Force-induced breakdown of flexible polymerized membrane
aa r X i v : . [ c ond - m a t . s o f t ] F e b Force-induced breakdown of flexible polymerized membrane
J. Paturej , , H. Popova A. Milchev , , and T.A. Vilgis Max Planck Institute for Polymer Research, 10 Ackermannweg, 55128 Mainz, Germany Institute of Physics, University of Szczecin, Wielkopolska 15, 70451 Szczecin, Poland Institute of Physical Chemistry, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
We consider the fracture of a free-standing two-dimensional (2D) elastic-brittle network to beused as protective coating subject to constant tensile stress applied on its rim. Using a MolecularDynamics simulation with Langevin thermostat, we investigate the scission and recombination ofbonds, and the formation of cracks in the 2D graphene-like hexagonal sheet for different pullingforce f and temperature T . We find that bond rupture occurs almost always at the sheet peripheryand the First Mean Breakage Time h τ i of bonds decays with membrane size as h τ i ∝ N − β where β ≈ . ± .
03 and N denotes the number of atoms in the membrane. The probability distributionof bond scission times t is given by a Poisson function W ( t ) ∝ t / exp( − t/ h τ i ). The mean failuretime h τ r i that takes to rip-off the sheet declines with growing size N as a power law h τ r i ∝ N − φ ( f ) .We also find h τ r i ∝ exp(∆ U /k B T ) where the nucleation barrier for crack formation ∆ U ∝ f − , inagreement with Griffith’s theory. h τ r i displays an Arrhenian dependence of h τ r i on temperature T .Our results indicate a rapid increase in crack spreading velocity with growing external tension f . I. INTRODUCTION
Fracture in engineering materials is a long-standing topic of research due to problems that arise with technologicalapplications and the ensuing economical implications. Thus, for decades a lot of attention has been focused onunderstanding the macroscopic and microscopic factors which trigger failure. Recently, the interest and the need forbetter understanding of the interplay between elastic and fracture properties of brittle materials has been revived dueto the rapidly developing design of advanced structural materials.Promising aspects for applications include reversible polymer networks [1, 2], and also graphene, that shows unusualthermomechanical properties [3, 4]. Among other things, graphene, which is a honey-comb lattice packed with C atoms can be used as anti-corrosion gas barrier protective coating [5], in chemical and bio-sensors [6], or as efficientmembrane for gas separation [7]. In all possible applications the temperature and stress-dependent fracture strengthof this 2D-network is of crucial importance. Graphene has been investigated recently by Barnard and Snook [8] using ab initio quantum mechanical techniques whereby it was noted that that the problems “has been overlooked by mostcomputational and theoretical studies”.An important example of biological microstructure is spectrin , the red blood cell membrane skeleton, which reinforcesthe cytoplasmic face of the membrane. In erytrhrocytes, the membrane skeleton enables it to undergo large extensionaldeformations while maintaining the structural integrity of the membrane. A number of studies, based on continuum-[9], percolation- [10–12], or molecular level [14, 15] considerations of the mechanical breakdown of this network,modeled as a triangular lattice of spectrin tetramers, have been reported so far. Many of these studies can be viewedin a broader context as part of the problem of thermal decomposition of gels [16], epoxy resins [17, 18] and other 3Dnetworks both experimentally [16–18], and by means of simulations [19] in the case of Poly-dimethylsiloxane (PDMS).The afore-mentioned examples illustrate well the need for deeper understanding of the processes of failure in brittlematerials. Besides analytical and laboratory investigations, computer simulations [20–22] have provided meanwhile alot of insight in aspects that are difficult for direct observations or theoretical treatment - for a review of previous workssee Alava et al. [23]. Most of these studies focus on the propagation of (pre-existing) cracks, relating observations tothe well known Griffith’s model [24] of crack formation. A number of important aspects of material failure have foundthereby little attention. Thus only a few simulations examine the rate of crack nucleation which involves long timescales necessary for thermal activation - see, however, [25–28]. Effects of system size on the characteristic time for bondrupture have not been examined except in a recent MD study by Dias et al. [29]. Also recombination of broken bondshas not been considered. These and other insufficiently explored properties related to fracture have motivated ourpresent investigation of a free-standing 2D honeycomb brittle membrane by means of Molecular Dynamics simulation.In view of the possible applications as anti-corrosion and gas barrier coating, we consider a radially-spanned sheetof regular hexagonal flake shape so as to minimize effects of corners and unequal edge lengths that are typical forribbon-like sheets. Tensile constant force is applied on the rim of the flake, perpendicular to each edge. By varyingsystem size, tensile force and temperature, we collect a number of results which characterize the initiation and thecourse of fragmentation in stretched 2D honeycomb networks.The paper is organized as follows: after a brief introduction, we sketch our model in Sec. II where we considerinteractions between atoms in the brittle honeycomb membrane, define the threshold for bond scission, and alsointroduce some basic quantities that are measured in the course of the simulation. In Sec. III we present oursimulation results, presenting briefly the results on recombination of broken bonds - III A, the distribution of bondscission rates over the membrane surface, the dependence of the Mean First Breakage Time (MFBT) before a bondscission takes place and of the mean failure time until the 2D sheet breaks apart on applied tensile force, and examinehow these times depend on membrane size and temperature - III B. The formation of cracks at different cases ofapplied stress as well as their propagation in a 2D honeycomb brittle sheet are briefly considered in subsection III C.We end this report by a brief summary of results in Section IV. II. MODEL AND SIMULATION PROCEDUREA. The model
We study a coarse-grained model of honeycomb membrane embedded in three-dimensional (3D) space. The mem-brane consists of N spherical particles (beads, monomers) of diameter σ connected in a honey-comb lattice structurewhereby each monomer is bonded with three nearest-neighbors except for the monomers on the membrane edgeswhich have only two bonds (see Fig. 1 [left panel]). The total number of monomers N in such a membrane is N = 6 L where by L we denote the number of monomers (or hexagon cells) on the edge of the membrane (i.e., L characterizesthe linear size of the membrane). There are altogether N bonds = (3 N − L ) / symmetric hexagonal membranes (i.e., flakes ) so as to minimize possible effects due to the asymmetric ofedges or vortices at the membrane periphery. FIG. 1: [left panel] A membrane with honeycomb structure that contains a total of N = 54 beads and has linear size L = 3 ( L is the number of hexagonal cells on the edge of the membrane). [right panel] A snapshot of a typical conformation of an intactmembrane with L = 30 containing 5400 monomers after equilibration with no external force applied. Typical wrinkles are seento form on the surface. For the analysis of our results we find it appropriate to divide the two-dimensional membrane network so that allbonds fall into different subgroups presented by concentric “circles” with consecutive numbers (see Fig. 1 [right panel])proportional to their radial distance from the membrane center. To odd circle numbers thus belong bonds that arenearly tangential to the corresponding circle.
Even circles contain no encompass radially oriented bonds (shown tocross the circle in Fig. 1). The total number of circles C in a membrane of linear size L is found to be C = (2 L − B. Potentials
The nearest-neighbors in the membrane are connected to each other by breakable anharmonic bonds described bya Morse potential, U M ( r ) = ǫ M { − exp[ − α ( r − r min )] } . (1)where r is the distance between the monomers. Here α = 1 is a constant that determines the width of the potentialwell (i.e., bond elasticity) and r min = 1 is the equilibrium bond length. The dissociation energy of a given bond, ǫ M = 1, is measured in units of k B T where k B denotes the Boltzmann constant and T is the temperature. Theminimum of this potential occurs at r = r min , U Morse ( r min ) = 0. The maximal restoring force of the Morse potential, f max = − dU M /dr = αǫ M /
2, is reached at the inflection point, r = r min + α − ln(2) ≈ .
69. This force f max determinesthe maximal tensile strength of the membranes bonds. Since U M (0) ≈ .
95, the Morse potential, Eq. (1), is onlyweakly repulsive and beads could partially penetrate one another at r < r min . Therefore, in order to allow properlyfor the excluded volume interactions between bonded monomers, we take the bond potential as a sum of U M ( r ) andthe so called Weeks-Chandler-Anderson (WCA) potential, U WCA ( r ), (i.e., the shifted and truncated repulsive branchof the Lennard-Jones potential), U WCA ( r ) = ( ǫ h(cid:0) σr (cid:1) − (cid:0) σr (cid:1) i + ǫ, for r ≤ / σ , for r > / σ (2)with parameter ǫ = 1 and monomer diameter σ = 2 − / ≈ .
89 so that the minimum of the WCA potential tocoincides with the minimum of the Morse potential. Thus, the length scale is set by the parameter r min = 2 / σ = 1.The nonbonded interactions between monomers are taken into account by means of the WCA potential, Eq. (2). Thus,the nonbonded interactions in our model correspond to good solvent conditions whereas the bonded interactions makethe bonds breakable when subject to stretching. External stretching force f is applied to monomers at the membranerim in direction perpendicular to the respective edge - Fig. 2a.Before we turn to the problem of membrane failure under constant tensile force, we show here some typical elasticproperties of the intact honeycomb network sheet that is used in our computer experiments - Fig. 2. In Fig. 2b 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L = 12 (a)
Strain S t r e ss Uniaxial stretchingRadial stretching Y r = . ⋅ - Y u = . ⋅ - b) FIG. 2: (a) A protective honeycomb network is spanned at the orifice of a prism whose size may vary due to thermal expansion.Tensile forces acting on the membrane periphery are indicated by red arrows. (b) Mean strain of a honeycomb membrane ofsize L = 10 as a function of external tensile stress f at T = 0 .
01 and γ = 0 .
25. Depending on the way in which externalforce is applied: (i) radial stretching - c.f. Fig. 9f, or (ii) uniaxial stretching - Fig. 9b the observed Young modulus is Y r = 2 . × − [ k B T /a ] or Y u = 2 . × − [ k B T /a ]. one can see an S − shaped variation of the stress - strain relationship with initial significant elongation at vanishingstress due to the straightening of the membrane wrinkles (ripples) that are typical for an unperturbed membrane - cf.Fig. 1b. This behavior is followed by a linear stress - strain elastic relationship where we measure the Young modulus Y r = 2 . × − [ k B T /a ] (or Y u = 2 . × − [ k B T /a ]), depending whether radial of uniaxial loading is applied.Eventually, for stronger stretching the elasticity of the network decreases as the anharmonicity of the bond potentialcomes into play. Moreover Fig. 2b indicates that the destructive strain of the whole membrane is considerably less inthe case of uniaxial stretching.In our work we have tried to develop the model which should serve as generic one for all kinds of 2D brittle-elasticnetworks with honeycomb orientation. We have been anxious to emphasize the common features of failure in materialswith similar architecture but largely varying elasticity properties, e.g., from 1000 GPa graphene’s Young modulus [4]compared to 4 × − GPa for spectrin [15]. Putting the value of a Kuhn segment ( σ = 1 .
44 ˚A) and taking the thermalenergy k B T = 4 × − J at T = 300 K, we get from our simulation a Young modulus ∼ .
03 GPa which is rangedbetween typical values for rubber-like materials 0 . ÷ .
1. Compared to ab initio simulations of graphene with linearsize L = 6 which corresponds to 216 network nodes [8], our objects are about an order of magnitude larger, L = 50and 2500 nodes, in units of elementary cells. C. MD algorithm
As in our previous studies concerning scission kinetics of linear chains [33, 34] and bottle-brushes [35] we use aLangevin dynamics which describes the Brownian motion of a set of interacting particles whereby the action of thesolvent is split into slowly evolving viscous (frictional) force and a rapidly fluctuating stochastic (random) force. TheLangevin equation of motion is the following: m −→ ˙ v i ( t ) = −→ F i ( t ) − mγ −→ v i ( t ) + −→ R i ( t ) (3)where m denotes the mass of the particles which is set to m = 1, −→ v i is the velocity of particle i , −→ F i = ( −→ F M + −→ F WCA ) i is the conservative force which is a sum of all forces exerted on particle i by other particles in the system, γ is thefriction coefficient and −→ R i is the three dimensional vector of random force acting on particle i . The random force −→ R i , which represents the incessant collision of the monomers with the solvent molecules, satisfies the fluctuation-dissipation theorem h R iα ( t ) R jβ ( t ′ ) i = 2 γk B T δ ij δ αβ δ ( t − t ′ ) where the symbol h ... i denotes an equilibrium average andthe greek-letter subscripts refer to the x , y or z components. The friction coefficient γ of the Langevin thermostat isset to γ = 0 .
25. Our simulation was performed in the weakly damped regime of γ = 0 .
25 where effects of inertia areimportant. This value of γ is more or less standard in Langevin MD. However, we carried out additional simulationin the strongly damped regime for γ = 10. No qualitative changes were discovered except an absolute overall increaseof the rupture times τ which is natural for a more viscous environment. The integration step is 0 .
002 time units (t.u.)and the time is measured in units of r min p m/ǫ M . We emphasize at this point that in our coarse-grained modelingthe solvent is taken into account only implicitly. In this work the velocity-Verlet algorithm is used to integrate theequations of motion.Our MD simulations are carried out in the following order. First, we prepare an equilibrated membrane confor-mation, starting with a fully flat configuration, Fig. 1, where each bead in the network is separated by a distance r min = 1 equal to the equilibrium separation of the bond potential ( U M + U WCA ) [see Eq. (1) and (2)]. The externalconstant force is switched on from the very beginning of the simulation. Then we start the simulation with thisprepared conformation and let the membrane equilibrate with the applied force in the heat bath for sufficiently longtime ( ≈ t.u.) at a temperature that is low enough so that the energy barrier for scission is high and the membranestays intact. This equilibration is done in order to prepare different starting conformations for each simulation run.Once the equilibration is finished, the temperature is raised to the working one and we let the membrane equilibrateat this temperature for roughly ∼
20 t.u. We have checked that this time interval is sufficient for equipartition anduniform distribution of temperature to be established throughout the membrane sheet. Then the time is set to zeroand we continue the simulation with this well-equilibrated membrane conformation checking for scission of the bonds.We measure the elapsed time τ until the first bond rupture occurs and repeat the above procedure for a largenumber of runs (10 ÷ ), starting each time with a new equilibrated conformation so as to sample the stochasticnature of rupture and determine the mean h τ i which we refer to as the mean first breakage time. In the course ofsimulation we also calculate properties such as the probability distribution of breaking bonds regarding their positionin the membrane (a rupture probability histogram), the probability distribution function of the first breakage time W ( τ ) (i.e., the MFBT probability distribution), the strain (extension) of the bonds with respect to the consecutivecircle number in the membrane, as well as other quantities of interest.In separate runs each simulation is terminated as soon as the honeycomb sheet disintegrates into two separate partswhereby the time it takes to “rip-off” the sheet is termed “mean failure time h τ r i and measured. In order to monitorthe propagation of cracks, we perform also individual runs labeling breaking bonds in succession and reconstructingthe crack trajectory which is a laborious and rather involved problem. D. Rupture criterion
An important aspect of our simulation is the recombination (self-healing) of broken bonds. The constant stretchingforce acting on the monomers at the membrane edges creates a well-defined activation barrier for bond scission. Directanalysis of the one-bond potential with external force, U M ( r ) − f r indicates that the positions of the (metastable)minimum r − and of the barrier (or hump) r + are given by [36] r − , + = 1 a ln " ± q − ˜ f (4)where the dimensionless force ˜ f = 2 f /aǫ M . For the range of tensile forces used in the present study one has typically r + ≈ r min . The activation energy (barrier height) for single bond scission is itself given by [36] E b = U ( r + ) − U ( r − ) = ǫ M q − ˜ f + ˜ f " − q − ˜ f q − ˜ f (5)One can easily verify that E b decreases with ˜ f . Since a bond may get stretched beyond the energy barrier andnonetheless shrink back again, i.e. recombine, in our numeric experiments we use a sufficiently large value forcritical extension of the bonds, r h = 5 r min , which is defined as a threshold to a broken state of the bond. Thisconventions is based on our checks that the probability for recombination (self-healing) of bonds, stretched beyond r h , is vanishingly small, as demonstrated below. In our model we deal with E b / ( k B T ) = 20 which at 300 K andbond length r min = 0 .
14 nm corresponds to ultimate tensile stress ∼ . ∼
100 GPa [4] and is ranged between typical values forrubber materials 0 . ÷
14 GPa.
III. MD-RESULTS
We examine the scission of bonds between neighboring nodes in the network sheet with honeycomb topology,assuming thermal activation as a driving mechanism in agreement with early experimental work by Brenner [37] andZhurkov [38]. In Fig. 3 we show a series of representative snapshots of a membrane of size L = 10 with N = 600monomers taken at different time moments during the process of decomposition. Typically, the first bonds that breakare observed to belong to the last (even) most remote circle as, for example, at t ≈ t.u. in Fig. 3. As mentionedabove, these are the radially oriented bonds which belong to concentric circles of even number. Gradually a line ofedge beads is then severed from the rest of the membrane and a crack is formed which propagates into the bulk untileventually a piece of the network sheet is ripped off, as in Fig. 3 at t ≈ t.u. As we shall see below, this mechanismof membrane failure, whereby an initial crack is formed parallel to the edge monomers, yet perpendicular to the tensileforce, dominates largely the process of disintegration under constant tensile force. The process is, therefore, mainlydescribed by two characteristic times, τ and τ r , which mark the occurrence of the first scission of a bond (MFBT)and that of the eventual breakdown of the flake into two distinct parts. A. Bond recombination
As mentioned in Section III, throughout in our studies of the brittle sheet breakdown we use a threshold for criticalbond stretching (rupture criterion) r h = 5 r min . In the right inset of Fig. 4 we display the function Q h ( h ), whichrepresents the probability distribution of bond stretching h beyond the hump position r + , given that a subsequentrecombination has taken place. To this end one monitors for 10 integration steps the length of each bond once thebond expands beyond r + and stores its maximal expansion, h , provided such a bond contracts again to r < r + .Then Q h ( h ) is computed as the fraction of extensions to h over the total number of recombination events. For eachbond recombination one measures also the distribution of the respective self-healing times, P h ( t ), which is shownin Fig. 4 too. Both distributions are characterized by exponentially fast decaying tails, indicating that successfulrecombinations are possible after very short time interval ≈ . t.u. , and the possible stretching of a bond in suchcases is minimal - about 0 . ÷ . r + ≈ .
96, that is, significantly shorter than r h ≈
5. We also find that recombination of bonds takes place seldom (roughly 1.5% over 5 · runs of average length ≈ t.u. for a membrane composed of N = 600 beads). Yet as indicated below, allowing for self-healing events maysignificantly change the observed kinetics of membrane destruction. The left inset in Fig. 4 indicates that self-healingof bonds happens most frequently at the membrane periphery, C = 19, where bond stretching occurs most frequently. B. Mean First Breakage Time
These conclusions, based on visual evidence from snapshots taken in the course of membrane decomposition, arecorroborated in Fig. 5a where we show the probability distribution of a first rupture for all bonds in the honeycombmembrane flake as a 3D plot. It is seen that the scission rate is localized in the outer-most circle of radial bondswhereas bonds in the inner part of the membrane practically hardly break. Note that this is not a trivial effect sincetension is distributed uniformly over all bonds in the equilibrated membrane so there is no additional propagation of
FIG. 3: Snapshots illustrate the process of bond breakage (crack generation) in different time moments for a membrane with N = 600 particles subject to external tensional force f = 0 .
15 at T = 0 .
05 and γ = 0 .
25. The force is applied to peripherymonomers only and stretches the network perpendicular to its original edges. the tension front from the rim towards the center. Fig. 5b also indicates a qualitative change in the rupture PDFwhen self-healing is not allowed for (by reducing the threshold position to that of the energy barrier - r h = 3 .
1) incontrast to results where self-healing was fully taken into account - r h = 5. Moreover, a closer inspection the newFig. 5b indicates that scission of bonds with no self-healing takes place almost uniformly throughout the membranewhile with self-healing it is concentrated only at the membrane periphery.One can try to relate this finding to the distribution of strain within the network as shown in Fig. 6a and sampledfor several strengths of the external stretching force f . In the case of strongest pulling, f = 0 .
15, the variation ofthe mean-squared bond length h l i with distance from the membrane center (i.e., with consecutive circle number C )displays a well expressed saw-tooth behavior whereby the peaks correspond to bonds with radial rather than tangentialorientation (odd C ). The alternation of strongly / weakly stretched bonds modulates the overall gradual increase ofthe mean bond length with growing distance from the center. Evidently, the amplitude of the mean-squared bondlength attains a pronounced maximum on the last circle of radially oriented network bonds. This distribution of strainis found to persist down to vanishing tensile force f = 0 - Fig. 6a. The distribution of first scission events is clearlyseen in Fig. 6b where we show it for several strengths of f . Evidently, with growing value of f bonds happen to breakalso deeper inside the membrane although such events remain much less probable. t -3 -2 -1 P h ( t ) P h (t) ~ exp(-0.69 ⋅ t) Bond expansion h -3 -2 -1 Q h ( h ) Q h (h)~ exp(-5.1 ⋅ h) Circle number C × -3 × -3 × -3 × -3 R h ( C ) f = 0.15 FIG. 4: Probability distribution P h ( t ) of maximal times (full circles), and Q h ( h ) of maximal bond lengths h (circles, rightpanel of inset) before a recombination event in the stretched membrane with N = 600, T = 0 . γ = 0 .
25 takes places. Theexponential tail of P h ( t ) is fitted by blue line. The exponential decay of Q h ( h ) is given by red line. The left panel of the insetshows the healing probability R h vs. circle number C . The healing events under applied stress occur roughly 10 times lessfrequently than for f = 0. Consecutive circle number R up t u r e P D F threshold = 3.1threshold = 5.0 b) FIG. 5: (a) Rupture probability histogram of flexible hexagonal membrane subjected to external tensile stress f = 0 .
15. (b)Scission probability histogram vs consecutive circle number for membrane pulled with f = 0 .
125 displayed for different rupturethresholds r h as indicated. Real rupture events ( r h = 5 .
0) are concentrated at the periphery whereas fictitious ones ( r h = 3 . N = 600, T = 0 .
05 and γ = 0 . The variation of the MFBT τ with system size N (i.e., with the number of monomers in the membrane N = 6 L where L denotes the linear size of the flake) is shown in Fig. 7. For sufficiently large membranes one observes a powerlaw decline of the MFBT, τ ∝ N − β with an exponent β ≈ . ± .
03 for the tensile forces studied.. If thermallyactivated bonds break independently from one another and entirely at random, then the MFBT τ measures theinterval before any of the available intact bonds undergoes scission, that is, either the first bond breaks, or the secondone, and so on which, at constant rate of scission, would reduce the MFBT τ ∝ /N as observed for instance in thecase of thermal degradation of a linear polymer chains [34]. A more comprehensively this simple result can be derivedby means of the classical theory of Weibull. In the present system of a honeycomb membrane the bonds that undergorupture are nearly all located at the rim of the flake and their number is proportional to L so that with β ≈ . N ∝ L , one obtains eventually the important result τ ∝ /L . This observation is in agreement withrecent results of Grant et al. [29] who studied the nucleation of cracks in a brittle 2 D -sheet. We should like to point Consecutive circle number < l > f = 0.15f = 0.125f = 0 (a) C on s e c u t i v e c i r c l e nu m be r E x t e r n a l f o r c e f R up t u r e P D F (b) f=0.125f=0.150f=0.175f=0.200 0 2 4 6 8 10 12 14 16 18 20 0.125 0.15 0.175 0.2 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 FIG. 6: (a) Variation of the mean squared bond length h l i with consecutive circle number in a membrane with N = 5400beads subjected to different strengths of the external force f (as indicated in the legend). (b) Probability distribution of thefirst bond scission event vs circle number in a membrane with N = 600 beads at different strengths of the external force f asindicated. For a force f ≤ .
15 the bonds from the last two outer circles ( f ≥ .
175 the bonds which are located in the circles T = 0 .
05 and γ = 0 . out at this place that without self-healing, c.f., Fig. 5b, rupture time goes as h τ i ∝ N − β , with β ≈ β ≈ . β ≈ . τ on temperature T in our elastic-brittlehoney-comb network (see below) while the exponent β depends on the external load f and on T giving rise to anon-Arhenian τ vs. T relationship.Note that the decline of MFBT τ in a topologically connected brittle system is by no means a trivial one. Ina recent study [33], using Molecular Dynamics (MD) simulation of a single anharmonic polymer chain subject toconstant external tensile force, we found a rather complex interplay between the polymer chain dynamics and theresulting bond rupture probability distribution along the chain backbone. In a breakable chain (rather than 2Dnetwork) it was observed that the corresponding power β → N → ∞ . A major role in this was attributedto nonlinear excitations as the possible origin for the observed increasing insensitivity of rupture time with respectto polymer length as the pulling force grows. One may thus conclude that nonlinear effects in bond scission aresuppressed in 2D honeycomb networks.One can also see from the inset in Fig. 7a that the MFBT τ decreases rapidly with growing stress f , that is, theenergy barrier for rupture declines with f in agreement with Eq. (5) and Zhurkov’s experiments [38]. The probabilitydistribution of MFBT W ( t ) is shown in Fig. 7b. It is well described by a Poisson probability distribution function W ( t ) = 5 . · − t / exp( − t/ . C. Cracks and Mean Failure Time
The variation of τ r , the mean failure time of the membrane with system size N , shown in Fig. 8a, displays alsoa power-law dependence on system size N , h τ r i ∝ N − φ , whereby φ undergoes a cross-over to a lower value beyondroughly N > τ r has different physical meaning. Following Pomeau [39], the failure time can beapproximately identified with the nucleation of a crack of critical size l c given by Griffith’s critical condition [24, 40]assuming that crack propagation is much faster than the nucleation time. For a 2 D -geometry consisting of a flatbrittle sheet with a crack perpendicular to the direction of stress, the potential energy per unit thickness of the sheet N < τ > f = 0.125 β=0.56 f = 0.15 β=0.47 f = 0.175 β=0.47 External force < τ > N = 294 ~ e xp ( - ⋅ f) < τ > ∼ Ν −β t W ( t ) W(t)~ t n e -t/ τ FIG. 7: (a) Mean first breakage time h τ i vs. number of particles N in the membrane pulled with different tensile stress f as indicated. Symbols represent simulation data whereas solid lines stand for fitting functions h τ i ∼ N − β . The inset showsforce-dependent h τ i for a membrane composed of N = 294 beads. (b) MFBT probability distribution W ( t ) for the first scissionof a bond in a flake with N = 600 particles at stress f = 0 .
15. Symbols denote result of simulation and full line stands for thefitting function W ( t ) ∝ t n exp( − t/τ ) with n = 1 / τ = 291 . t.u. Parameters of heat bath are T = 0 .
05 and γ = 0 . N < τ r > f = 0.15 f = 0.175 n S ( n ) < τ> ∼ Ν −φ N = 150 φ = . φ = . (a) φ = . φ = . Tensile force f F a il u r e ti m e τ r τ r
16 18 20 22 T -1 τ r (b) - e x p ( . / k B T ) FIG. 8: (a) Mean failure time h τ r i (time needed to split membrane into two pieces) vs. number of particles in the membrane fortwo values of the external pulling force f at T = 0 .
05 and γ = 0 .
25. Symbols denote simulation results and solid line representspower law fitting function h τ r i ∼ N − φ . The inset shows PDF of number of particles in the moment of splitting for a membranecomposed of N = 150 beads. (b) Failure time h τ r i vs f in the case of N = 294. reads U = − πl f Y + 2 εl + U where Y is the Young modulus, ε is the surface energy needed to form a crack of length l , and U is the elastic energy in the absence of stress ( f = 0). This energy reaches a maximum for a critical cracklength l c = εYπf beyond which no stable state exists except the separation of the sheet into two broken pieces. Thus,with a crack nucleation barrier ∆ U = ε Yπf (in 3 D ∆ U ∝ f − ), the failure (rip-off) time τ r = τ exp(∆ U /k B T ) asfound in experiments with bidimensional micro crystals by Pauchard and Meunier [41] and in gels by Bonn et al. [42].In Fig. 8b we present the variation of τ r for membrane failure with stress f in good agreement with the expectedrelationship ∆ U ∝ f − . In addition, we show the variation of τ r with temperature, see inset in Fig. 8b, which is foundto follow a well expressed Arhenian relationship with inverse temperature, in agreement with earlier studies [29, 40].The end of the sheet rupturing process is marked as a rule by disintegration into two pieces of different size so it isinteresting to asses the size distribution of such fragments upon failure. In the inset in Fig. 8a we show a probability0distribution S ( n ) of the sizes of of both fragments upon membrane rip-off. In a membrane composed of N beads oneobserves a sharp bimodal distribution with narrow peaks at sizes N ≈
10 and N ≈ FIG. 9: Overview of observed cracks in a honeycomb membrane composed of N = 600 particles for different orientation ofthe applied external pulling force. Green arrows show indicate the orientation of the applied force ( f = 0 . T = 0 .
05 and γ = 0 .
25. The typical cracks are markedin color on the geometrically undistorted arrangement of network nodes for better visibility.
One can readily verify from the typical topology of the observed cracks in the membrane, presented in Fig. 9, that f v c Time [t.u.] N u m b e r o f b r o ke n bond s f=0.145f=0.15f=0.16f=0.18f=0.20 (a) Time [t.u.] N u m b e r o f b r o ke n bond s (b) FIG. 10: (a) Crack propagation velocity (number of broken bonds per unit time) for a membrane with N = 600 beads atdifferent strength of the external force f as indicated. (b) Three different realizations of cracks at applied force f = 0 .
14. Theinset shows a variation of the mean crack propagation velocity with f . Here T = 0 .
05 and γ = 0 . (i) cracks emerge as a rule perpendicular to the direction of applied stress, and (ii) it is almost always the first rowof nodes to which the tensile force is immediately applied that gets ripped off upon failure. Cracks that break thenetwork sheet in the middle occur very seldom, in compliance with the sampled distribution of fragment sizes, S ( n )1in the inset of Fig. 8a. One would, therefore, predict a breakup of a protective cover spanned on the orifice of tubelike the one shown in Fig. 9 to proceed immediately at the fixed orbicular boundary where the tensile force appliesto the network. It is interesting to note that the geometry of cracks in the membrane shown in Fig. 9 appears verysimilar to the one observed in drying induced cracking of thin layers of materials subject to structural disorder [32].The emerging cracks are expected to propagate with speed that increases as the strength of the external force isincreased as the inset in Fig. 10a indicates. In fact, in Fig. 10a one observes typical curves comprising a series of shortintervals with steep growth of the number of broken bonds per unit time and longer horizontal ’terraces’ precedingthe nucleation of a new crack. Even though the data, presented in Fig. 10a, is not averaged over many realizations,and, as Fig. 10b suggests, individual realizations of propagating cracks may strongly differ even at the same stress f , a general increase of the propagation velocity with growing external force f - see inset - can be unambiguouslydetected, in agreement with earlier observations [20].For our model membrane with computed Young modulus Y ≈ .
02 we get for the Rayleigh wave speed c R ≈ . c R - inset in Fig. 10b.As argued by [43] propagation speed cannot exceed c R because crack splits off into multiple cracks before reaching c R .In contrast, Abraham and Gao in Ref. [44] have reported on cracks that can travel faster than the Rayleigh speed.Thus, our rough estimates (inset in Fig. 10) agree well with data from literature. Converting our results to propermetric units, with bond length σ ≈ .
144 nm and energy ≈ k B T which yields 1 MD t.u. ≈ − s, we estimatethe typical crack propagation speed v c ≈
50 m/s. Note that mean crack speed for natural latex rubber was given as56 m/s [45].
IV. SUMMARY
In the present work we have studied the bond rupture and ensuing fracture of a honeycomb brittle membrane subjectto uniform radially applied external stretching forces for different values of force f , temperature T , and membrane size N . The most important conclusions that can be drawn from our Molecular Dynamics simulation can be summarizedas follows: • bonds scission in hexagonal 2D sheets with honeycomb structure of the underlying network under subjected toexternal pulling perpendicular to flake’s edges take place overwhelmingly at the sheet periphery • The Mean First Breakage Time of breaking bonds depends on membrane size N as a power law, τ ∝ N − β with β ≈ . ± . • The failure time τ r until a brittle sheet disintegrates into pieces follows a power law too, τ ∝ N − φ ( f ) , and anexponential decay τ r ∝ exp(const /f ) upon increasing strength of the pulling force, in agreement with Griffith’scriterion for failure. • cracks emerge in the vicinity of membrane edges and typically propagate parallel to the edges, splitting thesheet in two pieces of size ratio of ≈ • crack propagation speed is observed to increase rapidly with tensile forceWe believe that these findings can be seen as generic also for 2D network brittle sheets of different geometry(hexagonal lattices, or quadratic lattices with second nearest-neighbor bonding) where similar interplay betweenelastic and fracture behavior is expected to take place. It is clear, however, that more investigations are needed beforea full understanding of fracture in such systems is achieved. V. ACKNOWLEDGMENTS
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