
 nonconcatenated ring polymers differ from their counterparts in linear 
 chains . for d = 3 , np motion in a melt of 
 rings is comparable to but slightly faster than in a melt of linear 
 chains . the ratio of the msds in rings and linear chains is 1.4 
 in the fickian regime ( see figure 2b ) . more remarkable differences between msds in entangled 
 ring and linear polymers are observed for d a  5. min for nps in rings 
 decreases slowly with d and is 0.6 for d = 15 ( see the inset of figure 2a ) . unlike in entangled linear chains , the 
 msds are not strongly suppressed in entangled rings even for the largest d. the ratio of the msds in rings and linear chains is already 
 2 at t  e and increases 
 with t for t > e as shown in figure 2b . the trajectories of a np with d = 10 > a during a time period 10  25e in entangled rings and linear chains are illustrated in the 
 inset of figure 2b . the absence of strongly suppressed np motion in entangled rings demonstrates 
 that there are no long - lived confining tubes in entangled nonconcatenated 
 rings at least for n / ne up to 30 , consistent with experiments and simulations of entangled ring melts . to understand the subdiffusive motion of nps in ring polymers , 
 we developed a scaling model for the dynamical coupling between nps 
 and entangled rings . entangled ring polymers relax in a self - similar 
 manner for t > e through relaxations 
 of progressively larger sections until the entire rings of size r are relaxed . the motion of 
 nps with size a < d < r is coupled to the relaxations of ring sections with sizes 
 up to d. this coupling gives rise to the subdiffusive 
 motion of nps for e < t < s , where s is the relaxation 
 time of a ring section with size d . for e < t < 
 s , a ring section that has just relaxed contains g monomers . the scaling ansatz is that the effective viscosity 
 eff experienced by a np at t is 
 the same as the melt viscosity of ring polymers with n = g. annealed lattice tree model and fractal 
 loopy globule ( flg ) model have been proposed 
 to describe the dynamics of multiple entangled nonconcatenated ring 
 polymers . we use the flg model that self - consistently accounts for 
 the complete tube dilation with an appropriate renormalization of 
 monomeric friction at different time scales . in the flg model , the conformation of a ring polymer the loops at different length scales 
 overlap with each other in a self - similar manner such that the overlap 
 parameter is the same at all scales and equal to the kavassalis the constant overlap 
 parameter at the entanglement threshold okn for length scales above the entanglement spacing has been confirmed 
 in the simulations by rosa and everaers . during the relaxation of ring polymers , topological constraints 
 associated with rearranged smaller loops affect the dynamics of larger 
 loops through increasing the effective friction coefficient . nevertheless , 
 these topological constraints do not contribute to the tubes confining 
 the larger loops and thus are dynamically released . the tube diameter 
 as the average spacing between dynamically relevant topological constraints 
 increases with time and is comparable to the size of a characteristic 
 loop that has just been rearranged . this picture of complete tube 
 dilation self - consistently describes the dynamics of entangled rings 
 without long - lived tubes . the flg model predicts that the relaxation 
 time of a ring section containing g monomers scales 
 as relax(g )  e(g / ne ) . note that the exponent 7/3 is between 2 for rouse dynamics of unentangled 
 polymers and 3 for reptation dynamics of entangled linear polymers . according to the flg model , the number of monomers in a relaxed 
 ring section increases with time as g  t. the terminal relaxation time for a ring 
 with n monomers is relax  e(n / ne ) . the stress relaxation 
 modulus g(relax )  kbt / vn at t  relax is on the order of thermal 
 energy kbt per volume vn of a ring polymer , where v is the volume 
 of a monomer . as a result , the melt viscosity of ring polymers  
  relaxg(relax )  n / n  n. using g  t and   n , we obtain eff  g  t for e < t < s . the effective 
 diffusion coefficient deff  kbt/(effd )  t , and thus r(t)  defft  t for e < t < s . the resulting scaling exponent for the power - law subdiffusion 
 of nps in entangled rings is  = 3/7  0.43 . recently , 
 grosberg et al . have theoretically studied 
 the scale - dependent viscosity in polymer fluids . for the flg model 
 of nonconcatenated ring polymer melts , they obtained ( ) 
   for 1/relax <  < 1/e . since  
  1/t , this agrees with our 
 result eff  t. the msd subdiffusive exponent  is identical to the 
 exponent 
  for the power - law time dependence of the stress relaxation 
 modulus associated with the self - similar dynamics of entangled rings . 
 because of the finite polymer size , the relaxation time of ring polymers 
 is not well separated from e even for n / ne  30 , and thus the asymptotic 
 entangled dynamics is not well developed . previous analysis has shown that   0.48 for n / ne  30 , and the asymptotic 
 theoretical value 3/7  0.43 could only be observed for n / ne > 100 . for nps in rings 
 with n / ne  30 , 
 it is expected that     0.48 for sufficiently 
 large nps that can be completely coupled to all the relaxation modes 
 of rings . simulation results of min for nps 
 in ring polymers 
 ( see the inset of figure 2a ) are all above the expected value 0.48 ( as indicated by 
 the thick blue line in figure 2a ) for n / ne  
 30 . this is attributed to the finite sizes of nps , which are not large 
 enough to allow the complete dynamical coupling between nps and relaxation 
 modes of rings . the effects of d on the dynamical 
 coupling between nps and rings can be quantified by the dependence 
 of min on d. min decreases as d increases , indicating that the degree 
 of the dynamical coupling increases with d. extrapolation 
 of the d dependence of min yields 
 that the expected value 0.48 for complete coupling would occur for d > dc  45. dc is about 2.6 times the root - mean - square 
 average spanning distance r  15.3 
 between pairs of monomers separated by n/2 monomers 
 ( see figure s1 ) . the exact value of dc / r may be related to the shape 
 of nps and the boundary conditions between nps and polymers . future 
 work is needed to elucidate the factors affecting dc / r . as shown 
 in figure 2a , the nps 
 have reached the fickian diffusive regime except for the largest nps 
 in linear chains . results for diffusion coefficient , computed as d = limtr/(6 t ) , as a function 
 of d are shown in figure 3 by the solid symbols . for nps with d > a in linear polymers , we can estimate 
 the upper bounds of d ( indicated by the arrows in figure 3 ) . as in previous 
 simulations , for the sake of consistency , 
 we do not correct d for the long - range hydrodynamic 
 effects associated with the finite sample sizes . in the limit of very large d , it is expected that 
 np mobility is completely coupled to the melt viscosity  of 
 polymers , and thus d = kbt/(2d ) assuming the 
 stokes the slip lengths for nps in the simulations are anticipated 
 to be much larger than d , resulting in the slip np - polymer 
 boundaries . previous simulation results of nps in linear polymer melts have also been analyzed using the 
 slip boundary condition . for d  40 , 
 the results of d calculated using the stokes results for  in simulations are 
 presented in section 2 of the supporting information . simulation data for the d dependence of d can be extrapolated to match the stokes diffusion coefficients d as functions of diameter d for monomers 
 ( d = 1 ) and nps ( d  
 3 ) in ring polymers with n = 800 ( blue squares ) 
 and n = 100 ( green triangles ) 
 and in linear polymers with n = 800 ( red circles ) . < a , the decrease 
 of d is almost the same for np - ring and np - linear 
 systems and 
 exhibits d  d scaling ( see figure 3 ) , which agrees with the theoretical prediction for the coupling between np motion and unentangled rouse 
 dynamics . however , for d > a , 
 the 
 decrease is much steeper for nps in linear chains as the nps are confined 
 by the entanglement network before they are able to freely diffuse . 
 for d = 15 , even the estimated upper bound 
 of d in linear chains is smaller than d in rings with the same n by more than 2 orders 
 of magnitude . recently , nahali and rosa have compared the motion of nps in concentrated solutions of semiflexible 
 linear polymers and ring polymers . they observed that d for d  a is about 3 times 
 larger in ring polymers than in linear polymers . in our simulations , 
 the ratio of msds in ring and linear polymers is also  3 at d  a. however , our simulations 
 go further up to d  3 and show much stronger 
 effects of polymer architecture on d. there 
 is a crossover of the d dependence of d from d  d to d  d for nps with d > a in linear 
 polymers . the crossover is broad and extends at least 
 up to d  3a . according to 
 cai et al . , hopping diffusion of nps 
 with d moderately larger than a results 
 in d  exp(d / a ) that broadens the crossover to the stokes the results of d for np - linear systems are 
 fitted by a crossover function d = c1(/)(/d ) exp(c2d / a ) + kbt/(2d ) , in which c1 and c2 are fitting parameters , and the exponential 
 term is motivated by the theory of hopping diffusion . the best - fit 
 result with c1 = 0.081  0.018 and c2 = 2.2  0.1 is shown by the red line 
 in figure 3 ( see figure s2 for the details of fitting ) . for nps with a < d < r in ring polymers , our scaling analysis predicts d  d. this 
 prediction is based on the ansatz that the effective viscosity eff that a np with a < d < r experiences in the fickian regime is equal 
 to the melt viscosity of ring polymers with size r  d. a ring polymer with r  d contains n  d monomers , and the melt viscosity   n  d. therefore , eff  d and d  kbt/(effd )  d. grosberg et al . found that the wave 
 vector q - dependent effective viscosity (q )  q for 1/r < d < 1/a in the 
 flg model . based on their result , the effective viscosity for fickian 
 diffusion of a np with a < d < r is (q  
 1/d )  d , in 
 agreement with our result eff  d. the d dependence of d for np - ring 
 systems with a < d < r in our simulations is weaker than the theoretical prediction d  d. the crossovers 
 between the scaling regime for a < d < r and two neighboring regimes contribute to 
 the weakening of the d dependence . the results of d for nps in rings with n = 800 and n = 100 are fitted simultaneously using a crossover function d = b1(/)(/d)/(1 + b2a / d ) + kbt/(2d ) , in which the fitting parameters b1 and b2 are independent of n. the fitting function describes the crossover from d  d for small 
 nps coupled to the unentangled dynamics of local ring sections to d  d for nps 
 coupled to the self - similar entangled dynamics of rings and then to d  d in the 
 stokes the best - fit results with b1 = 0.039  0.003 and b2 = 0.05  0.03 are shown in figure 3 ( see figure s3 for the details of fitting ) . d  d dominates over d  d only for d > 25. 
 this crossover occurs in the regime where d  d dominates the overall d dependence . therefore , there is no apparent d  d regime . for n = 100 , the crossover to d  d occurs at smaller d , and even the d  d regime 
 is not well developed . in conclusion , 
 our simulations show that the motion of nps can 
 be strongly suppressed prior to fickian diffusion in entangled linear 
 polymers . such a strong suppression occurs progressively as the np 
 diameter d becomes increasingly larger than the entanglement 
 spacing a. the motion of nps with d > a in entangled nonconcatenated ring polymers 
 is not strongly suppressed as in entangled linear polymers . the decrease 
 of diffusion coefficient d with increasing d in entangled rings is more gradual compared to the steep 
 reduction of d in entangled linear chains as d exceeds a. np motion for d > a in entangled linear chains is understood 
 through 
 a hopping diffusion mechanism . for nps 
 with d > a in entangled rings , 
 we 
 develop a scaling model for the coupling between np motion and the 
 self - similar dynamics of entangled rings . our results demonstrate 
 the role of polymer architecture in the dynamical coupling between 
 nps and polymers in nanocomposites .