Distinguishing failure modes at the molecular level by examining heterogeneity in local structures and dynamics
DDistinguishing failure modes at the molecular level by examiningheterogeneity in local structures and dynamics
Emily Y. Lin, Robert A. RigglemanUniversity of Pennsylvania, Philadelphia, PA
Brittle failure is ubiquitous in amorphous materials that are sufficiently cooled below theirglass transition temperature, T g . This catastrophic failure mode is limiting for amorphousmaterials in many applications, and many fundamental questions surrounding it remainpoorly understood. Two challenges that prevent a more fundamental understanding of thetransition between a ductile response at temperatures near T g to brittle failure at lowertemperatures are i) a lack of computationally inexpensive molecular models that capturethe failure modes observed in experiments and ii) the lack of quantitative metrics that candistinguish various failure mechanisms. In this work, we use molecular dynamics simula-tions to capture ductile-to-brittle transition in glass-forming short-chain polymer systems byusing a modified Lennard-Jones potential to describe non-bonded interactions between themonomers. We characterized the effects of this new potential on macroscopic mechanicalproperties as well as microscopic structural and dynamical differences during deformation.Lastly, we present quantitative metrics that distinguish between different failure modes. Glass-forming polymers play a crucial role in many applications, and amorphous polymersremain one of the few glass-forming materials commonly found in applications. Other familiesof glasses, such as metallic glasses, are too brittle for many potential applications, andunderstanding the dynamics in these systems during deformation has become an active topicof research in the last few decades. Glassy materials tend to fail catastrophically in a brittlefashion under high loading conditions, which can be problematic in applications where the1 a r X i v : . [ c ond - m a t . s o f t ] M a r tructural integrity of these materials is critical to their function. In crystalline systems,defects propagate during deformation to provide ductility, and the motion of the defects canbe easily monitored. However, in disordered solids defects cannot be easily identified, andparticle rearrangements that lead to failure become much more complicated to parse. Plasticevents are present even when the macroscopic properties indicate the material is still in theelastic regime of the deformation, and they eventually organize into the shear band wherethe material ultimately fails.
Failure in non-polymeric disordered solids after yielding can take several forms. For ductilematerials, the shear band draws down to a neck near the yield, at which the sample breaksafter significant plasticity. For brittle materials, this necking behavior is not observed; in-stead, cavities form in and near the shear band, and the sample catastrophically fails viafracture with little additional strain. The transition between ductile and brittle behaviorcan be tuned through both sample preparation protocols and experimental conditions, suchas temperature at which deformation occurs and the strain rate, and because glasses arenon-equilibrium systems, the vitrification process and the age of the specimen can also playa role in the failure mechanism.
At low temperatures, brittle failure is expected for essentially all amorphous solids, includingglassy polymers. However, previous studies using the common coarse-grained model polymerwith Lennard-Jones (LJ) interaction sites connected by rigid bonds have not observed brittlefailure, even for small chain lengths. While many of the phenomena produced via thismodel agree qualitatively very well with experiments,
20, 21 the inability to capture a transitionfrom homogeneous plastic flow to brittle failure modes remains a significant shortcoming.Previous studies of cavitation and crazing have required imposing triaxial loading;
11, 12, 22–25 however, this method increases the volume of the specimen, and cannot be used to observe thetransition between cavitation and crazing to homogeneous plastic flow. Even the commonbinary monomer Lennard-Jones glass, which is nominally a model for a metallic glass Clearly an inexpensive molecular model, or family of models, that exhibit the expected failuremodes would provide a platform to study failure in amorphous solids.Another challenge facing molecular simulation studies of failure in amorphous solids is thelack of quantitative metrics that can distinguish the various failure modes. Since simulationstypically use relatively small sample sizes (with notable exceptions
28, 29 ) where the failure isnot as sudden as in macroscopic samples, the strain to failure is often of limited utility. Thissituation often leads to describing transitions between various failure modes in a qualitativemanner and often through visual inspection.In this work, we take a first step towards addressing these two issues by characterizing a seriesof model glass-forming polymers. Beginning with the standard Lennard-Jones models, wesystematically modify the temperature, cooling rate used to prepare the glass, and the formof the potential (following ideas proposed in previous work
30, 31 ) and study the failure modeof nanopillars under tensile loading. Across temperature and the potentials considered, weobserve a spectrum of failure modes including brittle fracture, ductile necking, homogeneousplastic flow, and several intermediate modes. In addition, we develop and examine somequantitative metrics that distinguish the various failure modes. In general, we find that thetransitions between the different failure modes are gradual as the system parameters arechanged, and we demonstrate that surface roughness alone does not capture the changesobserved.
For our Molecular Dynamics (MD) simulation studies, we used coarse grained model oligomerswith chain length of N = 5 monomers per chain. The non-bonded interactions take the form3f U nbij = 4 (cid:15) (cid:34)(cid:18) σ (cid:48) r ij − D (cid:19) − (cid:18) σ (cid:48) r ij − D (cid:19) (cid:35) − u cut (1)where D is a parameter that allows us to modify the range and curvature at the minimum inour potential, and when D = 0 this reduces to the standard LJ potential and has been wellestablished previously. We adjust σ (cid:48) = 1 − D/ / to fix the location of the minimum,as shown in Figure 1 for the four values of D used in this study. The bonded interactionin our simulations is a stiff harmonic bonding potential, U bij = ( k h / r ij − σ ) , where k h = 2000 (cid:15)/σ . All units reported in this study are in LJ reduced units: temperature T = kT ∗ /(cid:15) , and time τ LJ = t ∗ (cid:112) (cid:15)/mσ , where m represents the mass of a single LJ interactionsite, T ∗ and t ∗ represent temperature and time measured in laboratory units, and (cid:15) and σ are the parameters of the standard LJ potential ( D = 0). We used the LAMMPS simulationpackage with a small simulation time step of δt = 0 . τ LJ / timestep. We chosethis δt value to be commensurate with the increased curvature of the modified LJ potential(mLJ), and the mLJ potential is implemented in LAMMPS as the “lj/expand” pair styledue to its common use in simulating nanoparticles and colloids. r -1.00.01.02.02.0 U (r) LJ, (D=0)D = 0.25D = 0.50D = 0.75
Figure 1: Standard and modified LJ pair potentials. .1 Characterization of the mLJ potentials For comparing the T g of the standard and of the modified LJ systems, bulk polymer glasssimulations were used. Monodispersed systems consisting of 6000 monomers total were usedfor LJ and mLJ ( D = 0 .
75) potentials. Three independent configurations were generated byequilibrating at high temperature for different lengths of time, and uncertainties are takenas standard error calculated using these three configurations. After equilibration at T (cid:29) T g ,we rapidly quenched the systems to T = 0 .
05 at a cooling rate of ˙Γ = ∆
T / ∆ t = 1 × − .We collected volume change during cooling, which allowed us to obtain T g by locating thetemperature at which the thermal expansion coefficient α T = ( ∂ ln v/∂T ) P changes, shownin Figure 2. The T g for the LJ systems is 0 . ± . T g for the mLJ ( D = 0 . . ± . α T,l /α T,g )for the LJ and the mLJ systems are 2 . ± .
001 and 15 . ± .
09, respectively. While thisquantity for the LJ system matches well with experimental data of a typical polymer, thevalue for the mLJ system is significantly higher due to the large α T for the supercooledliquid. However, since our interest in this study is mainly to investigate the mechanics of theglass pillar failure mechanism and the transition from necking to fracture failure at constanttemperature below T g , properties of the supercooled liquid do not play a significant role inour simulations. Temperature l n ( V ) LJ (D = 0): Tg = 0.417 ± 0.003
D = 0.75: Tg = 0.398 ± 0.001
Figure 2: Characterization of LJ and mLJ ( D = 0 .
75) systems. (a). Cooling profiles of LJ and mLJ bulksystems using a quench rate of ˙Γ = 1 × − . The dotted lines are linear regression fits used to identify T g . .2 Strain localization studies Cylindrical pillars with a diameter of approximately 30 σ and an aspect ratio of approxi-mately 2 were generated; this diameter is large enough to allow for bulk like dynamics inthe center of the pillar. Four independent configurations of pillars for each D value weregenerated at high temperatures and equilibrated to erase previous thermal history. Pillarswere equilibrated in the NVT ensemble at T (cid:29) T g with periodic boundary condition alongthe length of the pillars, then quenched to T = 0 .
05. A repulsive wall potential was used athigh temperature to maintain pillar geometry. Configurations at a total of six temperaturesbelow T g ( T ∈ [0 . , .
3] with an increment of 0.05) were collected during cooling in orderto ensure a consistent sample history. We deformed each pillar at constant temperature andconstant true strain rate ( ˙ (cid:15) = 1 × − ) by applying uniaxial tension along the z direction.Using particle configurations during deformation, we calculated the local strain rate associ-ated with each particle
4, 36 J ( (cid:15), (cid:15) + ∆ (cid:15) ) = 1∆ (cid:15) (cid:115) d T r (cid:20) d ( J Ti J − I ) − d T r ( J Ti J − I ) (cid:21) (2)where d is the dimensionality of the system, J i is the best affine transformation matrix forparticle i at strain (cid:15) , given a lag strain of ∆ (cid:15) , and I is the identity matrix. This local strainrate is calculated for each monomer by calculating the best-fit local affine transformationmatrix, constructing the Lagrangian strain tensor, and extracting the deviatoric components.Particles with large J values have a higher non-affine strain rate in their local environment.While it is informative to see the variation of J for each monomer, we also develop metricsto characterize the mesoscale nature of the strain response in order to distinguish strainlocalization and homogeneous plastic flow. We divided the cylindrical pillar axially into 20slabs of equal thickness, and calculated the spatial fluctuations of average J values between6airs of slabs. We define S L as this quantity averaged over all pairs of slabs, S L ( T, (cid:15) ; (cid:15) w ) = 1 n b ( n b − n b (cid:88) i =1 n b (cid:88) j =1 ,j (cid:54) = i (cid:104) ( J ( i, (cid:15) ) − (cid:104) J ( j ) (cid:105) (cid:15) w ) (cid:105) (cid:15) w · (cid:104) ( J ( j, (cid:15) ) − (cid:104) J ( i ) (cid:105) (cid:15) w ) (cid:105) (cid:15) w (3)where n b denotes the number of blocks, the overhead bar represents averaging over all particle J values within the block, (cid:15) w represents the strain window over which we averaged J valuesin addition to slab average. In all of our calculations, we chose (cid:15) w to be the same as thelag strain used in calculations of J , ∆ (cid:15) = 1%. This additional average over strain windowallows us to focus on spatial fluctuations that are relatively long lived. When the strainis homogeneous, the averages of J in any two slabs i and j will be approximately thesame, and as a result (cid:104) ( J ( i, (cid:15) ) − (cid:104) J ( j ) (cid:105) (cid:15) w will be nearly zero and S L is small. In contrast,when there are large spatial variations in the strain rate, S L increases sharply, indicatingstrain localization. Each slab is approximately 5 monomers thick prior to deformation, andincreases in thickness affinely as strain increases. We used a C++ library (Voro++) to perform Voronoi tessellation of our nanopillar, and tocompute the resulting Voronoi volumes. For the tessellation routine, we treated the centerof our monomer as the centroid of the Voronoi cell. After the calculation of the Voronoivolumes for all particles in our nanopillars for all strains, we discard the particles that are onthe surfaces of the pillars by discounting any particle that has a Voronoi volume larger than8 σ . We examined the distributions of Voronoi volumes before and after strain is applied fordifferent D values and temperatures. Finally, to identify particles near a fracture surface,we sorted all of the non-surface particle Voronoi volumes by their magnitude, and defineda cutoff Voronoi volume for each D value and (cid:15) to be the 90 th percentile of the Voronoivolumes. 7 Results4.1 Macroscopic Mechanical Response
We began our analyses by comparing the mechanical response of the LJ and mLJ (with D = 0 .
75) systems. First, we used engineering stress data collected during our deformationsimulations to plot the stress-strain relationship for both systems to investigate the effect ofmodifying the LJ potential on the yield strength and the elastic modulus. As shown in Fig.3, both the yield strength and the elastic modulus are higher in the mLJ system, and theydecrease modestly with increasing temperature, as expected. The effects of temperatureare more pronounced in the mLJ system with D = 0 .
75. At the highest temperature inour study, the LJ and mLJ pillars exhibit similar stress-strain behavior, suggesting that theeffect of modifying interaction potential on bulk mechanical properties becomes minimal athigher temperatures close to T g . The differences in behavior at low temperature are stark:LJ exhibits significant plasticity, while mLJ stress curve decreases sharply after yieldingat (cid:15) ≈ D considered; as D increases, the stress-strain curveshows less plasticity after yield.The effect of quench rate exhibits the expected behaviors as quench rate decreases. Toexamine the effect of quench rate, we constructed pillars using quench rate two orders ofmagnitude slower ( ˙Γ = 1 × − ) than the fast quenched samples. Figure 3d demonstratesthat the slowly quenched pillars at a given D and temperature tend to have higher yieldstress and higher elastic modulus than their fast quenched counterparts, consistent with thecommon observation that physical aging of a glass increases its modulus.8 Strain ( ε eng ) S t r e ss ( σ e ng ) + c T = 0.05T = 0.10T = 0.15T = 0.20T = 0.25T = 0.30 LJ T = 0.05 (a)
Strain ( ε eng ) S t r e ss ( σ e ng ) + c T = 0.05T = 0.10T = 0.15T = 0.20T = 0.25T = 0.30
T = 0.05mLJ(b)
Strain ( ε eng ) S t r e ss ( σ e ng ) + c D = 0 (LJ)D = 0.25D = 0.5D = 0.75 (c)T = 0.05
Strain ( ε eng ) S t r e ss ( σ e ng ) + c LJ (D = 0)D = 0.75Fast quenchedSlowly quenched
T = 0.05 (d)
Figure 3: Stress-strain curve of (a) LJ and (b) mLJ pillars at six temperatures below T g . Pillars at tem-peratures below T g were deformed uniaxially at a true strain rate of ˙ (cid:15) = 1 × − . Insets show variationsamongst four independent pillar configurations at the lowest temperature. (c). Pillars at T =0.05 using allfour values of D (0.00, 0.25, 0.50, 0.75). (d). Comparison between two quench rates for the LJ and mLJ (D= 0.75) pillars at T = 0.05. A stress offset is used by subtracting the initial system stress (constant c) tostart all curves at σ eng = 0 Figure 4 demonstrates the difference in failure mechanism between the two potentials with D = 0 and 0 .
75. Snapshots of our simulation were taken while the deformation is in elastic,yield, and post-yield regimes, and instantaneous J values calculated over a lag strain of 1%are employed to color each monomer. The color scale is compressed so that blue represents80th percentile and lower, and red represents 99.95th percentile and higher, and these per-centiles are defined based on J data collected from the entire simulation. Each framed (blueand red) set represents a configuration at the stated conditions: LJ vs. mLJ ( D = 0 . T = 0 .
05 vs. T = 0 .
30. The LJ system exhibits necking at the lowest temperature in ourstudy, implying the system retains significant ductility even far below its T g , while the mLJ9ystem at low temperature exhibits brittle failure with a fracture surface at approximately45 o to the normal. At T = 0 .
30, both potentials exhibit homogeneous plastic flow.
LJ, T = 0.05(a) mLJ, T = 0.05(b)LJ, T = 0.30(c) mLJ, T = 0.30(d) 99.95%80% J Figure 4: Visualization of the strain field of a subset of our polymer pillars. Pillars are colored with a selectedrange of J values in each frame (80th to 99.5th percentiles). The four framed sets show (a) LJ at T = 0.05,(b) mLJ at T = 0.05, (c) LJ at T = 0.30, and (d) mLJ at T = 0.30. Within each set, we visualized the strainfield on the pillars in the elastic (left: (cid:15) ≈ . (cid:15) ≈ . (cid:15) ≈ Figure 5 visualizes the post-yield strain field as a function of temperature for all of thepotentials considered. We can see that for a given D value and quench rate, the degreeof strain localization decreases and eventually vanishes as temperature increases. For thesame quench rate and temperature combination, as we increase D we transition from neckingdeformation to brittle fracture failure modes. Additionally, the slowly quenched pillars havemore pronounced strain localization than their fast quenched counterparts, especially athigher temperatures. 10 emperature D Fast quenched Slowly quenched
D = 0 (LJ)D = 0.25D = 0.50D = 0.75 J Temperature D Figure 5: Representative images of pillars at the end of our deformation simulations ( (cid:15) ≈ J values calculated comparing the configurations at the initial and theend of the entire simulation. Particle color scales are determined using the 80th and 99.95th percentile ofall J values for the cutoff for blue and red colors respectively. This means the blue particles have lowestcumulative J values and the red particles have highest cumulative J values. In each of the cooling rates,temperature increases from left to right, and D value increases from top to bottom. We quantify the extent of strain localization by examining the S L function defined above forall of the systems as a function of strain. In Figure 6, we plotted log S L as a function of totalstrain. In both the LJ and the mLJ systems, log S L at a given strain increases sharply astemperature decreases, and this increase is larger in the mLJ system. At higher temperatures,the log S L values remain relatively unchanged throughout deformation, consistent with thesnapshots shown in Figure 4 and the expectation for a homogeneous strain field. At lowertemperatures, the magnitude of the spatial variations in strain rate becomes substantiallylarger as strain increases due to the formation of either a neck or the shear band. As T is lowered, S L gradually increases with no sharp cross-over from homogeneous to localizedplastic flow. Figure 7 shows that for different values of D , the S L values differ by up to oneorder of magnitude between the two extreme cases of D values as the sample is deformed11hrough the yield point. While S L does distinguish homogeneous dynamics from strainlocalization, S L does not allow us to draw a distinctive boundary between necking andbrittle failures. ε eng l og ( S L ) T = 0.05T = 0.10T = 0.15T = 0.20T = 0.25T = 0.30
LJ (a) ε eng l og ( S L ) T = 0.05T = 0.10T = 0.15T = 0.20T = 0.25T = 0.30 mLJ (b)
Figure 6: Spatial fluctuation of local strain rate ( S L ) as a function of strain for (a) LJ and (b) mLJ pillarsdeformed at temperatures below T g using uniaxial deformation rate of ˙ (cid:15) = 1 × − . S L is calculated usinga time window that corresponds to (cid:15) = 1%, and each pillar is divided into 20 slabs in length ( z ) such thateach slab has a thickness of 5 . σ to 5 . σ , before and after 30% strain is applied, respectively. ε l og ( S L ) LJ (D=0)D = 0.25D = 0.50D = 0.75
Figure 7: S L as a function of strain for each D value at T = 0 . J values used to compute S L here arecumulative, i.e., between the current strained configuration and initial configuration. To distinguish between ductile necking and brittle fracture, we examine the Voronoi volumes12or the monomers in our nanopillars. When we plot the Voronoi volume distributions for thesamples experiencing brittle and ductile failure modes, we observe that the brittle sampleshave significant changes in Voronoi volume distributions after strain is applied. Figure 8ashows that for the most brittle nanopillar (mLJ, D = 0 . D = 0) after the same amount ofstrain is applied. Additionally, at the highest temperature ( T = 0 . V c ( (cid:15) ). We then plotted this cutoff volume as afunction of strain for the various D parameters we have used in our study; to better comparethese cutoff volumes across different systems, we normalized V c at each strain by its valuebefore deformation, ¯ V c = V c ( (cid:15) ) /V c ( (cid:15) = 0). Voronoi volume, V -1 P [ V ] LJ, ε = 0LJ, ε = 0.1D = 0.75, ε = 0D = 0.75, ε = 0.1Fast quenchedSlowly quenched (a) Strain ( ε eng ) V c LJD = 0.25D = 0.50D = 0.75Fast quenchedSlowly quenched (b)
Figure 8: (a). Voronoi volume distributions for LJ and D = 0 .
75 before deformation and after yield. (b).Cutoff Voronoi volume for the four D values tested in our study, at T = 0.05, on samples generated byusing different quench rates. Solid lines represent fast quenched samples ( ˙Γ = 1 × − ), while dashed linesrepresent slowly quenched samples ( ˙Γ = 1 × − ). Error bars for all samples are generated using standarderrors over three independent configurations. Figure 8b shows the cutoff volume vs. strain for all four D parameters used, at the lowesttemperature we have studied (T = 0.05). Because of the normalization procedure, the ¯ V c V exhibits a modest increase of approximately 1%while the more brittle pillars exhibit a more rapid and larger increase, up to approximately5%. For more slowly quenched pillars, ¯ V c values are higher at a given strain for each D ,suggesting that the slowly quenched samples are embrittled due to their more rapid increasein the particles’ relative free volume. The mechanical properties of glass are highly dependent on surface treatment, and previousstudies have argued that the response of different materials can be understood throughchanges in surface roughness.
38, 39
Here we attempt to quantify whether our observed failuremodes correlate with changes in surface roughness. From our pre-deformation configurations,we calculated the surface roughness as σ R = (cid:80) ni =1 (cid:16) R i − (cid:104) R (cid:105) pillar (cid:17) , where we summed thesquared differences between i th slab radius, R i , and the averaged pillar radius, (cid:104) R (cid:105) pillar , forall of our pillars, and plotted them against the fluctuation in spatial correction of local strainrate, S L . As demonstrated above in Figure 6, lower temperature and higher D values resultin higher S L values. For systems with D >
0, the pillars tend to have a rougher surface.However, the variation in roughness is not significant between samples with the same D value and temperature, even when we compare the coldest pillar with the hottest pillar,which clearly have different failure modes and S L values. This result suggests that surfaceroughness alone cannot adequately differentiate the different degrees of strain localization.A similar lack of correlation is also observed between ¯ V c and the surface roughness.14 .1 1.0 roughness ( σ ) S L LJ (D=0)D = 0.25D = 0.50D = 0.75
Fast quenchedSlowly quenched
Figure 9: Spatial fluctuation of local strain rate as a function of pillar surface roughness for fast (closedcircles) and slowly (open circles) quenched pillars in our studies. Each data point is an average of threeindependent configurations.
We used Molecular Dynamics (MD) systems of glassy polymer nanopillars to simulate uni-axial tensile deformation. In order to simulate brittle failure, we incorporated a previouslyproposed change to the Lennard-Jones (LJ) pair potential and further characterized theimpact of this modification on macroscopic material properties as well as structural anddynamic information on the length scale of merely a few Kuhn lengths. Using an order pa-rameter that quantifies the localized strain rate spatial fluctuation, we found that the mLJpotential can be used to describe significant differences in the degree of strain localization asa function of D values and temperature, which allows it to differentiate from homogeneousplastic flow in the cases when no shear band forms. From calculating the Voronoi volumesof each monomer, we found that the brittle nanopillars have drastically different Voronoivolume distributions from the ductile samples, and we devised a cutoff Voronoi volume at90th percentile to quantitatively characterize the nanopillars with different failure modes.15e believe this metric would enable us to quantitatively understand different failure modesusing features in the sample structure at the molecular scale.
The authors would like to gratefully acknowledge our funding sources: National ScienceFoundation (NSF) Civil, Mechanical and Manufacturing Innovation (CMMI15-3691), partialsupport from Materials Research and Engineering Center (MRSEC) at the University ofPennsylvania (DMR-1720530). This work made use of computational resources providedthrough NSF Extreme Science and Engineering Discovery Environment (XSEDE) awardDMR-150034. We also thank Robert J. S. Ivancic for helpful discussions.
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