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Growth, microstructure, and failure of crazes in glassy polymers Jörg Rottler∗ and Mark O. Robbins Department of Physics and Astronomy, The Johns Hopkins University, 3400 N. Charles Street, Baltimore, Maryland 21218 (Dated: February 2, 2008) 3 We report on an extensive study of craze formation in glassy polymers. Molecular dynamics 0 simulations of acoarse-grained bead-springmodelwere employedto investigatethemolecular level 0 processes duringcraze nucleation,widening, andbreakdown fora widerangeof temperature,poly- 2 merchainlengthN,entanglementlengthNe andstrengthofadhesiveinteractionsbetweenpolymer n chains. Crazewideningproceedsviaafibril-drawingprocessatconstantdrawingstress. Theexten- a sionratioisdeterminedbytheentanglementlength,andthecharacteristiclengthofstretchedchain J segmentsinthepolymercrazeisNe/3. Inthecraze,tensionismostlycarriedbythecovalentback- 3 bonebonds,andtheforcedistributiondevelopsanexponentialtailatlargetensileforces. Thefailure 2 mode of crazes changes from disentanglement to scission for N/Ne ∼ 10, and breakdown through scission is governed by large stress fluctuations. The simulations also reveal inconsistencies with ] previoustheoretical models of craze widening that were based on continuum level hydrodynamics. i c s PACSnumbers: PACS:81.05.Lg,62.20.Fe,83.10.Rs - l r t I. INTRODUCTION m . t a The failure of glassy polymers such as polystyrene m (PS) or polymethylmethacrylate (PMMA) under exter- - nal stresses occurs either through shear deformation or d throughcrazing[1,2]. Whileshearyieldingoccursessen- n tially at constantvolume, crazinghas a strongdilational o component, and the volume of the material increases to c [ several times its original value before catastrophic frac- tureoccurs. Crazingisafailuremechanismuniquetoen- 1 tangledpolymericmaterialsandusuallyprecedesacrack v 7 tip (see Fig. 1). The fundamental and technological im- FIG. 1: Craze fracture of glassy polymers. The craze is a 4 portanceofcrazesisthattheyareinpartresponsiblefor deformedregion(shaded)thatgrowsinwidthandlengthun- 4 thelargefractureenergyGc ofpolymerglasses[3,4,5,6] der an applied vertical stress S. Its density is reduced with 1 that makes them useful load-bearing materials. They respect to the undeformed polymer by a constant extension 0 controlthe cracktipadvanceandrequirealargeamount ratio λ. Characteristic values for width d and length l are 3 ofenergydissipationup to the point ofcatastrophicfail- indicated. During growth, S acts perpendicular to the sharp 0 interfacebetweenundeformedpolymerandcraze. Alsoshown ure. Crazescanreachseveralµm inwidthandconsistof / is an advancingcrack tip from theleft that breaksthecraze. t an intriguing network of fibrils and voids that spans the a Small representative volumes of each region are studied with m entire deformed region. molecular simulations. - Despitethefrequentappearanceofcrazes,thereisstill d comparatively little theoretical understanding about the n conditions and mechanisms of craze nucleation, growth o experimentsandofferanopportunitytotestanddevelop and ultimate breakdown[1, 2, 3, 4, 7, 15]. In this paper, c theoretical models of crazing. : we present an extensive set of nonequilibrium molecular v dynamics (MD) simulations that address these various A fundamental limitation on molecular level treat- Xi phenomena. In this approach, polymers are modeled on ments is of course the finite system size. The largest a coarse-grained scale that takes into account van-der- volumes accessible at present are 100nm3, while the r ∼ a Waals (vdW) and covalent interactions without specific crazespans many µm. We arethus limited to a study of reference to chemical detail. The effect of chain length, craze widening in a small representative region and can- temperatureT,wideningvelocityv andvdWinteraction not include e.g. the entire crack tip. Craze tip advance strength on the craze structure can be studied over a processes [15] are beyond the scope of the present work. wide range of parameters. The molecular simulations Several aspects of craze physics have already been allow insight into microscopic details not accessible to adressed with simulations in previous papers. Baljon andRobbins[8,9]demonstratedtheimportanceofchain length for the onset of craze growth. Rottler et al. studied the elastic properties and fracture stresses of fully ∗Electronicaddress: [email protected] evolvedcrazesandusedthemincombinationwithlinear 2 fracture mechanics to calculate the macroscopicfracture zoneh(see alsoFig.2)isusuallybetween D and D . 0 h i h i energy of glassy polymers that fail by crazing [6]. Rot- Crazewideningisasteady-stateprocess,inwhichacon- tler and Robbins also investigated how polymer entan- stant“drawingstress”Srangingbetween20and100MPa glementsaffectthecrazestructureonamicroscopiclevel is applied. Typical experimental values are 35 MPa and argued that they “jam” the expansion of the glass (polystyrene)[14]and70MPa(polymethymethacrylate) under tension [10]. [38]. The value of S is of the same order as the shear Thispaperextendsthepreviousworkandisorganized yieldstressofthe polymer, andis foundto increasewith as follows. In Section II, we briefly summarize the key the entanglement density. experimental observations and review existing theoretical models of crazing. Section III gives the technical details of the molecular models used in this study. We B. Theory then analyze results for craze nucleation (Section IV), growth (Section V), microstructure (Section VI), and A theory of crazing has to explain the molecular ori- failure (Section VII), and compare our findings to pre- gin of the craze structure and the interdependencies of vious models and experiments. Final conclusions are of- the various quantities measured in experiments. Despite fered in Section VIII. a wealth of experimental data on crazing, there is cur- rently no theoretical description that addresses all aspects of craze physics. The following models have been II. CRAZE PHENOMENOLOGY AND THEORY proposed to explain the extension ratio λ and the rela- tionship between fibril spacing D and drawing stress 0 A. Experiments S. h i Crazeshavebeenstudiedexperimentallyformorethan 30 years [11, 12, 13, 14, 15]. The techniques most com- 1. The extension ratio λ monly used to analyse the craze structure are transmis- sion electron microscopy (TEM) [17, 18], low angle elec- The extension ratio λ has been successfully explained trondiffraction(LAED)andsmallanglex-rayscattering byasimplescalingargument,thatrelatesλtothemicro- (SAXS)[19,20,21]. Comprehesivereviewsoftheoretical scopic entanglement network in the polymer glass. En- andexperimentalresultshavebeenpresentedbyKramer tanglements arise in dense polymeric systems from the and Berger [14, 15] and Creton et al. [16]. topologicalconstraintsthatthechainsimposeuponeach The density in the undeformed polymer ρ and craze i other. Themobilityofthechainsisgreatlyrestricted,be- ρ is obtained from TEM measurements. The increase f causetheycannotpassthrougheachother. The starting involumeduringcrazeformation,orextensionratioλ ≡ pointforthepresentargumentistheassumptionthatthe ρ /ρ , is found to have a characteristic value for a given i f glass inherits these entanglements from the melt, where polymerthatisindependentofmolecularweight. Typical anentanglementmolecularweightisgivenbytheplateau valuesofλfordifferentpolymersrangefromtwotoseven. modulus under shear, G(0): Realspaceimagesofthecrazeshowthatthepolymers N are bundled into fibrils that merge and split to form an intricatenetwork. The fibrils arehighly alignedwith the Me =ρ4RT/5G(N0). (1) applied tensile stress and vary in diameter and length. However,this complex structure is normally idealized as This result can be derived from the microscopic tube a set of uniform vertical cylinders connected by short model [24], which relates the rheological response of the cross-tie fibrils [15]. The characteristic fibril diameter polymer melt to the deformation of a tube to which D andseparation D (seeFig.2)arethendetermined the polymer chain is confined. With repeat units of 0 fhromi aPorodanalyshisoifscatteringexperiments(seeSec- weight M0, one can define a typical number of steps tion V). Measured values range between 3 30 nm for Ne = Me/M0 (entanglement length) between entangle- D and20 50nmfor D [14,15,22,23]. F−orexample, ments along the polymer backbone. 0 h i − h i forpolystyreneoneobtains D 6nmand D 20nm These entanglements are assumed to act like perma- 0 h i∼ h i∼ [1]. nent chemical crosslinks during crazing, which implies Nucleation ofcrazes [12] occurs preferentially near de- that the expansion ends when segments of length Ne fects in the polymer. These produce large local tensile are fully stretched. The initial separation of entangle- stresses that lead to the formation of microvoids that ment points is di = (lpl0Ne)1/2, according to standard evolve into a craze. Once nucleated, the craze grows random walk (RW) scaling, where l0 is an elementary in length and width (see Fig. 1). It is well established step length and lp the persistence length. The length [15] that the craze widens by drawing material from the of this segment rises from di to a maximum final length dense polymer into new fibrils. This deformation is con- df =λmaxdi =Nel0, and thus fined to a narrow active zone at the interface between the dense polymer and craze. The width of the active λ =(N l /l )1/2. (2) max e 0 p 3 Experimentally, Eq. (2) is well confirmed, but Section VD shows that the picture motivating this expression is oversimplified. 2. The drawing stress S The value of the drawing stress S has traditionally been related to the craze microstructure ( D , D ) via 0 h i h i capillarymodels[14,15]. Inthesemodels,thepolymerin theactivezoneistreatedasaviscousfluidwithasurface tension Γ and a viscosity η. Figure 2 shows an idealized picture of the craze geometry, where craze formation is modelled as the propagation of void fingers with a char- FIG. 2: Surface tension model of craze widening. Void fin- gerswithcharacteristicspacinghD ipropagateintoastrain- acteristicspacing D intothestrain-softenedfluid. The 0 0 h i softened layer of polymer fluid of width h, leaving behind applied stress S required to advance the interface has a fibrils of a characteristic diameter hDi. The externally ap- dissipative contribution arising from a suitable flow law plied stress S acts perpendicular to the fluid-glass interface. (e.g. power-law fluid) and an energy penalty contribu- Thecharacteristic radiusof thefingercaps is on the orderof tion due to the surface tension. The tension is S in the hD i/2. (SeeRefs. [14, 15] for an analogous figure.) 0 polymer glass and the Laplace pressure 2Γ/( D /2) at 0 h i the ceiling of the finger, where D /2 is the character- 0 h i istic radius of curvature (see Fig. 2). By estimating the III. SIMULATIONS AND MOLECULAR width of the active zone as h D /2, Kramer calcu- ∼ h 0i MODELS lated a stress gradient between glassy polymer and the finger void ceiling, Westudycrazeformationbyperformingmoleculardy- ∆σ S 4Γ/ D namicssimulationsofastandardcoarse-grainedpolymer 0 σ − h i. (3) ∇ ∼ h ∼ D /2 model[25], where eachlinear polymer contains N spher- 0 h i ical beads of mass m. Models of this kind have a long Since σisproportionaltotheinterfacevelocity,hethen traditioninpolymerresearchandhaveverifiedtheoriesof ∇ predicted that the system will select a value of polymer dynamics [24] in the melt. They have recently been employed by other researchers to study failure in D 8Γ/S, (4) networkpolymer adhesives[27]andend-graftedpolymer 0 h i∼ chains between surfaces [28]. which maximizes the stress gradient between finger ceil- In this bead-spring model, van der Waals interactions ing and bulk polymer and thus will lead to the fastest between beads separated by a distance r are modeled propagationvelocity of the fingers. with a truncated Lennard-Jones potential: Morerecently,KrupenkinandFredrickson[7]havefor- V (r)=4u (a/r)12 (a/r)6 (a/r )12+(a/r )6 mulatedatheoryofcrazewideningthatissimilarinspirit LJ 0 c c − − (6) toKramer’sargumentsandalsoequatesthecrazewiden- (cid:2) (cid:3) forr r ,whereu 20 40meVanda 0.8 1.5nm ing stress with a viscous and a surface tension contribu- c 0 ≤ ∼ − ∼ − arecharacteristicenergyandlengthscales[26]. Asimple tion. However,theseauthorssuggestadifferentinterpre- analytic potential [28] tation of Γ. They introduce an effective surface tension that begins to rise above the vdW value when the finger V (r)= k (r R )3(r R ) (7) br 1 0 1 radiusrisesabovethermsspacingbetweenentanglement − − − lengths d . This ansatz is motivatedby the idea that ex- is used for covalent bonds between adjacent beads along i panding the random walk between entanglements gener- the chain. The form of this potential was chosen to al- ates an additional energy penalty. An upper bound to low for covalent bond breaking, which is not possible Γ is provided by the energy required for chain breaking, with other standard bond potentials such as the popu- which sets in once the finger radius exceeds the maxi- lar FENE potential [25]. Bonds are permanently broken mum elongation between entanglement points, l0Ne. By when r exceeds R0 = 1.5a. The constant R1 = 0.7575a minimizing the finger propagation stress, they conclude waschosentosettheequilibriumbondlengthl0 =0.96a, that the fibril spacing will always be whichis the “canonical”valueforthe bead-springmodel with the FENE potential [25]. This allows us to use D d , (5) results from previous studies, most importantly the en- 0 i ∼ tanglement length. The constant k determines the ra- 1 independent of surface tension. In their model, the fibril tio of the forces at which covalent and van der Waals spacing is determined exclusively by the entanglement bonds break. We find that this ratio is the only impor- network. tantparameterinthecovalentpotentialandsetitto100 4 based on data for real polymers [27, 28], which implies k = 2351u /a4. Tests with other analytical forms of 1 0 the bond potential showedno appreciableimpact onour results as long as the bonds break before the chains can pass through each other. In order to vary the entanglement length, we include a bond-bending potential [28, 29] N−1 (~r ~r ) (~r ~r ) i−1 i i i+1 V =b 1 − · − (8) B − (~r ~r ) (~r ~r ) i=2 (cid:18) | i−1− i || i− i+1 |(cid:19) X that stiffens the chain locally and increases the radius of gyration. Here, ~r denotes the position of the ith bead i alongthechain,andbcharacterizesthestiffness. Wewill FIG. 3: Octahedral shear stress τy at yield as a function of considertwocasesherereferredtoasflexible(b=0)and oct pressurepattwodifferenttemperaturesT =0.3u0/kB (open semiflexible (b = 1.5u ) polymers. The corresponding entanglement lengths a0re Nefl ≈70 and Nesfl ≈30 beads, sayremfibtoslst)oaEnqd.T(9=)a0n.d01tuh0e/dkBash(fiedlleldinseysmshboowls)t.hTehoensseotliodflcinaves- respectively [25, 26, 29]. itation. Values of α are indicated for the two temperatures. We consider three temperatures T = 0.01u /k , T = Alsodrawnisadottedlinethroughthetransitionpointsthat 0 B 0.1u /k and T = 0.3u /k , where the last tempera- separates the regions of shear and cavitational failure. Here 0 B 0 B ture is close to the glass transition temperature. The yield is associated with thestrain where τoct peaks. amount of adhesive interaction between beads is varied by changing the range r of the LJ potential from 1.5a c to 2.2a. tensilestresseswillfavorcavitation. Cavitationandcraz- The equations of motion are solved using the veloc- ingarecloselyrelated,becausecrazesusuallyrequirethe ity Verlet algorithm with a timestep of dt = 0.0075τ , LJ initial formation of microvoids [15]. We therefore first whereτ = ma2/u isthecharacteristictimegivenby LJ 0 address the initial failure of the polymer glass through the LJenergyand lengthscales. Periodicboundarycon- p either shear yielding or cavitation, and later discuss the ditionsareemployedinalldirectionstoeliminateedgeef- formation of crazes. fects. The temperature is controlledwith a Nosé-Hoover The loading conditions that lead to shear yielding in thermostat (thermostat rate 1τ−1), and the thermostat LJ manyexperimentalpolymers[31,32]aremostaccurately is only employed perpendicular to the direction of craze representedbythepressure-modifiedvonMisesyieldcri- growth. SimulationswithaLangevinthermostatshowed terion. It is formulated in terms of simple stress invari- no appreciable difference between the two methods. ants, the hydrostatic pressure p = (σ + σ + σ )/3 1 2 3 In all simulations of crazing, an initial isotropic state − and the deviatoric or octahedral shear stress τ = oct inacubicsimulationcellofedgelengthLiscreatedusing (σ σ )2+(σ σ )2+(σ σ )2 1/2/3, where the standardtechniques[26]. Polymerchainsareconstructed 1− 2 2− 3 3− 1 σ denote the three principal stress components. The as ideal RWs with a suitably chosen persistence length (cid:0)i (cid:1) pressure-modified von Mises criterion states that yield l . l is fixed by matching the radius of gyration of the p p will occur at an octahedral yield stress τy given by chainstotheequilibriumvalueinthemelt,andthevalues oct are lfl = 1.65a and lsfl = 2.7a for flexible and semiflex- p p τy =τ +αp, (9) ible chains, respectively. Subsequently, the interaction oct 0 potentials are imposed and the system is cooled at con- where τ is the yield stress at zero hydrostatic pressure stant volume from a melt temperature T = 1.3u /k , 0 m 0 B and α is a dimensionless constant. Its physical motiva- to the desired run temperature. tion is that the elastic free energy stored in shear defor- Allruns beginatzerohydrostaticpressure. Strainsǫ ii mation is proportional to τ2 and failure should occur arethenimposedbyrescalingthesimulationboxperiods oct when this energy exceeds a threshold that rises slowly L andallparticlecoordinatesproportionately[30]. This i with p. allowsarbitrarystressstatestobe studiedinSectionIV. In ref. 33, we examined a much larger range of stress states than in previous experimental studies and showed thatthepressure-modifiedvonMisescriterionprovidesa IV. CRITERIA FOR CAVITATION AND CRAZE gooddescriptionofshearyieldinourbead-springmodel. NUCLEATION Data for two extremal temperatures are replotted in Fig.3alongwithsolidlinesshowingfitstoEq.(9). Shear Theloadingconditionsonthepolymerglassdetermine yieldwasobservedtotherightofthedot-dashedline,and whether it will fail initially by shear yielding or the for- these data points follow Eq. (9) quite accurately. To the mation of voids and cavities. In general, strong triaxial left of the line cavitation was observed. The deviation 5 from the von Mises fits is very sharp, and τ quickly constant velocity while maintaining the simulation box oct dropsto zero. The values ofτc wherecavitationoccurs periods in the perpendicular x y plane. This leads to oct − are well described by a straight lines an initial stress state where all three principal stresses are tensile. The initial voids formed during cavitation τc =τc+αcp (10) oct 0 expand upon further straining, but their growth rapidly with new constants τc and αc. This new “cavitationcri- becomes arrested [9]. Instead of forming new voids, ad- 0 ditional material is drawn out of the uncavitated poly- terion” can be motivated in analogy to the von Mises mer, andstable crazegrowthoccurs. In our simulations, criterion by assuming that the elastic free energy F as- V growthcontinues until allmaterialin the simulationbox sociatedwithvolume changesmustreacha criticalvalue for cavitation to occur. F is proportional to p2, which is converted into the craze. V givesacriterionoftheformp=p . Onecanthenassume 0 thatshearcomponentsinthestresstensoraidcavitation A. Images of crazes in a linear fashion, i.e. p = p +τ /α , which can be 0 oct c rearrangedto give Eq. (10) with τc =α p . 0 c 0 Noclearexperimentalconsensusexistsaboutthestress A good impression of the crazing process can be ob- state required for crazing, partly because of the impor- tained by inspecting the snapshots of the simulation cell tance of surface defects in nucleating crazes. However, shown in Figs. 4 - 6. Each slice has a lateral width of severalcriteriaforcrazenucleationwereproposedalmost 64a, and three different strains are shown. In all im- 30 years ago. They all try to take into account the crit- ages,the chainlength N =512. Previousstudies [9]had ical role of tensile stress components. Sternstein et al. shownthatN hastobetwicetheentanglementlengthor [35] suggested a craze yield criterion of the form greaterinordertoformstablecrazes. Forshorterchains, thematerialcavitates,butthenrapidlyfailsduetochain 1 τ σ σ =A+B/p, (11) pullout. In the following, we only consider chains with max i j max ≡ 2| − | N 2N . e ≥ where A and B are constants that depend on tempera- Note first that in all cases, there is a sharp inter- ture. With respect to our criterion Eq. (10), p has been face between dense polymer and crazed material. This replacedby1/pandτ bythelargestdifferencebetween narrow ’active zone’ is one of the key features of craze oct any two stress components. Bowden and Oxborough [1] phenomenology found in experiment. In the craze, the formulated a similar criterion, where τ is replaced by polymerchainshavemergedintofibrilsthatarestrongly max σ νσ νσ and ν is Poisson’s ratio for the polymer aligned. However, the structure is quite complicated, as 1 2 3 gla−ss. Th−is expression is another possibility to describe there are many lateral connections between fibers. the shear components of the stress state, and it reduces One can also observe that the fine structure of the to τ when ν = 1/2 and σ = σ . The Sternstein crazes in the three sequences varies greatly. Fig. 4 with max 2 3 and Bowden and Oxborough expressions could in prin- flexible chains at the low temperature of T = 0.1u0/kB ciple also be fitted to the rather narrow range of pres- and the weak adhesive interaction (cutoff distance rc = sure in Fig. 3 where cavitation occurs. However, we are 1.5a) shows many thin fibrils, whereas the fibrils in unaware of a convincing physical motivation for the 1/p Fig. 6 at the higher temperature of T = 0.3u0/kB and term,whichleadstoobviousanalyticalproblemsatsmall the stronger adhesive interaction rc = 2.2a are much p. In addition, the experimental results that motivated thicker in diameter. These trends are not surprising,be- Eq. (11) are sensitive to surface defects [12]. cause increased chain mobility at higher temperatures The above considerations pertain to the initial mode and stronger adhesive interactions should drive the sys- of failure of the polymer glass at strains typically less temtolargerfibrildiameters,whichminimizethesurface than 10%. However, crazing is a large strain deforma- area. tion with strains of several hundred percent. Although we find voiding to be a neccessary precursor to crazing, it is not guaranteed that a loading state that leads to B. The drawing process and stress-strain curves cavitational failure according to Eq. (10) will ultimately produce stable crazes. Likewise, we have observed that A second characteristicfeature of craze growthis that aninitialfailurethroughsheardeformationcanstilllead deformation occurs at a constant plateau or drawing to later void formation and crazing. One should thus stress S. This plateau can be easily identified in the strictly call Eq. (10) a cavitation failure criterion and stress-strain curves shown in Fig. 7. The curves can be not a craze yielding criterion. separated into three different regimes. In regime I, the stressrisesto apeakof 2.6u /a3 andthendropswhen 0 ∼ the polymer yields by cavitation. Following cavitation, V. GROWTH OF CRAZES the stress rapidly relaxes and remains at the plateau value S in regime II, the growth regime. Regime II is Inordertoinducecrazing,weenforcecavitationbyex- much shorter in the semiflexible case Fig. 7(b) than in pandingthe periodicsimulationboxinthe z-directionat the flexible case Fig. 7(a) (note different lateral scales). 6 FIG. 6: Three snapshots of craze growth for semiflexible chains with T =0.3u0/kB, rc =2.2a, and 262144 beads. FIG. 4: Three snapshots of craze growth for flexible chains with T =0.1u0/kB andrc =1.5a. Thetotalsystem contains 262144 beads, but only slices of thickness 10a normal to the page are shown in order to resolve the fine structure. The completionofcrazegrowthwhenL /Lreachesλandthe z lateraldimensionofeachsliceis64aandtheverticaldirection entire craze is strained. Baljon and Robbins [9] showed is to scale. Each dot represents one Lennard-Jones bead. that the peak stress remained constant for much shorter chains, but that regime II only appeared when N was 2N or longer. Another important fact to note is that e S is independent of system size. For example, values of S in systems ranging between 32768 and 1048576 beads are the same within a few percent. The biggest change with increasing system size is that temporalfluctuations in S decrease. InFig.8(a),weanalyzetrendsofS withT andr . The c drawingstressdecreaseslinearlywithincreasingtemper- atureandincreaseswith increasingadhesiveinteractions (i.e. increasing r ). Fig. 8 (b) shows that S varies log- c arithmically with the widening velocity v over two orders of magnitude, which is indicative of a thermally ac- tivated process. For the subsequent figures, we choose v = 0.06a/τ , which is at the upper end of the loga- LJ rithmic regime [9]. Similar behavior is alsofound for the shear yield stress of glassy polymers [33, 34]. FIG. 5: Three snapshots of craze growth for semiflexible chains with T =0.1u0/kB,rc =1.5a, and 262144 beads. C. Crazing under plane stress conditions TheresultsofSectionIVshowthatcavitationonlyoc- Regime II ends when the strain L /L reaches the exten- curs when all three principal stresses are tensile. Many z sionratioλ. Atthispoint,allthematerialinthesimula- experimental crazes grow in a thin film geometry under tioncellhasbeenconvertedintothecraze,andadditional plane stress conditions. However, in these experiments deformationstrainsthe entirecrazeuniformly. As acon- the crazeis oftenprenucleatedornucleatesnear a defect sequence,thestressrisesagaininregimeIII.Thisregime [12]. This situation can also be mimicked in our simu- finally ends in catastrophic failure either through chain lations. To this end, the periodic boundary conditions disentanglement or chain scission (see Section VII). in the x direction were replaced with free boundaries, so Note first that neither the peak stress at cavitation that the solid is free to relax in that direction. Initial nor the value of S depends on the chain length N. The failure is now nucleated by placing 1000purely repulsive curves for different N in Fig. 7 only split apart after LJbeadsinthecenterplaneofthesimulationcelllocated 7 FIG. 8: (a) Trends of S with T and rc at v = 0.06a/τLJ for flexible ((cid:4)) and semiflexible (N) chains and rc = 1.5a FIG. 7: Stress σzz in the widening direction during craze (lower curves) and rc = 2.2a (upper curves). (b) Velocity growth at T = 0.1u0/kB, rc = 1.5a for (a) flexible and (b) dependence of S for flexible chains at T = 0.1u0/kB. The straightlineisafittoalogarithmicvelocitydependence,S = semiflexible chains of length N = 128, N = 256,N = 384, and N = 512. Three characteristic regimes of (I) cavity nu- 1.085u0/kB+0.048u0/kBlnv. Uncertaintiesarecomparable to symbolsizes. cleation, (II) craze growth and (III) craze failure are also indicated. The two perpendicular stress components σxx and σyy also peak at cavitation (see text), but then rapidly drop to zero. Qualitatively identical curves are obtained at other ρ inthecraze. Ascanbeseen,ρ ishigherforthesemi- values of T and rc. f f flexible chains, which have a smaller value of N 30. e ≈ Remarkably, we find that λ is a function of N only and e decreases with decreasing N . For instance, while in- e atz =Lz/2[42]. Thisweakensthe solidlocallyandcon- creasing T and rc produces dramatic coarsening of the strains the location of initial failure, while not affecting fibrilstructureinFig.6relativetoFig.4,λisunchanged. subsequent craze growth. We obtain values of λfl = 6.0 0.6 and λsfl = 3.5 0.3 ± ± independent of N, T, and adhesive interaction strength. Fig. 9 shows three snapshots of a craze in this geometry. As in experiments, necking is observedat the craze- In order to understand the dependence of the macro- bulk interface. Although σ vanishes in the rest of the xx scopic quantity λ on N , we analyze the structural film,theneckproducesstrongtensilestressesinallthree e changesinthepolymerglassduringdeformationonami- directionsintheactivezone. Thecrazegrowsinthesame croscopiclevel(seealsoref.[10]). Figure11(a)showsthe fashion as in the simulations with 3D periodic boundary averagefinal position of beads in the completely evolved conditions. Since the latter yield better statistics for the craze as a function of their initial positions along the di- crazestructure,wehavefocusedonthismethodologyfor rectionofthe expansion(z-axis). The averagewas taken our analysis. over all beads with initial heights in a bin of width 1a. Although the strain rate is strongly localized during the crazeprocess,the ultimate displacementprofileis linear, D. The extension ratio z =λz . f i To measure deviations from a purely affine (uniform) The extension ratio λ can be calculated from the av- deformation, we evaluated the rms variation δz in z for f erage densities of crazed and uncrazed material. Fig. 10 beads in each bin. This quantity is indicated by error shows how the density drops from the initial value ρ to bars in Fig. 11(a). Note that the variation in each bin is i 8 FIG. 11: (a) Final bead heights z as a function of initial f heights zi for flexible (large slope) and semiflexible (small slope) chains (T = 0.1ǫ/kB, rc = 1.5a). Averages were calculated over z-intervals of width a. Straight lines have slope λ = 5.9 and λ = 3.5, respectively. Error bars represent a standarddeviationfromtheaveragesineachlayerandareon the order of 19a (flexible) and 9a (semiflexible). (b) Square oftheheight change∆z asa function ofthenumberofcova- lentbonds∆N betweenabeadandthechaincenter. Dashed straightlineshaveslopeλ2lpl0/3withλfrom(a). Deviations from the RW scaling occur in the vicinity of the chain ends (notshown). OthersystemsatdifferentT,rc andN showthe FIG. 9: Cross-sections through a craze with a free interface same results. at T = 0.1u0/kB, rc = 1.5a, and 262144 beads. Periodic boundary conditions were maintained in the direction into the plane. The location of initial cavitation was constrained ations δx that are indicated by error bars. These lateral by placing repulsive beads in the center plane at z = Lz/2. displacements allow chains to gather in fibrils at the ini- Thelateral dimension is47aandtheverticaldimensionisto scale. tial density to minimize surface area. Unlike the verti- caldisplacements δz,these lateraldisplacements depend stronglyonT andr . Forexample,δx 2.5aforthefine c ∼ structureshowninFig.4,wheremanythinfibrilscanbe seen, while δx 5.6a for the much coarser structure of ∼ Fig.6. Ingeneral,δxcorrelateswiththespacingbetween fibrilsasdiscussedinSectionVIandislessthand . Kru- i penkinandFredrickson[7]suggestedthat d providesan i upper bound for the lateral chain deformations. We now examine changes in the conformation of individual chains. In the initial state, the polymer chains exhibit an ideal random walk (RW) structure inherited fromthe melt. The averageend-to-endvector R2 thus h i scaleswiththe numberofcovalentbondsconnectingtwo beads ∆N as R2 = l l ∆N. The component along p 0 h i each direction is 1/3 of that value since the initial state isisotropic. Fig.12(b)showsthisinitialscalingbehavior FIG. 10: Density profile through the active zone for crazes for ∆x2 (dashedline)andthat ∆x2 isnotaffectedby with flexible chains (Ne ≈70) and semiflexible chains (Ne ≈ crazhing (isolid line). h i 30). Horizontallinesindicatetheaveragedensityinthecraze After an affine deformation by λ along z, one would for the two cases. have an anisotropic RW with no change in ∆x or ∆y, but ∆z2 = λ2l l ∆N/3. Fig. 11(b) shows the actual p 0 h i behavior (solid lines) of ∆z2 in the craze. At large h i veryreproducible. Wefindthatδz isnearlyindependent scales, it exhibits the expected scaling for an affine de- ofT andrc andhasvaluesontheorderof19aand9afor formation (dashed lines). However, the separation be- flexible and semiflexible chains, respectively. tweenbeads is fixed by the length ofthe covalentbonds, Since no strainis appliedin the perpendicular x andy so the deformation of individual polymers along z can- directions,one wouldassume that there is onaverageno not be purely affine. At small scales, the linear scaling displacement in these directions. That this is indeed the behavior of ∆z2 crosses over into a quadratic behav- h i case is shown in Fig. 12, which repeats the analysis of ior, which indicates that the polymer has been pulled Fig. 11 for the x-direction. Average final bead positions taut on this scale. The typical number of beads in areidenticaltoinitialpositions,buttherearelateralvari- such a straight segment N˜ can be calculated by let- st 9 FIG. 12: Analysis of bead positions analogous to the previous figure (same systems), but for the x-positions. No strain isappliedinthisdirection,andthestraight linesinpanel(a) have slope one. The curves for the semiflexible chains in (a) weredisplacedverticallyupwardby10atoavoidoverlap. Er- rorbarsrepresentastandarddeviationδxfromtheaverages, andareontheorderof3.7a(flexible)and2.2a(semiflexible). (b) Bead displacements as a function of distance from the center in bond lengths, ∆N, along the chain. Dashed lines haveslope lpl0/3. ting (N˜ l )2 = ∆z2 = λ2l l N /3 at the crossover st 0 p 0 st h i point, which yields N˜ = λ2l /3l . Inserting the ob- st p 0 served values of λ, l , and l , we arrive at N˜fl = 21 4 p 0 st ± and N˜sfl = 12 2, respectively. These values are com- parablestto the ±values of δz found in Fig. 11(a). On this FIG. 13: (a) Probability distribution of straight segments of lengthN forflexibleandsemiflexiblechains. Thicklinescor- scale, the deformation is non-affine. st respond to simulations at T = 0.1ǫ/kB, rc = 1.5σ,N = 512 The length of taut sections can also be determined by with 1048576 beads. Dotted lines were obtained at T = direct analysis of the chain geometry. To this end, we 0.3ǫ/kB, rc = 2.2σ,N = 512 with 262144 beads, and long calculate the angle between every covalentbond and the dashed lines correspond to T = 0.1ǫ/kB, rc = 1.5σ,N = z-axis and label a bond as pointing up (down) if the an- 256 with 262144 beads. The straight lines show fits to gle is within 45◦ of the z (-z) axis. We then count the exp(−N /N˜(s)fl). (b) z-component of the bond-bond corre- st st number N of consecutive up (down) steps. The prob- lation function for the same systems. Thin solid lines show st ability P(N ) of finding a straight segment containing exponential fitswith theindicated decay lengths. st N steps is shown in Fig. 13(a). For both flexible and st semiflexible chains, the distribution develops an expo- nentialtail. Likeλ, this tail is independent of N, T,and r . The characteristiclength scales that arise fromthese noted already in earlier work [14] that, due to this ge- c tailsareN˜fl 21andN˜sfl 13,ingoodagreementwith ometric factor, λ should be √3λ for fully stretched st ∼ st ∼ max the prediction from the RW argument. Fig. 13(b) shows chains. However,this resultis little cited since λ λ max ≈ that very similar length scales arise from an equivalent inmanysystemsand,untilourwork,therewasnoreason analysis of the decay of the correlation function for the to expect the length of straight segments to be N /3. e z-component of successive bonds. The emergence of the length scale N /3 is a conse- e In section IIB1, we introduced the standard scaling quence of the random nature of the entanglement mesh. argumentEq.(2) that relates extensionratio and entan- Clearly,allstrandswouldbeexpandedsimultaneouslyby glement length, which has been verified experimentally the same factor in a regular mesh as reported in a simu- with great success. In our cases, it predicts λfl = 6.5 lationstudybyStevens[27]. Bycontrast,inthepolymer max and λsfl =3.5, which agree with the observed values of glassonlythesegmentsthatareinitiallyalignedwiththe max λ. However,theargumentwasmotivatedbytheideathat stretching direction become fully stretched. These fully segments between entanglements become fully stretched stretchedsegmentsareabletopreventfurtherextension, andthusappearstobeatoddswiththefindingofanav- because the entanglements act like chemical crosslinks. erage straight segment length of only N /3 rather than Barsky and Robbins have confirmed the equivalence be- e N . This discrepancy is resolved by realizing that since tweenentanglementsandcrosslinksbyaddingpermanent e thedeformationisuniaxial,onlytheprojectionofd onto crosslinks randomly to the system [37]. The length be- i the z-axis, d cos(Θ ), is expanded, where Θ is the an- tween constraints then decreases from N , and λ de- i z z e max gle between d and the z-axis. The average projection creases accordingly. They found λ λ in all cases i max ≈ is thus only 1/√3 of the total length. Indeed, it was andthattheaveragestretchedlengthN remainsat1/3 st 10 TABLEI:Dissipation duringcraze growth and covalent contribution to the crazing stress S for several different systems of size 262144. T rc N δQ/δW % cov stress fl. 0.1 1.5 256 0.88 87 fl. 0.1 1.5 512 0.88 88 sfl. 0.1 1.5 256 0.71 95 sfl. 0.1 1.5 512 0.67 97 fl. 0.1 2.2 512 0.92 69 sfl. 0.1 2.2 512 0.71 75 fl. 0.3 2.2 512 0.87 61 sfl. 0.3 2.2 512 0.78 67 of the distance between constraints. The success of the scaling argument Eq. (2) and the constancyoftheextensionratioimplythatthereisnoap- preciablelossofentanglementsinoursimulationsduring craze growth. Chains do not disentangle once N >2N , e and chain scission (see also section VII) is not observed duringgrowthforanychoiceofparametersinourmodel. E. Energy dissipation and stress transfer during FIG. 14: (a) Density profile through a craze simulation at crazing T =0.1u0/kB,rc =1.5a with flexiblechains. (b)vdWstress (solid line), covalent stress (dashed line) and total stress S The work done in transforming a volume dV of poly- (thick line) as a function of position along z. The kinetic mer into a craze is δW = S(λ 1)dV. This work can contributiontothestressissplitevenlybetweenthecovalent − either increase the potential energy dU or be dissipated and vdW stress here and in Table I. asheatδQ. Thedivisionbetweenenergyandheatishard to determine experimentally, but simulations with short chainsfoundthatbothcontributionsweresubstantial[8]. (a) displays the density profile in order to identify dense WehavemeasuredδW andtheenergychangedirectlyin polymer regions (high density) and craze regions (low our simulations and calculated δQ using the first law of density). In Panel (b) one observes that in the dense re- thermodynamics: δQ=dU δW. dU can be calculated − gionallthetensioniscarriedbythevanderWaalsbonds directly from the bead positions and interaction poten- and the covalent bonds are under slight compression. In tials. Table I shows the fraction δQ/δW of dissipated thecraze,between60 95%ofthetotalstress(seeTable total work for a number of large systems. In all cases, a − I)iscarriedbythecovalentbonds,andthevanderWaals large percentage, 80%, of the total work is dissipated, ∼ bonds only contribute a small fraction. and only 20% is stored as potential energy. Since the ∼ craze drawing stress varies logarithmically with velocity (see Fig. 8(b)), these percentages could change with velocity. However, we find that dU also decreases with F. Problems with the capillary models decreasing velocity, and there is no measurable change in the percentage ofwork convertedto heatoverat least The results presented so far reveal serious difficulties two orders of magnitude in velocity. with the surface tension models discussed in the Intro- Stress in the craze can also be partitioned into two duction. The firstevidence ofthis comesfromthe obser- components that originate either from van der Waals vationthatS isindependentofsystemsize. Inoursmall- (LJ) interactions Eq. (6) or from covalent interactions est simulations of lateral width 32a the simulation box Eq. (7). The two contributions are very different in only contains a few fibrils at T =0.1u /k ,r =2.2a. If 0 B c the uncrazed and crazed material. In the undeformed S were controlledby Eq. (4), one should expect that the polymer, the tensile stress is mainly carried by the vdW simulation box would need to contain a statistically sig- bonds. This stress is transferred in large part to the nificant number of fibrils of spacing D for S to reach 0 h i covalent bonds as they pass through the active zone. its steady state value. However, the value of S does not Evidence for this is provided in Fig. 14, which shows fluctuate as the lateral dimensions are increased to 64a the covalent and LJ contributions to the total stress as or 128a. a function of height in the widening direction. Panel Thesecondandmoresevereproblemconcernsthe dis-