the structural relaxation of polymer melts 
 has been the subject 
 of considerable experimental and theoretical investigation due to 
 its relevance to the processing of these viscoelastic materials .
the 
 relaxation of chains strongly depends on the degree of polymerization 
 ( or chain length ) , n , in a manner that is qualitatively 
 captured by the rouse 
the question 
 that logically follows is how the internal relaxations of the chains 
 depend on the length of the subchain under consideration . for long 
 enough chains ,
it is thought that short subchains relax according 
 to rouse dynamics and the longest by reptation with an intermediate 
 crossover behavior .
the crossover between these two limiting behaviors 
 occurs around the critical molecular weight nc  ne 
2ne , which depends on the details of the polymer 
 local architecture and packing and is generally attributed 
 to the topological constraints of entanglement that arise for large n. perez - aparicio et al .
studied the dynamics 
 of unentangled pep melts using neutron spin echo ( nse ) spectroscopy 
 and molecular dynamics ( md ) simulations .
they found that the relaxation 
 of the chains deviates from rouse behavior at short length scales 
 ( i.e. , below 2 nm ) , which are still larger than the kuhn length of 
 the chains ( 1.1 nm ) .
analogous deviations from rouse behavior 
 at time scales shorter than rouse time were also observed for 1,4-polybutadiene , 
 polyethylene , and poly(ethylene oxide ) .
these results are now 
 thought to be due to chain stiffness , other local packing effects , 
 and local frictional variations . for longer chain lengths , richter 
 et al .
found that there is a crossover where chain 
 relaxation is significantly slowed down due to the presence of entanglements . 

clearly , it is important to understand how topological constraints 
 control the relaxation of different subsections of a chain in this 
 crossover regime .
this is the focus of the present work . to 
 provide context for our studies , we begin with the simplest 
 model for the dynamics of short polymer chains in a homopolymer melt , 
 the rouse model .
a chain is represented 
 by a sequence of beads connected by harmonic springs , each spring 
 with mean - squared bond length b. this 
 model is known to describe the dynamics of short , unentangled melts 
 reasonably well , though deviations appear at shorter length scales . 

while these deviations have been thought to be due to local excluded 
 volume interactions and chain stiffness , more recent work has suggested that these effects could also arise 
 from hydrodynamic interactions operating over a range of length scales . for long
, entangled chains , the rouse model 
 describes the dynamics at intermediate time / length scales even though 
 the longer scale dynamics are strongly affected by constraints formed 
 by surrounding chains . the rouse modes ( p = 0 , 1 , 
 2 , ... ,
1 ) of a chain of length n are defined asxp = ( 2/n)i=1nri cos[(p/n)(i  1/2 ) ] .
the p = 0 mode describes 
 the motion of the chain center - of - mass , while the modes with 1 
n  1 describe internal 
 relaxations with a mode number p corresponding to 
 a subchain of ( n  1)/p segments . 

the autocorrelation of the rouse modes is predicted to be xp(t)xp(0) 
 = xpe ; i.e. , each mode should decay exponentially 
 and the modes are independent . here
the square of the pth mode amplitude is xp = b/[4 sin(p/2n ) ] , and its reciprocal relaxation time is p1 = 
 ( 12kbt/b ) sin(p/2n ) , where  is the monomeric friction coefficient . 

in contrast to these predictions , the simulation determined autocorrelation 
 function of rouse modes for chains in a homopolymer melt has been 
 found to be well fit by a stretched exponential relaxation : xp(t)xp(0) = xpe padding and briels have conducted 
 md simulations 
 of polyethylene using a coarse - grained model in which one bead represented 
 20 methylenes . using this model , they simulated melts as long as 50 
 beads ( or 1000 methylenes ) and found that the effective exponent p is dependent on mode number  for the 
 shortest modes ( largest p ) they found p  0.8
. with decreasing mode number 
 p decreases , reaching a minimum 
 0.5 for modes that are in the vicinity of n / p  ne/3 , independent 
 of chain length . for small p
found similar results for the value 
 of p but with its minimum value 
 occurring close to n / p  ne for a bead  spring model with chain 
 length n = 500 .
shaffer suggested that this minimum 
 ( as well as the presence of stretched exponential relaxations ) results 
 from kinetic constraints experienced by long chains .
several questions become apparent from this set of simulations : 
 ( i ) are these results general ? ( ii )
is the minimum of 0.5 in the vicinity 
 of ne a general result , and how 
 does it correlate with the concept of entanglements ?
more recently , 
 likhtman has examined 
 several aspects relative to the applicability of the rouse model and 
 a rouse mode analysis to melts of short , unentangled chains and longer 
 entangled analogues .
first , the normalized amplitude 
 of the static rouse modes , xp2 sin(p/2n ) , are dependent on mode number , p  this result 
 reflects the power law decay of intramolecular correlations in melts . 

second , there is coupling between the dynamics of the different modes , 
 for example the first and the third modes ( of a chain of length n ) , and this coupling only relaxes on the time scale of 
 the slowest mode .
while these results thus clearly violate the fundamental 
 principle of mode decoupling in the rouse model , we shall continue 
 to use this description for the following two reasons .
first , as we 
 shall discuss below , experimentalists still tend to model chain dynamics 
 in the language of the rouse model .
understanding experimental results 
 therefore require us to analyze the simulations in the same manner . 

second , our overall goal is to understand the role of nanoparticles 
 in chain melts . exploring these concepts based on the rouse model 
 thus seems to be the easiest means , at least in the context of current 
 experimental practice . here
, we study the internal relaxation 
 of a series of monodisperse 
 polymer melts with different lengths ( n = 10500 ) 
 using molecular dynamics ( md ) simulations .
grest ( kg ) bead  spring 
 model and chain - crossing model ( in which the chains are unentangled 
 for all
comparison of these two 
 models shows unequivocally that chain uncrossability for long - enough 
 chains gives rise to the minimum in p and also a significant slowing down in the relaxation of the 
 longest chain modes .
nonbonded 
 monomers interact through the lennard - jones ( lj ) 126 potential : u(r ) = 4[(/r )  ( /r ) ] 
 for r  rc , where 
  is the lj energy scale and  is the monomer diameter . 

the lj time scale is  = ( m/ ) , where m is the mass of 
 a monomer . for most of the studies
the lj interaction was cut off 
 at rc = 2.5 , though some simulations 
 were run with shorter cutoffs for comparison .
two successive segments 
 in a chain are connected by a finitely extensible nonlinear elastic 
 ( fene ) potential with k = 30/ and r0 = 1.5. these parameter values ensure 
 noncrossing of the chains . in addition
, a three - body bending potential 
 of the form ubend = k(1 + cos  ) is used to control the stiffness 
 of the chains with k = 0 , 0.75 , 
 and 1.5 in three separate sets of simulations .
the entanglement 
 length ne for these three values of k are 85 , 45 , and 28 , respectively .
we have considered chain lengths n = 10500 
 to study the relaxation of chains in both unentangled and entangled 
 homopolymer melts . to evaluate the role of entanglements
, we 
 also performed a series 
 of simulations that allow for chain crossing ( cc)these latter 
 class of models have no entanglement effects , even though their static 
 properties ( e.g. , chain dimensions ) closely match their entangled 
 analogues .
seg(r ) = a[1 + cos(r / rc ) ] , with a = 6.5 and rc = 1.6 and switching the bonds from their fene form 
 to a softer harmonic form vspring(r ) = ( k/2)(r  r0 ) with k = 20/ and r0 = 1.222. these 
 parameter values were chosen so that there is no significant effect 
 on the static properties of the melts while still enabling chain crossing . 

( a detailed calculation for two bonds crossing at 90 yields 
 a minimum energy of 2kbt per bead , corresponding to a bond crossing probability 
 of 0.15 .
) most of the simulations are carried out on 
 the nersc cray xe6 hopper 
 using the large - scale atomic / molecular massively parallel simulator 
 ( lammps ) .
the initial configurations 
 of the systems are prepared at random at a constant number density 
 while allowing for overlaps among beads .
the overlaps are removed 
 by initially using a soft potential between monomers and then by gradually 
 increasing the strength of the potential .
after all overlaps are removed , 
 the lj interactions between monomers are turned on and the volume 
 of the simulation cell is allowed to adjust at a constant pressure p * = 0 .
systems of chain length n = 100500 
 are equilibrated following the double - bridging algorithm .
the shorter n melts are equilibrated 
 by running isobarically and then at constant volume until the chains 
 have moved their own size .
after equilibration , the systems 
 are run at constant volume with 
 a langevin thermostat with damping constant  = 0.1. all simulations are run at temperature t * = kbt/ = 1.0 
 with a time step of 0.01. for longer chain lengths we find 
 that the average pressure p * = 0  0.05 / , whereas for shorter chains p * = 0  
 0.1 /. we simulated homopolymer melts of m chains of length n for [ m , n ] as shown in table 1 for k = 0.75 . additional systems with k = 0.0 , 1.5 for chain length n = 500 were also simulated to study the effect of bending 
 stiffness on the relaxation . normalized autocorrelation 
 function of different rouse modes p for a melt of 
 chain length n = 500 with 
 varying chain stiffness : ( a ) k = 
 0 with rc = 2 , ( b ) k = 0 , ( c ) k = 0.75 , and ( d ) k = 1.5 
 with rc = 2.5. the dashed black 
 lines are fits to stretched exponentials .
we first present results for 
 the effect of chain stiffness on the structure and relaxation of entangled 
 chains of length n = 500 . for rc = 2.5 , the monomer number density  = 0.89 for p * = 0 , n = 
 500 and for all three k = 0 , 0.75 , 
 and 1.5 .
these simulations were run for a total time of 5  10 for k = 0 and 0.75 
 and 8  10 for k = 1.5. for comparison , we also ran the system of chains of 
 length n = 500 with a purely repulsive lj potential 

( rc = 2 ) for k = 0 at the same density . ( a )
amplitudes of the 
 autocorrelation function of the rouse modes 
 for chains of length n = 500 .
( b ) effective monomeric relaxation rates of melts of chain length n = 500 with different stiffness . ( c ) effective monomeric 
 relaxation rates for chains of different lengths with stiffness k = 0.75 for high n / p scaling . ( d )
exponent p from fitting a stretched exponential to the autocorrelation 
 function of the rouse modes for n = 500 .
all autocorrelation functions , i.e. , for the different 
 rouse modes 
 of index p , different stiffness and cutoffs ( figure 1 ) decay to zero , implying that the chains are fully 
 relaxed .
attraction seems to have a relatively minor effect ( figures 1a vs 1b , also open squares 
 vs closed squares in figure 2 ) , but the curves 
 for a given mode p are shifted toward longer time 
 with increasing stiffness ( from figures 1b 
 to 1d ) .
the scaled amplitudes of the rouse 
 modes , xp sin(p/2n ) = b/4 representing the mean - squared 
 bond distances of chains , asymptotically approach the characteristic 
 ratio , which increases with chain stiffness ( figure 2a ) . for small p , xp sin(p/2n ) = ar[1  c(n / p ) ] , where the constant 
 of proportionality , ar , is found to be 
 linearly proportional to the characteristic ratio , c. these results simply reflect the importance 
 of local intramolecular correlations in the chains and that the chains 
 follow gaussian statistics at long length scales .
( a ) effective monomeric 
 relaxation rates for chains of different 
 lengths with stiffness k = 0.75. 
 ( b , c ) effective relaxation times of melt chains of different length 
 for k = 0.75. ( d ) longest 
 relaxation time ( p = 1 ) of chains of different length 
 is compared with cc model .
( e ) stretching 
 parameter p for chains of different 
 lengths . our results in figure 2b for the monomeric 
 relaxation rates are not consistent with the rouse model prediction 
 that w should be independent of n / p . indeed , only on the shortest length 
 scales ( n / p < 20 ) is the polymer 
 relaxation rouse - like .
the effective monomeric relaxation rates reach 
 a plateau value at large n / p , which 
 we postulate as being due to the crossover to reptation - like scaling 
 arising from constraints to chain motion from the neighboring chains . 
 to lend credence to our postulate for smaller p modes , 
 we note that the relaxation time for the longer chain modes should 
 scale as p  [ n / p][1 + ( n / ne ) ] . from here we can show for 
 small p that the rouse rates follow wpeff  [ 1 + ( n / ne ) ] .
this scaling is well 
 followed by long chain segments ( figure 2c ) . 
 for intermediate length scales ( i.e. , for 20 < n
/ p < 90 ) , we conjecture that there is a crossover 
 between these two behaviors .
the stretching exponent p from 
 fitting a stretched exponential form to the autocorrelation function 
 is a nonmonotonic function of n / p with larger values in the rouse and reptation regimes and a minimum 
 at the crossover around the entanglement length ne . for large n / p ,
p decreases from 0.9 to a value 0.5 around ne ( figure 2d ) with decreasing n / p , independent of chain stiffness .
. also found that the minimum in 
 p occurs around n / p  ne for k = 0 .
previous works by padding and briels and by shaffer suggest 
 that this minimum in p is due to 
 kinetic constraints on the chains .
our results 
 agree with this general conclusion , and the location of the minimum 
 of p does track qualitatively with 
 the reduction in entanglement length with increasing stiffness .
that 
 is the minimum occurs for n / p  
 40 for k = 1.5 and n / p  90 for k = 0 .
( for comparison , the entanglement lengths are 45 and 
 85 , respectively . ) previously , almost all simulations 
 of this model have used a purely 
 repulsive ( rc = 2 ) 
 interaction between beads to reduce computational resources necessary 
 to reach long times .
our results in figure 2 clearly show that the interaction cutoff has almost no effect on 
 the rouse modes of the chain .
we also carried out a primitive path 
 analysis following everaers et al . and 
 found that the entanglement length ne is 
 independent of rc
. the effective monomeric relaxation 
 rate of chains with different n is presented in figure 3a for k = 0.75. 
 for n < 40 the rate w is essentially independent of n / p ( or that the relaxation time follows rouse scaling , figure 3b ) , except for the largest p where 
 chain stiffness presumably enters .
chains of length n = 100 show that w changes by a factor 
 of 2 over the p values studied . as ne  45
the behavior of longer chains is reminiscent of the 
 behavior that we found for n = 500 in figure 2a .
in addition to a plateau for large p ( i.e. , small n / p ) there is a plateau 
 for small p ( i.e. , large n / p ) . in this longer chain length
small p regime , the effective monomeric relaxation rate is expected to scale 
 as wpeff  [ 1 + ( n / ne ) ] , which 
 is what is found when chain length is varied ( figures 3a and 2c ) .
the longest relaxation time 
 ( p = 1 ) ( figure 3d ) shows 
 that its dependence on chain length changes from n ( unentangled , rouse - like chains ) for short chains to n ( entangled ) for long chains consistent with 
 experimentally observed behavior .
chain crossing 
 ( cc ) model : ( a ) effective monomeric relaxation rates 
 of melts of different chain lengths with k = 0.75. ( b ) comparison of effective monomeric relaxation 
 rates of chains with different lengths between cc model ( upper set 
 of curves ) and kg model ( lower set of curves ) .
( c ) effective relaxation 
 times of chains in melts for different chain length for k = 0.75. ( d ) stretching parameter p for melts with cc chains of different lengths .
consistent with these ideas , the effective relaxation 
 times of 
 the different modes , especially for large n / p fall on a reptation - motivated master curve for different 
 chain lengths as shown in figure 3c , while 
 the shorter length scale modes follow rouse scaling ( figure 3b ) .
figure 3c also suggests 
 that this constraint dominated regime is applicable for n / p > 10 , which 
 implies that only chains longer than n = 100 (  
 2ne ) are in this regime .
the stretching 
 exponents p ( figure 3e ) increase monotonically with n / p for short chains , while for longer chains there is a well - defined 
 minimum in the vicinity of ne .
results for the chain - crossing ( cc ) 
 model are shown in figure 4 . in the cc model , 

chain relaxations are rouse - like for any given n as 
 topological constraints are eliminated by chain crossing .
the effective 
 monomeric relaxation rates are constant for n / p > 5 ( figure 4a ) . for smaller n / p ,
the results are similar to those for 
 the kg model ( figure 4b ) .
the effective relaxation 
 times for different length scales fall on a master curve which scales 
 as p  ( n / p ) ( especially for small p ) as shown in figure 4c .
the stretching exponent 
 p monotonically increases with n / p ( see figure 4d ) as for unentangled noncrossing chains ( figure 3e ) .
comparison of the cc chains to the chains that can not 
 cross ( figure 4b ) clearly shows that entanglements 
 affect the relaxation starting at the intermediate length scales around ne .
the extent of the effect depends on n as w decreases gradually 
 with n / p in the kg model , whereas 
 it is constant in the cc model . for large n / p
, the plateau in w for the 
 cc model shows that it better satisfies the assumptions inherent in 
 the rouse model than the kg model .
the much lower plateau values of w for the entangled chains clearly show that 
 the presence of topological constraints serve to significantly slow 
 down chain motion .
the major 
 point of our analysis is that the rouse modes of the 
 chains show characteristic trends in the regime of chain lengths where 
 entanglement effects become prominent .
the relaxation of chain segments 
 smaller than this topologically defined size follow rouse - like behavior , 
 although local correlations also lead to departure from rouse behavior 
 at the shortest length scales .
our results , especially on the monomeric 
 relaxation rates w of the chains , and 
 their mode number dependence , are in good agreement with the experimental 
 results of richter et al .
these workers found 
 that the effective relaxation rates decrease dramatically when one 
 goes through the entanglement crossover ( see figure 3a ) .
however , these workers did not see any minimum in the 
 apparent stretched exponents as a function of mode number  rather , 
 the mode relaxations of these very modestly entangled chains were 
 described by a normal exponential relaxation .
the reasons for this 
 difference in behavior between our simulations and the experiments 
 might arise from the fact that the experiments were conducted for 
 chains which are only weakly entangled .
( previous theories on short 
 chains , below the entanglement length , also show deviations from rouse 
 behavior due to local packing effects and local friction .
the experimental 
 evidence for such non - rouse behavior is weaker , and we do not understand 
 the source of these discrepancies between simulations and experiments 
 at this time . )
one question is why we choose to use the rouse 
 modes of the chains 
 to describe motion rather than true normal modes , which are guaranteed to be orthogonal to each other .
first , as discussed above , experiments 
 are normally analyzed in terms of the rouse model .
using the same 
 language as the experiments is a prudent means to make connections 
 of our models to reality .
second , previous works by hess , ronca , and edwards have shown that the equations of motions for 
 long , entangled chains in a polymer melt can be reduced to a generalized 
 rouse form , but with spectra at short and high frequency showing distinctly 
 different dependences .
this is precisely what is seen in figure 3a , in good agreement with previous theoretical works .
our rouse mode results for polymer melts show very interesting 
 trends which greatly expand our understanding of relaxation of different 
 length scales of chain and the role of entanglements on the dynamics 
 of polymers . for all chain lengths , the effective monomeric relaxation 
 rate ( friction ) for p 
n is not affected as w is same for 
 all n. the relaxation of intermediate length scales 
 is greatly affected by entanglements and depends on n in the studied crossover regime .
 we use molecular dynamics simulations 
 of the kremer  grest 
 ( kg ) bead  spring model of polymer chains of length between 
 10 and 500 , and a closely related analogue that allows for chain crossing , 
 to clearly delineate the effects of entanglements on the length - scale - dependent 
 chain relaxation in polymer melts . 
 we analyze the resulting trajectories 
 using the rouse modes of the chains and find that entanglements strongly 
 affect these modes . 
 the relaxation rates of the chains show two limiting 
 effective monomeric frictions , with the local modes experiencing much 
 lower effective friction than the longer modes . 
 the monomeric relaxation 
 rates of longer modes vary approximately inversely with chain length 
 due to kinetic confinement effects . 
 the time - dependent relaxation 
 of rouse modes has a stretched exponential character with a minimum 
 of stretching exponent in the vicinity of the entanglement chain length . 
 
 none of these trends are found in models that allow for chain crossing . 
 
 these facts , in combination , argue for the confined motion of chains 
 for time scales between the entanglement time and their ultimate free 
 diffusion . 
Introduction
Model
Results
Discussion and Conclusion