Microscopic Perspective ofPolymer Glasses DuringPhysical Aging and MechanicalDeformationA computational study of dynamicalheterogeneity, plasticity, and soft vibrationalmodes as the link to the molecular structure.byAnton SmessaertDiplom-Physiker (M.Sc. equivalent), Technical University Berlin, 2010A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2015c© Anton Smessaert 2015AbstractMicroscopic dynamics and mechanical response of polymer glasses are studiedin four projects using molecular dynamics simulations of a simple bead-springmodel. The first project studies the interplay between physical aging andmechanical perturbation. Structural, dynamical and energetic quantities aremonitored in the recovery regime following aging and uniaxial tensile defor-mation periods. The total engineering strain is found to control a continuoustransition from transient to permanent mechanical rejuvenation: After defor-mation in the pre-yield regime all quantities quickly reset to pre-deformationvalues, while deformation around the yield point results in the erasure of ag-ing history. Deformation in the post-yield regime, however, drives the systeminto a distinct thermodynamic state.In the second project, I introduce an efficient algorithm that detectsmicroscopic relaxation events, which are the basis of aging dynamics andplasticity. I use this technique to calculate the density-density correlationsfrom the spatio-temporal distribution of so called hops in quiescent polymerglasses at different temperatures and ages. Correlation ranges are extractedand I analyze the size distributions of collaboratively rearranging groups ofparticles. Furthermore, I spatially resolve dynamical heterogeneity (DH) ashop-clusters, and I compare cluster growth, as well as volume distributionduring aging with the four-point dynamical susceptibility χ4 as the estab-lished measure of DH.The third and fourth project use the hop detection technique to investi-gate the link between relaxation events and local structure. Quasi-localizedlow-energy vibrational modes, called soft modes, are found to correlate withthe location and direction of hops. In the third project, I analyze thetemperature- and age-dependence of this correlation in quiescent polymerglasses, and I show that the soft modes are long lived structural features.The fourth project extends the analysis to mechanically deformed polymerglasses. I find that the spatial correlation of hops and soft modes is reducedto pre-aging values after deformation in the strain softening regime. ThisiiAbstractreveals an additional perspective on mechanical rejuvenation and substanti-ates the findings from the first project. In the strain hardening regime thecorrelation increases, and this novel effect is linked to a growing localizationof the soft modes.iiiPrefaceA version of chapter 3 has been published as A. Smessaert, and J. Rottler,Recovery of Polymer Glasses from Mechanical Perturbation, Macromolecules45, 2928 (2012) [84]. The thesis author was responsible for all of the nu-merical work as well as the analysis of the results. Prof. Rottler providedthe initial idea for the project and he gave suggestions for the analysis andinterpretation of the results. The thesis author wrote the manuscript whichwas then edited by Prof. Rottler.A version of chapter 4 has been published as A. Smessaert, and J. Rottler,Distribution of local relaxation events in an aging three-dimensional glass:Spatiotemporal correlation and dynamical heterogeneity, Physical Review E88, 022314 (2013) [85]. The thesis author initiated the development of thehop detection algorithm and he was responsible for all of the numerical workas well as the analysis of the results. Prof. Rottler provided suggestions forthe analysis and interpretation of the results. The thesis author wrote themanuscript which was then edited by Prof. Rottler.A version of chapter 5 as well as part of chapter 7 have been publishedas A. Smessaert, and J. Rottler, Structural relaxation in glassy polymers pre-dicted bz soft modes: a quantitative analysis, Soft Matter 10, 8533 (2014) [86].The thesis author initiated the analysis of the alternative softness field def-initions and he was responsible for all of the numerical work as well as theanalysis of the results. Prof. Rottler provided the initial idea for the projectand he gave suggestions for the analysis and interpretation of the results. Thethesis author wrote the manuscript which was then edited by Prof. Rottler.A manuscript for a peer refereed journal is being prepared from the resultsin chapter 6. The thesis author was responsible for all numerical work as wellas the analysis of the results. Prof. Rottler was involved in the analysis andinterpretation of the results. The manuscript is being written primarily bythe thesis author with contributions by Prof. Rottler.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . xviiList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xxDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The glass transition . . . . . . . . . . . . . . . . . . . . . . . 31.2 Key features of glassy matter . . . . . . . . . . . . . . . . . . 61.2.1 Intermittent microscopic dynamics: the caging effect . 61.2.2 Non-equilibrium effects: aging . . . . . . . . . . . . . 91.2.3 Heterogeneous dynamics . . . . . . . . . . . . . . . . . 131.2.4 Soft modes . . . . . . . . . . . . . . . . . . . . . . . . 161.3 Polymer glasses under deformation . . . . . . . . . . . . . . . 191.4 Objectives of this work . . . . . . . . . . . . . . . . . . . . . 222 Simulation and measurement techniques . . . . . . . . . . . 252.1 Polymer model . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2 Molecular dynamics simulations . . . . . . . . . . . . . . . . . 29vTable of Contents2.3 Mechanical deformation . . . . . . . . . . . . . . . . . . . . . 332.4 Preparing the polymer glass . . . . . . . . . . . . . . . . . . . 342.5 Interpreting simulation results . . . . . . . . . . . . . . . . . 372.6 Hop detection . . . . . . . . . . . . . . . . . . . . . . . . . . 382.7 Vibrational modes and the softness field . . . . . . . . . . . . 432.7.1 Alternative definitions . . . . . . . . . . . . . . . . . . 453 Recovery from mechanical deformation . . . . . . . . . . . . 473.1 Immediate impact of creep . . . . . . . . . . . . . . . . . . . 483.2 Recovery of the relaxation time . . . . . . . . . . . . . . . . . 513.3 Recovery of the local structure . . . . . . . . . . . . . . . . . 573.4 Recovery in the potential energy landscape . . . . . . . . . . 603.5 Recovery after constant strain rate deformation . . . . . . . . 613.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634 Spatio-temporal correlation of structural relaxation events 654.1 Statistical properties of hops . . . . . . . . . . . . . . . . . . 674.2 Dynamical heterogeneity and clustering of hops . . . . . . . . 744.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825 Soft modes predict structural relaxation . . . . . . . . . . . . 855.1 Softness field . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.2 Spatial correlation . . . . . . . . . . . . . . . . . . . . . . . . 905.3 Directional correlation . . . . . . . . . . . . . . . . . . . . . . 945.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 966 Soft modes and local plastic events during deformation . . 986.1 Correlation between hops and softness field . . . . . . . . . . 1006.2 Examining the strain hardening regime . . . . . . . . . . . . 1026.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116viList of Tables2.1 Reduced units used in this work in terms of bead diameter a,bead mass m as well as the characteristic energy u0. . . . . . . 292.2 Densities in the melt used for the different target temperaturesin the glass state. Densities in the left two columns are usedin the projects discussed in chapters 3 and 4, while the righttwo columns are the parameters used in chapters 5 and 6. . . . 342.3 Hop identifier function thresholds used for the different targettemperatures in the glass state. . . . . . . . . . . . . . . . . . 41viiList of Figures1.1 (a) Sketch of viscosity as function of temperature when ap-proaching Tg from above. Viscosity is shown on a logarithmicscale and temperature is rescaled such that 1 indicates T = Tg.(b) Sketch of volume and enthalpy dependence as function oftemperature. Melting and glass transition temperatures areindicated with vertical dashed lines. . . . . . . . . . . . . . . . 31.2 (a) Sketch of a typical mean square displacement in glassymatter. Ballistic, caging and diffusive regimes are indicatedand cartoons illustrate caging effect and hop process. (b)Square displacement of individual particles measured in sim-ulations of the model polymer glass introduced in section 2.1.Particles mostly vibrate around fixed mean positions and thesharp changes are indications for hops. . . . . . . . . . . . . . 71.3 Sketch of the potential energy landscape in the glass state.The x-axis represents all configurational coordinates. . . . . . 101.4 Sketch of the PEL in the quiescent state and during externaldeformation. The x-axis represents all configurational coordi-nates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.5 Snapshot of the simulated model polymer glass, see section 2.1.The coloring indicates the squared displacement measured overa time interval τα/5, here τα is the structural relaxation time.On the right side only the 15% fastest particles are shown. . . 141.6 The upper half of the figure shows the eight lowest energyvibrational modes of an example polymer glass simulation,and the participation ratio is indicated above each snapshot.The modes are visualized by their polarization vector fieldsand coloring indicates depth. (a) Density of states as functionof the mode frequency, rescaled to reveal the boson peak. (b)Participation ratio as function of mode frequency. The curveis an average over 20 independent realizations of the system. . 17viiiList of Figures1.7 Stress versus strain observed in our polymer glass model. Thedeformation regimes are indicated (left to right): Elastic, yield,strain softening, strain hardening. The horizontal dashed lineindicates steady state flow of a non-polymeric glass. . . . . . . 212.1 Visualization of a bead-spring polymer and the filled simula-tion box. The beads are indicated as spheres. . . . . . . . . . 262.2 The LJ-potential between non-bonded beads is given by theblue curve, with energy- and force-shifted corrections in redand orange. For better comparison, the FENE spring potentialis scaled by a factor of 10−2 and indicated by the green curve. 272.3 Sketch that illustrates periodic boundary conditions and mini-mum image convention. The central square with the blue par-ticles is the simulation box, while image-particles are shown ingray. The arrow illustrates the position update of the particle.The dashed lines indicate the distance vectors from the centralparticle to all particles within a cut-off (dashed circle). . . . . 302.4 Mean square internal distance at different times during theequilibration run, as indicated in the legend. . . . . . . . . . . 362.5 Sample trajectory of a particle and hop identifier functionPhop. (a) shows the localization into two cages and the ’hop’is marked by rapid changes in the trajectory (e). Correspond-ing to this, Phop in (d) is sharply peaked at the transition andthe maximum defines the hop time thop. Plots (b) and (c) areoverlays of Phop and the trajectory just before and after thehop (z-comp. only for better visibility). The colored fields (Aand B) in both plots indicate the evaluation window for Phop[see eq. (2.5)]. Initial and final positions are calculated fromtime averages of the trajectory parts that are highlighted aszoom, i.e., (b) initial position (orange) and (c) final position(cyan). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39ixList of Figures2.6 Histograms of hop identifier Phop peak heights calculated atglass temperature T = 0.3 and time window parameters (orderof legend):(Nhist, Nobs) = (20, 100), (40, 50), (10, 200), (10, 100),(40, 100). The data for the three parameter sets with teval = 15are nearly identical (blue D overlay other markers). The solidline is an exponential fit of the distribution tail and the dashedline indicates the resulting parameter choice Pth. The insetshows the mean square displacement as function of time andPth is indicated by the horizontal line. . . . . . . . . . . . . . . 422.7 Distribution of softness in the system T = 0.3, tage = 750 forthree definitions of the softness field. The softness is rescaledin terms of the mean softness to allow an easier comparison. . 453.1 Schematic protocol of the simulation with waiting time periodtw, uniaxial tensile creep (a) or fixed strain rate (b) deforma-tion period ts, and recovery period tr. . . . . . . . . . . . . . . 483.2 Panel (a) shows the creep compliance of a glass with age tw =75000 for different external stresses σ. The indicated strainsare calculated at ts = 15000. Panel (b) shows the intermediatescattering function CSq of the perturbed glasses shown in (a) inthe recovery regime together with a just-quenched glass (F).The main graph shows CSq just after unloading (tr = 0) andthe inset is calculated at tr = 75000. The dashed lines definethe α-relaxation time. . . . . . . . . . . . . . . . . . . . . . . 493.3 Evolution of α-relaxation time of a just-quenched “new” glasscompared to those of mechanically perturbed glasses in therecovery regime. The glasses are identical in the initial agingregime tw = 75000, the creep time is ts = 15000, and the timeregimes are indicated by the dashed lines. Other results fromsimulations with these parameters are shown in fig. 3.2. . . . . 513.4 Relaxation time dynamics of unperturbed glasses with a rangeof wait times 103 ≤ tw ≤ 105 in a double logarithmic plot. Thecontinuous black line is the “generic” aging dynamics τ 0α andin the inset (axes: τα over tr) I show the shift of τα withincreasing tw. The main graph shows a data collapse with trmeasured in units of glass age ta = tw and τα being rescaledby τ 0α(ta). The dashed lines are guides to the eye. . . . . . . . 52xList of Figures3.5 Relaxation time of mechanically perturbed glasses in the re-covery regime. The coloring represents the total strain at timeof unloading. The continuous line indicates the generic agingdynamics and the dashed lines are guides to the eye. . . . . . 533.6 Sketch of possible recovery paths of the α-relaxation time.Line (o) is the path of an unperturbed glass and lines (a)-(c)are possible paths of glasses after mechanically perturbation.The diagonal black line indicates the generic aging dynamics. . 553.7 Percentage change of the rescaled average Voronoi cell volumewith respect to the “generic” value ∆〈Vv〉 = 〈Vv(t)〉−〈V 0v (ta)〉at glass age ta = tw + ts. Shown in the main graph is theevolution of mechanically perturbed glasses in the recoveryregime. The coloring represents the total strain at time ofunloading. The continuous line indicates the generic agingdynamics and the dashed lines are guides to the eye. Theinset shows the evolution of the average cell volume for just-quenched glasses and the black line is a logarithmic fit thatyields the generic aging dynamics. . . . . . . . . . . . . . . . . 563.8 (a) Difference in the coordination number ∆〈cv〉 = 〈cv(t)〉 −〈c0v(ta)〉, (b) triangulated surface order parameter ∆〈S〉 =〈S(t)〉 − 〈S0(ta)〉, and (c) evolution of the eigenvalue differ-ence ∆〈λ1,3〉 = 〈λ1,3(t)〉 − 〈λ01,3(ta)〉. All three quantitiesare rescaled percentage changes with respect to the “generic”value at glass age ta = tw+ts. The coloring represents the totalstrain at time of unloading (see e.g., fig. 3.7). The continuousline indicates the generic aging dynamics and the dashed linesare guides to the eye. . . . . . . . . . . . . . . . . . . . . . . . 583.9 Percentage change of the difference between the minimizedpotential energy of a mechanically perturbed glass ∆〈umin〉 =〈umin(t)〉 − 〈u0min(ta)〉 and the “generic” behavior at glass ageta = tw + ts. Shown is the evolution of the potential energylandscape in the recovery regime. The coloring represents thetotal strain at time of unloading. The continuous line indicatesthe generic aging dynamics and the dashed lines are guides tothe eye. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60xiList of Figures3.10 Evolution of (a) α-relaxation time, (b) Voronoi cell volume,and (c) inherent structure energy in the recovery regime afterdeformation with constant strain rates ˙ = 10−5 and 10−6 andtwo waiting times (glass age ta = tw + ts). (b) and (c) arepercentage changes as defined in fig. 3.7 and fig. 3.9, and thecoloring represents the total strain at time of unloading. Thecontinuous lines indicate the generic aging dynamics and thedashed lines are guides to the eye. . . . . . . . . . . . . . . . . 624.1 Snapshots of a single configuration showing (a) all particlesand (b) only those particles that are in the middle of a hop.There are only four hops at that time step and their positionsare highlighted by arrows. (c) shows all hops that are detectedin a time window of 3000τLJ . The configurations are takenfrom a glass at T = 0.2 and age ta = 20000. . . . . . . . . . . 664.2 (a) Distribution of persistence time τ of cages at three temper-atures. The solid lines indicate power laws. (b) Distributionof hop distances |d|, i.e., the distance between old and newcage. Exponential fits are indicated by the solid lines and thevertical dashed lines indicate√Pth and 2√Pth for the respec-tive temperatures; see legend in panel (a). Error bars in bothplots are smaller than the markers and are omitted for visibility. 674.3 The hop frequency fhop in units 1/τLJ over the duration of thesimulation run in a log-log plot. Solid lines indicate power-lawfits with exponents µ given in the legend. . . . . . . . . . . . . 684.4 Hop displacement autocorrelation for three glass temperaturesat age tage = 105 (a) as function of number of hops separationand (b) as histogram, that is calculated using normalized dis-placement vectors (T = 0.25 and age as above) and reveals ananisotropy in the direction of consecutive hops. The lines areguides to the eye. . . . . . . . . . . . . . . . . . . . . . . . . . 69xiiList of Figures4.5 (a) Probability density surface for spatio-temporal separationof two different hopping particles based on Eq. (4.3) for aglass at T = 0.25 and tage = 105. The color scale is logarith-mic (scale at the top-right corner) and the dashed lines indi-cate integration limits used to calculate the one-dimensionalprobability functions (b,c). Center plots show the probabilityfunction of (b) separation and (c) time delay between hops forT = 0.2(blue ©), T = 0.25(green 5), and T = 0.3(red 3) atthe same age. The gray vertical dashed lines in (b,c) illustratethe correlation ranges and the black dashed curve in (b) indi-cates the radial distribution function. (d) Probability densitysurface following Eq. (4.3) for the same glass as the left panel,but with r∗ calculated from initial position (at the origin) tothe final position of the second particle after the hop. Thecolor scale is again logarithmic. . . . . . . . . . . . . . . . . . 714.6 Size distribution of cooperatively rearranging particles. Themain panel shows distributions at six ages for a glass at tem-perature T = 0.3; see legend in fig. 4.7. The inset shows thesize distribution of three glasses at age tage = 105 and temper-atures T = 0.2 (blue ©), 0.25 (green 5), 0.3 (red 3). Bothplots have the same axes ranges, and the solid black lines in-dicate P (s) ∝ exp (−s). . . . . . . . . . . . . . . . . . . . . . . 744.7 The top panel shows the number of caged, i.e. not yet hopped,particles Ncaged averaged over independent simulations as afunction of time for six glass ages. The four-point suscepti-bility χ4 shown in the bottom panel is calculated from thevariance of Ncaged, eq. (4.4). . . . . . . . . . . . . . . . . . . . 754.8 Snapshots of the growth of a single cluster over time. Theparticles are visualized at their initial positions (before thehop) and the coloring indicates depth. The plot on the rightshows the cluster volume of 15 example clusters as a functionof time. Examples were recorded at glass age tage = 105. . . . 774.9 Mean cluster volume (top) given as a fraction of the total sim-ulation box volume and the number of simultaneous clusters(bottom) as a function of time. Results for six ages are shown,see legend in fig. 4.7. The insets show data collapse when timeis rescaled by the time of the χ4 peak. . . . . . . . . . . . . . 78xiiiList of Figures4.10 Mean hole volume (top) given as a fraction of the total simu-lation box volume and the number of simultaneous holes (bot-tom) as a function of time. Results for six ages are shown, seelegend in fig. 4.7. The insets show data collapse when time isrescaled by the time of the χ4 peak. . . . . . . . . . . . . . . . 794.11 Collection of cluster volume distributions of a glass at agetage = 105 measured at various times in double-log scale. Eachdistribution is plotted in the Vcl-p(Vcl) plane and placed alongthe t axis according to the size of the time window used forthe cluster analysis. The back wall shows an overlay of dis-tributions at small times in a single plane. I include data attimes . 103 (up to and including the second blue distribution)and the same colors as the separate distributions are used toindicate the origin of the data points. The solid line on theback wall indicates a power law with exponent −2. The blacksolid curve on the floor wall indicates χ4 as a function of time. 804.12 Main panel shows the mean fractal dimension of the hop clus-ters over time for six ages, see legend in fig. 4.7. The insetshows data collapse when time is rescaled by the time of theχ4 peak. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.1 (a) Snapshot of the softness field. The right side shows onlythe 10% softest regions and the solid black spheres (size equalsparticles) indicate the first 100 hopping particles detected afterthe measurement of the softness field. (b) Distribution of thesoftness field for three temperatures and three ages. Errorbars are omitted and smaller than the symbols. . . . . . . . . 865.2 Autocorrelation of the softness field for three temperatures (a-b) and three ages (c-d). Panel (a)[(c)] shows Ca as functionof time, and the dotted lines indicate the ISF for the sametemperatures [ages]. Panel (b)[(d)] shows Ca as function ofnumber of hopped particles Nh, with dotted lines again indi-cating the ISF and dashed lines mark when 50% of the systemhas hopped. Error bars are omitted and smaller than the sym-bols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87xivList of Figures5.3 Number of holes as a function of the fraction of hopped parti-cles. The sketch illustrates the definition of a hole: a continu-ous volume (green) that is surrounded by the union of spheresthat approximates the cages that hopping particles have es-caped and are “broken”. The dashed line indicates 50% of thesystem has undergone rearrangements, and the solid lines areguides to the eye. See fig. 5.1 for a legend. . . . . . . . . . . . 895.4 Cross-correlation between softness field and cumulative mapof hopped particles (a) as function of time and (b) as functionof number of hopped particles. See fig. 5.1 for a legend, anderror bars are omitted and are smaller than the symbols. . . . 915.5 (a) Probability of a particle to hop as function of its softness,rescaled by the average hop probability. The solid lines indi-cate the averaged saturation probability and the dotted lineis a guide to the eye. Success rate of predicting hops to oc-cur in the softest regions of the system Θ (b) as function ofcoverage fraction of the softest region f and (c) as function oftime rescaled to the number of hopped particles at constantcoverage fraction of f = 30%. The solid lines in (b,c) indi-cate the success rate based on randomly chosen regions andthe dashed line indicates 50% of the system has undergonerearrangements. To evaluate (a) and (b) the first 100 (1%)hopping particles after the softness field measurement wereused. See fig. 5.1 for legend, and error bars are omitted whensmaller than symbols. . . . . . . . . . . . . . . . . . . . . . . . 935.6 Directional correlation between softness field and hops (a) asfunction of softness and (b) as function of time rescaled tothe number of hopped particles. The dashed line indicates50% of the system has undergone rearrangements and the hopdirection is measured as the vector between initial and finalposition of the particle. To evaluate (a) the first 100 (1%)hopping particles after the softness field measurement wereused. See fig. 5.1 for a legend. . . . . . . . . . . . . . . . . . . 95xvList of Figures6.1 Snapshots of the system are shown at beginning (a) and end(b) of the deformation. To better visualize polymer configura-tions, 15 polymers are colored separately and only beads thatbelong to these polymers are displayed on the right side of thesimulation box. (c) Stress along the deformation axis es func-tion of total strain. Vertical colored lines indicate the investi-gated deformation states = 0.0, 0.04, 0.1, 0.5, 1.0, 2.0, 3.0, 4.0and the inset shows the peak at the yield point in more detail. 996.2 (a) Fraction of hops in soft spots as function of the coveragefraction of soft spots measured in three deformation regimes:elastic ( = 0.0), strain softening ( = 0.1), and strain harden-ing ( = 4.0). The dashed line indicates no correlation. Theevolution of the predictive success rate reached at f = 0.3is shown during aging (b) and during deformation (c) - thedashed lines are guides to the eye. . . . . . . . . . . . . . . . . 1006.3 Mean inherent structure energy during aging (a) and (b) asfunction of strain during deformation. . . . . . . . . . . . . . . 1026.4 (a) Mean participation ratio as function of eigenfrequency. (b)Mean participation ratio of all Nm modes used for the softnessfield calculation as function of total engineering strain. . . . . 1036.5 Relative change observed during deformation in the strainhardening regime. Lines are guides to the eye. . . . . . . . . . 105xviList of AbbreviationsDH . . . . . Dynamical HeterogeneityFENE . . . . Finitely Extensible Nonlinear Elastic bead-spring polymer modelIC . . . . . . Iso-Configurational ensembleISF . . . . . self-part of the Intermediate Scattering FunctionLJ . . . . . . Lennard-JonesLT . . . . . . Langevin ThermostatMD . . . . . Molecular DynamicsMSD . . . . Mean Square DisplacementMSID . . . . Mean Square Internal DistanceNHT . . . . Nose´-Hoover ThermostatNPT . . . . Isothermal-isobaric ensembleNVT . . . . Canonical ensemblePEL . . . . . Potential Energy LandscapePMMA . . . Poly(methyl methacrylate), e.g. PlexiglasPVC . . . . PolyVinyl ChlorideSD . . . . . . Square DistancexviiList of SymbolsH . . . . . . Hessian matrixN . . . . . . number of particlesNh . . . . . . Number of hopsNhist . . . . . hop detection trajectory length parameterNm . . . . . number of vibrational modes parameterNobs . . . . . hop detection observation frequency parameterP . . . . . . Participation ratioPhop . . . . . hop identifier functionPth . . . . . . hop identifier function threshold parameterT . . . . . . temperatureTg . . . . . . glass transition temperatureU . . . . . . potential energyULJ(r) . . . 6-12 Lennard-Jones potentialUbond(r) . . . bond potentialUpair(r) . . . pair potential∆t . . . . . . time stepχ4 . . . . . . dynamical susceptibilityd . . . . . . . hop displacement vectorxviiiList of Symbolse . . . . . . . polarization vectorr . . . . . . . location in spacerfinal . . . . final particle position after a hoprinit . . . . . initial particle position before a hopP . . . . . . pressureω . . . . . . . angular frequencyφ . . . . . . . softness fieldρ . . . . . . . densityσ . . . . . . . stressτα . . . . . . structural α-relaxation timeθ(f) . . . . . predictive success rate of a softness fieldf . . . . . . . coverage fraction of soft regionsm . . . . . . particle massrc . . . . . . cut off distancet . . . . . . . timeta, tage . . . glass agets . . . . . . deformation timetw . . . . . . waiting timeteval . . . . . hop detection time window parameteru0 . . . . . . Lennard-Jones energy well depthumin, UIS . . Inherent structure energyxixAcknowledgementsI thank my supervisor Prof. Jo¨rg Rottler, who helped me develop and shapemy professional abilities as a researcher and physicist. I am grateful for hisguidance and continuous support as well as for giving me the freedom todevelop my own research path and goals.I thank the members of my supervisory committee James Feng, ScottOser and Steven Plotkin for providing feedback and benchmarks during theprogression of my research. My fellow graduate students and friends DarrenSmith, Amanda Parker and Kevin Whyte I thank for sharing in the frustra-tions and successes of my research.For creating and maintaining the balance between professional and per-sonal life that allowed me to succeed in my research and to draw strengthand happiness from my work, I thank my heroes and unicorns. For unendingsupport, love and friendship I thank Lisa Hansen, my sister Angela as wellas my parents Ute and Pieter Smessaert.xxTo my dad.xxiChapter 1IntroductionIn the last century a new class of matter has become a key manufacturingmaterial: polymer glasses also known as plastics. Rather than a specific ma-terial, the term encompasses a variety of polymeric substances in the samestate, the glass state. The glass state is reached by cooling a glass form-ing material below a characteristic, material dependent temperature that iscalled glass transition temperature Tg. Already in Roman times glassy mat-ter was known in the form of silicon dioxide SiO2 (window glass), whichis a non-polymeric glass former. Nowadays, especially plastics are of enor-mous industrial importance, because they are cheap and easy to produce.Furthermore they are easy to manipulate and shape by exploiting the glasstransition, i.e. a plastic is poured at high temperature into a mold and thencooled to below Tg. Polymers are often used in the glass state, and promi-nent examples of polymer glasses are Nylon, PVC (polyvinyl chloride) andother carbon-hydrogen based polymers like polyethylene (shopping bags),polyethylene terephthalate (PET - bottles) and polypropylene (PP - pack-aging). Not all plastics are in a purely glassy state, but rather contain adegree of crystallinity which is introduced to alter the material characteris-tics. Although the use of glasses permeates most manufacturing branches,from water bottles and office material to medical instruments, houses andaircraft, our knowledge of the physical processes governing this state of mat-ter is limited. Key mechanical properties like the yield point and onset offlow in Plexiglas (PMMA) or for a rope of Nylon are estimated largely basedon empirical data. In contrast to crystalline solids, we do not have accessto theoretically well founded models for the propagation and interaction ofstructurally weak regions. We can therefore not predict at which locationin the material critical failure might occur. Furthermore, the mechanicalproperties of glasses are changing over time. A plastic ruler will be morebrittle in a year than it is today, yet we have no model that describes thisevolution. This lack of knowledge stands in the way of confidently utilizingglasses in critical components. Furthermore, a better understanding of the1Chapter 1. Introductionprocesses in the glass state might give new insights into how to better uti-lize or manipulate the material in order to perform new tasks or increase itsefficiency.This thesis explores the underlying physics in the glass state. This isdone by analyzing the molecular scale behavior of a model polymer glassusing computer simulations in order to better understand the processes thatcontrol macroscopic behavior like mechanical failure under load. In thischapter I introduce the conceptual ideas behind our current understanding ofthe glass state and I highlight how this work contributes to open questions.My research focuses on the more specific class of polymer glasses, whichincludes industrially important plastics, and I elaborate on specific propertiesof polymer glasses.Plasticity of glasses Why can we not describe plasticity in glasses in thesame way in that we understand it in crystalline solids? Crystalline solids un-dergo a phase transition upon cooling below the melting temperature, wherethe material rearranges on the atomic level into an ordered lattice. This long-range order is the basis of our understanding of plasticity in crystalline solids:The solid is modeled as a set of springs that attach atoms to their latticepositions. Elastic, or reversible deformation is understood as small displace-ment of atoms from the lattice and the resulting pendulum-like relaxationsback towards the equilibrium positions. Plastic, or irreversible deformationis modeled as irregularities in the long-range order, where the lattice sym-metry is locally broken. An example is the so called edge-dislocation whichis located at the end of an extra plane that extends into one half-space andends in the lattice. The dynamics and interactions of such dislocations is thebasis of plasticity models in crystalline solids.The glass state combines rigidity and mechanical properties of a solidwith a liquid-like structure. Unlike the freezing phase transition, the glasstransition is not accompanied by long-range ordering on the atomic scale. Inthe liquid phase, molecules do not form a lattice but are essentially randomlydistributed in space. Repulsive and attractive interactions (typically van-der-Waals interactions) drive the self-organization of nearest neighbor shells,yet this ordering is only short-range. As the temperature is lowered belowTg, the material vitrifies and the molecules are stabilized in their currentpositions, i.e. the liquid-like structure is preserved in the solid. The transitionis discussed in more detail below, yet the key point is that without long-21.1. The glass transitionviscosity [Pa s]temperature(a) (b)101310-4strongfragileTg/T 1volume, enthalpyglasscrystalsupercooledliquidliquidTmTgFigure 1.1: (a) Sketch of viscosity as function of temperature when approach-ing Tg from above. Viscosity is shown on a logarithmic scale and temperatureis rescaled such that 1 indicates T = Tg. (b) Sketch of volume and enthalpydependence as function of temperature. Melting and glass transition tem-peratures are indicated with vertical dashed lines.range order plastic deformations can not be understood in terms of latticedefects. In the absence of a lattice, dislocations can not be defined andthe machinery developed for plasticity in crystalline solids is incompatiblewith glasses. Indeed, being mechanically solid without long-range orderedstructure is the defining feature of a larger class of matter called amorphoussolids. Polymer glasses are amorphous solids with disorder on the nanometerscale, yet the term also encompasses systems like colloidal glasses, foam andgranular media that show disorder on the micro- to millimeter scale and eventectonic faults on the kilometer scale. All of these systems share a commondynamical behavior that arises from the absence of long-range order, and theresults of this work can be considered in the context of this broader class ofmaterials.1.1 The glass transitionThe glass state is reached by cooling a glass forming material in the liquidstate to the glass transition temperature Tg. As the temperature approaches31.1. The glass transitionTg one observes a sharp increase in the viscosity by many orders of magni-tude. The viscosity η measures the ability of a liquid to withstand shear ortensile stress, i.e., a liquid with large viscosity requires a large force to createflow. The glass transition is reached when the viscosity is 1013Pa ·s, which isan empirical threshold that originates from practices in (silicon based) glass-manufacturing. Figure 1.1(a) shows a sketch of the viscosity as functionof temperature in a semi-log plot. The temperature is rescaled by Tg anddecreases from left to right. Two types of behavior are distinguished: mea-surements of SiO2 show exponential behavior η ∝ exp (E/kBT ) known fromthermally activated processes with activation energy E, kB is the Boltzmannconstant and T is the temperature. Materials that exhibit such Arrhenius-type behavior are known as strong glass formers. The second type of behavioris called super-Arrhenius, because of the much steeper increase of viscositynear Tg. It can be expressed with the empirical Vogel-Tammann-Fulcherequation η ∝ exp [E/(kBT − kBT0)] [14]. The polymer glasses discussed inthis work fall into this category and are called fragile glass formers. Thereasons underlying this separation of glasses into two groups is still activelydiscussed [30]. The research presented in this work is based on a fragile glassmodel and generalization of the results to strong glass formers should beconsidered with care. However, the exponential dependence on temperatureindicates that thermally activated processes dominate structural relaxationand molecular mobility in the glass state.Fingerprints of the glass transition can also be found in the change of slopeat Tg of state variables like volume or enthalpy, as sketched in fig. 1.1(b). Asimilar behavior is known from first order phase transitions, yet the changeof slope is continuous in case of the glass transition. The crossover betweenliquid- and glass-slope is often used to measure the glass transition temper-ature more accurately than based on the threshold criterion of the viscosity.The glass state is furthermore characterized by another feature: ther-modynamic state variables and mechanical properties are time- and historydependent. In other words, a glass is not in thermodynamic equilibrium,because its state variables and properties do not fluctuate around historyindependent mean values, but change over time. Glasses are therefore non-equilibrium systems, and an alternative definition of the transition is when aglass former falls out of equilibrium upon cooling. The reason behind the lossof equilibrium is connected to the mobility in the glass state. As the mobilitydecreases, the system requires more time to find its optimal configuration,which is the thermodynamic mean. Upon cooling, the system is eventually41.1. The glass transitionunable to keep up with the change of this optimal configuration, and thenon-equilibrium glass state is reached. This inability to fully explore theaccessible configurational- or phase-space is also called a loss of ergodicity.Glass forming matter To understand which materials are glass formers,it is helpful to contrast the glass transition with the freezing transition ofcrystalline solids, where the material self-assembles on an atomic-scale lat-tice. This process is driven by the existence of an ordered ground state thatis energetically favorable. The crystallization transition is sharp, meaningthat a material will rapidly transform from an unordered liquid-like struc-ture with high entropy to an energetically stable structure with long-rangeorder. In contrast to this, the glass transition is not sharp, but gradual.Motion on the atomic scale rapidly slows until diffusion becomes negligibleand the atoms become arrested in their current position. The unorderedliquid-like structure is essentially frozen into the material. Why does thecompetition between a thermally stabilized high entropy state and an ener-getically favorable ground state not result in a first order phase transition?There are two possible reasons: Either such a ground state does not exist, orthe temperature quench is too rapid for the system to self-assemble onto alattice before the existing structure is frozen into place. Both possibilities areobserved in nature. From a simplified perspective, order on a lattice requiresthat the atomic interactions are dominated by a common length scale, whichbecomes the lattice constant. Systems that exhibit two competing lengthscales might not have a well defined ground state, because no structure sup-ports both lengths at the same time. Many polymers are such systems, withPMMA (Plexiglas) being a prominent example: the average distance betweenbonded monomers along the backbone is different from the average distancebetween non-bonded monomers. Systems without a well defined crystallinestate are also known as good glass formers, because the cooling rate doesnot play a critical role. However, it is also possible to produce a glass stateby cooling a liquid so rapidly, that crystallization is avoided. Without goinginto any detail, the dynamics in first order phase transitions requires nucle-ation processes. If nucleation does not occur, then one can cool a liquid totemperatures below its freezing point and the system is then called a super-cooled liquid. Nucleation is a diffusive process, and further cooling reducesthe mobility on the atomic scale and stabilizes the supercooled state. Theglass transition occurs when the system falls out of equilibrium (see above).51.2. Key features of glassy matterNature of the glass transition Whether the glass transition can beunderstood as a phase transition is still a matter of active research [11].One prominent microscopic theory is the Mode Coupling Theory [6, 39, 60](MCT), which predicts a phase transition from a liquid state to a dynamicallyarrested state where particles stop diffusing and an amorphous density profileis frozen into the system. The theory successfully describes aspects of thesuper-cooled liquid state, yet it fails to predict the Arrhenius-type behaviorof the viscosity near Tg and leads to a complete freezing of the particle con-figuration which is clearly not observed in glasses. Another view on the glasstransition is given by kinetically constrained models (KCM, see ref. [23] fora review), which assume that dynamical activity is facilitated by nearby dy-namical events and does not decay or arise spontaneously. This leads to thepropagation of dynamical activity via “defect-like” objects, and such mod-els successfully predict Arrhenius and super-Arrhenius behavior [11]. Onedrawback of this approach is the lack of a microscopic derivation and thatdetails of different models lead to different qualitative features. Rather thanadding to the debate about the glass transition, this thesis aims at exploringthe physical processes inside the glass state. I use computer simulations asnumerical experiments that give insights into dependences, correlations andtrends, rather than providing accurate quantitative estimates for specific ma-terial properties.1.2 Key features of glassy matter1.2.1 Intermittent microscopic dynamics: the cagingeffectThe molecular scale dynamics in glasses is separated into three regimes:the ballistic regime at small timescales, the caging regime at intermediatetimescales and the diffusive regime at large timescales. Figure 1.2(a) showsa sketch of the mean square displacement (MSD)〈(r(t0)− r(t0 + t))2〉 typ-ically found in glasses, which measures the average distance that a particlemoves between times t and t0. The three regimes are visible in the MSD:On very small time scales a particle moves ballistically, like a free particlein empty space. When the corresponding length scales become of order themean separation between particles, the MSD flattens to a plateau that ischaracteristic for the glass phase. On these intermediate time scales a par-61.2. Key features of glassy matter(a)012 34 567t/τα01234|r(t)−r(0)|2(b)Figure 1.2: (a) Sketch of a typical mean square displacement in glassy matter.Ballistic, caging and diffusive regimes are indicated and cartoons illustratecaging effect and hop process. (b) Square displacement of individual particlesmeasured in simulations of the model polymer glass introduced in section 2.1.Particles mostly vibrate around fixed mean positions and the sharp changesare indications for hops.71.2. Key features of glassy matterticle becomes effectively trapped by the shell of neighboring particles due tohigh densities at which the glass state exists. The surrounding particles actas a cage that confines the motion of the central particle to vibrations aroundthe center of that cage. The left sketch in fig. 1.2(a) illustrates this cagingeffect. The MSD does not increase until such times at which the probabilityfor a particle to escape the cage becomes large. The cage escape is a cooper-ative and therefore nonlinear process, since particles in the neighboring shellhave to move to “open the cage” and allow the central particle to escape. Icall this process a hop and it is visualized in the central sketch of fig. 1.2(a).Indications of particle hops can be found by inspecting the squared displace-ment (SD) of individual particles as shown in fig. 1.2(b). The SD has longperiods of relative stagnation where the particle is caged that are interruptedby short jumps in the SD that correspond to hop events. Finally, at largetime scales diffusive motion is realized as a succession of hops as indicated bythe right sketch in fig. 1.2(a). The MSD shows a departure from the cagingplateau and a transition to diffusive behavior. Vollmayr-Lee [95] was amongstthe first to exploit this essential feature of glassy dynamics to measure theeffective motion in a glass and additional detection methods for hops havebeen developed for glasses [98] and amorphous solids [19]. A key technicalaccomplishment of this work is the improvement of an algorithm introducedby Candelier et al. [19] such that hop events can be measured during com-puter simulations of glasses with high spatial and temporal resolution. Thehop detection algorithm is discussed in detail in section 2.6.Hop events are of key importance not only to explain motion in glasses onthe molecular scale, but they are also vital for the understanding of plasticityin glasses. Plastic events are irreversible rearrangements of atoms driven byexternal mechanical deformation. Vibrational motion is not irreversible, sinceparticles merely fluctuate around their static mean positions and stay withintheir cages. Plastic events can therefore be identified with changes of theconfiguration of cages. Assuming that in the presence of external load themolecular mobility remains dominated by the caging effect, then hops areelementary plastic events. In the absence of mechanical load, hop eventsplay the role of structural relaxation events. The non-equilibrium nature ofthe glass state drives such structural relaxation processes to evolve the glassinto more favorable configurations. Hops are therefore also at the center ofnon-equilibrium effects discussed in the next section.81.2. Key features of glassy matter1.2.2 Non-equilibrium effects: agingMatter in the glass state is not in thermodynamic equilibrium but slowlychanges its properties over time. The state of the system can thereforenot be described by a small set of equilibrium mean values and fluctua-tions around them. The reason for this is that the molecular mobility hasdecreased so far at the glass transition, that the glass can not explore phasespace sufficiently to relax to the equilibrium state. Instead, the glass be-comes trapped in the transition, resulting in a non-equilibrium system thatis slowly evolving towards an “ideal equilibrium state”. This time evolutionis known as physical aging, which I simply call aging in this work. The dy-namical processes governing aging are of key importance in making reliablepredictions of mechanical and other properties of glasses. Aging effects areparticularly important in polymer glasses [48]: density, enthalpy and yieldstress increase, but also the tendency for shear localization, which limits thelifetime to failure. The seminal work of Struik [88] is a striking presentationof the age-dependence of mechanical properties of polymer glasses. Struikperformed creep experiments (deformation at constant stress, see next chap-ter) of PVC, Polystyrene (PS) and other glass samples repeatedly over up tofour years of time between measurements. He measured the creep compli-ance, which is the ratio of strain over the applied stress as function of timeduring deformation. His results show a shift towards smaller compliance forincreasing age, indicating that the plastics become stiffer with time. Thesize of this shift increases like a power law with the glass age with exponent. 1. In other words, the mechanical response of plastic to an applied stressis strongly dependent on aging effects.Potential energy landscape To better understand the aging dynamicsand indeed glasses themselves, it is helpful to describe the state of a systemas a single point on a potential energy landscape (PEL). For a 3D systemwith N particles, the landscape exists in a dim = 3N + 1 dimensional spacewith the added dimension representing the current total potential energy. Asketch of the PEL in the glass state is shown in fig. 1.3. While all particlesare trapped in their respective cages in the glass state, the system resides ina minimum in the PEL. A hop event changes the local structure and with itthe potential energies, which moves the system from the original minimuminto another one close by. In the remainder of this work, I will therefore referto hops as structural relaxation events.91.2. Key features of glassy matterFigure 1.3: Sketch of the potential energy landscape in the glass state. Thex-axis represents all configurational coordinates.Let us first consider the glass transition from the PEL perspective. Attemperatures much higher than Tg, the kinetic energy in the system is higherthan the maxima in the PEL and the system is in a molten state. Whencooled into the supercooled regime T & Tg, then the kinetic energy becomesso low, that the system evolves by switching from one minimum to another.The transitions are thermally activated, which corresponds to the exponen-tial behavior of the viscosity discussed above. At the glass transition, thethermal transition processes become sufficiently unlikely, that the system canno longer fully explore the PEL on experimental timescales. The dynamicsbecome non-ergodic and the system gets stuck in the process of finding theequilibrium state. It falls out of equilibrium and the material transitions intothe glass state. The key reason for this is that the PEL is very rough, withmany metastable states. Inside the glass state, the process of finding theequilibrium state continues, and the glass evolves towards regions of lowerpotential energy and higher barriers between minima. This non-equilibriumrelaxation is the process causing physical aging or structural recovery.Hops are processes that allow the system to explore the PEL. Vollmayr-Lee et al. [95, 97] measured hops directly during aging, finding indicationsfor an age- and temperature-independent distribution of persistence times,i.e. times that particles stay inactive between hops. A closer investigation ofthe persistence time distribution by Warren and Rottler in quiescent poly-101.2. Key features of glassy mattermer and binary Lennard-Jones glasses [98] showed that physical aging canindeed be explained by hops on the microscopic level. They found that thepersistence times are distributed following a broad power-law with exponentµ ∼ −1.3 that is independent of the age of the glass. Power-law distributionswith exponent µ > −2 have no defined mean value, which means that withincreasing number of draws from the distribution the mean persistence timeincreases. During the evolution of a glass, hops occur and because of thebroad power-law, the persistence times increase with age. This increase wasdirectly measured as an age-dependence of the time distribution until the firsthop occurs for all particles. Hops are therefore intimately tied to the non-equilibrium dynamics in glasses. Chapter 4 explores the hop process beyondthe level of persistence time distributions by investigating spatio-temporalcorrelations between hop events.Mechanical rejuvenation Mechanical deformation interacts with the in-trinsic relaxation (non-equilibrium) dynamics in complicated ways. Numer-ous experimental observations [40, 42, 88, 89, 93] suggest that the history ofa glass can be altered by mechanical perturbation in such a way that the ma-terial appears “younger”. Struik [88] observed that after a short but strongtensile stress pulse the creep compliance of PVC was shifted towards largercompliance, thereby effectively reversing effects of previous aging. Continuedmeasurements of the creep compliance after the pulse then showed shifts withage that paralleled that of a younger glass. Mechanical deformation therefore“erased” part of aging and the phenomenon was coined mechanical rejuve-nation. However, the nature and extent of this “mechanical rejuvenation” inaging glasses is far from understood and frequently debated [64]. The oftenreported [22, 59, 61, 74, 75] increase of molecular mobility during deforma-tion may suggest that a “material clock” has indeed been turned back, butthe thermodynamic state reached by mechanical perturbation could be verydifferent from that of a younger version of the system. In other words, isthe “rejuvenated” state indeed comparable to a younger glass with measuresother than molecular mobility and mechanical response?Studies of “rejuvenation” tend to differentiate between deformation in thesub-yield regime, without yield and material flow, and the post-yield regime,where the material has undergone significant plastic deformation (the defor-mation regimes are discussed in more detail further below). Experimentsin the sub-yield regime observe an initial increase in mobility in agreement111.2. Key features of glassy matterwith the rejuvenation hypothesis [56, 79]. However, further measurementsafter the deformation has ended indicate that the mobility rapidly returnsto values found for samples of the same age but without deformation. Itis therefore often suggested that sub-yield deformation only transiently per-turbs the aging dynamics and that there is no direct connection to changesin material properties [56, 64, 79]. For post-yield deformation, however, reju-venation experiments on polystyrene [42] and polycarbonate [40, 93] glassesdemonstrate more convincingly thermal and yield stress properties akin to amuch younger glass.A recent series of experiments on PMMA glasses by Lee and Ediger pro-vides a very detailed insight into the interaction between physical aging andplastic deformation [57, 58]. These authors employ a fluorescence microscopytechnique that permits measurements of segmental relaxation times simulta-neously with macroscopic mechanical response [59]. During creep deforma-tion at constant stress, they report significantly increased molecular mobilityindicating partial erasure of aging. Full erasure, however, was only found inthe post-yield regime [57]. In a second study the authors went a step fur-ther, and performed detailed measurements of the molecular mobility in therecovery regime immediately following the deformation. Their results quali-tatively distinguish two impacts of deformation: in the pre-flow regime onlytransient rejuvenation effects were found and relaxation times rapidly re-turned to those of an unperturbed glass. Glasses that experienced post-yieldhowever, did not return to the pre-deformation aging trajectory but wereinstead found to age in the same way as a thermally quenched glass. Basedon these observations, the authors suggest that mechanical rejuvenation doesoccur in the post-yield regime [58].From the perspective of PEL, mechanical deformation can be abstractedto a tilting of the PEL towards the directions that are favored by the de-formation, see fig. 1.4. Barriers and adjacent minima are lowered by thedeformation, and the system evolves away from the aging-favored region inthe PEL. At the end of the deformation, the tilt is removed and the systemstate in the PEL is in a region with higher potential energy and lower barriers.The system again begins evolving towards lower minima and higher barriers,which manifests as a renewed physical aging. Simulations that investigatethe energy landscape of strongly deformed glasses [54, 62, 92] find indeedthat the material has been taken to higher inherent structure energies. Thedeformation trajectory, however, does not appear to return the material tothe exact same position on the landscape as a thermal quench, but rather to a121.2. Key features of glassy matterFigure 1.4: Sketch of the PEL in the quiescent state and during externaldeformation. The x-axis represents all configurational coordinates.different state. One focus of this work is to investigate mechanical rejuvena-tion on the molecular level, see chapters 3 and 6. Computer simulations giveaccess to experimentally inaccessible quantities like local structural features,and this allow me to quantify whether a deformed system is indeed similar toan equivalent glass of less or equal age. Furthermore, I explore the transitionfrom transient to permanent impact of deformation, and which deformationvariable is governing this transition.1.2.3 Heterogeneous dynamicsThe rapid slowing of the molecular mobility that is observed when approach-ing the glass transition is accompanied by a characteristic feature that furtherdistinguishes glasses from other systems. Instead of a homogeneous distri-bution of the particle mobility known from simple liquids, the dynamics ofglasses are spatially heterogeneous. This dynamical heterogeneity (DH) man-ifests itself as “fast” regions in a glass sample with dynamics that are ordersof magnitude faster than other “slow” regions which are only a few nanome-ters apart [34]. Computer simulations are especially useful to visualize DH,and fig. 1.5 shows an example snapshot of a simulated model polymer glasswith squared displacements indicated by the coloring (blue - slow, red - fast).In the left half of the simulation box one can see large areas of blue particles131.2. Key features of glassy matterFigure 1.5: Snapshot of the simulated model polymer glass, see section 2.1.The coloring indicates the squared displacement measured over a time inter-val τα/5, here τα is the structural relaxation time. On the right side only the15% fastest particles are shown.141.2. Key features of glassy matterindicating slow regions with a largely static structure. In the right half ofthe simulation box I only show the 15% fastest particles. One can see thatthese fast particles are clustered in small fast regions that undergo substantialstructural relaxation.Dynamical heterogeneity in glasses is now supported by a large number ofexperimental [9, 28, 34] as well as computational results [24, 47, 51, 53], andit explains the non-exponential structural relaxation observed for examplein fluorescence microscopy experiments [57, 58]. It furthermore directed thesearch for growing correlation effects near Tg away from static correlationfunctions known from conventional phase transitions, towards the study ofdynamical correlations. Since fast and slow particles spatially cluster intodistinct groups, such a correlation is present in glassy systems. First insightswere gained by computational studies that monitored a subset of “fast” parti-cles, which revealed a heterogeneous distribution and string-like cooperativemotion [33, 51]. The dynamical correlation itself can be directly measuredby calculating a four-point correlation function [7] that quantifies how manypairs of particles (two points in space) have moved by a similar distance overthe same time window (two points in time). This allows the measurementof a “dynamical susceptibility” χ4 [9, 53]. Strikingly, it was found that χ4increases when Tg is approached from above, indicating a growing correlationlength. Diverging correlation lengths play a crucial role in phase transitiontheory, and it is currently believed that the correlation between the dynam-ics of particles is driving the glass transition [11, 12]. The measurement ofthe distribution of hops gives a direct picture of DH on the level of struc-tural relaxation events. In chapter 4, I explore this strategy of observingDH with high spatial and temporal resolution. My analysis focuses on thespatio-temporal correlation between hops and the aggregation of hops intoclusters that as the system evolves grow until percolation and bulk structuralrelaxation is reached.As a consequence of DH, structural relaxation is concentrated in someregions and this has important consequences for the plasticity of glasses. Mi-croscopic plastic events are structural relaxation events under external load,and one can therefore expect that DH leads to a partition of glassy matterinto soft (hard) regions of fast (slow) particles. In order to create predictivemodels of plasticity, it is of key importance to understand what makes certainregions structurally weaker than others. In crystalline solids, the location ofplastic events is intimately tied to the local structure via dislocations. Inglasses, the situation is less clear and conventional structural indicators like151.2. Key features of glassy matterlocal geometric order (e.g. hexagonal order) or density fail to correlate wellto the location of plastic events. A first key insight that a link betweenstructure and dynamics exists was found using computer simulations in theiso-configurational (IC) ensemble [101]. In the IC ensemble the kinetic contri-bution to DH is suppressed by averaging displacement maps over many real-izations of a system, where the same spatial configuration of particles is usedwith velocities drawn at random from the appropriate Maxwell-Boltzmanndistribution. The resulting map for the propensity of motion, that dependsonly on the initial structure, still shows DH with distinct regions of high andlow likelihood of particle rearrangements. In a later study [13] the relativeimpact of kinetic and structural contributions to DH were quantified for asimple metallic glass model, and the results suggest that on the order 50%of DH is linked to the molecular structure. Understanding this link in termsof structural features that determine whether a certain region in a glass issoft or hard is an important open question that is addressed in chapter 5 forthe case of a quiescent glass during aging and in chapter 6 for a glass underload.1.2.4 Soft modesA promising candidate of a structural feature that predicts relaxation andplastic events are anomalous modes in the vibrational spectrum of glasses.The vibrational motion in a solid can be understood in terms of normalmodes, which describe vibration of particles as a collective dynamical process.The modes are given by the eigenmodes of the HessianH(ri)k(rj)l =∂2U({ri})∂(ri)k∂(rj)l. (1.1)Here U({ri}) is the interaction potential between pairs of particles and (ri)kis the k’th component of the position of particle i. For the case of crystallinesolids the Debye model describes vibrational modes via planar waves knownas phonons. In amorphous solids the description of vibrational properties isless understood, yet the spectrum characteristically shows an excess of modesin the low energy range when compared to the Debye expectation, which isknown as the Boson peak [18, 31, 83, 105].161.2. Key features of glassy matterP=0.12 P=0.29 P=0.33 P=0.33P=0.60 P=0.11 P=0.63 P=0.62D(ω)/ω2(a)0.5 1.01.52.0 2.5 3.0 3.5 4.0ω0.10.20.30.4P(ω)(b)Figure 1.6: The upper half of the figure shows the eight lowest energy vibra-tional modes of an example polymer glass simulation, and the participationratio is indicated above each snapshot. The modes are visualized by theirpolarization vector fields and coloring indicates depth. (a) Density of statesas function of the mode frequency, rescaled to reveal the boson peak. (b)Participation ratio as function of mode frequency. The curve is an averageover 20 independent realizations of the system.171.2. Key features of glassy matterAlready over 20 years ago [55], it was observed that many modes in thelow energy range are “quasi-localized” in the sense that most of the activityis concentrated on a small number of particles that are clustered in space.This is caused by the scattering of phonons at the local structure. Quasi-localized modes have been observed in a variety of glasses and super-cooledliquids in computer simulations [55, 80] as well as experiments [18], and theyhave been linked to the Boson peak [31]. At the top of fig. 1.6 I show theeight lowest energy eigenmodes of an example simulation of a polymer glass(details of the model and simulation are discussed in the next chapter). Ineach mode j a particle i is represented by the polarization vector e(i)j , whichis the projection of the eigenvector on the degrees of freedom of particle i.The extent of localization of mode j can be calculated via its participationratioPj =(∑Ni=1(e(i)j )2)2N∑Ni=1(e(i)j )4,where N is the total number of particles in the system. A small participationratio indicates that only few particles are active and that the mode is there-fore quasi-localized, while P = 0.66 is the value found for a planar wave. Theparticipation ratio of each mode in fig. 1.6 is indicated above the snapshot,and localized as well as planar waves are shown. Figure 1.6(a) shows thedensity of state rescaled by the Debye-expectation, and the Boson peak islocated at ω ∼ 2.3. The peaks at ω ∼ 1.0 and ω ∼ 1.3 are due to the finitevolume of the simulation box. In fig. 1.6(b) I show the participation ratioas the function of eigenfrequency, and one can see that the low participationmodes are concentrated before and around the Boson peak and that at higherfrequencies the modes are less localized.In 2008, Widmer-Cooper et al. [102] reported a qualitative spatial corre-lation between quasi-localized, low energy vibrational modes or soft modesand irreversible molecular rearrangements. The correlation was observed incomputer simulations of a metallic glass model in 2D [102] and 3D [103] inthe supercooled regime, and it successfully linked DH to a structural prop-erty, the vibrational spectrum. Further indications for this link were found insimulations of hard spheres [16], a kinetically constrained lattice glass [3], aquasi-statically sheared binary glass in 2D [90], and experiments on colloidalglasses [27]. Indications for the reason behind the connection of curvaturein the PEL (Hessian, see eq. (1.1)) and particle rearrangements in amor-181.3. Polymer glasses under deformationphous solids was identified in a study of a binary mixture of jammed packedspheres [104]. The authors found that soft modes identify the directions inthe PEL with the lowest-energy barriers to adjacent minima. This links theharmonic vibrational modes to the anharmonic event of particle rearrange-ments.A striking quantitative correlation between soft modes and rearrangementevents was verified by Manning and Liu [63] in a study from 2011. In com-puter simulations of a metallic glass in 2D they calculated the lowest energyvibrational modes and derived a map of “soft spots” by overlaying the mostparticipating particles of the lowest modes in a binary map. Starting fromthe initial configuration for which the soft spot map was calculated, the au-thors applied very small step strains using a quasi-static shear protocol whilemonitoring where the first rearrangements would occur. They found that therearrangements strongly overlap with one of the identified soft spots. Thiscorrelation between soft spots and plastic events was recently found to holdalso in thermal binary glasses at finite shear rate [81]. Chapters 5 and 6 dis-cuss in how far this correlation holds in the case of a more realistic thermalpolymer glass in 3D.1.3 Polymer glasses under deformationPolymers are macromolecules that consist of many individual segments thatare chemically bonded via covalent bonds. Due to their importance in indus-trial applications, a multitude of polymers have been developed, especiallyin the family of oil-based carbon-hydrogen polymers. If upon cooling of apolymer melt no chemical crosslinks are formed, then the polymer is ther-moplastic and it undergoes a glass transition into a polymer glass.In this work, I study an idealized model of such polymer glasses in orderto better understand these crucial manufacturing materials in particular andglassy matter in general. Besides their huge practical importance, I chose tofocus on polymer glasses, because they are good glass formers without ten-dencies to poly-crystallization. It is important to note, that having the chaintopology of polymers in our model does introduce characteristic processeslinked to polymer physics into the analysis of the glass state, specifically inthe context of mechanical deformation. This section discusses the origin andimportance of these processes in the context of this work.191.3. Polymer glasses under deformationReduced mobility in the melt To understand the dynamics of polymers,the Rouse model [32] proposes a simple view on polymers as a set of Ncbeads that are connected via Nc-1 linear springs. This is also known as theGaussian chain model. The connectivity between the beads constrains thediffusive motion, and one finds that the mean square displacement (MSD) ofthe polymer increases more slowly in time (∝ t1/2) than a single bead basedon Brownian motion (∝ t). However, this behavior is only observed for smallchain lengths. As the number of beads increases, multiple chains becomeentangled, meaning that they are wrapped around each other. Since polymerscan not cross through one another, this further constrains the chain motionto a process called reptation [32]. De Gennes showed that at chain lengthsbeyond a material dependent entanglement length the MSD is reduced to∝ t1/4 [29]. The dependence of the diffusion coefficient to chain length Ncalso changes from ∝ Nc in the Rouse model to ∝ N3c for entangled chains [32].Mobility in the glass state As discussed above, the dynamics in the glassstate is dominated by the caging effect and the MSD exhibits a characteristicplateau beyond the ballistic time scale. In glassy matter, the important dy-namical processes are cage-escapes that I call hop events. Although diffusivemotion is possible as a series of hops by the same particle, the key glassy fea-tures of aging and heterogeneous dynamics are governed by the hop processand not by very long time scale diffusion. The diffusion constraining effectsof polymer topology are therefore unimportant for the glassy dynamics [1].This is true for polymer glasses in the quiescent state, yet one has to be morecareful in the case of mechanical deformation.Mechanical response A schematic stress-strain curve of a polymer glassis shown in fig. 1.7. The first three regimes are common to all glassy matterand amorphous solids: First, at small strains the stress increases linearlywith the strain. In crystalline solids this linear response is a consequenceof displacements of particles away from their lattice positions without re-arrangements to other lattice points. When the external load is removed,the system returns to its original state and the deformation is elastic. Inamorphous solids this elastic regime is marked by an analog linear macro-scopic response, yet close observation reveals some plastic activity even atvery small strains [10]. Viewed from the perspective of hops as elementaryplastic events, this is not surprising however, since hops are present even in201.3. Polymer glasses under deformationviscoty [Pvisavvviscoty]cseatotmapcvior viscotvu(iatotm)oapeFigure 1.7: Stress versus strain observed in our polymer glass model. Thedeformation regimes are indicated (left to right): Elastic, yield, strain soft-ening, strain hardening. The horizontal dashed line indicates steady stateflow of a non-polymeric glass.the quiescent state (no external load) due to the non-equilibrium dynamics.As the strain increases the linear response smoothly transitions to a max-imum stress, the yield point. This is accompanied by an increase in plasticevents until the yield stress is reached and the system begins to flow. Theheight of the yield stress indicates the stiffness of the glass, is history de-pendent and increases during aging. The onset of flow is followed by astrain softening regime, where the stress decreases with increasing strain.The steepness of this decrease is again dependent on the material historyand aging.In non-polymeric glasses, flow drives the system to a steady state andthe stress reaches a plateau value. In polymer glasses, the situation is dif-ferent due to the chain-topology, and polymeric effects become importantfor the macroscopic mechanical response. As shown in fig. 1.7, instead ofa steady state plateau, polymer glasses enter the strain hardening regime,where the stress increases non-linearly with strain. To increase the strain,the orientation of the polymer backbones have to more and more align withthe deformation axis. As this happens an increasing amount of stress isstored in the bond interactions along the polymer backbone, which are muchstronger than the monomer pair-interactions. Furthermore, if the chains aresufficiently long, entanglement effects come into play which further increase211.4. Objectives of this workthe stress necessary to apply additional strain on the system.1.4 Objectives of this workThe central goal of this thesis is to illuminate the microscopic processes thatgovern the mechanical properties of glassy matter in general, and polymerglasses in particular. Although plastics are ubiquitous industrial materials,our understanding of their plasticity is limited. The macroscopic responsecan be measured in experiments, yet the underlying microscopic processeshave proven difficult to resolve. To bridge the gap between the microscopicbehavior, the mesoscopic phenomena of dynamical heterogeneity and themarcroscopic response, I performed large scale molecular dynamics simula-tions of a widely used polymer model. The computer simulations allow theobservation of each particle with perfect detail and are therefore uniquelypositioned to provide insights on the microscopic level. In chapter 2, I ex-plain in detail the simulation techniques and the polymer model, as wellas the measurement techniques used to identify the driving processes. Inthe remainder of this chapter, I highlight the individual questions that I amaddressing in this work. The results of the four projects are discussed inchapters 3-6. This is followed by concluding remarks in the final chapter.Recovery from mechanical deformation The design of my first projectis inspired by the recent experimental study of Lee and Ediger [58] of polymerglasses in the recovery regime, discussed in detail in section 1.2.2. I give acomplementary picture of the recovery regime by simultaneously measuring avariety of structural quantities that are inaccessible to experimental studies.To this end, I mirror the creep experiment in computer simulations as wellas performing constant strain rate deformation. In addition to structuralquantities I monitor the structural relaxation time (molecular mobility) aswell as the inherent structure energy in the recovery regime. The key questionis, whether mechanical deformation indeed leads to erasure of aging and toan “as newly quenched” glass beyond the perspective of molecular mobility.Computational studies of the PEL [54, 62] suggest that deformation leads toa state that is distinct from that of a younger glass. This study provides newinsights by comparing the full aging behavior in the recovery regime with thatof undeformed glasses. Furthermore, I explore the transition from transientto permanent rejuvenation and which deformation parameter is controlling221.4. Objectives of this workthis transition. This question is important, since the driving deformationparameter has to play a prominent role in any theory of plasticity and thereis a debate whether stress, strain or strain rate should be considered.Spatio-temporal correlation between structural rearrangementsThe aim of the second project is to spatially resolve DH and to quantify thecorrelated dynamics for a 3D glass in the aging regime, where only a fewstudies exist [71]. Does physical aging change DH in terms of correlationrange or geometry of the regions that undergo significant structural relax-ation? To perform our study on large-scale systems over extended simulationperiods that are sufficient to analyze aging effects, I implement a novel hopdetection algorithm that is based on a technique introduced by Candelieret al. [19]. This new algorithm allows the monitoring of hop events for allparticles in large scale computer simulations with high spatial and temporalresolution. I use this method to create a detailed map of hops and to directlyquantify the density-density correlation between hops. I study dependenceon temperature, aging and infer correlation ranges. In the second part of thisstudy, I focus on DH in the aging regime. This regime has only recently beenexplored using three- and four-point correlators [17, 71], and indications werefound for an increase of dynamical correlation with age. The detailed mapof hops allows me to spatially resolve DH and to measure its evolution asgrowing clusters of hops. A cluster algorithm is used to calculate the clustervolume distribution and I compare its evolution directly with the four-pointdynamical susceptibility χ4 as the standard measure for DH.Soft modes link structure and rearrangements in polymer glassesThe third project broadens the understanding of the link between the loca-tion of rearrangements and the local structure. Recent studies [63, 81] showedthat quasi-localized modes in the vibrational spectrum of glasses identify softspots in model binary glasses, discussed in much detail in section 1.2.4. Tofully understand the role of soft spots in plasticity of amorphous solids, itis important to test the robustness of the soft spot picture in other glassformers. For the first time, I quantify this link in the more realistic andindustrially relevant case of a thermal polymer glass in 3D. Are particle re-arrangements indeed predominantly occurring at soft spots and what is theimpact of temperature and aging on this correlation? Furthermore, do softmodes hold additional information about the direction of individual rear-231.4. Objectives of this workrangements, and what is the lifetime of soft spots? As a first step, I exploredifferent approaches on how to extract a map of the structural softness fromthe soft modes. In this third project, I focus on the quiescent state and Ianalyze the correlation between the calculated softness field with the like-lihood to undergo hops. My results show that a positive correlation existsnot only between the location of hops and soft modes, but also that the hopdirection is correlated to the soft mode direction. Systems above (super-cooled regime) and below (aging regime) the glass transition temperatureare analyzed, and I discuss the impact of temperature as well as aging onthe correlation. Furthermore, I analyze the lifetime of the softness field andmy results show that the identified “soft” and “hard” regions are long livedcompared to elementary vibrational timescales. This is important, becauseit shows that the softness field is a meaningful structural variable that couldplay a role in plasticity of amorphous solids similar to dislocations in the caseof crystalline solids.Predicting plastic events with structural features As a fourth and fi-nal project, I investigate the robustness of the found link between soft modesand hops in the case of deformed polymer glasses. The hops represent lo-cal plastic events and predicting their location based on the softness mapis a central result of this project. Is this correlation affected by mechani-cal deformation and what are the processes underlying such a change in thecorrelation? To answer these questions, I quantify the correlation of soft-ness field and hops at different points during uniaxial tensile deformation,from the elastic regime immediately after loading until far into the strainhardening regime. Prior to the deformation the polymer glass is aged for anextended time period, and I investigate indications for mechanical rejuvena-tion in the spatial correlation between softness field and hops. Interestingly,I find that the spatial correlation is increasing with growing strain in thestrain hardening regime, and I explore possible explanations of this novelpolymeric effect.24Chapter 2Simulation and measurementtechniquesThe results of this thesis are based on molecular dynamics (MD) simulationswith a simple glass-forming polymer model. The simulation approach hasthe advantage of giving access to microscopic details that are inaccessiblevia experimental techniques. Many experiments only give access to bulkaveraged quantities, and this makes the study of inherently local phenomenalike dynamical heterogeneity and plastic events challenging. With computersimulations it is possible to measure the microscopic dynamics with perfectdetail, so that connections between microscopic processes and macroscopicresponse can be explored.In the first half of this chapter, I introduce the polymer model, and elab-orate on the MD techniques, the simulation of mechanical deformation, andthe preparation of glass states. I also discuss general issues of this typeof computer simulations, and how to interpret the results correctly. In thesecond half of this chapter, I explain the key measurements that I use toprobe the polymer glass: First, I give details on the hop detection algorithmand second, I elaborate how vibrational modes are calculated and how thesoftness field is defined.2.1 Polymer modelThis work is based on a coarse-grained bead-spring polymer model that cap-tures the chain-topology of a linear polymer without including the full atom-istic detail. The polymer is represented as a sequence of identical beads,and neighboring beads are bonded to a linear chain that forms the polymerbackbone. Figure 2.1 shows an example polymer and simulation box. Orien-tational correlations along the polymer backbone decay with increasing sep-aration, and the characteristic decay length is called persistence length [32].A polymer can therefore be partitioned into segments that are independent252.1. Polymer modelFigure 2.1: Visualization of a bead-spring polymer and the filled simulationbox. The beads are indicated as spheres.262.1. Polymer modelFigure 2.2: The LJ-potential between non-bonded beads is given by the bluecurve, with energy- and force-shifted corrections in red and orange. Forbetter comparison, the FENE spring potential is scaled by a factor of 10−2and indicated by the green curve.from each other. In the present model a single bead represents such a seg-ment.In a classical MD simulation, one defines potentials that describe theinteraction between beads, as well as the covalent bonds along the polymerbackbone. This work uses the finitely extensible nonlinear elastic (FENE)bead-spring polymer model [52]. It has been studied extensively in the glassstate and shows the characteristic behavior expected for polymer glasses [8,38, 70, 76, 94, 99]. The pair-potential between non-bonded beads is modeledas a 6-12 Lennard-Jones (LJ) potentialULJ(ri,j) = 4u0[(ari,j)12−(ari,j)6]. (2.1)Here, a is the bead diameter, u0 is the depth of the LJ energy-well and ri,j isthe distance between beads i and j. To simplify the notation, I neglect thisindex in the following. The potential is shown in fig 2.2. It has a repulsivepart at short range r < 21/6a, which ensures that beads don’t occupy the samevolume. For r > 21/6a the potential becomes attractive and it models the272.1. Polymer modelvan der Waals force. To improve computational efficiency [2], the potentialis cut off and set to zero for distances r > rc together with an energy shift,which ensures that the potential is continuous at rc. In the projects discussedin chapters 3 and 4 the pair potential is given byUpair(r) ={ULJ(r)− ULJ(rc) r < rc0 r ≥ rc(2.2)with rc = 1.5a chosen at the first minimum in the radial distribution func-tion, i.e., only particles that are neighboring each other are interacting. Theprojects that study soft modes, chapters 5 and 6, require the calculation ofsecond derivatives of the potentials. To ensure that the forces are differen-tiable at rc I include a force-shift termUpair(r) ={ULJ(r)− ULJ(rc) + (rc − r) · FLJ(rc) r < rc0 r ≥ rc. (2.3)In this case I use the cut-off rc = 2.5a, which is beyond the second minimumof the radial distribution function. This choice makes the potential morenumerically comparable to eq. (2.1), and both energy- and force-shifted ver-sions are compared to eq. (2.1) in fig. 2.2. The main difference between bothmodels is the extended attractive range of eq. (2.3), which results in smalldifferences in the glass transition temperature and densities (see below).A non-linear stiff spring-like interactionUbond(r) ={KR202 ln[1− (r/R0)2]+ ULJ + r < 21/6aKR202 ln[1− (r/R0)2] r ≥ 21/6a(2.4)acts as covalent bonds along the polymer backbone and it consists of twoparts: The last two terms are the repulsive part of the energy-shifted LJpotential with a cut-off rc = 21/6a. The first term is attractive and divergesat a distance R0, which models unbreakable chains. My choice of K = 30u0and R0 = 1.5a [52] ensures that chains can not cross through each other.Reduced units The above defined potentials allow a scaling of distancesand energies with the bead diameter a and energy-well depth u0 respectively.In this thesis all results are given in reduced LJ-units [2] that are listed intable 2.1. For simplicity the ∗ is omitted henceforth.282.2. Molecular dynamics simulationslength r∗ = ra density ρ∗ = ρ a3energy U∗ = Uu0 temperature T∗ = kBTu0time t∗ = t√u0ma2 pressure P∗ = P a3u0force F∗ = F au0Table 2.1: Reduced units used in this work in terms of bead diameter a, beadmass m as well as the characteristic energy u0.2.2 Molecular dynamics simulationsThe principal idea behind MD simulations is to solve Newtons equations ofmotion simultaneously for N particles in a fixed volume. The simulations forthis thesis were done with the open-source package LAMMPS [72] that usesthe velocity-Verlet algorithm [2]. The key approximation is the discretizationof time, and the simulation updates particle locations r(t), velocities v(t),and accelerations a(t) to the next time step t+ ∆t in four stages:1. Velocities are updated to their mid-step valuesv(t+ ∆t/2) = v(t) + a(t)∆t/2 .2. New positions are calculatedr(t+ ∆t) = r(t) + v(t)∆t+ a(t)∆t2/2 .3. New forces and accelerations are calculated for all particles i with massm using the model potentials and Newtons 2nd lawm ai(t+ ∆t) =∑j 6=iFij(t+ ∆t) = −∑j 6=i∇U(rij(t+ ∆t))4. The velocity-step is completedv(t+ ∆t) = v(t+ ∆t/2) + a(t+ ∆t)∆t/2 .This procedure is repeated to evolve the system of N particles forward intime. For the simulation results shown in this thesis I used ∆t = 0.0075.292.2. Molecular dynamics simulationsFigure 2.3: Sketch that illustrates periodic boundary conditions and mini-mum image convention. The central square with the blue particles is thesimulation box, while image-particles are shown in gray. The arrow illus-trates the position update of the particle. The dashed lines indicate thedistance vectors from the central particle to all particles within a cut-off(dashed circle).302.2. Molecular dynamics simulationsBoundary conditions This thesis explores the physics of polymer glassesin the bulk, and confinement effects near the edges of the simulation volumeare avoided by imposing periodic boundary conditions. When a particleleaves the rectangular simulation box through one surface, a copy of it (animage-particle) reenters on the opposite surface. This effectively creates aninfinite lattice of simulation boxes as indicated in fig. 2.3. As sketched in thesame figure by a circle, particle interactions and other distance dependentcalculations are performed by considering image-particles whenever they arecloser than the particles in the simulation box. This technique is known asminimum image convention.Thermostat Conventional MD simulations are energy conserving, sincethey are based on Newtons equations of motion. Under experimental condi-tions, however, the polymer glass is interacting with its environment. It istherefore more realistic to simulate the system at constant temperature. Avariety of approaches have been developed to constrain the kinetic tempera-ture in the system in agreement with the NVT ensemble (constant numberof particles, volume and temperature). These methods are called “ther-mostats”, and I use the LAMMPS [72] implementation of two algorithms. Iam using the Langevin thermostat [2] during equilibration in the melt, thetemperature quench and during mechanical deformation. The Nose´-Hooverthermostat [46, 69] is used during the simulation of a quiescent glass.The Langevin thermostat(LT) simulates the existence of a heat bath by in-troducing random and dissipation forces into the equations of motion, whichbecome Langevin equationsm a(t) = F(t)−mγv(t) + fr .Here, γ is a dampening constant and I used γ = 1. The random forcesare independent for each particle as well as time step and have zero mean.They couple the system to a fixed external temperature T via a fluctuation-dissipation relation and the variance of the random force is 2mkBT/γ∆t,with kB being the Boltzmann constant.In the Nose´-Hoover thermostat (NHT), the heat bath is introduced as anadditional degree of freedom, that controls the flow of energy between thesystem and the reservoir. This approach leads to an additional friction forcem a(t) = F(t)−mξ v(t) ,312.2. Molecular dynamics simulationswith a coefficient ξ that is coupled to the fixed external temperature [2] viaξ˙ =1Q(∑imv2i − f kBT).Here, the dot above ξ represents a time derivative, f = 3N is the number ofdegrees of freedom in the system, and Q = f kBT τ 2NH controls the strengthof the coupling via a relaxation time τNH . The simulations discussed in thisthesis were performed with τNH = 0.5.The preparation of the polymer glass and mechanical deformation (dis-cussed in the next two sections) as well as all simulations for the projectdiscussed in chapter 3 were performed using the LT. During the implemen-tation of the hop detection algorithm (see below) I realized that the LT canhave a diminishing impact on the spatially heterogeneity of the dynamicsin the glass state. The LT applies to each particle an independent randomforce and thus has a homogenizing effect on the dynamics. In contrast tothis, the friction coefficient ξ of the NHT is a global property of the system(there is only one degree of freedom added for the heat bath), and in thissense the thermostat affects all particles in the same way. A comparison ofthe hop frequency and cluster size measured in the glass state shows thatsimulations with the NHT have slightly more hops that are more clusteredin space. I emphasize that the observed differences are small, and to assesswhich simulation describes the glassy dynamics more accurately goes beyondthe scope of this thesis. However, for the projects discussed in chapters 4-6I chose to simulate the quiescent glass state (no external deformation) usingthe NHT, because I believe it to be the “safer” alternative that avoids toartificially drive the system into a dynamically less heterogeneous state.Barostat The pressure is controlled by a Nose´-Hoover barostat [45], whichadds an additional degree of freedom that simulates a pressure-bath [2]. Thisallows simulations in the NPT ensemble (constant number of particles, pres-sure and temperature). The bath couples to the pressure by rescaling thesimulation box L˙α = ναLα in dimension α with coefficient να that acts as astrain rate. Physically, the coupling resembles the action of a piston, whichadjusts the accessible volume depending on the balance between internal and322.3. Mechanical deformationexternal pressure. The equations of motion are modified tor˙α(t) = vα(t) + να(rα −R0,α)maα(t) = Fα(t)−mναvα(t) ,and in analogy to the thermostat discussed above, the coupling coefficientevolves according to [65]να =1τ 2BN kBTV (Pαα(t)− Pext,αα) .Here, Pαβ(t) = (∑Ni vi,αvi,βm/2+∑Ni ri,αFi,β)/3V is the instantaneous pres-sure tensor, while Pext is the fixed external pressure. All deformation sim-ulations are performed with a relaxation time τB = 7.5, and I use theLAMMPS [72] implementation of the algorithm that controls each dimen-sion independently.2.3 Mechanical deformationUniaxial tensile deformation is simulated with two different protocols. Inboth protocols the pressure perpendicular to the deformation axis is kept atPxx = Pyy = 0. A creep experiment is simulated by applying a constant stressσz in the z-direction for a time period ts, using the barostat with Pzz = −σz.To avoid discontinuities, the pressure is ramped to −σz over the time period37.5. At the end of the deformation period, the pressure is ramped back tozero in the same way. This creep-protocol is used in the project discussedin chapter 3. A limitation of this protocol is that deformation can only besimulated up to the yield point. At the onset of flow the barostat becomesnumerically unstable, since it cannot keep up with the rapid response of thesystem to the applied stress.An alternative protocol that allows the simulation of mechanical defor-mation beyond the yield point is the deformation at constant strain rate˙ = L˙z/Lz, where strain is defined as engineering strain = ∆Lz/Lz. Thesimulation box size along the z-direction is continuously increased at a con-stant rate for a time period ts by directly rescaling space in that dimension.During the deformation period, the z-component of the pressure is not con-trolled by the barostat, while the other pressure components remain fixed atzero.332.4. Preparing the polymer glassEnergy-shifted model, eq. (2.2) Force-shifted model, eq. (2.3)T ρ T ρ0.2 1.005 0.2 1.0590.25 0.995 0.3 1.0430.3 0.983 0.4 1.023Table 2.2: Densities in the melt used for the different target temperaturesin the glass state. Densities in the left two columns are used in the projectsdiscussed in chapters 3 and 4, while the right two columns are the parametersused in chapters 5 and 6.2.4 Preparing the polymer glassThe polymer glass is prepared in two steps [99]: First, I create and equilibratea polymer melt at the temperature T = 1.2. Second, this melt is quenched ata rate 6.7 ·10−4 to below the glass transition temperature Tg. In the projectsdiscussed in chapters 3 and 4 I simulated chains of length Nc = 50 beadswith a total number of beads in the simulation N = 50, 000. The simulationbox volume was chosen such that the pressure at the end of the quench isclose to P = 0, and the densities used for the different final temperaturesare given in the first two columns of table 2.2. The calculation of the softmodes (see further below) in chapters 5 and 6 is computationally expensiveand required a reduction of the system size to N = 10, 000. The changes inthe model potential discussed above result in slightly different melt densitiesand are given in the two right columns in table 2.2. The spatial size of thesystem is around 20 particle diameters in each dimension (for N = 10, 000,otherwise larger), which is much larger than the physically important lengthscales. As discussed in section 1.2.1, glassy dynamics are dominated by thecaging effect caused by neighboring particles, and particle distances are onthe order of the particle diameter. The largest length scale present in thesystem is the end-to-end distance of a polymer, which is around 8.5 particlediameters (see fig. 2.4 below).342.4. Preparing the polymer glassPolymer melt The creation of the polymer melt closely follows the proce-dure introduced in Ref. [4]. The positions of the beads of each polymer areinitialized as a random walk in the simulation box with step length 0.92a,and consecutive steps are constrained by ‖ri−ri+2‖ ≥ 1.02a [4]. This resultsin polymer configurations that are close to the equilibrium statistics, yet itallows beads to overlap, which is unphysical and result in diverging forcesdue to the repulsive part of the LJ potential (see eq. (2.1)). This problem isresolved in two stages:1. Slow push - Instead of using the LJ pair-potential from the beginning,one simulates the first 3000 time steps using the soft potentialUpair(r) = A [1 + cos(pir/rc)] , if r < rc .The coefficient A is increased linearly 1 ≤ A ≤ 100 over the run, drivinga reduction of the overlap.2. Fast push - The soft potential is now replaced by the LJ potential andthe simulation is run for an additional 3000 time steps. To keep theparticle velocities from diverging due to the remaining overlap, onerescales the velocities every 50 time steps such that the temperature is∑Ni=1 v2i /2m = 1.2.The system is then equilibrated by running it in the NVT ensemble for107 time steps. To ensure that an equilibrium state is reached, I measurethe mean-square end-to-end distance of chain segments of length n along thepolymer backbone〈R2〉(n) =〈1NnNn∑|i−j|=n(ri − rj)2〉.Here, 〈.〉 indicates an average over all polymers. This mean square internaldistance (MSID) has a signature peak at intermediate n that vanishes whenequilibration is reached [4]. Figure 2.4 shows the MSID at different equilibra-tion run lengths N∆t for a short-chain system. One can see a characteristicpeak around n = 10, which reduces with increasing run time. After 107time steps the MSID does not change any more and the system has reachedequilibrium.352.4. Preparing the polymer glass100101segment size n1.01.21.41.61.8〈R2〉(n)/nN∆t=102N∆t=103N∆t=104N∆t=105N∆t=106N∆t=107N∆t=108Figure 2.4: Mean square internal distance at different times during the equi-libration run, as indicated in the legend.Glasses at different temperatures and ages The polymer glass is cre-ated from the melt via a rapid quench to below the glass transition tempera-ture at constant volume and quench rate T˙ = −6.7·10−4. The glass transitiontemperature for the energy-shifted model (chapters 3, 4) is Tg ' 0.35 [76].For the force-shifted model (chapters 5, 6) Tg ' 0.4 is estimated from thetemperature dependence of the pressure during cooling, see also fig. 1.1(b).To study the temperature dependence of measured quantities, glasses at dif-ferent final temperatures are created, and a list of all temperatures used inthis thesis is given in table 2.2.As discussed above, the density is chosen for different target temperaturessuch that the hydrostatic pressure is close to zero at the end of the quench. Ina final step after the quench, the simulation is switched to the NPT ensembleusing a barostat (see above) and the pressure is quickly ramped to zero overthe time period 37.5. The resulting system is what I call an “as quenched”or “newly quenched” glass in the quiescent state. To study the effects ofnon-equilibrium dynamics, the system can then be evolved to different agesby simulating it at zero pressure in the NPT ensemble for the desired timeperiod.362.5. Interpreting simulation results2.5 Interpreting simulation resultsThere are two fundamental problems in correctly interpreting results fromthese simulations. The first challenge is: how can one ensure that the nu-merical model accurately simulates the physical glass? Fully atomistic simu-lations with quantum-mechanical detail are computationally very expensiveand restrict the system size to a few hundred atoms. As explained in theintroduction, glassy dynamics can be found in many diverse physical sys-tems that stretch many orders of length scales. Glassy physics is thereforenot critically dependent on microscopic details, but is instead driven by theinteraction of many particles in a crowded environment and without an ac-cessible ordered ground state. As mentioned above, the bead-spring polymermodel has been successfully used in other studies [8, 70, 76, 94, 99] to explorethe glass state. To ensure that my simulations capture the glassy physics, Ihave checked that the key signatures of glassy dynamics (see introduction)are present in the simulation. I identified the expected power-law age depen-dence of the relaxation time and logarithmic aging of structural properties(chapter 3), as well as dynamical heterogeneity (chapter 4).The second fundamental problem is the numerical accuracy of the pre-dicted dynamics. MD simulations are based on Newtons equations of motion,and temperature as well as pressure are controlled via well established tech-niques. Although this work is not aiming at characterizing a specific physicalsystem with high numerical detail, but rather to explore general character-istics, trends and dependencies of glassy dynamics, it is nonetheless crucialto accurately estimate the equations of motion in order to successfully sim-ulate the physical processes that drive glassy dynamics. The key numericalapproximation of MD is the discretization of time and a typical time stepis of order 10−14s. Although computational resources are growing exponen-tially, it is currently only possible to simulate on time scales up to ∼ µs. Incontrast to this, at the viscosity threshold for the glass transition structuralrelaxation happens on time scales ∼ 100s. In order to use MD techniques tostudy glasses, it is therefore necessary to choose a model that exhibits glassybehavior on the computationally accessible, rather than realistic time scales.Furthermore, to simulate experimental perturbations like mechanical defor-mation and cooling, one has to increase the rate of change by 4-8 orders ofmagnitude. These non-physical rates, which are used in this study, are a com-mon and legitimate cause of concern for numerical studies of glasses and thereis no simple solution for this problem. The “accelerated dynamics” that is372.6. Hop detectionintroduced via the simple bead-spring model might equalize the disparity be-tween physical and simulated perturbation rates, i.e., the system is deformedmuch faster than in the laboratory, but due to the faster than physical dy-namics it can respond in kind and in a physically correct way. Validation ofthe simulation results again relies on a comparison with experimental results.The project discussed in chapter 3 was designed in close relation to a recentexperimental study, and our results for the structural relaxation time all thequalitative features identified by the experiment. Despite the challenges ofsimulating glassy dynamics with MD there is a large body of computationalwork that study binary [50, 81, 82, 98] and polymeric glasses [38, 70, 74, 99],supercooled liquids [21, 101] and granular media [19, 20] using MD.2.6 Hop detectionA key measurement used in this thesis is the detection of hops, see cagingeffect in section 1.2.1. The detection is based on an adaptation of an algo-rithm proposed by Candelier et al. [19]. The algorithm measures the averagesquared distance between two adjacent parts of the trajectory, and a hopis detected when this distance is bigger than a threshold, which is relatedto the height of the caging regime plateau in the MSD [19, 20] (see fig. 1.2in section 1.2.1). The method was successfully used in studies of agitatedgranular media [19, 20], a supercooled liquid in 2D [21] and more recently ina cyclically sheared glass in 3D [73].The original algorithm is based on a recursive scheme on saved trajecto-ries, and my adaptation is to run the detection “on-the-fly” in the spirit ofa running average. This adaptation is an important technical contributionof this thesis, because it allows the measurement of hops with much bettertemporal resolution.382.6. Hop detectionxzA B A B0.00.20.4Phopthop450 500 550 600 650 700t19.019.419.8zy x(a)(b) (c)(d)(e)Figure 2.5: Sample trajectory of a particle and hop identifier function Phop.(a) shows the localization into two cages and the ’hop’ is marked by rapidchanges in the trajectory (e). Corresponding to this, Phop in (d) is sharplypeaked at the transition and the maximum defines the hop time thop. Plots(b) and (c) are overlays of Phop and the trajectory just before and after thehop (z-comp. only for better visibility). The colored fields (A and B) in bothplots indicate the evaluation window for Phop [see eq. (2.5)]. Initial and finalpositions are calculated from time averages of the trajectory parts that arehighlighted as zoom, i.e., (b) initial position (orange) and (c) final position(cyan).392.6. Hop detectionDuring the simulation run the most recent section of the trajectory of aparticle is stored in Nhist = 20 data points, and every Nobs = 100 time stepsthe oldest point is replaced by the current position. This gives access to atime window teval = NobsNhist ∆t = 15 (the parameter choices are discussedbelow), and the trajectory is separated into two parts A (t− teval, t− teval/2]and B (t− teval/2, t] of equal size. A hop identifier functionPhop(t−teval2)=√〈(rA − r¯B)2〉A·〈(rB − r¯A)2〉B(2.5)is calculated every Nobs time steps that measures the averaged squared dis-tance between the mean position in part A, r¯A, and all trajectory points inpart B, rB, and vice versa. It is large when the trajectory changes rapidly att− teval/2, and a hop is detected when Phop exceeds a temperature dependentthreshold Phop > Pth. The thresholds used in this thesis are given in table 2.3and the parameter choices are discussed further below.In Figs. 2.5(a) and (e) I show a sample trajectory of a particle togetherwith the calculated Phop, fig. 2.5(d). A hop is clearly identified and thealgorithm records particle id, hop time thop and position of the particle beforeand after the hop rinit and rfinal. The hop time is defined at the maximumof Phop, and the locations are calculated from averages as the threshold iscrossed: before the peak rinit = 〈r〉A [see fig. 2.5(b)] and after the peakrfinal = 〈r〉B [see fig. 2.5(c)]. Using this algorithm on each particle via aparallel implementation in LAMMPS [72] allows the monitoring of all hopsin the system for the full duration of the simulation.Irreversible hops In the projects discussed in chapters 5 and 6, I usean additional evaluation step in order to exclude back-and-forth hops of aparticle between the same two positions. This is implemented as a post-processing procedure at the end of the simulation. All hops of a single particleare read from the output of the hop detection algorithm, and two consecutivehops are removed, if the final position of the second hop is within a distanceof√Pth/2 of the initial position of the first hop. With this adaptation, theremaining hops give a full map of the irreversible structural rearrangementsmeasured during the simulation. To assess the robustness of this procedure,the results in chapter 5 were recalculated with a distance threshold√Pth,and no significant changes were found.402.6. Hop detectionT 0.2 0.25 0.3 0.4Pth 0.15 0.18 0.21 0.27Table 2.3: Hop identifier function thresholds used for the different targettemperatures in the glass state.Parameter choices Candelier et al. [19] used a threshold Pth that wassomewhat bigger than the plateau height in the MSD, which I denote asa2c . I reduce the ambiguity in the choice of Pth by measuring a histogram ofpeak-heights in the hop identifier function. If one assumes that a particle ina glass has only two well defined modes of movement: vibration around afixed mean position (caged) and instant jump by a distance > ac (hop), thenthe hop detection would only pick up peaks of height > a2c .In fig. 2.6 the D data (standard time window parameters, see below) is themeasured histogram. We find a distribution of peak heights that is largestat small heights, with a long exponential tail towards larger peaks. Thesmall peaks are due to shifts of the mean particle position within the cages.This is expected, because each member of the surrounding shell that formthe cage is not fixed in space, but moves within a small area (its own cage).Cages are therefore not static, but instead change slightly over time. Themeasured distribution shows a relatively sharp transition to an exponentialtail, which indicates the existence of an additional source of larger peaks, thecage-escapes or hops. I therefore use the crossover point as the threshold,and based on the data shown in fig. 2.6 for the polymer glass at temperatureT = 0.3 the threshold is Pth = 0.21 (dashed line). For comparison, the insetshows the mean square displacement and the horizontal line indicate Pth.Analog measurements were performed for the other glass temperatures andall thresholds used in this thesis are given in table 2.3.The time window teval = NhistNobs ∆t used in the algorithm is definedby two parameters, the number of stored trajectory points Nhist and thefrequency in time steps Nobs at which the trajectory is updated and Phoprecalculated. The update frequency used in this thesis is Nobs ∆t = 0.75,which is the mean time between particle collisions indicated by the onset ofthe caging plateau in the inset of fig. 2.6 (dotted line). This also sets themaximal temporal resolution for the hop time. The time window teval is themain influence on the detection algorithm. It sets the maximal resolution of412.6. Hop detection0.0 0.2 0.4 0.6 0.8Phop101102103104105number of hopsteval=15teval=15teval=15teval=7.5teval=3010-210110410-410-2100Figure 2.6: Histograms of hop identifier Phop peak heights calculatedat glass temperature T = 0.3 and time window parameters (order oflegend):(Nhist, Nobs) = (20, 100), (40, 50), (10, 200), (10, 100), (40, 100). Thedata for the three parameter sets with teval = 15 are nearly identical (blueD overlay other markers). The solid line is an exponential fit of the distri-bution tail and the dashed line indicates the resulting parameter choice Pth.The inset shows the mean square displacement as function of time and Pthis indicated by the horizontal line.two consecutive hops (teval/2) and also acts as an upper limit for the durationof a hop. To maximize the resolution but also ensure that meaningful aver-ages are possible [see eq. (2.5)], I choose Nhist = 20 and therefore teval = 15.In fig. 2.6 I illustrate the impact of this parameter. Note that the histogramfor the present parameter values (D) coincides with two other parameter setswith equal teval (histograms are nearly identical and therefore difficult to dis-tinguish). This shows that only the window size has a direct impact on thehop detection. Consequently, a smaller time window yields the detection ofmore peaks with small height, and a larger teval results in fewer small peaks,but also in a lower resolution.422.7. Vibrational modes and the softness field2.7 Vibrational modes and the softness fieldThe low energy vibrational spectrum is used to calculate a softness field thatquantifies the participation of particles in quasi-localized vibrational modes,see discussion in section 1.2.4. During the simulation run, snapshots of thesystem are stored in logarithmically spaced time intervals. The softness fieldat time t is measured in four steps:1. Starting with a snapshot of the system at time t, a combination ofgradient descent and damped dynamics (FIRE [15]) algorithms are usedto minimize the forces present in the system. Both algorithms areimplemented in LAMMPS [72]. The resulting spatial configuration ofthe particles is the inherent structure.2. The Hessian [see eq. (1.1)] is calculated with the model potentials andthe inherent structure. The result is a 3N by 3N symmetric sparsematrix.3. The low energy vibrational spectrum is calculated by partially diago-nalizing the Hessian using ARPACK [87]. I am using the solver for areal symmetric matrix in the shift-inverted mode with shift parameterset to zero. Arpack is an implementation of the Implicitly RestartedLanczos Method [87], which is a numerically reliable algorithm for theestimation of eigenmodes that is based on the power method for findingeigenmodes. It exploits the idea that the result of repeated multiplica-tion xn = A · xn−1 of an initially random vector x0 with the matrix Aconverges to the eigenvector of the largest eigenmode of A.4. The Nm lowest energy eigenmodes with non-zero eigenvalues are usedto calculate the softness field. The definition of the softness field isexplained in detail below.An analysis of the maximal cross-correlation (see eq. (5.1) in chapter 5)between hops and softness field (see eq. (2.6) below) as a function of Nmreveals a broad and weak maximum between 300 > Nm > 900 (1-3% ofthe modes). For the results discussed in chapters 5 and 6 I use Nm = 600.Already Nm = 300 yields 95% of the quantitative accuracy.432.7. Vibrational modes and the softness fieldSoftness field definition The softness of a particle is defined as thesuperposition of the participation fractions in the low energy vibrationalmodes [68, 102, 103],φi =1NmNm∑j=1|e(i)j |2 . (2.6)Here, the polarization vector e(i)j is the projection of the eigenvector of modej on the degrees of freedom of particle i, see section 1.2.4 for more details.The softness field φ depends on a single parameter, the number of includedlow energy modes Nm, and the scaling factor is added to make the softnessan intensive quantity in terms of Nm. A particle i is considered “softer” thelarger φi is, and it is used to rank the particles according to their relativesoftness. The absolute value of φ is not in itself meaningful, since the partici-pation fractions are normalized quantities, i.e.,∑Ni=1 |e(i)j |2 = 1. The softnessof a particle therefore describes its average participation fraction in the Nmlowest energy vibrational modes.It is worth pointing out that the contribution of each mode should beweighted by its energy. A particle that is involved in a low energy moderequires less energy to be excited to the same extent than a particle withidentical projection |e(i)|2 in a higher frequency mode. However, I find thatthis weighting does not improve the predictive strength of the softness fieldfor the present system, since the frequency does not vary strongly for thecontributing modes. More details about the effect of this weighting as wellas alternative definitions of softness are discussed below.Direction of the softness field Directional information can be added tothe scalar softness field in a similar superposition scheme. The eigenvectorof mode j defines the mode direction for particle i via the polarization vectore(i)j . This vector is defined up to its sign due to the harmonic nature of thedescription. In order to find the dominating direction in the Nm modes, Icalculate nematic tensors from the unit length projection vectors (Q(i)j )α,β =(32 eˆ(i)j,αeˆ(i)j,β −12δα,β)and perform a weighted averageQ(i)φ =∑Nmj=1 |e(i)j |2Q(i)j∑Nmj=1 |e(i)j |2.442.7. Vibrational modes and the softness field05101520φ/〈φ〉10-710-510-310-1101P(φ/〈φ〉)eq. (2.6)eq. (2.7)eq. (2.8)Figure 2.7: Distribution of softness in the system T = 0.3, tage = 750 forthree definitions of the softness field. The softness is rescaled in terms of themean softness to allow an easier comparison.The eigenvector of the largest eigenvalue of Q(i)φ then defines the directionof the softness field e(i)φ . The direction of the softness of a particle thereforeindicates the mean direction of the Nm lowest energy vibrational modes.2.7.1 Alternative definitionsThe definition of a field that quantifies the participation of particles in softmodes is not unique. The superposition of participation fractions used hereand in several other studies [68, 102, 103] essentially measures the averagepotential energy of the particles. To see this, note that the participationfractions distribute the total potential energy 〈Uj〉 of mode j over particles iso that 〈U (i)j 〉 = 〈Uj〉|e(i)j |2, and 〈...〉 denotes a thermal rather than disorderaverage. The softness field defined in eq. (2.6) is therefore proportional to themean potential energy of particle i in the Nm lowest energy modes, assumingequipartition of the mode energies. However, as mentioned above, participa-tion in lower energy modes should be more important than in higher energymodes, since less energy is needed to displace a particle from its inherentstructure position. In this view, the softness of a particle becomes propor-tional to its mean squared vibrational amplitude 〈x(i)2j 〉 = 2〈U(i)j 〉/mω2j , and452.7. Vibrational modes and the softness fieldthe softness field becomesφi =1NmNm∑j=1|e(i)j |2mω2j. (2.7)Finally, one may ask why the mean squared vibrational amplitude should beconsidered rather than the mean absolute or root-mean-squared amplitude.This leads to a third alternative for the softness fieldφi =1NmNm∑j=1|e(i)j |√mω2j. (2.8)Figure 2.7 compares the distributions of softness resulting from these threealternative definitions. The weighting w.r.t. mode energy given in eq. (2.7)stretches the distribution obtained from eq. (2.6) without changing it qual-itatively. Using the average displacements as measures on the other handyields a qualitatively different, purely exponential distribution of softness.This exponential form reminds of the self-part of the van Hove function,which measures the distribution of particle displacements over a fixed timewindow. A signature of dynamical heterogeneity is the non-Gaussian, expo-nential tail found in the van Hove function, and the softness field based oneq. (2.8) seems to hold a fingerprint of this characteristic feature of glassydynamics. I also compared the spatial and directional correlation of the soft-ness based on eqns. (2.6)-(2.8) to hops. The results are remarkably insensitivequalitatively as well as quantitatively, with eq. (2.8) yielding a slightly bet-ter spatial correlation. I chose to use the definition in eq. (2.6) for ease ofcomparison with previous studies in other systems.46Chapter 3Recovery from mechanicaldeformation1This first project studies the complex interplay between physical aging andmechanical deformation in glasses, specifically the phenomenon called me-chanical rejuvenation. A detailed explanation is given in section 1.2.2, insummary: aging leads to a slowing of the molecular mobility, which can bemeasured as an increasing bulk structural relaxation time τα with growingage. One can therefore think of τα as an internal “material clock”. Mechan-ical deformation decreases τα and this reversal of the aging effect results ina system that is dynamically equivalent to a younger glass. Whether thedeformed glass is truly comparable to a younger glass is still an open ques-tion [64], with computational studies of the potential energy landscape [54,62] suggesting that deformation drives the system into a state that is differentfrom a glass of less age.A recent experiment [58] explored the behavior of polymer glasses in therecovery regime after the end of a creep deformation. A fluorescence mi-croscopy technique was used to observe changes of the molecular mobilityover time and the results in the recovery regime were compared to the agingbehavior of a quiescent glass without deformation history. The experimentfound that pre-yield deformation results in a transient reduction of τα fol-lowed by a quick return to pre-deformation behavior, while the molecularmobility of a glass after deformation far in the post-yield regime is compara-ble with an as quenched glass.This project uses computer simulations to give a complementary pictureof glasses in the recovery regime. The glass was simulated in the agingregime at the temperature T = 0.2, which is far below the glass transitiontemperature Tg ∼ 0.35 [76]. In addition to the molecular mobility I monitorstructural as well as energetic quantities to clarify whether deformation drivesglasses to a state that is comparable to a younger glass without deformation1Large parts of this chapter have been published in ref. [84]473.1. Immediate impact of creepFigure 3.1: Schematic protocol of the simulation with waiting time periodtw, uniaxial tensile creep (a) or fixed strain rate (b) deformation period ts,and recovery period tr.history. In analogy to the experiment, the simulations are designed in threestages, shown in fig 3.1(a): The newly quenched glasses are aged at zeropressure in the quiescent state for a waiting time tw. This is followed bya creep period ts at constant tensile stress σ, and the recovery regime trafter unloading. At the beginning and end of the creep period, the stress isquickly ramped up and down to avoid unphysical discontinuities. I study thequasi-adiabatic limit of negligible aging during deformation by restricting thedeformation time to ts < tw. To study the effect of deformations beyond theyield point at strains ∼ 6%, I also use the constant strain rate ˙ protocolshown in 3.1(b).3.1 Immediate impact of creepWhen tensile stress is applied to a glass, the material reacts first with anelastic response followed by plastic deformation. In fig. 3.2(a) I show theplastic part of the creep complianceJ(t, tw) =(t, tw)σof the glass for a range of imposed stress amplitudes σ. The engineeringstrain = ∆L/L is calculated along the z-direction and the values at ts =483.1. Immediate impact of creep101102103104ts0.000.020.040.060.08Jz(a)σ=0.1 ǫ=0.0%σ=0.2 ǫ=0.2%σ=0.3 ǫ=0.4%σ=0.4 ǫ=0.7%σ=0.5 ǫ=1.6%σ=0.6 ǫ=5.5%10-1101103105t0.50.60.70.80.91.0CSqtr=0(b)tr=7.5 ·104Figure 3.2: Panel (a) shows the creep compliance of a glass with age tw =75000 for different external stresses σ. The indicated strains are calculatedat ts = 15000. Panel (b) shows the intermediate scattering function CSq ofthe perturbed glasses shown in (a) in the recovery regime together with ajust-quenched glass (F). The main graph shows CSq just after unloading(tr = 0) and the inset is calculated at tr = 75000. The dashed lines definethe α-relaxation time.493.2. Recovery of the relaxation time15000 are given in the legend. It is clearly visible that larger stress yields amuch larger creep compliance, indicating a non-linear response. Figure 3.2(a)shows results for a glass with age tw = 75000, and for increasing wait timesI observe (not shown here) the well known double logarithmic shift towardssmaller compliance as discussed for instance in ref. [99].Figure 3.2(b) shows the self part of the intermediate scattering function(ISF)CSq (t, ta, tr) = 〈exp(iq · [r(t+ ta + tr)− r(ta + tr)])〉 ,which is a standard measure of relaxation times in aging glasses [50]. Here,the average is performed over all beads at a certain time, r(t) is the position ofa bead at time t and the wave vector q = 2piez corresponds to a displacementof a along the deformation axis. The glass age after unloading is given byta = tw + ts. In glasses, the ISF shows three regimes that are identicalto those discussed for the mean square displacement in section 1.2.1: Aballistic regime at very small time scales is followed by a caging plateauat intermediate times, and the decay at large times indicates the diffusiveregime. Aging increases the intrinsic time scale of the glass, and the resultinglengthening of the plateau is visible by comparing the inset and main plot offig. 3.2(b). I define the structural relaxation time τα as the time when the ISFhas decayed below the caging-plateau or CSq (τα, ta, tr) = 0.8. In fig. 3.2(b),this is indicated by dashed lines.The main graph in fig. 3.2(b) shows the ISF measured immediately afterunloading at tr = 0 for the systems in panel (a) that were deformed atstresses σ = 0.2, 0.4, 0.5, 0.6. Additionally, the ISF of a newly quenchedglass is shown (F). The mechanical perturbation yields a shortening of thecaging plateau and a corresponding shift of τα towards smaller times, whichagrees with previous studies [48, 88, 99]. The relaxation curve for = 5.5%indicates that this glass relaxes even faster than the just-quenched glass.The reason for this is that the later already has an effective age of ∼ 103,which it acquires during the quench. The inset shows that the differencesin the relaxation behavior disappear after a long recovery period. In thefollowing, I clarify whether the impact of mechanical perturbation is transientor permanent by studying the path of recovery.503.2. Recovery of the relaxation time05101520 25 30t (104)102103104105106ταtwtstrσ=0.2σ=0.4σ=0.5σ=0.6newFigure 3.3: Evolution of α-relaxation time of a just-quenched “new” glasscompared to those of mechanically perturbed glasses in the recovery regime.The glasses are identical in the initial aging regime tw = 75000, the creeptime is ts = 15000, and the time regimes are indicated by the dashed lines.Other results from simulations with these parameters are shown in fig. 3.2.3.2 Recovery of the relaxation timeThe evolution of the α-relaxation time after unloading holds key informa-tion about the change in the aging dynamics. Lee and Ediger showed ex-perimentally that τα measured in the recovery regime can be shifted ontothe relaxation time of an unperturbed sample [58], and this behavior wasassociated with mechanical rejuvenation. Furthermore, they repeated theexperiment with a sample deeper in the glass phase and found merely tran-sient changes. The relaxation time did not behave like that of a new glass,but rather returned to the original unperturbed evolution after a short time.In fig. 3.3 I show the relaxation time of an unperturbed glass together withτα in the recovery regime after mechanical perturbation with several stressamplitudes. In all cases I find that the relaxation time immediately afterunloading decreases with increasing stress amplitude. For σ < 0.5 my resultsshow a rapid return to the unperturbed evolution, which suggests a non-permanent change of the aging dynamics. In agreement with ref. [58], ταfor larger stresses σ ≥ 0.5 exhibits dynamics very similar to a just-quenchedglass but shifted in time.513.2. Recovery of the relaxation time10-210-1100101102tr/ta100101102103τα/τ0α(ta)102103104105106103104105106twFigure 3.4: Relaxation time dynamics of unperturbed glasses with a rangeof wait times 103 ≤ tw ≤ 105 in a double logarithmic plot. The continuousblack line is the “generic” aging dynamics τ 0α and in the inset (axes: τα overtr) I show the shift of τα with increasing tw. The main graph shows a datacollapse with tr measured in units of glass age ta = tw and τα being rescaledby τ 0α(ta). The dashed lines are guides to the eye.523.2. Recovery of the relaxation time10-410-2100102tr/ta10-310-210-1100101102τα/τ0α(ta)0123456ǫ(%)Figure 3.5: Relaxation time of mechanically perturbed glasses in the recoveryregime. The coloring represents the total strain at time of unloading. Thecontinuous line indicates the generic aging dynamics and the dashed lines areguides to the eye.To illuminate this transition from transient to permanent change, I com-pare the recovery paths of perturbed glasses in a wide parameter range tothose of unperturbed control samples. I calculate the τα dynamics of a just-quenched glass by averaging results of six simulations. I find the knownpower law behavior with aging exponent µ ' 0.89 and refer to it as the“generic” aging scenario τ 0α ∝ tµw [26, 50] (0 indicates generic behavior). Infig. 3.4 the generic aging is indicated as a continuous black line and the insetshows the relaxation time of unperturbed glasses: The initial value dependson the wait time and the power law behavior is seen for t > tw. In the maingraph I collapse this data by rescaling with the total age ta = tw + ts and τ 0αat this age. The dashed lines are guides to the eye and illustrate the idealpath of unperturbed glasses until the power law behavior is resumed at theirintersection. Changes introduced by mechanical perturbation can thereforebe seen as departure from this expected path. The applied rescaling allowsme to compare glasses of all ages and with varying time intervals of stressapplication ts.533.2. Recovery of the relaxation timeIn fig. 3.5 I show the evolution of the relaxation time of mechanicallyperturbed glasses in the recovery regime. The results were gathered fromsimulations with parameters 103 ≤ tw ≤ 105 and 0.1 ≤ σ ≤ 0.6. I find thatthe magnitude of change from the unperturbed behavior is best characterizedby the total strain at the end of creep - not the stress amplitude. Note thatthe further away τα is from the horizontal dashed line, the stronger the impactof the perturbation, and the larger the corresponding strain. Indeed one canidentify three recovery behaviors in fig. 3.5 which are indicated by coloring:The green curves ( ≤ 1%) are in close proximity to the dashed line andthe relaxation time stays constant at timescales smaller than the glass ageta. This means that creep with small total strain has little impact on theglass and the behavior essentially equals that of an unperturbed sample. Atintermediate strains 1% ≤ ≤ 3% (blue curves) the initial τα is at a lowervalue but it begins to evolve before reaching the black line. The merger withthe generic glass behavior however, only takes place at the intersection ofthe dashed lines. This indicates timescales around the original glass age andthe implications are discussed in the next paragraph. Finally, the red curvesindicate that the largest strains ≥ 3% yield the lowest τα. Remarkably, therelaxation times of these strongly perturbed glasses do not begin to evolveimmediately but stay approximately constant until reaching the generic agingdynamics which is then followed. This is the behavior that is expected for ayounger glass, i.e., I observe mechanical rejuvenation.To better understand the recovery behavior observed in fig. 3.5, I showa sketch of idealized paths in fig. 3.6. As before, the diagonal black lineindicates the generic aging dynamics and path (o) would be followed by anunperturbed glass. I have shown that mechanical perturbation results in ashift towards lower τα, which is the starting point of possible recovery scenar-ios: path (a) is a fast return to the original (unperturbed) aging trajectory.This is and idealized example of a memory effect and not observed in thesimulations. The glass remembers its age and the appropriate relaxation timeis recovered on timescales that are small compared to the aging dynamics.Path (b) also exhibits dynamics on smaller timescales, but it merges with theoriginal aging trajectory at a later time. Indeed the intersection is at timesof the order of the glass age, whereupon the power law behavior is resumed.This timing shows that the original age is still remembered by the perturbedglass. By comparing fig. 3.5 and fig. 3.6 it is easy to see that glasses withintermediate total strain (blue) follow this path. Similarly, the recovery fromlarge strains (red) is found to follow the third idealized trajectory (c). The543.3. Recovery of the local structureFigure 3.6: Sketch of possible recovery paths of the α-relaxation time. Line(o) is the path of an unperturbed glass and lines (a)-(c) are possible pathsof glasses after mechanically perturbation. The diagonal black line indicatesthe generic aging dynamics.relaxation time essentially stays constant until the generic aging behavior isreached. The merger happens at times that are smaller than the original age,which apparently indicates a loss of memory, i.e., mechanical rejuvenation.However, it is important to realize that no curve in fig. 3.5 truly enters theregion to the right of the black line. The reason for this lies in the non-equilibrium nature of glasses. The evolution of τα is driven by the relaxationtowards equilibrium and a given value can only become stable when thermo-dynamic equilibrium is reached. Since for an ideal glass this state is neverreached, the black line acts as an upper time limit for the impact of memoryeffects, which is the recovery of remembered age and corresponding τα. Fur-thermore, dynamics on very small timescales (ballistic regime) can not alterthe relaxation time, because collective motion is required. This means thata lower initial τα gives the glass less time to act on its memory. As a result,a strongly perturbed glass that has a small initial relaxation time but thatstill remembers its original age behaves very similar to a truly rejuvenatedglass when viewed only from this dynamical perspective.553.3. Recovery of the local structure10-410-2100102tr/ta-0.3-0.10.10.3∆〈Vv〉〈V0v(ta)〉(%)0123456ǫ(%)Figure 3.7: Percentage change of the rescaled average Voronoi cell volumewith respect to the “generic” value ∆〈Vv〉 = 〈Vv(t)〉 − 〈V 0v (ta)〉 at glass ageta = tw + ts. Shown in the main graph is the evolution of mechanicallyperturbed glasses in the recovery regime. The coloring represents the totalstrain at time of unloading. The continuous line indicates the generic agingdynamics and the dashed lines are guides to the eye. The inset shows theevolution of the average cell volume for just-quenched glasses and the blackline is a logarithmic fit that yields the generic aging dynamics.563.3. Recovery of the local structure3.3 Recovery of the local structureIdentifying structural quantities that are equally sensitive to aging has provento be difficult. No power-law behaviors similar to that of the α-relaxationtime are known. I am, however, able to find measures of the local structurewith logarithmic dependencies on the glass age. I calculate a variety of suchquantities to better characterize the recovery path after deformation and theunderlying thermodynamic state. Similar to the relaxation time analysis, Iapproximate the “generic” aging behavior (indicated via superscript 0) andcompare this to the evolution after deformation. To perform a combinedanalysis for varying stress amplitudes and glass ages (see section 3.2 forparameter ranges) I use the same rescaling approach as for τα. Note thatin the case of logarithmic dependencies, the rescaling yields a change of thegeneric aging slope (continuous black line in fig. 3.7 and following). I stilluse the approach, however, because the evolution during aging and thereforethe change of the slope is small.I first study the decrease of local volume during aging in the NPT en-semble by performing a Voronoi tessellation on configuration snapshots andcalculating the average cell volume 〈Vv〉 [78]. The Voronoi tesselation dividesspace into cells around the particle locations, and each cell is that part ofspace that is closest to a single particle. In the main graph of fig. 3.7 Ishow the recovery paths after creep for the same simulations as in fig. 3.5.As before, the horizontal dashed line indicates the expected path of an un-perturbed glass until the intersection with the continuous black line, whichindicates the generic aging behavior. The calculation of the generic aging〈V 0v 〉 is shown in the inset of fig. 3.7 and this is exemplary for all quanti-ties with logarithmic dependency on the glass age: I combine the simulationdata of six just-quenched glasses and fit for the slope of the emerging loga-rithmic behavior. In the main graph, the coloring again indicates the totalstrain at time of unloading. One can clearly see that the deviation from theunperturbed path increases with increasing strain, which agrees with the re-sults from the dynamical perspective. I also observe the merger of red (largestrains) recovery paths with the continuous line at timescales smaller thanta, indicating erasure of memory.In the absence of long range order, which is the case for glasses, the localspatial order is important for the characterization of the thermodynamicstate. I calculate three measures of the local structure that are sensitiveto aging and their recovery paths are shown in fig. 3.8. Since I focus on573.3. Recovery of the local structure-0.10.00.1∆〈cv〉〈c0v(ta)〉(%)(a)-1.0-0.50.00.51.0∆〈S〉〈S0(ta)〉(%)(b)10-410-2100102tr/ta-1.5-0.50.51.5∆〈λ1,3〉〈λ01,3(ta)〉(%)(c)Figure 3.8: (a) Difference in the coordination number ∆〈cv〉 = 〈cv(t)〉 −〈c0v(ta)〉, (b) triangulated surface order parameter ∆〈S〉 = 〈S(t)〉 − 〈S0(ta)〉,and (c) evolution of the eigenvalue difference ∆〈λ1,3〉 = 〈λ1,3(t)〉 − 〈λ01,3(ta)〉.All three quantities are rescaled percentage changes with respect to the“generic” value at glass age ta = tw + ts. The coloring represents the totalstrain at time of unloading (see e.g., fig. 3.7). The continuous line indicatesthe generic aging dynamics and the dashed lines are guides to the eye.583.4. Recovery in the potential energy landscapemechanical rejuvenation and because the data is relatively noisy, only theresults for strains ≥ 2.5% are shown. In panel (a) one can see the evolutionof the average coordination number 〈cv〉, i.e., the number of beads that formthe nearest neighbor shell which is calculated using the Voronoi tessellation.I find that the number of participating beads is decreasing with increasingglass age and that an increase in strain leads to larger coordination numbers.In panel (b) I show the recovery paths of the triangulated surface orderparameterS =∑q(6− q)νq ,which is sensitive to short range order and often used in the study of amor-phous metals [82, 99]. Here q is defined for each bead in the coordinationshell as the number of nearest neighbors that are also part of the shell, andνq is the number of beads in the shell that have the same q. This orderparameter decreases during aging, which is a sign of increasing order andpacking fraction. I find that 〈S〉 is larger after mechanical deformation andthat the recovery path for large strains is similar to that of younger unper-turbed glasses. The third structural quantity that is shown is the averageddifference between the biggest and the smallest eigenvalue of the moment ofinertia tensor of the cages 〈λ1,3〉 = 〈λ1 − λ3〉. The Voronoi tessellation iden-tifies the cage for each bead and after setting the origin of the coordinatesystem to the position of the central bead, I calculate the tensor using thepositions of all beads that form the cage. I find that all eigenvalues decreasewith glass age, and more importantly that all differences exhibit a logarith-mic dependency as well. This means that the aging dynamics manifest ina change of the cages towards a more spherical shape. In fig. 3.8(c) I onlyshow the evolution of 〈λ1,3〉 and the just discussed “generic” aging behavioris again indicated by the continuous black line. The observed paths resem-ble the expected behavior of younger unperturbed glasses, which indicatesmechanical rejuvenation. All of these measures of local order show the nowfamiliar dependence on strain, i.e., increase in strain yields further distancefrom the unperturbed aging path.593.4. Recovery in the potential energy landscape10-410-2100102tr/ta-0.6-0.4-0.200.20.40.6∆〈umin〉|〈u0min(ta)〉|(%)0123456ǫ(%)Figure 3.9: Percentage change of the difference between the minimizedpotential energy of a mechanically perturbed glass ∆〈umin〉 = 〈umin(t)〉 −〈u0min(ta)〉 and the “generic” behavior at glass age ta = tw + ts. Shown isthe evolution of the potential energy landscape in the recovery regime. Thecoloring represents the total strain at time of unloading. The continuous lineindicates the generic aging dynamics and the dashed lines are guides to theeye.3.4 Recovery in the potential energylandscapeThe potential energy is another quantity that is sensitive to aging, and thedepth of the occupied minima in the potential energy landscape has beenthe focus of previous studies [54, 62, 92]. I explore this third perspective onthe recovery paths by investigating the evolution of the inherent structureenergy 〈umin〉 in the recovery regime. The inherent structure is the particleconfiguration in the zero temperature limit. The calculations are performedusing a gradient descent algorithm on spatial configuration snapshots, whichyields results in agreement with the potential energy after a very fast quenchto zero temperature. In analogy to the analysis of the local structure (seesection 3.3), I calculate the “generic” aging behavior 〈u0min〉 and rescale therecovery paths of simulations with varying stress amplitudes and glass ages603.5. Recovery after constant strain rate deformationto allow a direct comparison.In fig. 3.9 I show the recovery of the inherent structure energy of mechan-ically perturbed glasses after unloading. As before, the horizontal dashedline marks the expected path of an unperturbed glass and the continuousblack line is the generic aging behavior. The shown energies are calculatedsolely from contributions of non-bonded beads, since the stiff springs betweenbonded monomers do not age. I find that the depths of the minima decreasewith increasing total strain, which is indicated by color code. The greencurves ( ≤ 1%) closely follow the horizontal dashed line and cross over tothe generic behavior at times of the order of the glass age. This indicatesthat the mechanical perturbation was too weak to yield changes in the ag-ing dynamics. For intermediate strains 1% < < 5% (blue and purple) Iobserve a decreased initial depth of the minima 〈umin〉 and the onset of dy-namics at times smaller than ta. However, in agreement with the relaxationtime dynamics and the evolution of the local structure, I find that the mergerwith the generic dynamics takes place at timescales similar to the glass age.The recovery paths at larger strains (red) resemble the expected paths ofyounger unperturbed glasses: An initially constant behavior is followed bythe crossover to the generic aging at timescales smaller than the original ageof the glass, suggesting mechanical rejuvenation.3.5 Recovery after constant strain ratedeformationDeformations in the creep protocol are restricted to around the yield point,because the onset of flow introduces numerical instabilities in the barostat,see also section 2.3. I extend the range of analyzed strains by deformingat a constant strain rate as shown in fig. 3.1(b), and results in the recoveryregime are shown in fig. 3.10. In close analogy to fig. 3.5, the α-relaxationtime in panel (a) decreases with increasing total strain. Although results fordifferent strain rates and wait times again collapse for the same amount ofstrain, I note that the α-relaxation time begins to change noticeably onlyfor strains greater than ∼ 2%. This observation points to some differencesbetween the two deformation protocols that require further investigation.Focusing on the regime of strains larger than 6%, I find that the recoverypaths saturate and cannot enter the region to the right of the generic aging613.5. Recovery after constant strain rate deformation0 3 6 9 121518ǫ (%)10-310-210-1100101102τα/τ0α(ta)(a)-0.3-0.10.10.30.5∆〈Vv〉〈V0v(ta)〉(%)(b)10-410-2100102tr/ta-0.400.40.8∆〈umin〉|〈u0min(ta)〉|(%)(c)Figure 3.10: Evolution of (a) α-relaxation time, (b) Voronoi cell volume, and(c) inherent structure energy in the recovery regime after deformation withconstant strain rates ˙ = 10−5 and 10−6 and two waiting times (glass ageta = tw + ts). (b) and (c) are percentage changes as defined in fig. 3.7 andfig. 3.9, and the coloring represents the total strain at time of unloading. Thecontinuous lines indicate the generic aging dynamics and the dashed lines areguides to the eye.623.6. Conclusionscurve (thick solid line) as explained before. By contrast, I show in panel(b) that the Voronoi volume can be driven beyond the range covered byunperturbed aging dynamics. This is clearly visible for strains > 12%,as the recovery paths overshoot the generic aging curve. It signifies thatI find cell volumes at time scales at which unperturbed aging would haverelaxed them to smaller sizes. With regard to the parameters describing localorder (see fig. 3.8), I find a similar overshoot in the recovery curves of theeigenvalue differences, but not for the coordination number and triangulatedsurface order parameter (not shown). I suspect that these quantities wouldalso begin to show overshoots for even larger deformations.Finally, the inherent structure energy shown in panel (c) also exhibitsa recovery path that enters the regime inaccessible to unperturbed aging.This crossing of the generic aging behavior indicates that strongly perturbedglasses ( > 12%) remain at a point in the potential energy landscape for atime period that would be sufficient for an unperturbed system to relax toa lower energy minimum. This implies that the state after deformation isdistinct from that of a younger glass.3.6 ConclusionsI investigate the recovery of polymer glasses from uniaxial tensile creep de-formation via molecular dynamics simulations. The impact of mechanicalperturbation on the aging dynamics is analyzed by comparing the recoverypaths after deformations of different duration and stress amplitude to the be-havior of unperturbed control samples. The evolution of α-relaxation time,inherent structure energy and measures of local spatial order are monitored,and they give three different perspectives on perturbed glasses. In the regimeof negligible aging during deformation, all the data suggests that the impactof deformation on the aging behavior is described solely by the total engi-neering strain at the end of the deformation, i.e. an increase in strain yieldsa recovery path further away from the unperturbed control path. I find aclear progression from seemingly no impact at ≤ 1% to transient changesat 1% ≤ ≤ 3 − 5% and finally permanent alterations of the glass historyat larger strains. This transition from “transient rejuvenation” to full “era-sure” of aging is continuous and the onset of permanent changes is aroundthe yield strain [76].I propose to distinguish between transient and permanent changes of the633.6. Conclusionsglass memory by identifying the timescale at which a perturbed glass recov-ers the aging behavior known from unperturbed samples. If this timescale isaround the full age ta = tw + ts, then the glass has preserved its memory. Onthe other hand, if the timescale is smaller than ta, then the glass history hasbeen altered. I visualize this measure of memory effects in fig. 3.6. For sam-ples deformed via tensile creep to strains of 3−6%, all quantities indicate per-manent rejuvenation as defined above. This agreement of dynamical, struc-tural, and energetic criteria suggests that the underlying thermodynamicstate is indeed similar to that of a younger glass. Simulations at constantstrain rate were used to perform deformations beyond the yield strain. Theresults for the recovery after post-yield deformation show a distinct behavior:The evolution of the α-relaxation time still suggest rejuvenation and a stateequivalent to a younger glass. Structural and energetic quantities, however,can be can be driven beyond the range accessible to younger glasses. Thisindicates that a distinct thermodynamic state is reached after deformationsin the post-yield regime.The creep deformation protocol used here closely followed that of recentexperimental studies [58]. The results are in agreement with earlier studiesin so far, as in observing transient changes of the aging dynamics in thesub-yield regime and permanent changes in the post-yield regime. Addition-ally, I show that the transition towards permanent mechanical rejuvenationis continuous, and that its only control parameter is the total strain at theend of the deformation. This finding supports an earlier study that reportsthe importance of strain in describing accelerated dynamics during deforma-tion [100].64Chapter 4Spatio-temporal correlation ofstructural relaxation events2This second project explores dynamical heterogeneity (DH) in polymer glass,and this central feature of glassy dynamics is discussed in detail in sec-tion 1.2.3. The dynamical activity in glasses is heterogeneously distributed,with some regions undergoing rapid structural rearrangements, while otherregions remain structurally mostly static. The heterogeneity manifests in co-operative motion of rearranging particles [33, 51], and four-point correlationfunctions have been used to quantify DH in terms of a dynamical suscepti-bility χ4 [9, 53], which is proportional to the volume within the glass thatexhibits correlated dynamics.This study gives a new, spatially resolved perspective on DH by reduc-ing the particle motion to local structural relaxation events. As discussedin section 1.2.1, the particle dynamics in glasses is dominated by vibrationsin metastable cages that is interrupted by rearrangements of the local struc-ture. In section 2.6 I introduce a new algorithm that detects rapid changesin the particle trajectories, which is the signature of cage escapes or hops.Using this algorithm I can record a list of all hops that happen during asimulation run. Each hop is characterized by a time, the initial and final po-sition of the particle (see section 2.6 for definitions) and the particle index.In fig. 4.1(a) I show a snapshot of the whole system followed in fig. 4.1(b)by a reduced picture where only particles are shown that hop at that timestep. The comparison highlights the sparseness of the effective dynamics,showing a reduction from 50000 to four particles. By merging all hops thatwere detected in a time window of 3000τLJ I directly reveal in fig. 4.1(c) theheterogeneous distribution of hops and their grouping into clusters, i.e. Idirectly show the dynamical heterogeneity in the glass resolved to individualstructural relaxation events.The study focuses on glasses in the quiescent state at three tempera-2Large parts of this chapter have been published in ref. [85]65Chapter 4. Spatio-temporal correlation of structural relaxation eventsFigure 4.1: Snapshots of a single configuration showing (a) all particles and(b) only those particles that are in the middle of a hop. There are onlyfour hops at that time step and their positions are highlighted by arrows.(c) shows all hops that are detected in a time window of 3000τLJ . Theconfigurations are taken from a glass at T = 0.2 and age ta = 20000.tures T = 0.2, 0.25, 0.3 (Tg ' 0.35 [76]) in the aging regime. To increasethe accuracy of the analysis I performed 11 independent simulation runs fortemperatures T = 0.25, 0.3 and I report averaged results with error bars in-dicating the standard error. The results are separated into two parts. In thefirst section, I analyze the hop statistics and the dependence on temperatureand age. Hop frequency, persistence time in cages, hop distance as well asdirection are discussed, and a comparison to earlier studies is made that useddifferent detection algorithms and monitored hops on subsets of the parti-cles in the simulation. The main result of this section is the calculation ofthe spatio-temporal density-density correlation of hop events. I furthermorequantify collaborative motion in the polymer glass and identify temperature-and age-dependence. In the second part, I focus on DH in the aging regime.This regime has only recently been explored using three- and four-point cor-relators [17, 71], finding indications for an increase of dynamical correlationwith age. On the basis of our hop detection, I calculate the four-point dy-namical susceptibility χ4 as the standard measure for correlated dynamics664.1. Statistical properties of hops101102103104105106τ10-910-810-710-610-510-410-310-2p(τ)−1.3−1.6(a)T=0.2T=0.25T=0.30.0 0.5 1.01.5|d|10-410-310-210-1100101p(|d|)(b)Figure 4.2: (a) Distribution of persistence time τ of cages at three tempera-tures. The solid lines indicate power laws. (b) Distribution of hop distances|d|, i.e., the distance between old and new cage. Exponential fits are indi-cated by the solid lines and the vertical dashed lines indicate√Pth and 2√Pthfor the respective temperatures; see legend in panel (a). Error bars in bothplots are smaller than the markers and are omitted for visibility.and directly compare it with the aggregation of hops into clusters, which isthe manifestation of DH in the hop picture. With this spatial resolution ofDH I calculate the full volume distribution of hop clusters and give a measureof their compactness.4.1 Statistical properties of hopsThe non-equilibrium nature of glasses gives rise to a continuously slowingstructural relaxation with increasing age. The caging plateau in the meansquared displacement, see section 1.2.1, extends to longer times with increas-ing age, which suggests an increasing time that a particle remains in its cage.This persistence time is measured as the time between two consecutive hopsof the same particle. Aging is caused by a broad distribution of persistencetimes and previous studies have found a power law p(τ) ∝ τµ with exponent−1 > µ > −2 in simulations of polymer glasses and binary mixtures [98].In a very recent study this broad distribution was also found in simula-tions of a strong glass former [97], although with exponent −0.3 ≥ µ ≥ −1.674.1. Statistical properties of hops102103104105106tage10-1100101fhopT=0.2, µ=−0.12T=0.25, µ=−0.18T=0.3, µ=−0.19Figure 4.3: The hop frequency fhop in units 1/τLJ over the duration of thesimulation run in a log-log plot. Solid lines indicate power-law fits withexponents µ given in the legend.In fig. 4.2(a) I show the persistence time distribution for quiescent glassesat three temperatures T = 0.2, 0.25, 0.3. I find a power-law behavior withµ ' −1.5 insensitive to the glass temperature, which agrees well with thepreviously found value for polymer glasses of −1.23 [98]. The slightly smallervalue is probably due to the increased sensitivity of the detection algorithm,which increases the likelihood of shorter persistence times. The shortenedtail observed for T = 0.3 indicates that persistence times in this system aresampled from a finite distribution and hence the system will equilibrate atlong times (see also ref. [97]). However, since the turnover happens at timesof the order of the total simulation time, during the observation time windowthe system is still well within the aging regime.During a hop, the particle moves from one metastable local configurationto another, i.e., from one cage to the next. The detection algorithm estimatesboth locations rinit, rfinal and I can therefore calculate the hop distance|d| = |rfinal − rinit| . (4.1)In panel (b) of fig. 4.2 the distribution of the hop distance is shown forthree temperatures. For each glass, the main peak is located at√Pth (thefirst vertical dashed line), which is the minimal distance that the detectionalgorithm sets for the separation of two ideal cages. I find an exponential684.1. Statistical properties of hops0 246 8 101214∆i0.00.20.40.60.81.0Cauto(∆i)(a)T=0.2T=0.25T=0.3−1.0 −0.5 0.0 0.5 1.0Cauto(∆i)0.20.40.60.81.0P(Cauto)(b)∆i=1∆i=2∆i=5∆i=10ππ/20Figure 4.4: Hop displacement autocorrelation for three glass temperaturesat age tage = 105 (a) as function of number of hops separation and (b) as his-togram, that is calculated using normalized displacement vectors (T = 0.25and age as above) and reveals an anisotropy in the direction of consecutivehops. The lines are guides to the eye.decay following the peak, with a transition to a faster exponential decay ataround 2√Pth. The transition is expected, because at distances > 2√Pth itis possible for the detection algorithm to separate the particle motion intotwo hops, if the particle briefly stabilizes at an intermediate distance. Theform of the distribution is qualitatively unchanged for varying temperatures,suggesting that the hop process is unchanged inside the glass state. Indeed,in simulations of a strong glass former [97] a comparable distribution wasfound, indicating a similar role of the hop process.In fig. 4.3 I show the hop frequency of the glass, i.e. the number of hopsper time τLJ . I observe about six times more hops at T = 0.3 compared toT = 0.2. Furthermore, I find that the frequency decreases as a weak powerlaw with age. Aging is accelerated at higher temperatures, which indicatesthat the phase space is explored more quickly. Indeed, for the glass closestto the glass transition (T = 0.3), I observe a flattening of the curve, i.e. thesimulation reaches timescales close to the end of the aging regime.I can further characterize the hop process by calculating the autocorrela-694.1. Statistical properties of hopstionCauto(∆i) =〈di · di+∆i〉〈di · di〉(4.2)of the displacement vector d [see eq. (4.1)]. The average in the numerator istaken over all hop-pairs i of a particle with a separation ∆i, the average hopdirection is zero and the denominator is the variance of d with zero mean.Simple mean-field trap models [67] assume a solely temperature driven escapefrom the cage, which yields independent hops, and a previous study indeedfound a vanishing autocorrelation after about two hops [98]. Although theearlier study used a different hop detection algorithm with lower sensitivity,our results shown in fig. 4.4(a) principally agree with these findings. Weobserve a correlation that decreases below 0.2 after at most seven hops, witha more rapid decline at higher temperatures. The decay is slower than previ-ously found, because the detection algorithm used here is able to separatelypick up back-and-forth hops of a particle between the same two cages. Iconfirm this observation with the autocorrelation histogram in fig. 4.4(b),where the displacement vectors were normalized to unit length. This isolatesthe directional correlation of the hops and one can clearly see a pronouncedanisotropy. For consecutive hops I find that angles close to 180 ◦ are clearlyfavored, indicating that the particle is more likely to return to the locationwhere it came from. Furthermore, there is an increased probability for afollowing third hop to be in the same direction as the first, indicating a back-and-forth between the two cages. The anisotropy is subdued with increasingseparation of hops and vanishes at ∆i = 5 for a glass at T = 0.25.Up to this point the results took advantage of the increased sensitivity andgreater number of detected hops. However, since all particles are tracked,I am able to directly measure the spatio-temporal correlation of hops. Infig. 4.5 I present surface plots that visualize probable temporal and spacialdistances between hops. I calculate the density-density correlation of hopsalong the lines of the normalized distinct part of the van Hove function [41]Ghopd (r, t) =1N ′〈δ(r − |r(i)init(t′)− r(j 6=i)init (t′ + t)|)〉. (4.3)Here the average is taken over all N ′ hop pairs that involve two distinct parti-cles, and using the initial positions, i.e. the initial cages. Figure 4.5(a) showsan example probability distribution. I find a dominating peak at times closeto zero and at a distance ' 1, which is caused by hops in the surroundingshell of particles. The accumulation of near-simultaneous hops is a direct704.1. Statistical properties of hops0 10 20 30 40 50t0.00.51.01.52.02.5r∗(a)012r∗10-210-1phopcorr(r)(b)0 10 20 30t10-210-1phopcorr(t)(c)0 10 20 30 40 50t0.00.51.01.52.0r∗(d)10-410-310-2Figure 4.5: (a) Probability density surface for spatio-temporal separation oftwo different hopping particles based on Eq. (4.3) for a glass at T = 0.25and tage = 105. The color scale is logarithmic (scale at the top-right corner)and the dashed lines indicate integration limits used to calculate the one-dimensional probability functions (b,c). Center plots show the probabilityfunction of (b) separation and (c) time delay between hops for T = 0.2(blue©), T = 0.25(green 5), and T = 0.3(red 3) at the same age. The grayvertical dashed lines in (b,c) illustrate the correlation ranges and the blackdashed curve in (b) indicates the radial distribution function. (d) Probabilitydensity surface following Eq. (4.3) for the same glass as the left panel, butwith r∗ calculated from initial position (at the origin) to the final position ofthe second particle after the hop. The color scale is again logarithmic.714.1. Statistical properties of hopsindicator of the cooperative nature of the hop process. The area r . 0.7and t . 7 is empty, which is an effect of the excluded volume of the hoppingparticle at the origin. A secondary, at least an order of magnitude weakerpeak is located around r = 1/2 and t = 10. It is caused by particles that hopafter having entered the space that was vacated by the particle at the origin.A comparison of density-density correlations calculated at various ages (notshown) indicate that aging effects are minimal. To highlight the dependenceon temperature I partially integrate eq. (4.3) from the origin to the dashedlines, which are chosen such that the main features are included. In fig. 4.5(b)I show the spatial correlation phopcorr(r) =∫ 300 dtGhopd (r, t) for three tempera-tures. One can see that the main features are found at all temperatures andthat an increased temperature weakens the sharpness of the peaks, which isdue to the increased vibrational motion of the particles. Apart from the firstpeak at r = 1/2 (see above), I find peaks at positions that coincide withthe static shell structure of the glass as indicated by the radial distributionfunction (black dashed curve). The splitting of the peak at around r = 1is due to the different mean distance between particles that are neighborsin the same polymer backbone and particles that are not directly bonded toeach other. The existence of the double peak shows that both pairs take partin cooperative rearrangements. Figure 4.5(c) shows the temporal correlationphopcorr(t) =∫ 1.50 drGhopd (r, t). A sharp decay at small times is followed by a peakat around t = 9, which is due to the immediate re-hopping of particles; theback-and-forth hopping that is also discussed in connection with the hop au-tocorrelation (see above). The position of the peak, i.e. the secondary peak inthe probability density surface is directly linked to the maximal resolution oftwo consecutive hops, which is t = 7.5 for the used parameters (see sec. 2.6).I find only a very weak temperature dependence in the temporal correlation,suggesting that the fundamental mechanisms of cooperativity are the sameover the temperature range studied here. The data also does not show anyclear indication for Poisson processes like those found for supercooled liq-uids and granular matter [19, 21]. Based on the sharp drop following themain peak in the spatial probability distribution, I infer a correlation rangeof rcorr = 1.5 [fig. 4.5(b), vertical dashed line], i.e., the correlation does notextend beyond the nearest neighbor shell. From the temporal probabilitydistribution I determine a correlation range of tcorr = 2.5 [fig. 4.5(c), verticaldashed line], which is the time at which the initial peak has decayed to val-ues below the close to constant region after the second peak. These ranges724.1. Statistical properties of hopstherefore restrict “correlated” hops to near-simultaneous hops of neighboringparticles, as indicated by the primary peak in the density-density correlation.Knowledge of the final position of hopping particles also allows me toexplore the direction of correlated hops. The probability distribution surfacein fig. 4.5(d) shows a density-density correlation very similar to the one onthe left. Again I use eq. (4.3), but the distance is now calculated betweeninitial cage at the origin and the final position of other hopping particlesr(j 6=i)final. Therefore, high probability regions indicate where the particles endup after a correlated hop. By comparing the surfaces, I find that hops thatstarted in the first shell (the primary peak in left plot) mostly end at r . 1/2,see the primary peak in the right plot. Indeed this peak extends all the wayto r ' 0, indicating that it is possible for the cage at the origin to stay largelyintact with a new particle taking the place of the last one. This suggests thestring-like motion that was previously observed in a binary LJ mixture [33].We also find a secondary peak at a distance r ∼ 1.5, which suggests thatsome hops from the first shell are directed away from the cage at the origin.Please note that the color scale in the plot is logarithmic, and that thissecond hop destination is at least one order of magnitude less likely than thefirst one. The secondary feature in fig. 4.5(d) at t ∼ 10 is located aroundr = 1, confirming that the accumulation of hops at this time lag are due toback-hops of particles that return to the first shell after having hopped intothe vacated volume closer to the center of the cage (origin).The correlation ranges rcorr = 1.5 and tcorr = 2.5 allow me to identify “co-operatively rearranging” groups of particles and I perform a cluster analysisto measure their size. Two particles are in the same cluster if they are closerin space and time than rcorr and tcorr, which is in close analogy to Candelieret al.’s study [19]. In fig. 4.6 I show the measured cluster size distributions,which exhibit initial exponential decays with stretched tails that approacha power law. In the main panel I show results for a single temperature andvarying age, and one can see that as the age increases the distribution flat-tens and becomes more exponential-like. In an older glass the hop-activity isreduced (see hop frequency in fig. 4.3), and therefore the constant clusteringtime (tcorr) used here results in a lower likelihood of finding larger clusterswith growing age. Varying temperature at the same age has a similar effect:as temperature increases, I observe a broadening of the distribution awayfrom exponential and towards a power-law form.734.2. Dynamical heterogeneity and clustering of hops5101520 25s10-610-510-410-310-210-1100P(s)tageTFigure 4.6: Size distribution of cooperatively rearranging particles. The mainpanel shows distributions at six ages for a glass at temperature T = 0.3; seelegend in fig. 4.7. The inset shows the size distribution of three glasses atage tage = 105 and temperatures T = 0.2 (blue ©), 0.25 (green 5), 0.3 (red3). Both plots have the same axes ranges, and the solid black lines indicateP (s) ∝ exp (−s).4.2 Dynamical heterogeneity and clusteringof hopsIn previous studies the heterogeneous dynamics in glasses and supercooledliquids were mainly probed with four-point correlation functions. A standardapproach is to measure the number of particles that remain approximatelystationary as a function of time using overlap functions [53]. The variance ofthis quantity over a multitude of independent simulations is the four-pointdynamical susceptibility χ4, which quantifies how many particles are mov-ing substantially from their initial position in the same time window. Thedynamical susceptibility exhibits a peak at the time of maximal dynamicalcorrelation in the system. The peak height is connected to the number ofparticles with correlated dynamics, and the increase of that height whenapproaching the glass transition signifies growing dynamical length scales.With knowledge of the location and time of all hops in the system, I canprovide a new perspective on the correlated dynamics. Specifically, I am744.2. Dynamical heterogeneity and clustering of hops100030005000Ncaged102103104105t05001000χ4tage=103τLJtage=3 ·103τLJtage=104τLJtage=3 ·104τLJtage=105τLJtage=3 ·105τLJFigure 4.7: The top panel shows the number of caged, i.e. not yet hopped,particles Ncaged averaged over independent simulations as a function of timefor six glass ages. The four-point susceptibility χ4 shown in the bottom panelis calculated from the variance of Ncaged, eq. (4.4).able to spatially resolve the clustering of hops, directly revealing the hetero-geneous dynamics, and to study the cluster distribution as a complimentaryperspective to χ4. For this part of the study I focus specifically on the agingregime, for which few studies exist. The results shown below are calculatedfrom a glass at T = 0.3, yet an equivalent analysis for T = 0.25 (not shown)confirms our findings further inside the aging regime.In a first step, I calculate the number of particles that have not hoppeddirectly from the hop dataNcaged(t, tage) =N∑ibi(t, tage) (4.4)where bi(t, tage) = 0 if particle i has hopped in the time window [tage, tage + t]and bi(t, tage) = 1 otherwise. In the upper panel of fig. 4.7, I show results forsix ages. I employed simulations of N = 5000 particles that are otherwiseequivalent to the usual simulations with N = 50, 000 particles. I had todownscale the system, because converged measurements of χ4 required 300754.2. Dynamical heterogeneity and clustering of hopsindependent runs. As mentioned above, the four-point dynamical suscepti-bility is proportional to the variance of Ncaged [53]χ4(t, tage) =βVN2(〈N2caged〉 − 〈Ncaged〉2), (4.5)with V being the simulation box volume, and 〈·〉 representing an average overindependent realizations of the system. The bottom panel of fig. 4.7 showsχ4 as function of time for the same six ages. In an earlier study, Parsaeianand Castillo [71] investigated four-point correlations in the aging regime of abinary LJ glass, and I observe the same main features in fig. 4.7: a shift ofthe peak towards larger times with increasing age and an increase in height,indicating a larger volume of correlated dynamics.To obtain a complementary picture of the spatially resolved dynamics,I perform a spatial cluster analysis on the subset of hops in the same timewindow [tage, tage+t] that is used for the calculation ofNcaged. I use a standardsingle-linkage cluster criterion, i.e., hops i and j are part of the same cluster,if the distance between the initial positions is below a threshold|riinit − rjinit| < rcl .If a hop k is already in a cluster with i, then hops j and k will belong tothe same cluster even if they don’t fulfill the above criterion. As a thresholdI use the spatial correlation range that is obtained from the density-densitycorrelation rcl = rcorr = 1.5 (see previous section). As the time window isincreased, I include more hops into the analysis and the clusters grow andmerge.In fig. 4.8 I illustrate the observed growth via snapshots of an examplecluster at increasing time t. The cluster first consists only of a few hoppingparticles and successively grows into an extended structure. An animationof the growth of a single cluster over time reveals periods of near stagnationinterrupted by large bursts which are due to the merging of simultaneouslyexisting clusters. The plot in fig. 4.8 illustrates this intermittent growthprocess. It shows the cluster volume of 15 example clusters as a function oftime. I define the volume of a cluster as the total correlated space of all hopsthat comprise the clusterVcl = ∪iVsp(riinit, rcorr) .Each hop contributes a spherical volume Vsp with radius equal to the correla-tion range rcorr = 1.5 centered around the initial position of the particle. In764.2. Dynamical heterogeneity and clustering of hops0 500 1000t0200040006000800010000Vclt=110τLJt=210τLJt=310τLJt=410τLJt=510τLJt=610τLJFigure 4.8: Snapshots of the growth of a single cluster over time. The parti-cles are visualized at their initial positions (before the hop) and the coloringindicates depth. The plot on the right shows the cluster volume of 15 ex-ample clusters as a function of time. Examples were recorded at glass agetage = 105.other words, with each hop I associate the volume of the entire cage and thecluster volume is the union of all cages that have rearranged. To calculatethis joined volume, I use a voxel technique, i.e. I partition the simulationbox into small cubes (voxels) and count the number of voxels with a centercloser than rcorr to any hop of a given cluster.To explore the cluster configurations near the χ4 peak I calculate the meanvolume and number of clusters as a function of time, and results for six agesare shown in fig. 4.9. In the upper panel one can see that at long times Iobserve a single cluster that spans the simulation box, and its formation isshifted in time with increasing age. In the inset I show the same data with atime axis that is rescaled by the time of the χ4 peak, which collapses the dataonto a single master curve. Note that the dominating cluster emerges just774.2. Dynamical heterogeneity and clustering of hops0.00.20.40.60.81.0〈Vcl〉/Vbox101102103104105t0200400Ncl10-210010210-410-2100Figure 4.9: Mean cluster volume (top) given as a fraction of the total sim-ulation box volume and the number of simultaneous clusters (bottom) as afunction of time. Results for six ages are shown, see legend in fig. 4.7. Theinsets show data collapse when time is rescaled by the time of the χ4 peak.when the four-point susceptibility reaches its peak. The success of the scalingcollapse indicates that aging merely delays, but does not otherwise alter theformation of this dominating cluster. The amount of clusters (bottom panel)peaks at much earlier times, and the rescaled data in the inset also showsan approximate collapse with age. Additionally, I find an age dependenceof the peak height, showing that the maximal number of clusters decreaseswith increasing age.To gain a more complete picture of the formed structures, I also measurethe extent of the regions where no hops are detected. To quantify these“holes,” I use the same voxel partition and perform a nearest neighbor clusteranalysis on the subset of voxels that does not lie inside the volume of anycluster. In fig. 4.10 I show the mean volume of the holes (top panel) andthe number of holes (bottom panel) as a function of time. I find that withincreasing time window, the size of the holes shrinks, which is of course due784.2. Dynamical heterogeneity and clustering of hops0.00.20.40.60.81.0〈Vh〉/Vbox101102103104105t0100200Nh10-410-210010-2100102Figure 4.10: Mean hole volume (top) given as a fraction of the total sim-ulation box volume and the number of simultaneous holes (bottom) as afunction of time. Results for six ages are shown, see legend in fig. 4.7. Theinsets show data collapse when time is rescaled by the time of the χ4 peak.to the growing hop clusters. Again, I observe a shift with glass age towardslarger times, and the inset in the top panel reveals that the break up of thesingle dominating hole happens just when the number of clusters is largest(both at ∼ 10−2 in rescaled time). In analogy, I find that the number of holesNh is maximal just when the mean cluster volume diverges and thereforewhen χ4 reaches its peak (see bottom panel inset). The maximum in Nhappears when the probability of closing a hole by placing a new hop into thesystem becomes larger than the probability of splitting a hole with a newhop. Therefore, the majority of the holes have shrunk to a size on the orderof a single cage at the time of the χ4-peak. The time of this crossover showsthe same age-dependence as χ4, yet I also find that the maximal numberof holes decreases with age. The scaling behavior of the hole volumes withage mirrors the behavior of the cluster volumes, and further supports theinterpretation that the geometry of DH is unchanged by aging.794.2. Dynamical heterogeneity and clustering of hopsFigure 4.11: Collection of cluster volume distributions of a glass at agetage = 105 measured at various times in double-log scale. Each distribu-tion is plotted in the Vcl-p(Vcl) plane and placed along the t axis accordingto the size of the time window used for the cluster analysis. The back wallshows an overlay of distributions at small times in a single plane. I includedata at times . 103 (up to and including the second blue distribution) andthe same colors as the separate distributions are used to indicate the originof the data points. The solid line on the back wall indicates a power lawwith exponent −2. The black solid curve on the floor wall indicates χ4 as afunction of time.804.2. Dynamical heterogeneity and clustering of hopsThe mean cluster volume already suggests a single, dominating cluster inthe system when χ4 is maximal. I gain further insight by directly studyingthe full distribution of cluster volumes. In fig. 4.11 I show its evolutionfor a single glass age, where the distributions for increasing time windows[tage, tage + t] are stacked along the t axis. One can clearly see how thedistribution lengthens over time until ∼ 103. At this time (blue to green) thedominating cluster is formed, indicated by a detached peak at large volumeand the successive shortening of the remaining distribution. The solid blackcurve on the floor wall indicates χ4 as a function of time, and as the largecluster grows, so does χ4. The peak is reached when the dominating clusteressentially covers the whole volume. Furthermore, I analyzed the form ofthe distributions, which prior to the emergence of the dominating clusterfollow a power law. On the back wall of fig. 4.11 I show an overlay of theseearly distributions (see caption) in a single plane. The overlay shows thatthe cluster volume distribution lengthens until a power law with exponent ofapproximately −2 is reached.From the snapshots in fig.4.8, one can see that the clusters are not com-pact, but have a complex geometry. Parsaeian and Castillo investigated DHin an aging binary LJ glass using four-point correlators [71]. By assumingthat the height of the χ4 peak was proportional to the correlated volume andusing an identified correlation range ξ, they found a scaling of χpeak4 ∝ ξbwith b = 2.89 ± 0.03. Our access to spatially resolved clusters allows us todirectly calculate their fractal dimension and compare to this scaling result.I use the box counting method [36], in which one covers the cluster withsuccessively smaller cubes, counting each time how many boxes are needed.Here, a cluster is represented by its correlated volume, i.e., the union ofspheres with radius 1.5 centered at each hop. The dimensionality of thecluster is then calculated via power-law fitNl ∝ ldf ,where l is the side length of the cube and Nl the cube count. In fig. 4.12I show the fractal dimension as a function of time for six ages. I find amean fractal dimension around 2.88 for short and intermediate times and anincrease to 3 for long times. The inset shows that this increase happens asχ4 peaks and from the discussion of the volume distribution above, I knowthat the peak is accompanied by the emergence of a system spanning cluster.Since such a cluster has the dimensionality of the simulation box, the increase814.3. Conclusions101102103104105t2.852.902.953.00〈df〉10-2100102Figure 4.12: Main panel shows the mean fractal dimension of the hop clustersover time for six ages, see legend in fig. 4.7. The inset shows data collapsewhen time is rescaled by the time of the χ4 peak.to df = 3 is not surprising and it is clearly a finite size effect. Therefore, thefound value of 〈df〉 = 2.88 agrees remarkably well with the above mentionedresult that was solely based on four-point correlators.4.3 ConclusionsThe microscopic structural relaxation is studied in quiescent polymer glassesat three temperatures in the aging regime. A refined version of a detectionalgorithm initially introduced by Candelier et al. [19] was used to measurethe relaxation events defined as particle hops everywhere in the system andon-the-fly for the full duration of the simulation. An evaluation of the distri-bution of persistence times, hop distance, and hop autocorrelation at threetemperatures showed good agreement with previous studies that used othermethods [97, 98]. Since the detection algorithm allows hop detection for thefull system, I was able to directly analyze the spatio-temporal density-densitycorrelation between relaxation events. A strong correlation was observed be-tween near-simultaneous hops of neighboring particles, which indicates coop-erative motion of groups of particles. I estimated correlation ranges and usedthese to analyze the size of the collaborative rearrangements as a functionof temperature and age. I found distributions that first have an exponen-824.3. Conclusionstial shape and then transition over to a power-law tail that becomes flatterduring aging. An increase in temperature broadened the power law, andthis trend connects well to the power-law distributions seen by Candelier etal. [19] in agitated granular media, where a very similar definition for therearranging groups was used. An earlier study of a binary LJ-glass in theaging regime on the other hand showed power-law distributions [96], both forvarious temperatures and ages. I believe that this disagreement is due to thevery different hop-time resolution, that was about three orders of magnitudesmaller than what was used in this work.In the second part of this study, I compared the standard χ4 measure ofdynamical heterogeneity (DH) with a direct geometric analysis of hop clus-ters, which gives a spatially resolved picture to complement the bulk averagedχ4. My results show that χ4 reaches its peak when a single dominating clus-ter is developed that extends throughout the system and is accompanied bymostly single-cage-sized pockets of inactive particles. I also observed a de-layed cluster aggregation in older glasses that mirrored the shift of the χ4peaks towards larger times with increasing age. Therefore, the geometricformation of DH is continuously slowed but otherwise unchanged by physicalaging. We furthermore observed increasing χ4 peak heights, which indicatea growing dynamical correlation range during aging. Both the shift of χ4with increasing age and the increasing χ4 peak height were also reported byParsaeian and Castillo [71] in simulations of a binary LJ mixture. Recently,further evidence for growing dynamical correlations was obtained via experi-mental measurements of the nonlinear dielectric susceptibility in glycerol byBrun et al. [17]. Parsaeian and Castillo also identified a power-law scaling be-tween an estimated growing correlation range and the χ4-peak height, whichis connected to the total correlated volume. I showed that this scaling is inexcellent agreement with the fractal dimension of the hop clusters. The meancluster volume did not directly reveal the aging correlation range, as it is notproportional to χ4 in the range of its peak, yet a clear age dependence wasobserved for the maximal number of clusters and inactive regions (holes).The shape of the evolving distribution of hop cluster volumes helps tounderstand the somewhat surprising success of mean-field models of aging.Despite the presence of heterogeneous dynamics, aging continuous time ran-dom walk descriptions [98] based on the trap model of aging [67] are verysuccessful in capturing the evolution of mean squared displacements, dynam-ical structure factors and van Hove functions while entirely neglecting DH.My measurements of the cluster volume distribution prior to the merging834.3. Conclusionsinto a single dominating cluster (fig. 4.11) showed a power-law form withexponent ≤ −2. This observation indicates that fluctuations in the size ofthe DH are sufficiently small that average quantities such as mean clustersize do not behave anomalously.84Chapter 5Soft modes predict structuralrelaxation3This third project investigates the link between heterogeneous dynamics(DH) and local structure in polymer glasses. In other words, which structuralfeature determines whether the particles in a certain region are dynamicallyactive and undergo substantial rearrangements, while other regions are struc-turally static? Recent studies [63, 102] indicate that so called soft modes,low energy vibrational modes in disordered solids that are spatially localized,could provide the key to answering this question, and section 1.2.4 discussessoft modes in great detail.So far, quantitative evidence for the correlation between soft modes andstructural rearrangements is restricted to studies of model metallic glassesin 2D under deformation [63, 81]. This project quantifies the correlation ina 3D polymer glass in the quiescent state and I investigate temperature andage dependence. Participation of particles in soft modes is measured in theform of a softness field and the definition as well as implementation detailsare given in section 2.7. I use the force-shifted model potential discussed insection 2.1, since differentiable forces are required for the calculation of theHessian. The Hessian is used to find the low-energy vibrational spectrumand I use a system size of N = 10, 000 for numerical efficiency. Structuralrearrangements are measured as irreversible hops as discussed in section 2.6.In addition to the spatial correlation of softness field and hops, the di-rection of hops is found to directly correlate to the direction of soft modes.I furthermore quantify the lifetime of the softness field and compare it totime scales of structural relaxation. Two glasses are investigated in the ag-ing regime, at temperatures T = 0.2, 0.3, and one system at T = 0.4. At thislatter temperature, the relaxation times are short enough so that the systemreached equilibrium shortly after the quench and it is therefore a supercooledliquid. To evaluate aging effects, I analyzed the glass at T = 0.3 at three3Large parts of this chapter have been published in ref. [86]855.1. Softness field012 34 56φ[10−4]100101102103104105P(φ)T=0.2, tage=750T=0.3, tage=750T=0.3, tage=7500T=0.3, tage=75000T=0.4(a)(b)Figure 5.1: (a) Snapshot of the softness field. The right side shows only the10% softest regions and the solid black spheres (size equals particles) indicatethe first 100 hopping particles detected after the measurement of the softnessfield. (b) Distribution of the softness field for three temperatures and threeages. Error bars are omitted and smaller than the symbols.ages: tage = 7.5 ·102, 7.5 ·103, 7.5 ·104. The results shown below are averagedover 20 realizations of each system with independent initial configurationsand error bars indicate the standard error.5.1 Softness fieldIn fig. 5.1(a) I show an example snapshot of the softness field. One canclearly see the heterogeneous spatial distribution of soft regions across thesimulation box. The black spheres are the first 100 hopping particles detected865.1. Softness field0.20.40.60.81.0Ca(t,tage)10-1101103105107t0.00.20.40.60.81.0Ca(t,tage)T0.20.30.410-310-210-1100Nh/Ntage750750075000(a) (b)(c) (d)Figure 5.2: Autocorrelation of the softness field for three temperatures (a-b)and three ages (c-d). Panel (a)[(c)] shows Ca as function of time, and thedotted lines indicate the ISF for the same temperatures [ages]. Panel (b)[(d)]shows Ca as function of number of hopped particles Nh, with dotted linesagain indicating the ISF and dashed lines mark when 50% of the system hashopped. Error bars are omitted and smaller than the symbols.immediately after the measurement of the softness field, and some overlap isvisible between hops and soft areas. The following sections are dedicated toa quantitative analysis of this correlation, yet I first focus on characteristicsof the softness field itself. Fig. 5.1(b) shows the softness distribution in theentire polymer sample for different temperatures and ages. I find that thedistributions feature a strong peak at small values and a rapid, yet slowerthan exponential decay. The results show that the structural heterogeneityis remarkably similar for all studied systems.In order for the softness field φ to represent the molecular structure interms of “soft” and “hard” or stable and unstable regions, the lifetime of875.1. Softness fieldφ must be of order of the structural lifetime. I measure the lifetime of thesoftness field via the decay of its autocorrelation functionCa(t, tage) =〈[φ(tage)− φ¯(tage)] [φ(tage + t)− φ¯(tage + t)]σφ(tage)σφ(tage+t)〉Here, the average is over all particles, σφ(t) is the standard deviation of thesoftness field φ(t), and φ¯(t) is its average. In fig. 5.2 I show the autocor-relation for three temperatures (a-b) and three ages (c-d). All systems inthe glass state exhibit an initial plateau in the ballistic regime, followed atintermediate times by a shoulder that becomes more pronounced at lowertemperature and with increasing age. The final decay to zero has stretchedexponential form and the autocorrelation reaches over many orders of mag-nitude in time. These characteristics are also found in the self-intermediatescattering function (ISF)CSq (t, tage) = 〈exp [iq · (rj(tage + t)− rj(tage))]〉 ,which is the standard measure of structural lifetime [41]. Here, the average isover all particles and I use q = (0, 0, 2pi). A value of CSq (t, tage) close to zeromeans that most particles have moved further than their diameter away fromtheir initial position. In fig. 5.2(a),c the ISFs are indicated as dotted lines,and I observe two main differences when compared to Ca: First, the plateauand associated shoulder of the ISF are more pronounced and reach further intime. Second, the ISF decays to zero at later times than the autocorrelation.Before these differences are discussed in detail, I clarify the role of tem-perature and age on the autocorrelation of the softness field: Panel 5.2(a)shows that the decay-time becomes larger with decreasing temperature, asCa shifts to the right. This is accompanied by the development of a shoulderat intermediate times, which is not present in the supercooled state (T = 0.4)but develops as the system becomes more glassy. This mimics the temper-ature dependence of the ISF, although a shoulder is already present in thesupercooled state. In panel 5.2(b) I show how much of the system has un-dergone rearrangements as Ca decays by re-parametrizing time in terms ofthe fraction of particles that have hopped at least once. I observe that com-plete decorrelation of φ occurs for T = 0.3, 0.4 after & 50% of particles haverearranged, and extrapolation suggests that this also holds true for T = 0.2.Panel 5.2(c) shows the autocorrelation for three ages at T = 0.3. I observe885.1. Softness field10-410-310-210-1100Nh/N010203040506070Nholes1.5σFigure 5.3: Number of holes as a function of the fraction of hopped particles.The sketch illustrates the definition of a hole: a continuous volume (green)that is surrounded by the union of spheres that approximates the cages thathopping particles have escaped and are “broken”. The dashed line indicates50% of the system has undergone rearrangements, and the solid lines areguides to the eye. See fig. 5.1 for a legend.that an increase in age results in a shift of Ca towards larger times via length-ening of the shoulder. The ISFs are shown as dotted lines and one can seea similar shift with increasing age. In panel 5.2(d) one can see that totaldecorrelation of φ again occurs when & 50% of the system has undergonerearrangements, independent of the glass age.A key difference between autocorrelation Ca and ISF is that the decorre-lation of the softness field begins as soon as particles hop, whereas the ISFremains at a high value for much longer. However, this is not surprising, sincethe ISF can only change after a substantial part of the particles has moved.The structurally soft regions on the other hand may very well only requirea few hops to transition into a more stable local configuration, which couldexplain the faster decay of Ca at intermediate times. It is also important torealize that the mean hop distance is of order half a particle diameter, whilewith a wavevector magnitude q = 2pi the ISF is sensitive to displacementsof order one particle diameter. A particle therefore has to undergo multiplerelaxation events to fully decorrelate the ISF, and in this sense the ISF decayprovides an upper bound on the structural relaxation time.895.2. Spatial correlationWhy does the softness field decorrelate after & 50% of particles havehopped? To answer this question, I first note that a particle that hopsby escaping its own local cage changes the local configuration of all theneighboring particles at the same time. To measure how much of the systemhas been affected by hops in this way, I place a sphere around each hoppingparticle with radius 1.5σ. This distance is the position of the first peak inthe pair correlation function and the sphere therefore approximates the cagearound each hopping particle. I then count the number of unconnected holesin the union of all spheres. The sketch in fig. 5.3 visualizes this: a hole isa continuous volume that is not part of any of the cages that are “broken”by the hopping particles. In the main panel of fig. 5.3 I show the numberof holes as function of the fraction of hopped particles. When only a fewparticles have hopped, then there is only a single hole. As more particleshop, the spheres form clusters [85], interconnect and eventually percolate,leading to a subdivision into many holes. A maximum is reached when theprobability of splitting a hole in two by including an additional sphere isequal to the likelihood of destroying a hole. In other words, the maximum isreached when the size of the holes is of the order of single cages. I find thatthis transition occurs when ∼ 50% of the particles have hopped (indicatedby the dashed line). At this time the total volume of the holes has droppedto ∼ 5% of the system size. Therefore, the decorrelation of the softness fieldat & 50% coincides with the change of the local configuration of nearly allparticles.5.2 Spatial correlationIn fig. 5.1(a) I show hops as black spheres together with the softness field,and one can see that hops appear at the center of soft regions as well as in thespace between them. Before analyzing the overlap of individual hops withthe softness field, I focus on the DH of the whole system. How well does asingle measurement of a softness field reflect the distribution of regions withhigh/low rearrangement activity? I create a map of DH by accumulatinghop events in a binary list hi of all particles. The map changes as the time-window [tage, tage + t] of included hops grows. The similarity of DH and thesoftness field is then quantified with the cross-correlationCDH(t, tage) =∑Ni=1(hi(t, tage)− h¯(t, tage)) (φi(tage)− φ¯(tage))Nσhσφ. (5.1)905.2. Spatial correlation101103105107t0.000.050.100.150.20CDH(t,tage)10-310-210-1100Nh/N(a) (b)Figure 5.4: Cross-correlation between softness field and cumulative map ofhopped particles (a) as function of time and (b) as function of number ofhopped particles. See fig. 5.1 for a legend, and error bars are omitted andare smaller than the symbols.Here h¯, φ¯ are averaged over all particles and σh, σφ are standard deviations.In fig. 5.4(a) I show the cross-correlation as a function of elapsed time afterthe φ measurement. A maximum is observed at times that grow with in-creasing age and decreasing temperature. The re-parametrization in termsof number of hopped particles in fig. 5.4(b) collapses the maxima at ∼ 20%rearrangement of the system. The degree of agreement is given by the max-imum value of the correlation, ranging from 0.11 for the supercooled systemto 0.21 for the oldest glass at T = 0.3. The absolute value of the correlation isnot very high, which can be expected from thermal systems. Simulations inthe iso-configurational (IC) ensemble [101] reduce the impact of kinetics onthe map of DH by averaging over many realizations of a single configurationwith randomly assigned velocity distributions. I performed such an analysison a single configuration of the system T = 0.3, tage = 75000 and found across-correlation between the softness field and 〈hi〉IC that is twice as strongat the peak. A systematic analysis using this technique, however, is beyondthe scope of the present study.The temperature and age dependence reveal the importance of the softmodes especially in the glass state. The increase of CDH with age showsthat the non-equilibrium state is important for the link between structure915.2. Spatial correlationand dynamics. From the perspective of the potential energy landscape [43](PEL): As the glass moves down the PEL towards more arrested, lower en-ergy configurations, the soft modes increasingly dominate the dynamics ofthe glass. In the supercooled state I observe a lower correlation, indicatingthat higher temperature increasingly washes out the effect of structural het-erogeneity defined by the soft modes. A lower temperature therefore yieldsa higher correlation. For the investigated temperature and age range botheffects are of comparable magnitude. The largest correlation was observerdin the oldest T = 0.3 glass, which was aged for two orders of magnitudelonger than the T = 0.2 glass.In fig. 5.5 I show two approaches that quantify the spatial correlationof relaxation events and softness field in greater detail: panel 5.5(a) showsthe probability for a particle of given softness to undergo a hop at timesimmediately after the φ measurement, rescaled by the total hop probabilityΩ(φ) =Nh(φ)N(φ)∫dφN(φ)∫dφNh(φ).Here N(φ) indicates the number of particles with given softness and Nh(φ)is the subset of those particles that have hopped at least once. I observe aclear increase of the hop probability with increasing softness for all tempera-tures and ages. Starting from a value below one at very low φ (hard region),the probability monotonically rises to a saturation plateau of up to 7 timesthe average probability of relaxation events. The correlation is temperaturedependent, being much more pronounced in the aging regime (T = 0.2, 0.3)than in the supercooled system (T = 0.4), where the soft regions undergorearrangements with three times the average probability. Furthermore, in-creased age yields a stronger correlation between soft modes and relaxationevents.I explore an alternative view on the spatial correlation by binarizing thesoftness field into a soft spot map, where the fraction of particles f withlargest softness are assigned a softness of φ(b)i = 1 and all other particles havea softness of zero. I then define the predictive success rate Θ of a softnessfield as the fraction of the first Nh = 100 hopping particles that are part ofa soft spot, orΘ(f) =∑Ni=1 φ(b)i hiNhwith hi = 1 if particle i is one of the first Nh particles to hop after themeasurement of the softness field, and hi = 0 otherwise.925.2. Spatial correlation012 34 56φ[10−4]0246810Ω(φ)(a)0.1 0.3 0.5f0.00.20.40.60.81.0Θ(f)(b)10-410-310-210-1100Nh/N0.30.40.50.60.70.8Θ(30%)(c)Figure 5.5: (a) Probability of a particle to hop as function of its softness,rescaled by the average hop probability. The solid lines indicate the averagedsaturation probability and the dotted line is a guide to the eye. Success rateof predicting hops to occur in the softest regions of the system Θ (b) asfunction of coverage fraction of the softest region f and (c) as function oftime rescaled to the number of hopped particles at constant coverage fractionof f = 30%. The solid lines in (b,c) indicate the success rate based onrandomly chosen regions and the dashed line indicates 50% of the systemhas undergone rearrangements. To evaluate (a) and (b) the first 100 (1%)hopping particles after the softness field measurement were used. See fig. 5.1for legend, and error bars are omitted when smaller than symbols.935.3. Directional correlationPanel 5.5(b) shows the predictive success rate as function of the coveragefraction and a comparison with a randomly chosen subset of the system assoft spots is indicated by the solid line. Clearly, the softness field is a muchbetter predictor, with the absolute difference being maximal at around 30%coverage fraction. Here, up to 70% of the first 100 hopping particles arepredicted. Again, I find that systems at lower temperature show a strongercorrelation and that increasing age also improves the predictive strength ofthe softness field. In panel 5.5(c) I show how the spatial correlation developsas a function of time between the φ measurement and hops, i.e., the predictivesuccess rate for 30% coverage fraction. Time is rescaled in terms of thenumber of particles that have hopped at least once, identical to the rescalingin fig. 5.2 and fig. 5.4. The correlation is long-lived, decays logarithmicallyand decorrelates only when & 50% of the system has undergone structuralrelaxation events.5.3 Directional correlationThe direction of the softness field, which is the average direction of the softmodes, contains information about the dynamics of the relaxation events.More precisely, the direction of the hops align with the direction of the soft-ness field in soft regions. I quantify this correlation via the second LegendrepolynomialCd = 〈32(dˆ · eφ)2−12〉 ,where dˆ = (rfinal − rinit)/|rfinal − rinit| is the unit vector between final andinitial position of a hopping particle, and eφ is the direction of the softnessfield for the same particle (see section 2.7). The average is taken over all hop-ping particles. A value of Cd = 1 means full alignment of hop and softnessfield direction, while Cd = 0 indicates that a random orientation with respectto each other. In fig. 5.6(a) I show the correlation immediately after the mea-surement of φ as function of softness. The alignment grows with increasingφ for all temperatures and ages until a saturation plateau is reached. Similarto the spatial correlation discussed above, increased temperature weakensthe link between the soft modes and the hops. The saturation value reachesfrom 0.4 for the supercooled system to 0.8 for the glass at T = 0.2, indicat-ing that hops in soft regions are nearly perfectly aligned with the softnessfield direction at low temperatures. The effect of aging, however, seems to945.3. Directional correlation012 34 5φ[10−3]0.00.20.40.60.81.0Cd(t,tage)10-410-310-210-1100Nh/N(a) (b)Figure 5.6: Directional correlation between softness field and hops (a) asfunction of softness and (b) as function of time rescaled to the number ofhopped particles. The dashed line indicates 50% of the system has undergonerearrangements and the hop direction is measured as the vector betweeninitial and final position of the particle. To evaluate (a) the first 100 (1%)hopping particles after the softness field measurement were used. See fig. 5.1for a legend.be negligible for the strength of the directional correlation. This behavioris qualitatively different from the age-dependent spatial correlation, whichgrows with increasing age. Our finding implies that the direction of molec-ular relaxation events is independent of the position on the PEL, becausethe latter is changed during aging. In other words, the non-equilibrium na-ture of the glass has no direct impact on the alignment of soft modes andhops. Temperature on the other hand acts as noise and reduces the degreeof alignment.In panel 5.6(b) I show the mean directional correlation as function ofnumber of hopped particles since the measurement of φ. I observe a slowlogarithmic decay of the correlation that vanishes only after & 50% of thesystem has hopped at least once, for all temperatures and ages. A closeinspection reveals that at increased age, the decay curve develops a smallshoulder around 1%−10% hopped particles. This behavior does not indicatea direct age-dependence, but rather is a consequence of the longevity of955.4. Conclusionsthe softness field itself. The autocorrelation data discussed in section 5.1shows how aging leads to an increased stability of φ in the range of < 50%hopped particles. The decay curves for the directional correlation reflect thislongevity, showing that the softness field direction has predictive strengthover the direction of hops until the local configuration of nearly all particleshas changed.5.4 ConclusionsThe correlation between soft modes and structural relaxation events wasquantified in a quiescent polymer glass at two temperatures below Tg in theaging regime and one temperature above Tg in the supercooled regime. Onesystem in the aging regime was analyzed at three ages and I identified theimpact of temperature as well as aging on the correlation. The structuralrelaxation events were identified as hops in the particle trajectories, and theparticipation of particles in soft modes is quantified in terms of a softnessfield φ, which was constructed from a superposition of low energy vibrationaleigenmodes [102]. The softness field is closely related to the binary soft spotapproach introduced by Manning and Liu [63], but here the sole adjustableparameter is the number of included modes.For all temperatures and ages the softness field was found to be heteroge-neous, with small regions of large softness. I showed that a strong correlationexists between the softness of a particle and its likelihood of undergoing astructural relaxation event. Starting from a much decreased probability atsmall φ, I found that with growing softness the hop probability increases toup to 7 times the average value. The spatial correlation is stronger at lowertemperature and also grows with increasing age. I showed that a binarysoft spot map based on φ with 30% coverage fraction predicts up to 75% ofthe hops immediately following the φ-measurement. The predictive strengthwas found to decrease slowly with increasing time separation between φ-measurement and hops. The correlation vanishes for all temperatures andages only after & 50% of the polymer glass has undergone rearrangements,which coincides with the decay of the softness autocorrelation function. Thesoftness field and the binarized soft spots that can be derived from it aretherefore long lived features that capture the heterogeneity of the amorphousstructure.In addition to the spatial correlation of hops to soft regions in the glass,965.4. ConclusionsI showed that the soft modes also correlate to the dynamics of relaxationevents. The direction of hops, measured as displacement vector betweeninitial and final position of the particle, are correlated to the soft mode di-rections. I find an increasing alignment with increasing softness that reachesvalues of 70% for the lowest temperature glass. The correlation is againstronger at lower temperature, yet it appears to be independent of the glassage. An older soft spot will attract more hops and hence has a larger spatialcorrelation than a younger soft spot, but hops actually occurring on a softspot follow the soft directions independent of age.The findings are in good quantitative agreement with a recent study [81]on a sheared 2D binary mixture at finite temperature. This study foundrearrangements to be 2-3 times more probable at soft spots than at ran-dom locations. A detailed analysis of the individual soft spot dynamicsshowed that they are robust structures that reach lifetimes of up to the bulkstructural relaxation time scale, which agrees with my analysis of the φ au-tocorrelation function. In the driven systems, there is a closer relationshipbetween self-intermediate scattering function and soft spot decay as in thepresent quiescent case. This may be due to extremely long local persistencetimes which here are not bound by an imposed external drive. Both spatialand directional correlations were furthermore identified in a recent study [77]that investigated the role of soft modes at the crossover between ordered todisordered systems. Soft modes were observed to predict the direction andlocation of rearrangements in a hierarchy of systems: from a crystal with asingle dislocation, to a polycrystal and a binary glass in 2D.97Chapter 6Soft modes and local plasticevents during deformationIn this final project, I extend the analysis of the correlation between softmodes and particle rearrangements to polymer glasses under deformation.The previous chapter verified a strong correlation in the quiescent state, anddiscussed temperature- as well as age-dependence. For the case of mechani-cally driven systems, this link between local structure and plastic events wasonly quantified in metallic glasses in 2D [63, 81]. These studies showed thatlocal plastic deformation is concentrated at so called soft spots, which aredefined from a binarized superposition of quasi-localized soft modes.This project focuses on two questions: First, do soft modes predict thelocation and direction of individual plastic events during mechanical defor-mation in polymer glasses? In analogy to the methods used in chapter 5,plastic events are measured as irreversible particle hops using the detectionalgorithm introduced in section 2.6, and I use the softness field definitiongiven in section 2.7 to measure the participation of particles in soft modes.Uniaxial tensile deformation is simulated at a constant strain rate and Ianalyze the correlation at different stages during the deformation.Second, does the correlation quantitatively change with the extent of thedeformation, and is this change in agreement with the concept of mechanicalrejuvenation? From the study presented in the last chapter, it is known thatthe spatial correlation between softness field and hops increases during aging.Furthermore, the analysis in chapter 3 showed that mechanical deformationin the pre-yield regime leads to a transient rejuvenation of the system in termsof dynamical as well as structural quantities, which becomes a permanenterasure of history at around the yield point. In this project, I first age aglass at T = 0.3 in the quiescent state for a time 7.5 · 106 and it is thendeformed using the constant strain rate protocol discussed in section 2.3at a rate ˙ = 10−5. The deformation ends at a final engineering strain of = ∆lz/l0 = 4, with ∆lz being the change in the simulation box length98Chapter 6. Soft modes and local plastic events during deformation(a)(b)0.0 0.5 1.01.52.0 2.5 3.0 3.5 4.0ǫ0.00.51.01.52.02.5〈σz〉(c)0.0 0.1 0.20.00.51.0Figure 6.1: Snapshots of the system are shown at beginning (a) and end(b) of the deformation. To better visualize polymer configurations, 15 poly-mers are colored separately and only beads that belong to these polymersare displayed on the right side of the simulation box. (c) Stress along thedeformation axis es function of total strain. Vertical colored lines indicatethe investigated deformation states = 0.0, 0.04, 0.1, 0.5, 1.0, 2.0, 3.0, 4.0and the inset shows the peak at the yield point in more detail.along the deformation axis and l0 is the box length before the deformation.All results shown below are averages of 20 independent simulation runs andI use the force-shifted model potential discussed in section 2.1 to ensurethat the forces are differentiable, which is required for the calculation of theHessian. In analogy to the study discussed in chapter 5 I use a system sizeof N = 10, 000 for numerical efficiency.The snapshots in fig. 6.1 show the system at the beginning (a) and end(b) of the deformation. In fig.6.1(c) I show the stress along the deformationaxis as function of total engineering strain. The deformation is separated intothree regimes: After an elastic deformation at very small strains, the stressreaches a maximum at the yield strain = 0.04 (see inset). This is followedby a strain softening regime, where the stress decreases until at ∼ 0.1 aplateau is reached and the stress remains constant until = 0.2. At this996.1. Correlation between hops and softness field0.0 0.1 0.20.30.4 0.5f0.00.20.40.60.81.0Θ(f)(a)strain ǫ0.00.14.0102104106tage0.600.620.640.660.680.700.720.74Θ(30%)(b)012 34ǫ(c)Figure 6.2: (a) Fraction of hops in soft spots as function of the coveragefraction of soft spots measured in three deformation regimes: elastic ( =0.0), strain softening ( = 0.1), and strain hardening ( = 4.0). The dashedline indicates no correlation. The evolution of the predictive success ratereached at f = 0.3 is shown during aging (b) and during deformation (c) -the dashed lines are guides to the eye.point, the stress starts to increase with strain, a polymeric effect known asstrain hardening, and at the end of the deformation at = 4.0 the stress hasincreased to about twice the yield stress.6.1 Correlation between hops and softnessfieldThe spatial correlation between softness field and hops is quantified as thepredictive success rate Θ that is also used in chapter 5. Here, the softnessfield is binarized into a soft spot map by assigning a softness of one to thefraction f of particles with largest softness and zero to the other particles. I1006.1. Correlation between hops and softness fieldthen calculate the fraction of the first Nh = 100 hopping particles that arepart of a soft spot, orΘ(f) =∑Ni=1 φ(b)i hiNhwith hi = 1 if particle i is one of the first Nh particles to hop after themeasurement of the softness field, and hi = 0 otherwise.Figure 6.2(a) shows the predictive success rate measured at three differentstrains during the deformation, and the dashed line indicates Θ for randomlydistributed soft spots (no correlation). For all deformation regimes, I find apositive correlation between softness field and the occurrence of hops. Com-pared to the strong correlation in the elastic regime, measured immediatelyupon loading at = 0.0, the correlation is decreased in the post-yield strainsoftening regime at = 0.1. Interestingly, however, in the strain hardeningregime at = 4.0 the correlation is again at a value comparable to the elasticregime.I furthermore calculated the directional correlation between softness fieldand hops (not shown) in analogy to the analysis presented in section 5.3. Ifind that the alignment found in the quiescent state is also present duringdeformation and that it is unchanged by the extent of the deformation.To explain the striking change in the spatial correlation, from decrease inthe pre-yield and strain softening regime to an increase in the strain harden-ing regime, I first compare to the evolution of the correlation during aging.To simplify the analysis I focus on the predictive success rate at a coveragefraction of 30%. This is the fraction where the difference between measuredΘ and uncorrelated value (dashed line) is maximal. In fig. 6.2(b) I showΘ(30%) at five ages. In agreement with the results reported in chapter 5,the correlation increases from 0.61 to 0.71 as the age grows by four ordersof magnitude. The position in the potential energy landscape (PEL), whichis changed in the direction of lower minima during aging, therefore playsan important role for the spatial correlation of soft modes and hops. Panel(c) shows the predictive success rate measured at different points during thedeformation. One can see that the spatial correlation in the elastic regime at = 0 is roughly equal to the value found in the quiescent state immediatelyprior to the deformation. At the yield strain = 0.04 the correlation hasdecreased to 0.64 and at the end of the strain softening regime ( = 0.1)Θ(30%) has reached the pre-aging value 0.61. This reversal of the agingeffects is consistent with the picture of mechanical rejuvenation, which is1016.2. Examining the strain hardening regime102104106tage14.3014.3214.3414.3614.3814.4014.42UIS(a)012 34ǫ(b)Figure 6.3: Mean inherent structure energy during aging (a) and (b) asfunction of strain during deformation.discussed in much detail in chapter 3.Interestingly, the spatial correlation does not remain constant upon fur-ther deformation in the strain hardening regime. Figure 6.2(c) shows thatΘ(30%) monotonically increases with growing strain. The increase in Θ(30%)accelerates at large strains > 2, reaching a value of 0.72 at = 4., which isabove the predictive success rate measured in the quiescent state prior to thedeformation. Is this strengthening spatial correlation the result of the sameprocesses that drive the increase during aging?6.2 Examining the strain hardening regimeAging is the non-equilibrium evolution of the system towards lower energystates in the PEL, and it is discussed in detail in section 1.2.2. The positionin the PEL can be measured by minimizing the energy in the zero tempera-ture limit. The particle configuration at the minimum is called the inherentstructure, and its potential energy UIS, previously introduced in section 3.4,is measured during the calculation of the softness field. In fig. 6.3 I showthe mean total inherent structure energy, the summation of pair and bondpotential energy averaged over all particles, during aging [panel (a)] and me-chanical deformation [panel (b)]. As expected, panel (a) shows that UISdecreases logarithmically during aging. Other than the predictive success1026.2. Examining the strain hardening regime0.0 1.0 2.0 3.0ω0.00.10.20.30.40.50.60.70.8P(ωi)(a)strain ǫ0.00.14.0012 34ǫ0.260.270.280.290.300.310.320.33〈P(ω)〉(b)Figure 6.4: (a) Mean participation ratio as function of eigenfrequency. (b)Mean participation ratio of all Nm modes used for the softness field calcula-tion as function of total engineering strain.rate Θ, which depends on the first Nh = 100 hops after the softness fieldmeasurement, the inherent structure is an instantaneous quantity, and UISat time of loading ( = 0.0) is therefore identical to the quiescent state justprior to loading. At the yield strain, = 0.04, UIS has increased nearly topre-aging values, and in the strain softening regime, = 0.1, the increase ofUIS slows and full erasure of history (mechanical rejuvenation) is reached.Importantly, in the strain hardening regime, the inherent structure energy isfound to increase with the extent of the deformation. In the PEL picture,this evolution is in the opposite direction than that occurring during aging,and “over-aging” can therefore not be the reason for the increase in spatialcorrelation.To better understand why the softness field correlates more strongly withhops at large strain deformation, I investigate changes in the low energy vi-brational spectrum, which is the basis for the softness field calculation. Theextent of localization of a vibrational mode j can be calculated as participa-tion ratioPj =(∑Ni=1(e(i)j )2)2N∑Ni=1(e(i)j )4.Here, e(i)j is the polarization vector of particle i, see also section 1.2.4. A1036.2. Examining the strain hardening regimevalue of Pj = 1 means that all particles are participating equally in modej, whereas a small value indicates that the mode is quasi-localized around afew active particles.In fig. 6.4(a) I show the participation ratio P (ω) as function of mode fre-quency. Since the vibrational spectrum is a feature of the inherent structure,P (ω) at = 0.0 is identical to that in the quiescent state at the same age.The participation ratios at the end of the strain softening regime ( = 0.1)are nearly unchanged, with a slight shift of the extended modes (large par-ticipation ratio) towards smaller frequencies. In the strain hardening regimeat = 4.0, the vibrational spectrum has changed in two ways: First, Ifind modes with large participation ratio at much smaller frequencies. Thischange is due to the large (400%) elongation of the simulation box along thedeformation axis, which allows modes with larger wavelength to “fit” intothe simulation volume. The participation ratios of these modes suggest thatthey are extended and do not scatter at the structurally weak regions in theglass. Second, the participation ratios of the modes near the boson peakω ∼ 2.0 (see discussion in section 1.2.4) is reduced compared to the unde-formed system. In fig. 6.4(b) I show the average participation ratio 〈P (ω)〉of all Nm = 600 modes used for the calculation of the softness field. In thestrain hardening regime I find a reduction of the average participation ratio,indicating that the soft modes become more localized as the deformationgrows. In the elastic and strain softening regime 〈P (ω)〉 is nearly constant,which is also the behavior observed during aging (not shown).The increase in predictive success rate in the strain hardening regime iswell correlated with the decrease of the average participation ratio. To showthis, I calculate the relative change of these quantities in the strain range0.1 ≤ ≤ 4.0. For a measured quantity O() the relative change is definedas∆O() =O()−O( = 0.1)O( = 4.0)−O( = 0.1).In fig. 6.5 I compare the relative change of predictive success rate Θ(30%)(black) and average participation ratio 〈P (ω)〉 (red) in the strain hardeningregime. Definition and scaling of the relative change sets the value at = 0.1to zero, while the final value at = 4.0 is set to one. The spatial correla-tion increases slowly at smaller strains and more rapidly at larger strains.This increase is closely tracked by the change in the participation ratio. Lo-calized (low participation ratio) soft modes, as discussed in more detail insection 1.2.4, are caused by the scattering of phonons at structurally weak1046.2. Examining the strain hardening regime0.5 1.01.52.0 2.5 3.0 3.5 4.0ǫ0.00.20.40.60.81.0Relative changeΘ(30%)〈P(ω)〉UbondISFigure 6.5: Relative change observed during deformation in the strain hard-ening regime. Lines are guides to the eye.regions. That more strongly localized soft modes lead to a better predictivestrength of the softness field is therefore not surprising. However, the inter-esting question must be: What process drives the increase of localization inthe strain hardening regime?Strain hardening is a polymeric effect, where elastic energy is stored in thecovalent bond interaction along the polymer backbone. The stored energyincreases with growing strain, which leads to larger external stresses. Infig. 6.5 I also show the relative change of the average covalent bond energy(blue) during strain hardening, calculated from the inherent structure. Thebond energy is near constant until = 1.0 and then increases more andmore rapidly with growing strains. The evolution matches the behavior ofparticipation ratio and predictive strength reasonably well, and there is aclear correlation between all three quantities. This correlation suggests thatthe process responsible for the increase in spatial correlation of softness fieldand hops might indeed be tied to the polymeric nature of the glass.A known process that is present during the deformation of polymer glassesis the alignment of polymers along the deformation axis [32]. The extent ofalignment can be calculated by projecting the bond orientations, that is theunit length vector connecting two bonded particles, on the deformation axisand averaging over all bonds in the system. I monitored the alignment of thepolymer with the deformation axis over the full deformation (not shown) and1056.3. Conclusionsfound that it does not correlate to the data shown in fig. 6.5. The polymeralignment starts to increase from zero immediately upon loading and growsmost rapidly in the elastic and strain softening regime. It then continuouslyslows until reaching a value of 0.6 at strain = 4.0 (zero means no corre-lation and one indicates full alignment). In addition to this, I investigatedanisotropy effects in the softness field direction and hop direction duringstrain hardening, yet both quantities remain isotropically distributed.6.3 ConclusionsThe correlation between soft modes and local plastic events is quantified for awell aged polymer glass under uniaxial tensile deformation. A constant strainrate protocol was used and the correlation is analyzed at various points ofthe deformation: in the elastic regime, at the yield strain, during strain soft-ening and far into the strain hardening regime with a maximum engineeringstrain of = 4.0. The plastic events were monitored as hops in the particletrajectories, and the participation of particles in soft modes is quantified as asuperposition of low energy vibrational eigenmodes [102] in a scalar softnessfield. The mean soft mode direction was calculated from a weighted averageof the eigenvectors [86].At all points of the deformation, I find a strong spatial correlation betweenlarge softness of a particle and the occurrence of plastic events. The correla-tion is quantified as the overlap between a binarized soft spot map [63, 81]of the softest particles with the occurrence of hops immediately after thesoftness measurement. At a coverage fraction of 30% soft spots, the locationof 71% of hops is predicted immediately after loading in the elastic regime.The overlap decreases to 61% after yield in the strain softening regime. Inthe following strain hardening regime the overlap increases, and at a strainof = 4.0 it has reached a value of 72%, indicating that the correlation hasbecome stronger than in the pre-deformation polymer glass. In agreementwith the results in chapter 5, I find that the direction of hops is well alignedwith the direction of the soft modes in the soft spots, and deformation didnot alter this directional correlation.The study discussed in chapter 5 and published in ref. [86] showed thataging increases the spatial correlation of soft modes and particle hops. Thisstudy measured the increase in the overlap of soft spots and hops in theaging period prior to the deformation, and I directly show that the decrease1066.3. Conclusionsduring deformation in the elastic and strain softening regime fully reversesthe impact of prior aging. This erasure of history indicates mechanical re-juvenation and the results are consistent with the more detailed study ofmechanical rejuvenation in polymer glasses discussed in chapter 3 and pub-lished in ref. [84].I showed that the increase of overlap between soft spots and hops inthe strain hardening regime is accompanied by a growing localization of thesoft modes. I furthermore showed that this localization is correlated to theincrease of elastic energy stored in the covalent bonds of the polymers. Thisis an indication for a polymeric effect, tied to the chain-topology of thepolymer glass. However, what process is driving the localization could notbe clearly identified from the gathered data. The alignment of polymerswith the deformation axis [32] was found to not correlate to the increase inlocalization and no indications for anisotropy in the softness field directionor hop direction were found in the strain hardening regime. It is also possiblethat the increase in correlation is merely due to the change in box geometryduring deformation, which as discussed above alters the vibrational spectrum.Resolving this question requires further study.107Chapter 7ConclusionsThe research presented in this thesis aims at creating a better understandingof the physical processes that govern plasticity in polymer glasses. Althougha large amount of research has explored the physics of glasses [5, 12, 25, 35,88], the challenge remains to connect the macroscopic effects of mechanicaldeformation with microscopic scale processes that are prevalent in glassymatter. This thesis uses large scale molecular dynamics simulations of awell known bead-spring polymer model [52] with documented glass formingcapabilities [8, 38, 70, 76, 94, 99] to investigate key aspects of glassy physicsin the industrially important case of polymer glass and to bridge the gapfrom microscopic scale to macroscopic plasticity. In four projects I studiedphysical aging, mechanical rejuvenation, dynamical heterogeneity and softmodes as the link between heterogeneous dynamics and local structure bothin the quiescent state and during deformation.The first project explored mechanical rejuvenation using bulk-averagedquantities that simultaneously captured the dynamical, structural and en-ergetic state of post-deformation polymer glasses, which are compared tosystems without deformation history. A key technical contribution of thiswork is the development of an effective algorithm for the detection of indi-vidual structural relaxation events called hops. The detection method wasfirst used to spatially resolve dynamical heterogeneity in the second project.Building on these results, maps of hop events were then used to investigatesignature features in the local structure of regions that undergo rearrange-ments in quiescent polymer glasses. Subsequently, the analysis was extendedto mechanically deformed systems to study the structural origin of localplastic events. This link between local structure and plastic events was fur-thermore found to be sensitive to physical aging as well as to mechanicalperturbation, substantiating the results on mechanical rejuvenation foundusing bulk-averaged quantities.108Chapter 7. ConclusionsMechanical rejuvenation The non-equilibrium nature of the glass drivesa continuous evolution towards lower energy states, which causes an increasein yield stress as well as brittleness of polymer glasses with increasing age [88].Mechanical deformation can reverse the impacts of physical aging, an effectknown as mechanical rejuvenation [88]. Whether deformation and agingare indeed directly coupled is a matter of scientific controversy [64] withone central question: Is the state that the glassy system is driven into bymechanical deformation indeed comparable to the state of a younger glass?In a recent series of experiments Lee and Ediger showed that the molecularmobility, which decreases during aging, is increased after the application ofmechanical stresses [57, 59]. Measurements in the recovery regime after thedeformation, however, revealed two different impacts [58]: After deformationat a small stress amplitude, the molecular mobility quickly recovered its pre-deformation value, while the application of a much larger stress amplitudepermanently altered the mobility and the system evolved in the recoveryregime comparable to a younger glass without deformation history.The study discussed in chapter 3 and published in ref. [84] was designedto complement these results with an experimentally inaccessible perspectiveof the evolution in the recovery regime (recovery path) by simultaneouslymonitoring the structural α-relaxation time, inherent structure energy andmeasures of local spatial order. In agreement with the experiment, my resultsshow that after weak deformation in the pre-yield regime the recovery pathreturns to the aging behavior of an undeformed polymer glass of equal age.The perturbation of the recovery path becomes stronger with increasing to-tal strain at the end of the deformation and the history of the polymer glassis permanently erased at around the yield strain. Here, structural, dynam-ical and energetic perspectives indicate a recovery path that is comparablewith the aging behavior of a younger glass, i.e. permanent mechanical re-juvenation. After deformation in the post-yield regime, however, the threeperspectives on the recovery path yield different indications on mechanicalrejuvenation: While the α-relaxation time is consistent with that of a youngerglass, the structural and energetic quantities reveal that the deformed systemis driven into a state that is distinct from a quiescent polymer glass of anyage. This observation supports the view of McKenna [64] that mechanicaldeformation does not only reset the internal clock of a glass, but that it candrive the system to a new thermodynamic state.Furthermore, my results discussed in chapter 6 give an additional per-spective on mechanical rejuvenation: The correlation between local plastic109Chapter 7. Conclusionsevents and soft modes was found to be sensitive to aging as well as me-chanical perturbation. The correlation became stronger during aging, yetdeformation up to the end of the strain softening regime resulted in a resetof the correlation to pre-aging values, indicating mechanical rejuvenation ofthis link between local structure and dynamics.My results for the pre- and post-yield deformation therefore resolve thequestion of mechanical rejuvenation in the following sense: Mechanical de-formation around the yield-point leads to a permanently altered glass statethat is comparable to a younger glass without deformation history. If thedeformation is limited to the elastic regime prior to the yield point, the sys-tem state is only transiently altered and the glass recovers quickly to thepre-deformation state. Deformation in the post-yield regime, beyond theend of strain softening as indicated by the results discussed in chapter 6,drives the glass to a new state that is structurally and energetically distinctfrom a younger glass without deformation history. My results furthermoreshow that, in the limit of no aging in the deformation period, the impact ofdeformation on the recovery path is controlled solely by the total engineer-ing strain at the end of the deformation. This finding supports an earlierstudy [100] that reports the importance of strain in describing accelerateddynamics during deformation. It would be interesting to further explore therole of strain as deformation parameter, because it could provide more insightinto the key components for models of plasticity in polymer glasses [25, 37].Dynamical heterogeneity in the aging regime One of the definingcharacteristics of glassy physics is the emergence of dynamical heterogene-ity (DH) near the glass transition [9, 24, 28, 34, 47, 51, 53]. In the glassstate particle rearrangements are correlated on the molecular level, leadingto cooperative motion of groups of particles [33, 51] and the partition ofthe system into transient regions of “faster” and “slower” structural relax-ation. The study of DH has been concentrated on supercooled liquids, yet itis clearly important in understanding plasticity in polymer glasses. Recentstudies that quantify DH via three- and four-point correlators indicate thatdynamical correlation is increasing during physical aging [17, 71].In the second project, discussed in chapter 4 and published in ref. [85],I analyzed the spatio-temporal distribution of structural relaxation in qui-escent polymer glasses at various temperatures and ages. Relaxation eventswere detected as hops in the particle trajectories and I introduced an adap-110Chapter 7. Conclusionstation of an algorithm by Candelier et al. [19]. This detection algorithm is akey technical contribution of this thesis, because it allows the measurementof relaxation events with high spatio-temporal resolution on-the-fly for thefull duration of the simulation. The technique allowed me to calculate thespatio-temporal density-density correlation and I found a strong correlationbetween near-simultaneous hops of neighboring particles. A cluster analysiswas used to measure the size of cooperatively moving groups of particles,and I found an exponential distribution with power-law tail that becomesless pronounced during aging and at lower temperatures. The latter trendconnects well with results for agitated granular media [19].I furthermore used the map of relaxation events to spatially resolve DHas hop clusters, and I compare their growth and volume distribution with thesimultaneously measured four-point dynamical susceptibility χ4 as the stan-dard measure of DH [9, 53]. In agreement with a study of a model metallicglass [71], I find a growing maximal dynamical correlation with increasingage that is shifted towards larger time scales during aging. A cluster analysisshowed that the time of maximal correlation coincides with the formationof a single hop cluster that encompasses nearly the whole system, with onlysingle cage sized pockets of particles that have not undergone structural re-arrangements. The cluster volume distribution prior to the merging into asingle dominating cluster follows a power-law with exponent −2, which in-dicates that fluctuations are sufficiently small so that averaged quantities donot behave anomalously. This result helps explain the success of mean-fieldmodels of aging [98], which capture the evolution of dynamical quantities likethe mean squared displacement while entirely neglecting DH.I introduced an efficient hop detection algorithm, that allowed me to givea new perspective on dynamical heterogeneity by simultaneously measuringDH using the established dynamical susceptibility measure as well as theaggregation of hops into mesoscopic clusters. For the first time, I measuredgrowing dynamical correlation and shift towards larger time scales duringphysical aging in a polymer glass, confirming recent results for model binaryglasses. The time of maximal correlation was found to coincide with theformation of a single system spanning cluster of hops with only single-cagesized regions that did not undergo structural relaxation.Link between local structure and particle rearrangements A keyopen challenge in developing a theory of plasticity in glasses is to understand111Chapter 7. Conclusionsthe link between the location of particle rearrangements and the local struc-ture. Which structural feature determines the spatial distribution of DHand distinguishes regions with rapid structural relaxation from quasi-stablelocal configurations of particles? The goal of this inquiry is to find a coarsegrained structural description for amorphous solids in the spirit of disloca-tions of crystalline solids. Recently, quasi-localized low energy vibrationalmodes have attracted much attention as a possible candidate [102, 103], anda strong correlation was quantified between these soft modes and particle re-arrangements in a mechanically driven model metallic glass in 2D and at zerotemperature [63]. My third and fourth projects were dedicated to verify thiscorrelation in a thermal, three-dimensional polymer glass in the quiescentstate and during mechanical deformation. Structural relaxation events, orelementary plastic events in the case of deformation, were spatio-temporallyresolved as particle hops using the detection algorithm developed in the sec-ond project, and I used a simple superposition scheme to construct a softnessfield from the low energy vibrational spectrum [102].In the third project, discussed in chapter 5 and published in ref. [86],I studied the correlation between softness field and hops in the quiescentstate at three temperatures and during physical aging. My results show thathops occur up to 7 times more often than average in the softest regions ofthe system. This indicates a strong spatial correlation that increases duringaging and decreases at higher temperatures. I furthermore found a strongdirectional alignment of hops in these soft regions with the mean directionof soft modes. These findings support a very recent study that investigatedthe role of soft modes at the crossover between ordered to disordered sys-tems [77]. Soft modes were observed to predict the direction and locationof rearrangements in a hierarchy of systems: from a crystal with a singledislocation, to a polycrystal and a binary glass in 2D. My analysis of thesoftness field autocorrelation showed that soft modes are long-lived struc-tural features compared to vibrational time scales, and the autocorrelationonly decays to zero when nearly the entire system has undergone structuralrelaxation. A very recent study of a sheared, thermal model binary glass in2D supports this finding by tracking the lifetime of individual soft regions,called soft spots [81]. The study showed that the lifetime is correlated to thestructural relaxation time and that individual soft spots can survive manyindividual rearrangements.The final project, discussed in chapter 6, quantified the correlation in anaged polymer glass under uniaxial tensile deformation at various points of112Chapter 7. Conclusionsthe deformation. Measurements were analogous to the study of the quies-cent case, and I verified the spatial and directional correlation of soft modesand hops in all deformation regimes. Interestingly, the spatial correlation,which increased during the aging period prior to the deformation, was foundto decrease to the pre-aging value when the deformation reached the strainsoftening regime. This finding showed that the link between structure anddynamics in glasses is sensitive to mechanical rejuvenation as discussed inthe first project. Monitoring of the position in the PEL confirmed this con-nection: After a decrease of the inherent structure energy during aging, thesystem was driven back towards higher energies by the deformation and thepre-aging value was reached in the strain softening regime, simultaneouslywith the resetting of the spatial correlation.Measuring the correlation of soft modes and hops during deformation inthe strain hardening regime revealed a novel effect with a link to the chainconnectivity in polymer glasses: The spatial correlation was found to increasewith increasing strain, reaching values higher than observed just prior to thedeformation at a strain of = 4. The increase in spatial correlation wasshown to be tied with a growing localization of the soft modes as well aswith the increase in potential energy stored in the covalent bonds, which isthe mechanism causing strain hardening. The latter link suggests a poly-meric origin of the effect. However, the orientation of polymers along thedeformation axis did not correlate to the increase in localization, nor did Ifind anisotropies in the softness field direction or hop direction during thedeformation.My results show a strong link between quasi-localized soft modes andthe location of particle rearrangements. For the first time, this correlationwas quantified in a polymer glass and I studied the impact of temperature,physical aging as well as mechanical deformation. Beyond the spatial cor-relation I also showed that the direction of individual rearrangements arealigned with the polarization of the soft modes. Finally, I showed that thestructural information encoded in the soft modes is long-lived compared tothe vibrational time scale and that the correlations fully decay on the orderof the structural relaxation time.Outlook A growing body of research is indicating that soft modes are in-deed linking irreversible rearrangements, plasticity and microscopic structurein amorphous solids. Evidence has been found in a diverse set of model sys-113Chapter 7. Conclusionstems mostly in 2D, and this thesis adds quantitative evidence in 3D: Frombinary supercooled liquids in the quiescent state [44, 102, 103], sheared bi-nary glasses at zero temperature [63, 68, 90] and finite temperature [81], topolycrystals[77], lattice models [3] and the present quiescent as well as de-formed polymer glass. Various measures for rearrangements have been used,ranging from the change of nearest neighbors [102], to maxima in the non-affine displacement field [81], and here hops in individual particle trajectories.Moreover, soft modes were quantified in different ways: correlations to rear-rangements were identified w.r.t. individual modes [90], the binary soft spotfield [63] and the superposition of participation fractions [102] used here.A recent study used information theory to directly measure the extent ofcorrelation between soft modes and the propensity of motion [49], which isthe part of the particle motion that is determined by the structure alone andthat can be isolated by averaging in the iso-configurational ensemble [13, 101].The study quantified the difference between joined probability distributionof soft modes and propensity of motion and the factorized distribution (in-dependent variables hypothesis), with a large difference indicating a strongcorrelation. Not surprisingly, the difference was maximal when the propen-sity was measured in a time interval on the order of the vibrational timescale and it decreased with increasing interval size. However, a significantdifference was still present at time intervals of the order of the structuralrelaxation time proving that a correlation exists. Despite these variationsin simulated models and analysis, the robustness of the correlation suggeststhat soft spots should play a prominent role in theories of plasticity for amor-phous solids in general, and the practically important case of polymer glassesin particular.A second approach to understand the cause of dynamical heterogeneityin amorphous solids has developed around the discovery of heterogeneouslydistributed local elastic moduli [66, 70, 91, 106]. Regions of small shearmoduli were found to be prone to plastic rearrangements, whereas areaswith high moduli tend to be more structurally stable. Both approachesare closely related as they are harmonic theories and some work has beendone to understand the link between them [31]. It would be interesting tofurther explore the relationship between soft modes and local elastic moduli,and directly analyze both spatial distributions. Do soft spots indeed havea small shear modulus, and are the decay timescales of the heterogeneousdistributions related?The softness field analysis during deformation in the strain hardening114Chapter 7. Conclusionsregime revealed a growing spatial correlation of hops to soft modes duringstrain hardening. Clearly identifying the process responsible for this increaseis an interesting question emerging from this thesis and should be the topic offuture work. To answer this question, it would be instrumental to understandthe true cause for the increase in localization of the vibrational modes. Strainhardening introduces a stiffening of the polymer backbones, and it wouldbe interesting to investigate more closely how this affects the vibrationalmodes. One possibility would be to model an amorphous solid as a randomnetwork of springs and to study the vibrational modes while increasing thestiffness of springs that are oriented along one dimension. Another possibleavenue would be to investigate whether strain hardening has an impact onthe distance to the glass transition in terms of molecular mobility. Thethird project showed that the spatial correlation is increasing with decreasingtemperature, i.e. the correlation is stronger deep in the glass state, where themolecular mobility is very low. Does the stiffening of the polymer backbonelead to a slowing of molecular mobility? A fruitful starting point of a futurestudy could be to measure the non-affine displacement, which excludes thedisplacement introduced by the change of the simulation box shape, and tocompare it at different points during deformation in the strain hardeningregime.Recently, Fielding et al. [37] proposed a simple model that explains theimpact of deformation-induced flow on the structural relaxation time in poly-mer glasses. The key idea is to separate relaxation events caused by theelastic energy stored in covalent bonds from those that relax the elastic fieldof neighboring but non-bonded monomers. This separation of “polymeric”and “solvent” degrees of freedom leads to a competition between these re-laxation types during flow. The solvent relaxes on smaller time scales andincreases the energy in the covalent bonds, until this forces polymeric relax-ation events. 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Microscopic perspective of polymer glasses during physical aging and mechanical deformation : a computational… Smessaert, Anton 2015
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Title | Microscopic perspective of polymer glasses during physical aging and mechanical deformation : a computational study of dynamical heterogeneity, plasticity, and soft vibrational modes as the link to the molecular structure |
Creator |
Smessaert, Anton |
Publisher | University of British Columbia |
Date Issued | 2015 |
Description | Microscopic dynamics and mechanical response of polymer glasses are studied in four projects using molecular dynamics simulations of a simple bead-spring model. The first project studies the interplay between physical aging and mechanical perturbation. Structural, dynamical and energetic quantities are monitored in the recovery regime following aging and uniaxial tensile deformation periods. The total engineering strain is found to control a continuous transition from transient to permanent mechanical rejuvenation: After deformation in the pre-yield regime all quantities quickly reset to pre-deformation values, while deformation around the yield point results in the erasure of aging history. Deformation in the post-yield regime, however, drives the system into a distinct thermodynamic state. In the second project, I introduce an efficient algorithm that detects microscopic relaxation events, which are the basis of aging dynamics and plasticity. I use this technique to calculate the density-density correlations from the spatio-temporal distribution of so called hops in quiescent polymer glasses at different temperatures and ages. Correlation ranges are extracted and I analyze the size distributions of collaboratively rearranging groups of particles. Furthermore, I spatially resolve dynamical heterogeneity (DH) as hop-clusters, and I compare cluster growth, as well as volume distribution during aging with the four-point dynamical susceptibility Χ₄ as the established measure of DH. The third and fourth project use the hop detection technique to investigate the link between relaxation events and local structure. Quasi-localized low-energy vibrational modes, called soft modes, are found to correlate with the location and direction of hops. In the third project, I analyze the temperature- and age-dependence of this correlation in quiescent polymer glasses, and I show that the soft modes are long lived structural features. The fourth project extends the analysis to mechanically deformed polymer glasses. I find that the spatial correlation of hops and soft modes is reduced to pre-aging values after deformation in the strain softening regime. This reveals an additional perspective on mechanical rejuvenation and substantiates the findings from the first project. In the strain hardening regime the correlation increases, and this novel effect is linked to a growing localization of the soft modes. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2015-04-30 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial 2.5 Canada |
DOI | 10.14288/1.0166245 |
URI | http://hdl.handle.net/2429/53018 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 2015-09 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc/2.5/ca/ |
Aggregated Source Repository | DSpace |
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