Johari-Goldstein heterogeneous dynamics in a model polymer

The heterogeneous character of the Johari-Goldstein (JG) relaxation is evidenced by molecular-dynamics simulation of a model polymer system. A double-peaked evolution of dynamic heterogeneity (DH), with maxima located at JG and structural relaxation time scales, is observed and mechanistically explained. {The short-time single-particle displacement during JG relaxation weakly correlates with the long-time one observed during structural relaxation}.

to the view that primary and JG relaxation are closely related also come from the evidence that the JG relaxation shows both a very broad distribution of relaxation times, 3,22 25 and cooperative dynamics, indicated by simulations in non-polymeric liquids, 26 experiments 8, 23 .
The existence of spatial correlations between dynamic fluctuations -in short, dynamic heterogeneity (DH) -has been revealed by experiments and numerical studies, e.g., see the reviews in. [27][28][29][30][31][32][33] In particular, the presence of DH in the α relaxation regime has been studied in bulk polymers, by e.g. multidimensional NMR 34 and simulations, 35 and tuned by nanoconfinement 32,33 . Of particular interest to the present study are the findings that DHs at both α and β relaxation time scales have been reported in colloids as clusters of fastermoving particles 36 and numerical studies found heterogeneous dynamics of JG relaxation, 37 supporting previous suggestions. 19 On even shorter time scales, picosecond DH -observed by incoherent quasielastic neutron scattering -allowed the evaluation of the characteristic time scale of primary relaxation of molecular liquids. 9 Furthermore, it has been noted that JG involves a broad distribution of processes with those occurring at longer times being characterized by a longer length scale. 8,23 A familiar tool to expose and quantify DH is the non-gaussian parameter (NGP): 38 α 2 (t) = 3 5 where r(t) and · · · denote the modulus of the particle displacement in a time t and the ensemble average, respectively. NGP vanishes if the displacement is accounted for by a single, i.e. spatially homogeneous, gaussian random process. 38 Instead, if dispersion is present and the individual particles undergo distinct motions (even if with gaussian features), NGP is positive, α 2 (t) > 0. 39 Notably, NGP is accessible to experiments e.g. by confocal microscopy 36 and neutron scattering. 39 Recently, JG β relaxation has been resolved by studying the isochronal NGP in simulations of metallic glasses 24 .
It must be pointed out that a distribution of relaxation times, as observed in JG relax-ation, 3,22 does not imply necessarily non-gaussian displacements -i.e. non vanishing NGP, the customary criterion to identify DH -, e.g., see the Rouse model which predicts multiple relaxation times of an unentangled single chain with gaussian particle displacements. 40 In this work we report results from extensive molecular-dynamics (MD) simulations of a polymer model melt proving, when JG relaxation is present, a bimodal double peaked NGP leading to two distinct DH growths in the β and α relaxation regimes. A mechanistic microscopic explanation is provided. Our findings capture a new correspondence between JG and structural relaxation in polymers.

Model and Numerical Methods
We adopt a variant of coarse-grained models of linear polymers having nearly fixed bond length and bond angles constrained to 120 • , Fig.1a. [41][42][43] Details are given as Supporting 3 Results and discussion

Detection of JG process by bond reorientation
Experiments and simulations demonstrated that orientational correlation functions are sensitive to detect and resolve secondary motions, 37,[44][45][46][47] in particular, the reorientation of the chain bonds. 37,47 Let us define the bond correlation function (BCF) C(t) as: 48 where θ(t) is the angle spanned in a time t by the unit vector along a generic bond of a chain.
An average over all the bonds is understood. Starting from the unit value, BCF decreases in time, finally vanishing at long times when the bond orientation has spanned all the unit sphere. with previous MD studies on similar model polymers, 42 it is seen that in the presence of shorter bonds BCF exhibits a characteristic two-step decay (in addition to the initial decay for t 1), signaling the presence of two distinct relaxation processes interpreted as the JG and structural relaxation. 41,42,47 The relaxation maps of the JG and the α processes for the present model were reported elsewhere 47 .  as also concluded by previous works. 37 The small peak at t ∼ 0.1 locates the average time needed by the monomer to hit the cage of the first neighbours. 49 For t > 0.1, DH is fully developed and NGP exhibits a complex pattern, strongly dependent on both temperature and bond length. • Region I: at short times (t 2) the NGPs are nearly coincident.

Correlation between relaxation and dynamic heterogeneity
There is a well-defined correlation between the DH evolution and relaxation. To deepen this aspect, in addition to BCF, dealing with bond reorientation, we also consider the correlation loss of the torsional angle (TACF), 41,42 see SI for rigorous definition. We also inspect quantities concerning the monomer dynamics: (i) the mean square displacement (MSD) δr 2 (t) where δr 2 (t) is the square modulus of the monomer displacement, δr(t), in a time t, (ii) the self-part of the intermediate scattering function (ISF) F s (q, t) = exp[iq · δr(t)] where i 2 = −1, q is the modulus of the wavevector q. ISF is negligibly small if the displacement exceeds the length scale 2π/q. We choose q = q max , where the static structure factor is maximum, so that 2π/q max ∼ σ , i.e. about the monomer diameter. We stress that, differently from TACF and BCF, both MSD and ISF are single-particle observables.  I  II  III   III  II  I  III  II  I   I II III

Negligible memory between particle displacements occurring in JG and α time scales
The sound assessment of the previous hinted correlation must consider if it holds not only on the whole system but on subsets too. 18 In non-polymeric liquids NMR experiments answered affirmatively by selecting subensembles of particles with given mobility, 18 whereas simulations on diatomic liquids concluded that there is no connection between properties associated to beta and alpha regimes of a single molecule. 37 On the other hand, the invariance of the ratio τ JG /τ α to P and T variations was reported in linear polymer melts represented by a simple bead-necklace model, 42 a variant of which is adopted in the present paper. We investigate this aspect by considering the system with JG relaxation (l 0 = 0.48) and focusing on the DH developed at t β , the time of the first peak of NGP (located in the β region) and DH still surviving at t α , the time of the second peak (located in the α region), see fig.4. We search for "memory" effects between t β and t α by: i) selecting a subset of n monomers (n = 0.05N , N being the total monomers) with smallest (or largest) displacement in a time lapse t β , and ii) evaluating the fraction f slow→slow (or f f ast→f ast ) of these monomers still belonging to the same mobility subset of n monomers with lowest (or highest) mobility after displacement in a time t α (in the case of full memory f slow→slow = f f ast→f ast = 1, whereas in the absence of memory f slow→slow = f f ast→f ast = f random with f random = n/N , with the latter result following if the initial subset in step i) is assembled by picking-up the n monomers randomly). Fig.6 shows the results. It is seen that both f slow→slow and f f ast→f ast are only slightly larger than f random , i.e. the correlation between the two subsets is quite low. The same conclusion is reached by considering the fractions f slow→f ast and f f ast→slow accounting for the cross-conversion between the two mobility subsets (note that in the case of full memory both quantities vanish). The pattern does not change by considering longer bonds, i.e. no JG relaxation, and different ratios n/N (n/N = 0.03, 0.1, not shown).

Conclusions
In where σ * = 2 1/6 σ is the minimum of the potential, U LJ (r = σ * ) = − + U cut . The potential is truncated at r = r c = 2.5σ for computational convenience and the constant U cut adjusted to ensure that U LJ (r) is continuous at r = r c with U LJ (r) = 0 for r ≥ r c .
It is important to note that the above model allows LJ interactions between all nonbonded monomers. This is the feature to build the torsional barrier up when l 0 < 0.5σ where |ϕ m,n (t)| is the modulus of the m-th dihedral angle of the n-th chain at a given time The temperature dependence of the TACF curves is reported in Fig.8.