Johari-Goldstein heterogeneous dynamics in a model polymer
Francesco Puosi, Antonio Tripodo, Marco Malvaldi, Dino Leporini
JJohari-Goldstein heterogeneous dynamics in amodel polymer
Francesco Puosi, ∗ , † , ‡ , § Antonio Tripodo, † , § Marco Malvaldi, † and DinoLeporini ∗ , † , ¶ † Dipartimento di Fisica “Enrico Fermi”, Universit`a di Pisa, Largo B.Pontecorvo 3,I-56127 Pisa, Italy ‡ INFN, Sezione di Pisa, Largo B. Pontecorvo 3, Pisa I-56127, Italy ¶ Istituto per i Processi Chimico-Fisici-Consiglio Nazionale delle Ricerche (IPCF-CNR),via G. Moruzzi 1, I-56124 Pisa, Italy § These authors contributed equally to this work.
E-mail: [email protected]; [email protected]
Abstract
The heterogeneous character of the Johari-Goldstein (JG) relaxation is evidenced bymolecular-dynamics simulation of a model polymer system. A double-peaked evolutionof dynamic heterogeneity (DH), with maxima located at JG and structural relaxationtime scales, is observed and mechanistically explained. The short-time single-particledisplacement during JG relaxation weakly correlates with the long-time one observedduring structural relaxation.
Avoiding crystallization, polymers and liquids freeze into a microscopically disordered solid-like state, a glass. On approaching the glass transition, molecular rearrangements occur via1 a r X i v : . [ c ond - m a t . s o f t ] F e b oth the primary mode, referred to as structural or α relaxation, and the faster secondary( β ) processes as evidenced by mechanical, electrical, and thermal properties of materials. There is wide interest in the β relaxation, as attested by the large number of experiments,as well as phenomenological and theoretical studies, and simulations. Here, we focus on linear polymers where there are no side groups and the secondaryrelaxations are usually ascribed to some movement of short lengths of the main chain, liketo limited vibrational oscillation about their mean position or crankshaft motion.
Thesecondary relaxation in linear polymers is thought to be a genuine manifestation of theJohari Goldstein (JG) β -relaxation, a special class of secondary relaxations having strongconnections to the α -relaxation in properties, and advocated to be a universal feature of theglass transition .It was early noted that the molecular rearrangements giving rise to JG relaxation pro-cess are similar to those that are responsible for the glass transition itself. In particular,correlation between JG and primary relaxation were reported in both molecular liquids andpolymers.
It was argued that the JG relaxation, like the alpha, involve transitionsbetween metabasins in the energy landscape , whereas other approaches suggest that JGprocesses can be interpreted in structural glasses as transitions between sub-basins belongingto a same metabasin. Furthermore, a number of experimental results indicate that the JG relaxation is sen-sitive to the thermodynamic variables underlying the glass transition, mimicking the α relaxation, being strongly pressure dependent and showing the invariance of the ratio τ α / τ β to variations of pressure and temperature, keeping τ α constant . This led to the conclusionthat the JG relaxation is precursor to structural relaxation and viscous flow, having a slowerdynamics due to cooperativity involving many body dynamics . This has been recentlysubstantiated by studying the invariance of the relation between α relaxation and β relax-ation in metallic glasses to variations of pressure and temperature by molecular dynamicssimulations combined with the dynamic mechanical spectroscopy method. Further support2o the view that primary and JG relaxation are closely related also come from the evidencethat the JG relaxation shows both a very broad distribution of relaxation times, andcooperative dynamics, indicated by simulations in non-polymeric liquids, experiments .The existence of spatial correlations between dynamic fluctuations - in short, dynamicheterogeneity (DH) - has been revealed by experiments and numerical studies, e.g., seethe reviews in. In particular, the presence of DH in the α relaxation regime has beenstudied in bulk polymers, by e.g. multidimensional NMR and simulations, and tuned bynanoconfinement . Of particular interest to the present study are the findings that DHsat both α and β relaxation time scales have been reported in colloids as clusters of faster-moving particles and numerical studies found heterogeneous dynamics of JG relaxation, supporting previous suggestions. On even shorter time scales, picosecond DH - observedby incoherent quasielastic neutron scattering - allowed the evaluation of the characteristictime scale of primary relaxation of molecular liquids. Furthermore, it has been noted thatJG involves a broad distribution of processes with those occurring at longer times beingcharacterized by a longer length scale.
A familiar tool to expose and quantify DH is the non-gaussian parameter (NGP): α ( t ) = 35 (cid:104) r ( t ) (cid:105)(cid:104) r ( t ) (cid:105) − r ( t ) and (cid:104)· · · (cid:105) denote the modulus of the particle displacement in a time t and the en-semble average, respectively. NGP vanishes if the displacement is accounted for by a single ,i.e. spatially homogeneous, gaussian random process. Instead, if dispersion is present andthe individual particles undergo distinct motions (even if with gaussian features), NGP is pos-itive, α ( t ) > Notably, NGP is accessible to experiments e.g. by confocal microscopy and neutron scattering. Recently, JG β relaxation has been resolved by studying theisochronal NGP in simulations of metallic glasses .It must be pointed out that a distribution of relaxation times, as observed in JG relax-3tion, 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. In this work we report results from extensive molecular-dynamics (MD) simulations ofa polymer model melt proving, when JG relaxation is present, a bimodal double peakedNGP leading to two distinct DH growths in the β and α relaxation regimes. A mechanisticmicroscopic explanation is provided. Our findings capture a new correspondence betweenJG and structural relaxation in polymers. We adopt a variant of coarse-grained models of linear polymers having nearly fixed bondlength and bond angles constrained to 120 ◦ , Fig.1a. Details are given as SupportingInformation (SI). All the data presented in this work are expressed in reduced MD units, inparticular lengths are in units of the length scale σ of the Lennard-Jones (LJ) potential. Aninteresting aspect of our variant is that, even if the force field does not include a torsionalpotential, thus saving computing times, an effective torsional barrier occurs for l < . σ ,see Fig. 1b, due to the LJ repulsion between the farther two monomers in a chain fragment offour monomers, see Fig. 1a. We focus on l = 0 . σ and l = 0 . σ leading to considerableor missing torsional barrier, respectively, see Fig. 1b. It is worth noting that the assessmentof the interplay of the intra-chain torsional barriers and other barriers, e.g. the ones arisingfrom packing constraints, is a difficult task and goes beyond the purposes of the presentwork. 4 / (cid:2) - - (cid:3) V ( (cid:1) , (cid:2) ) ` a)b) s * Figure 1: (a): fragment of the linear chain. The bond length is (cid:39) (cid:96) and the adjacent bondsform an angle (cid:39) ◦ . The monomer size is about σ (cid:63) = 2 / σ where σ is the length scaleof the LJ potential (see SI for details). The dihedral angle φ is indicated. (b): illustrationof the torsional barrier in the interacting potential V ( (cid:96) , φ ) between the first and the lastmonomer of the fragment due to the LJ repulsion in cis configuration when (cid:96) < . σ . Experiments and simulations demonstrated that orientational correlation functions are sen-sitive to detect and resolve secondary motions, in particular, the reorientation of thechain bonds.
Let us define the bond correlation function (BCF) C ( t ) as: C ( t ) = (cid:104) cos θ ( t ) (cid:105) (2)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 decreasesin time, finally vanishing at long times when the bond orientation has spanned all the unitsphere. 5ig. 2 plots BCF for the bonds with length l = 0 . σ and l = 0 . σ . In agreementwith previous MD studies on similar model polymers, it is seen that in the presence ofshorter bonds BCF exhibits a characteristic two-step decay (in addition to the initial decayfor t (cid:46) The relaxation maps of the JG and the α processes for thepresent model were reported elsewhere . C ( t ) -2 -1 t C ( t ) l = 0.48l = 0.55 TT Figure 2: BCF with bond length (cid:96) = 0 .
48 (top) and (cid:96) = 0 .
55 (bottom) at differenttemperatures. If l = 0 .
48, a clear two-step decay — evidencing two distinct relaxations—is observed.
Fig.3 compares the time evolution of DH of the melts of chains with different bond length byresorting to their NGP. At an early stage ( t (cid:46) . a -2 -1 t a l = 0.48l = 0.55 Figure 3: NGP with bond length (cid:96) = 0 .
48 (top) and (cid:96) = 0 .
55 (bottom) at differenttemperatures. Color codes as in Fig.2. In the presence of JG relaxation ( (cid:96) = 0 .
48) twopeaks increasingly grow by lowering T with positions close to the knees observed in BCFassociated to the β and the α relaxations, see Fig.2.7s also concluded by previous works. The small peak at t ∼ . For t > .
1, DH is fullydeveloped and NGP exhibits a complex pattern, strongly dependent on both temperatureand bond length. -2 -1 t NG P T = 0.85I II III
Figure 4: Time evolution of NGP at T = 0 .
85 for the two bond lengths.To reach better insight, Fig.4 compares the NGP time evolution of the chains with thetwo bond lengths at the same temperature (the lowest investigated). Three regions are seen. • Region I: at short times ( t (cid:46)
2) the NGPs are nearly coincident. • Region II: at intermediate times (2 (cid:46) t (cid:46) · ) in the presence of shorter bonds,switching the β relaxation on in this region, NGP is lower , i.e. DHs are weaker , and,after a maximum, decreases to a local minimum. Chains with longer bonds, with no apparent JG relaxation, do not show the minimum, as already known. • Region III: at long times ( t (cid:38) · ), where the α relaxation takes place, in the presenceof shorter bonds, NGP reaches a second maximum and finally decays, whereas the NGPof chains with longer bonds decreases monotonously.8 .3 Correlation between relaxation and dynamic heterogeneity There is a well-defined correlation between the DH evolution and relaxation. To deepen thisaspect, in addition to BCF, dealing with bond reorientation , we also consider the correlationloss of the torsional angle (TACF), see SI for rigorous definition. We also inspect quan-tities concerning the monomer dynamics : (i) the mean square displacement (MSD) (cid:104) δr ( t ) (cid:105) where δr ( t ) is the square modulus of the monomer displacement, δ r ( t ), in a time t , (ii)the self-part of the intermediate scattering function (ISF) F s ( q, t ) = (cid:104) exp[ i q · δ r ( t )] (cid:105) where i = − q is the modulus of the wavevector q . ISF is negligibly small if the displacementexceeds the length scale 2 π/q . We choose q = q max , where the static structure factor is max-imum, so that 2 π/q max ∼ σ (cid:63) , i.e. about the monomer diameter. We stress that, differentlyfrom TACF and BCF, both MSD and ISF are single-particle observables. -2 -1 t I SF BC F -2 -1 t M S D T A C F l a)c) b)d)I II IIIIIIIII IIIIIII II III Figure 5: from top left to bottom right: BCF, TACF, ISF and MSD at T = 0 .
85 for the twobond lengths. Panels are partitioned in the same three regions of Fig. 4. The dashed curvesin the TACF panel are best fits with the stretched function A exp[ − ( t/τ (cid:96) ) β ] with A = 0 . β = 0 . τ . = 190 (red dashed), τ . = 797 (blue dashed).Fig.5 presents the angular relaxation functions (BCF and TACF) together with monomerMSD and ISF, for the two bond lengths at the lowest temperature. Both BCF and TACF9learly show a larger decay in region I and II if the bond is shorter , i.e. the latter undergoes faster small-angle reorientation at short times, especially in the β region. In spite of thehigher angular mobility at short times, both BCF and TACF signal that shorter bonds slowdown in the α region (region III). While the reduced decay rate is apparent in BCF, to bettervisualise the effect in TACF, we first fitted the decay of TACF of chains with longer bondswith a stretched decay in regions II and III which, then, was log-shifted and superimposedto the TACF of chains with shorter bonds. We argue that the slowing down is due tothe torsional hindrance due to shorter bonds, see Fig.1b. The accelerated reorientation ofshorter bonds at short times and their slowing down at long times have strong influence onthe pattern of NGP. In fact, the crossover between regions I and II of NGP, see Fig.4, occurswhen the reorientation of shorter bonds becomes apparent, see BCF in Fig.5. This suggeststhat the reduction of NGP observed in polymer systems with shorter bonds follows by apartial averaging of DH. The latter effect is due to the fast bond reorientation which inhibitsthe DH increase, finally resulting in the bump observed in NGP at t ∼
10. Naturally, thecomplete DH erasure needs wide changes in the chain conformation and then full dihedraltorsions. This explains why NGP, like BCF, decays more slowly in region III in the presenceof shorter bonds. In particular, the peak of NGP in region III for shorter bonds is interpretedas due to the large scale structural relaxation triggered by the wide-angle torsions. Finally,by inspecting both ISF and MSD in Fig.5 one sees that the chains with shorter bonds exhibit smaller monomer MSD in regions I and III, and slower structural relaxation in region III, aspreviously reported. The role of ISF to model the memory kernel in a generalized Langevinequation theory dealing with both primary and JG relaxations has been emphasized.
The finding that, in the presence of JG relaxation, DH built up in the JG regime persistsup to the α relaxation, see Fig.4, conveys the impression of the possible interplay of DHs inthe JG and the long-time regimes, also on the basis that long-time DH is sensed at shorttimes, including JG time scale for non-polymeric systems, and even at vibrational timescales. .4 Negligible memory between particle displacements occurringin JG and α time scales The sound assessment of the previous hinted correlation must consider if it holds not onlyon the whole system but on subsets too. In non-polymeric liquids NMR experiments an-swered affirmatively by selecting subensembles of particles with given mobility, whereassimulations on diatomic liquids concluded that there is no connection between propertiesassociated to beta and alpha regimes of a single molecule. On the other hand, the in-variance of the ratio τ JG /τ α to P and T variations was reported in linear polymer meltsrepresented by a simple bead-necklace model, a variant of which is adopted in the presentpaper. We investigate this aspect by considering the system with JG relaxation ( l = 0 . T F r ac ti on f slow fi slow f fast fi fast f fast fi slow f slow fi fast f random l = 0.48 l = 0.55 Figure 6: Fraction of monomers of polymer system with JG relaxation ( l = 0 .
48) retainingmemory of their slow (or fast) mobility at JG time scale up to primary relaxation, f slow → slow and f fast → fast respectively. In the case of full memory f slow → slow = f fast → fast = 1. Dashedline sets the level f random = 0 .
05 corresponding to the absence of memory. Cross-conversionfractions are also plotted. The results point to very weak memory. See text for details. Inset:same analysis with l = 0 . 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 . N , N being the total monomers) with smallest (or largest) displacementin a time lapse t β , and ii) evaluating the fraction f slow → slow (or f fast → fast ) of these monomersstill belonging to the same mobility subset of n monomers with lowest (or highest) mobil-ity after displacement in a time t α (in the case of full memory f slow → slow = f fast → fast = 1,11hereas in the absence of memory f slow → slow = f fast → fast = 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 fast → fast are only slightly larger than f random , i.e. the correlation between the two subsets is quitelow. The same conclusion is reached by considering the fractions f slow → fast and f fast → slow accounting for the cross-conversion between the two mobility subsets (note that in the caseof 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 . , .
1, not shown).
In conclusion, we evidenced the heterogeneous character of the JG relaxation by MD simula-tions of a model polymer with secondary relaxation tuned by an effective adjustable torsionalbarrier. The DH evolution exhibits two distinct maxima at JG and structural relaxation timescales. We find that subsets of monomers lose memory of their mobility acquired in JG timescale before the occurrence of structural relaxation.
Acknowledgement
Simone Capaccioli is warmly thanked for discussions. We acknowledge the support fromthe project PRA-2018-34 (”ANISE”) from the University of Pisa. A generous grant ofcomputing time from IT Center, University of Pisa and Dell EMC (cid:114)
Italia is also gratefullyacknowledged. 12 upporting Information Available
Simulation Details
We study a melt of coarse-grained linear polymer chains with N c = 512 linear chains madeof M = 25 monomers each, resulting in a total number of monomers N = 12800. Adjacentbonded monomers belonging to the same chain interact via the harmonic potential U bond ( r ) = k bond ( r − l ) , where the constant k bond is set to 2000 (cid:15)/σ to ensure high stiffness. A bendingpotential U bend ( α ) = k bend (cos α − cos α ) , with k bend = 2000 (cid:15) and α = 120 ◦ , is introducedto maintain the angle α between two consecutive bonds nearly constant. Non-adjacentmonomers in the same chain or monomers belonging to different chains are defined as ”non-bonded” monomers. Non-bonded monomers, when placed at mutual distance r , interact viaa shifted Lennard-Jones (LJ) potential: U LJ ( r ) = (cid:15) (cid:34)(cid:18) σ ∗ r (cid:19) − (cid:18) σ ∗ r (cid:19) (cid:35) + U cut , (3)where σ ∗ = 2 / σ is the minimum of the potential, U LJ ( r = σ ∗ ) = − (cid:15) + U cut . The potentialis truncated at r = r c = 2 . σ for computational convenience and the constant U cut adjustedto 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 non-bonded monomers. This is the feature to build the torsional barrier up when l < . σ discussed in the paper. Alternatives reported in the literature exclude the LJ interactionsbetween atoms separated by three bonds or less. All the data presented in the work are expressed in reduced MD units: length in units of σ , temperature in units of (cid:15)/k B , where k B is the Boltzmann constant, and time in units of τ MD = ( mσ /(cid:15) ) / . We set σ = 1, (cid:15) = 1, m = 1 and k B = 1.Simulations were carried out with the open-source software LAMMPS. Equilibrationruns were performed at constant number of monomers N , constant vanishing pressure P = 013nd constant temperature T ( N P T ensemble). For each state the equilibration lasted atleast for 3 τ ee , being τ ee the relaxation time of the end-to-end vector autocorrelation func-tion. Production runs have been performed within the
N V T ensemble (constant numberof monomers N , constant volume V and constant temperature T ). Additional short equili-bration runs were performed when switching from N P T to N V T ensemble. No signaturesof crystallization were observed in all the investigated states.
Torsional correlation function
Alternatively to the BCF, the relaxation of the chain backbone arrangement has been char-acterized by the torsional autocorrelation function (TACF): T ACF ( t ) = 1 N c M − N c (cid:88) n =1 M − (cid:88) m =1 (cid:104)| ϕ m,n ( t ) || ϕ m,n (0) |(cid:105) − (cid:104)| ϕ m,n (0) |(cid:105) (cid:104)| ϕ m,n (0) | (cid:105) − (cid:104)| ϕ m,n (0) |(cid:105) , (4)where | ϕ m,n ( t ) | is the modulus of the m -th dihedral angle of the n -th chain at a given time t . ϕ is trivially related to the angle φ considered in Fig. 1 of the manuscript by therelation ϕ = φ − ◦ . The dihedral angle features the torsion of a given bond. Terminalbonds are not subject to torsion, therefore, in a chain of length M , there are M − ϕ is given from theintersection of the two planes defined by four consecutive monomers in a chain: the firstplane is defined considering the first set of three monomers while the second one is definedby the last set of three.The temperature dependence of the TACF curves is reported in Fig.8.14 Figure 7: Representation of the dihedral angle, ϕ , characterizing the torsion of the centralbond. -2 -1 t T A C F -2 -1 t l = 0.48 l = 0.55 T T
Figure 8: Temperature dependence of TACF in the two studied systems.
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OFF
Johari-Goldstein ON Evolution of mobility spatial distribution
Time D yy
Time D yy n a m i c h e t e r o g e n e i t yy