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A molecular view of bond rupture
Computational and Theoretical Polymer Science 9 (1999) 35±40
A molecular view of bond rupture Arlette R.C. Baljon a,*, Mark O. Robbins b a
Theory Center and Department of Chemical Engineering, Cornell University, Ithaca, NY 14850, USA Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD 21218, USA
b
Received 17 February 1998; received in revised form 11 May 1998; accepted 11 May 1998
Abstract Molecular mechanisms of rupture will be discussed in the light of recent computational studies. Bonds formed by materials as diverse as simple crystalline solids, glassy polymers, and biological ligands and receptors, reveal similar behavior. When these bonds are pulled apart at constant velocity, they rupture through a series of sudden yield events during which the material reorganizes. Yield events are separated by periods of elastic deformation where the stress builds until the system becomes unstable. The nature of the structural change at yield events varies from system to system. Small cavities form in the polymer ®lm, an additional atomic layer is formed in the crystal, and hydrogen binding sites rearrange in the biological system. The work required to rupture these bonds is determined by the full sequence of yield events. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: Molecular dynamics; Non-equilibrium processes; Plasticity; Energy dissipation
1. Introduction
2. Rupture of thin polymer ®lms
Understanding microstructure and how it changes during bond rupture is crucial to predicting, and thus tailoring, the macroscopic mechanical properties of bonds. Simulations have proven invaluable as tools to better understand bond rupture at a molecular level. They reveal spatial and temporal variations in structure, stress, and energy dissipation during the rupture process. Such details are hard to obtain from experiments, because the rupture process must be interrupted in order to probe the microstructure. In this paper, we review microstructural changes and energy dissipation during rupture of crystalline junctions [1], glassy oligomer bonds [2] and biological bonds between a ligand and receptor [3]. Although these systems vary widely, their mechanical response shares common features. Rupture occurs through a series of rapid structural rearrangements. In between rearrangements, elastic energy is stored and stresses build up. When the stress exceeds the yield stress, the microstructure changes, stress is released, and energy is dissipated. Similarities and dierences in the nature of the yield events, and the mechanisms of rearrangement will be discussed.
Fig. 1 shows several snapshots from a simulation of the rupture of a thin adhesive bond between two rigid walls [2]. The adhesive ®lm consists of small oligomer molecules that are modeled as chains of 16 beads connected by springs [4]. The beads of mass m interact through a Lennard-Jones (LJ) potential: V(r)=4[(/ energy r)12ÿ(/r)6], where and are characteristic p and length scales. The characteristic time t m= . In addition, a strong attractive potential connects adjacent beads on a chain. During the simulation chains are not allowed to break. Each of the walls consists of atoms coupled to lattice sites by sti springs [5]. The lattice sites form two (111) layers of an fcc lattice. The wall atoms are coupled to a heat bath that keeps the system at constant temperature by allowing energy dissipated during rupture to ¯ow out of the system. An LJ potential with slightly dierent energy and length scales is used to model the interaction between wall atoms and beads on the molecules. After equilibration, the walls are moved apart at constant velocity v. This causes stress in, and eventually rupture of, the oligomer ®lm. Fig. 2(a) shows the force per unit area needed to move the top wall outwards. The total work done, potential energy change, kinetic energy change, and heat ¯ow to the heat bath are shown in Fig. 2(b).
* Corresponding author.
0189-3156/99/$Ðsee front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S1089 -3 156(98)00049 -X
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A.R.C. Baljon, M.O. Robbins / Computational and Theoretical Polymer Science 9 (1999) 35±40
As the walls move apart, the ®lm at ®rst deforms elastically. As a result, the force needed to separate the walls increases linearly with time. The potential energy rises, and heat ¯ows from the heat bath into the system. The density decreases until a critical value is reached. Then the ®lm becomes unstable and small cavities begin to form [Fig. 1(b)]. These allow the rest of the ®lm to return to the equilibrium density and the force on the wall drops sharply [Fig. 2(a)]. Further separation of the walls causes plastic ¯ow in a sequence of discrete yield events that produce additional steps in the force curve [Fig. 2(a)]. Between steps, the ®lm behaves nearly elastically. When yield stresses are exceeded, rapid structural rearrangements occur which increase the cavity size. At some point the cavities coalesce and a few bridges are all that connect the walls [Fig. 1(c)]. Ultimately, these bridges snap back to the walls, leaving the rough surface shown in Fig. 1(d). As the rupture velocity decreases towards zero, more time is needed to elastically deform the polymer ®lm. However, the sequence of rapid structural rearrangements remains nearly unchanged. The amount of
dissipation is also constant at low velocities, suggesting that the rapid rearrangements cause almost all of the dissipation. We con®rmed this by studying hysteresis loops like that shown in Fig. 3(a). If the wall is moved out until just before cavitation occurs and then back to its initial position, all the work used to stretch the ®lm can be recovered. If cavitation occurs before the wall is moved back, almost none of the work can be recovered. Similar behavior is seen at other rapid yield events. In general, work done in a loop that includes yield events cannot be recovered. The local velocities during rearrangements remain comparable to thermal velocities even as the rupture velocity goes to zero. The rapid pop of atoms from one site to another excites random vibrations that eventually ¯ow out of the walls as heat. The atomic motions that occur during structural rearrangements are complicated and highly variable because of the disordered structure of the ®lm. As we now discuss, crystalline junctions also fail through a series of rapid rearrangements. However, the atomic motion and the condition for instability is much simpler to describe because of the greater symmetry.
Fig. 1. Several snapshots of a glassy ®lm during rupture at velocity v=0.003 / . The 800 atoms in each wall are black, and the beads of the 128 chains are gray. Periodic boundary conditions were used in the xy plane, and two periodic images are shown. In (b) only a thin crosssection of the ®lm is shown.
Fig. 2. Time dependence of (a) force on wall and (b) energy changes during rupture of the glassy ®lm shown in Fig. 1. Lines in (b) indicate the external work (dashed), the potential energy increase (dotted), and the total heat ¯ow (solid). The kinetic energy is proportional to the temperature and remains unchanged.
A.R.C. Baljon, M.O. Robbins / Computational and Theoretical Polymer Science 9 (1999) 35±40
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Fig. 4. Side views of atomic con®gurations in a thin slice through a wire at three dierent elongation stages. The left and right frames show con®gurations before and after a structural rearrangement. In the central frame the transformation stage is shown. The arrows in the frame denote the glide directions. (Reproduced, with permission, from Ref. [1].) Fig. 3. External work versus wall displacement z during cavitation of the glassy ®lm shown in Fig. 1 (dashed line). The cavitation point is indicated by a star. Solid lines show the amount of recovered work when the wall spacing is decreased either before or after cavitation.
3. Failure of crystalline junctions Several groups have used computers to study atomic rearrangements during fracture of solids and metals [1,6,7]. All of these studies show that rupture occurs through stepwise increases in the number of atomic layers along the direction of rupture. This is illustrated in Fig. 4, taken from simulations by Landman et al. [1]. In the unstrained state (a) the junction contains 16 layers and the order is crystalline. For small strains the order remains crystalline, but the spacing between layers increases. As the strain increases, the stress reaches a threshold value and the junction yields. Stress is released through multiple glide processes with (111) planes sliding over each other in the directions indicated by the arrows in the center of Fig. 4(b). There is an increase in disorder during sliding. After sliding the junction is in a well ordered crystalline state with one additional layer [Fig. 4(c)]. Further strain causes a sequence of similar elastic deformation and plastic yield events, as in the thin polymer ®lms. The reason that crystalline junctions fail through this speci®c mechanism is that they are highly anisotropic. The spacing between (111) planes is larger than that between planes with other orientations. Thus they slide over each other at a lower stress. Two invariant scalars can be used to describe the amount of dilational and shear stress on a system. The ®rst is the hydrostatic pressure p which is given [8] by the average of the diagonal elements of the pressure tensor P: p=Tr[P]/3. The second is the von Mises shear stress [9], J1/22 =[Tr(PÿpI)2 /2]1/2. Landman et al. studied the variation of J21/ 2 during rupture. They found that the system yielded to create a new layer when the local value of J21/2 exceeded a threshold value. This is typical of ductile crystalline materials where the easiest mode of internal motion
involves glide of lattice planes over each other, often through motion of dislocations. The oligomer ®lms discussed above are isotropic. There are no lattice planes where sliding would be easier than in other directions. It would be interesting to investigate whether p or J21/2 controls failure in these systems. While the issue has not been addressed for thin ®lms, it has been studied in simulations of cavitation and shear of bulk polymers. 4. Plastic deformations in bulk polymers Mott et al. have studied cavity formation and plastic deformation in bulk polymeric materials [10,11]. Their model for glassy polypropylene treats the backbone carbon and hydrogen atoms explicitly. Bond lengths and bond angles are ®xed and molecular movement can only occur by rotation around the C±C bonds. Thermal ¯uctuations are precluded. A single chain in a glassy state is con®ned in a cubic unit cell with periodic boundary conditions. The system is forced to dilate by expanding the size of the cell. As in the other systems described above, the system responds elastically at small dilations. Then cavities form suddenly when the density has dropped by about 12%. Just as in the simulations of oligomer ®lms, further dilation produces a series of elastic loading stages and sudden yield events at which all cavities grow simultaneously. Yield events produced jumps in both the pressure and the von Mises stress. Unlike the crystalline junctions described above, yield was more closely associated with release of large negative pressures than with relief of accumulated shear stress. In fact, the shear stress rises during some rearrangements. The pressure decreases in magnitude at each yield event, although not all the way to the unstrained value. The average magnitude of the pressure slowly increases during cavity growth and the magnitude at which yield occurs also rises. This indicates that the system strain hardens or toughens. One reason for the dierence in yield mechanism between polymers and crystals may be that random
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A.R.C. Baljon, M.O. Robbins / Computational and Theoretical Polymer Science 9 (1999) 35±40
polymers do not have lattice planes that facilitate shear. However, the polymer and crystal were also stressed in dierent ways. The polymer was under a uniform dilation in all three directions so that most of the stress was associated with the pressure. The crystal was under a uniaxial strain that also produced a shear stress. That this dierence is responsible for the dierence in yield mechanisms is supported by work of Mott et al. on the shear response of the same polymer system [11]. Here too the system ¯ows through a series of plastic loading and relaxation events. However, now the von Mises stress is larger in magnitude than the pressure and is more closely associated with the onset of yield instabilities. It remains to be seen whether p or J21/2 determines the yield threshold of uniaxially strained oligomer ®lms. One of the surprising results of the shear simulations [11] was that the clusters of atoms which rearranged when the system became unstable were up to 10 nm in size. All particles in these large clusters moved simultaneouslyÐat least within the time resolution of the simulation. Considering other studies of yield phenomena [12], one expects that transformation of the cluster starts when the yield stress is exceeded in some part of the cluster. Local rearrangements cause stress to be transferred to other bonds that may yield in turn. The resulting avalanche transforms the whole cluster. Mott et al. found that the early elastic loading stages of their dilation simulations could be described by the Rose±Smith±Guinea±Ferrante model [10]. This model relates the elastic moduli to a characteristic intermolecular spacing and the intermolecular binding energy. Its success implies that intramolecular bonds are not contributing to the modulii. Work on oligomer ®lms also shows that intermolecular bonds dominate the elastic behavior and yield stress. In particular, the values of the yield stress at cavitation and the ®rst few subsequent yield events are nearly independent of chain length. This is surprising because intramolecular features such as chain length are very important in determining the free energy, and one would expect that the free energy change would be directly related to bond strength. The importance of energy rather than free energy is also seen in studies of biological bonds. 5. Biological bonds Recent simulations of the rupture of a bond between a single ligand and receptor [3] have been motivated by atomic-force-microscope studies of streptavidin±biotin bond strengths [13]. Recognition mechanisms and dissociation pathways of this complex were investigated. Examination of atomic con®gurations shows that the ligand and receptor are held together by a network of hydrogen bonds. Fig. 5 shows that the force goes through a sequence of oscillations as the streptavidin±
Fig. 5. Pulling force as a function of displacement during rupture of a biotinstreptavidin bond at a velocity of 0.015 AÊ/ps. The dashed vertical lines mark the ruptures of hydrogen bonds (bold lines) and water bridges (thin lines). (Reproduced, with permission, from Ref. [3]).
biotin bond is stretched. As in the systems described above, the regions where the force rises are periods where the hydrogen bonds deform elastically. Some of the bonds rupture at each force maximum, allowing the atoms to rearrange rapidly to create a new hydrogenbond network with lower stress. The unbinding force, a measure for molecular recognition, is de®ned as the largest force along the unbinding pathway. Its value, as obtained from simulations by extrapolating to low rupture velocities, agrees nicely with experimental data. As yet unexplained is that the unbinding force is proportional to the enthalpy change during rupture, but independent of changes in free energy [13]. It has been suggested that this is because entropic: changes do not occur until after unbinding [14]. This conclusion is consistent with recent simulations of rupture of oligomer ®lms [15]. They also show that the largest force during the rupture process correlates with the enthalpy change. Moreover, failure occurs at the interface with the lowest enthalpy change rather than that with the lowest free energy change. When the wall±¯uid interaction is much weaker than the ¯uid± ¯uid interaction, the bond ruptures at the wall±¯uid interface (adhesive failure). When the ¯uid±¯uid interaction is weaker, the bond breaks within the adhesive (cohesive failure). The location of failure does not depend on chain length or temperature although both produce large changes in the entropic contribution to the free energy change. A lack of correlation between bond strength and free energy change has also been seen in other experimental systems [16]. 6. Outlook Computational capabilities have reached a point where it is feasible to study rupture processes through
A.R.C. Baljon, M.O. Robbins / Computational and Theoretical Polymer Science 9 (1999) 35±40
molecular dynamics simulations. Molecular mechanisms of rupture in materials as diverse as glassy oligomer ®lms, solid junctions, and biological molecules, all share common aspects. In each of them, rupture takes place through alternating stages of stress accumulation and relief. During the loading stages the system deforms elastically. Most of the work done on the system causes a reversible increase in the potential energy stored in intermolecular bonds due to a decrease in density. Yield starts through local imbalance of forces on one segment of a molecule and rapidly spreads through the system. Studies of oligomer ®lms and biological molecules indicate that the yield stress is determined primarily by energy rather than free energy changes. During the rapid yield stages, energy is dissipated, and stress is relieved. The total amount of dissipated energy is strongly aected by the factors that control the importance of entropy in the system, such as chain length and temperature. The separation between the factors that control yield stress and energy dissipation may open up opportunities to design improved adhesives. Intermolecular bond strengths control the yield stress and the location of failure. The macroscopic strength of bonds is usually determined by the force needed to open a crack. This scales with the work required to rupture a unit area of the bond rather than with the yield stress. In most cases the work is dominated by energy dissipation and thus depends most sensitively on chain length and temperature. The systems considered here were all at such low temperatures that atoms were not free to diuse and no motion occured until the yield stress was exceeded. As a result, the amount of energy dissipation and the entire sequence of stress accumulation and relief stages became independent of the rupture velocity at low velocities. Rupture may be very dierent at high temperatures. In the extreme limit, the material becomes a linear viscous ¯uid. There are no longer separate stages of stress accumulation and release, because molecules can ¯ow smoothly to relieve stress. Energy dissipation arises only from viscous eects, and vanishes linearly in the limit of low rupture velocity [2]. More interesting behavior may occur at intermediate temperatures where polymer ®lms exhibit visco-elastic behavior. Gent et al. [17] reported that the dependence of the work required to rupture a macroscopic polymeric bond on rupture speed and that of the shear modulus on shear rate is qualitatively similar in this regime. This is quite remarkable given the dierence in the nature of the two processesÐfor instance one expects much higher density ¯uctuations during rupture than during shear. The natural inference is that viscoelastic dissipation in the bulk dominates the dissipation mechanisms. However, a detailed quantitative analysis shows that the eective length of the fracture zone deduced from this correspondence is much too small. Further studies are required to explain this discrepancy.
39
Although all the systems described above undergo periodic cycles of stress buildup and release, they exhibit dierent types of microstructure. Larger simulations and more complex interactions may reveal more intricate spatial structures. For example, simulations in progress show that increasing the length of oligomers to four times the entanglement length produces rich spatial structure that is characteristic of polymer crazes. Cavities grow and merge to create an interconnected network of ®brils. Each ®bril contains several aligned chains and may merge with other ®brils or branch to create several ®brils. The craze grows by drawing more and more material into the ®brils from a thin, strain softened layer at the craze interface. We plan to use the simulations to study the details of this grow mechanism. Finally, experiments [18,19] and simulations [1] on ultrathin lubricant ®lms show that shear produces rich ¯uctuating patterns in the friction force. Analysis of the experimental data reveals chaotic behavior or broad band noise. Similar complex dynamic behavior has been found in rupture experiments. For instance, Gandur et al. [20] characterized stick-slip behavior during rupture of adhesive tape as a chaotic phenomenon. Non-equilibrium phase transitions, similar to those recently predicted to exist in Releigh-Bernard convection [21] might play a role as well. Hence, a complete understanding of dierent types of dissipation mechanisms and the range of processes that can cause stick-slip motion during rupture will require a better understanding of nonequilibrium phenomena. Acknowledgements The authors would like to thank D. Gersappe and E. Siggia for useful discussions and H. GruÈbmuller and U. Landman for allowing us to reproduce their ®gures. Support from the National Science Foundation through grants NSF96-25792, DMR-9634131, and PHY9407194 is gratefully acknowledged.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
Landman U, Luedtke WD, Gao J. Langmuir 1996;12:2514. Baljon ARC, Robbins MO. Science 1996;271:482. GruÈbmuller H, Heymann B, Tavan P. Science 1996;271:997. Kremer K, Grest GS. J Chem Phys 1990;92:5057. Thompson PA, Grest GS, Robbins MO. Phys Rev Lett 1992;68:3448. Lynden-Bell R. Science 1994;263:1704. Selinger RLB, Lynden-Bell RM, Gelbart WM. J Chem Phys 1993;90:9808. Allen M, Tildesley D. Computer simulation of liquids. Oxford: Clarendon Press, 1987. Johnson KL. Contact mechanics. Cambridge: Cambridge University Press, 1985. Mott P, Argon A, Suter U. Philos Mag 1993;68:537.
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[11] Mott P, Argon A, Suter U. Philos Mag 1993;67:931. [12] Nato advanced study institute on disorder and fracture. New York: Plenum Press, 1990. [13] Moy VT, Florin E, Gaub HE. Science 1994;266:257. [14] Berendsen HJC. Science 1996;271:954. [15] Gersappe D, Robbins MO. Phys Rev Lett (submitted). [16] Israelachvili JN, Berman A. Isr J Chem 1995;35:85.
[17] [18] [19] [20]
Gent A, Lai SM. J Polym Sci B 1994;32:1543. Demirel AL, Granick S. Phys Rev Lett 1996;77:4330. Israelachvili JN, Drumond G. personal communication. Gandur MC, Kleine MU, Galembeck F. J Adhesion Sci Technol 1997;11:11. [21] Cakmur RV, Egolf DA, Plapp BB, Bodenschatz E. Phys Rev Lett 1997;79:1853.
A molecular view of bond rupture Arlette R.C. Baljon a,*, Mark O. Robbins b a
Theory Center and Department of Chemical Engineering, Cornell University, Ithaca, NY 14850, USA Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD 21218, USA
b
Received 17 February 1998; received in revised form 11 May 1998; accepted 11 May 1998
Abstract Molecular mechanisms of rupture will be discussed in the light of recent computational studies. Bonds formed by materials as diverse as simple crystalline solids, glassy polymers, and biological ligands and receptors, reveal similar behavior. When these bonds are pulled apart at constant velocity, they rupture through a series of sudden yield events during which the material reorganizes. Yield events are separated by periods of elastic deformation where the stress builds until the system becomes unstable. The nature of the structural change at yield events varies from system to system. Small cavities form in the polymer ®lm, an additional atomic layer is formed in the crystal, and hydrogen binding sites rearrange in the biological system. The work required to rupture these bonds is determined by the full sequence of yield events. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: Molecular dynamics; Non-equilibrium processes; Plasticity; Energy dissipation
1. Introduction
2. Rupture of thin polymer ®lms
Understanding microstructure and how it changes during bond rupture is crucial to predicting, and thus tailoring, the macroscopic mechanical properties of bonds. Simulations have proven invaluable as tools to better understand bond rupture at a molecular level. They reveal spatial and temporal variations in structure, stress, and energy dissipation during the rupture process. Such details are hard to obtain from experiments, because the rupture process must be interrupted in order to probe the microstructure. In this paper, we review microstructural changes and energy dissipation during rupture of crystalline junctions [1], glassy oligomer bonds [2] and biological bonds between a ligand and receptor [3]. Although these systems vary widely, their mechanical response shares common features. Rupture occurs through a series of rapid structural rearrangements. In between rearrangements, elastic energy is stored and stresses build up. When the stress exceeds the yield stress, the microstructure changes, stress is released, and energy is dissipated. Similarities and dierences in the nature of the yield events, and the mechanisms of rearrangement will be discussed.
Fig. 1 shows several snapshots from a simulation of the rupture of a thin adhesive bond between two rigid walls [2]. The adhesive ®lm consists of small oligomer molecules that are modeled as chains of 16 beads connected by springs [4]. The beads of mass m interact through a Lennard-Jones (LJ) potential: V(r)=4[(/ energy r)12ÿ(/r)6], where and are characteristic p and length scales. The characteristic time t m= . In addition, a strong attractive potential connects adjacent beads on a chain. During the simulation chains are not allowed to break. Each of the walls consists of atoms coupled to lattice sites by sti springs [5]. The lattice sites form two (111) layers of an fcc lattice. The wall atoms are coupled to a heat bath that keeps the system at constant temperature by allowing energy dissipated during rupture to ¯ow out of the system. An LJ potential with slightly dierent energy and length scales is used to model the interaction between wall atoms and beads on the molecules. After equilibration, the walls are moved apart at constant velocity v. This causes stress in, and eventually rupture of, the oligomer ®lm. Fig. 2(a) shows the force per unit area needed to move the top wall outwards. The total work done, potential energy change, kinetic energy change, and heat ¯ow to the heat bath are shown in Fig. 2(b).
* Corresponding author.
0189-3156/99/$Ðsee front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S1089 -3 156(98)00049 -X
36
A.R.C. Baljon, M.O. Robbins / Computational and Theoretical Polymer Science 9 (1999) 35±40
As the walls move apart, the ®lm at ®rst deforms elastically. As a result, the force needed to separate the walls increases linearly with time. The potential energy rises, and heat ¯ows from the heat bath into the system. The density decreases until a critical value is reached. Then the ®lm becomes unstable and small cavities begin to form [Fig. 1(b)]. These allow the rest of the ®lm to return to the equilibrium density and the force on the wall drops sharply [Fig. 2(a)]. Further separation of the walls causes plastic ¯ow in a sequence of discrete yield events that produce additional steps in the force curve [Fig. 2(a)]. Between steps, the ®lm behaves nearly elastically. When yield stresses are exceeded, rapid structural rearrangements occur which increase the cavity size. At some point the cavities coalesce and a few bridges are all that connect the walls [Fig. 1(c)]. Ultimately, these bridges snap back to the walls, leaving the rough surface shown in Fig. 1(d). As the rupture velocity decreases towards zero, more time is needed to elastically deform the polymer ®lm. However, the sequence of rapid structural rearrangements remains nearly unchanged. The amount of
dissipation is also constant at low velocities, suggesting that the rapid rearrangements cause almost all of the dissipation. We con®rmed this by studying hysteresis loops like that shown in Fig. 3(a). If the wall is moved out until just before cavitation occurs and then back to its initial position, all the work used to stretch the ®lm can be recovered. If cavitation occurs before the wall is moved back, almost none of the work can be recovered. Similar behavior is seen at other rapid yield events. In general, work done in a loop that includes yield events cannot be recovered. The local velocities during rearrangements remain comparable to thermal velocities even as the rupture velocity goes to zero. The rapid pop of atoms from one site to another excites random vibrations that eventually ¯ow out of the walls as heat. The atomic motions that occur during structural rearrangements are complicated and highly variable because of the disordered structure of the ®lm. As we now discuss, crystalline junctions also fail through a series of rapid rearrangements. However, the atomic motion and the condition for instability is much simpler to describe because of the greater symmetry.
Fig. 1. Several snapshots of a glassy ®lm during rupture at velocity v=0.003 / . The 800 atoms in each wall are black, and the beads of the 128 chains are gray. Periodic boundary conditions were used in the xy plane, and two periodic images are shown. In (b) only a thin crosssection of the ®lm is shown.
Fig. 2. Time dependence of (a) force on wall and (b) energy changes during rupture of the glassy ®lm shown in Fig. 1. Lines in (b) indicate the external work (dashed), the potential energy increase (dotted), and the total heat ¯ow (solid). The kinetic energy is proportional to the temperature and remains unchanged.
A.R.C. Baljon, M.O. Robbins / Computational and Theoretical Polymer Science 9 (1999) 35±40
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Fig. 4. Side views of atomic con®gurations in a thin slice through a wire at three dierent elongation stages. The left and right frames show con®gurations before and after a structural rearrangement. In the central frame the transformation stage is shown. The arrows in the frame denote the glide directions. (Reproduced, with permission, from Ref. [1].) Fig. 3. External work versus wall displacement z during cavitation of the glassy ®lm shown in Fig. 1 (dashed line). The cavitation point is indicated by a star. Solid lines show the amount of recovered work when the wall spacing is decreased either before or after cavitation.
3. Failure of crystalline junctions Several groups have used computers to study atomic rearrangements during fracture of solids and metals [1,6,7]. All of these studies show that rupture occurs through stepwise increases in the number of atomic layers along the direction of rupture. This is illustrated in Fig. 4, taken from simulations by Landman et al. [1]. In the unstrained state (a) the junction contains 16 layers and the order is crystalline. For small strains the order remains crystalline, but the spacing between layers increases. As the strain increases, the stress reaches a threshold value and the junction yields. Stress is released through multiple glide processes with (111) planes sliding over each other in the directions indicated by the arrows in the center of Fig. 4(b). There is an increase in disorder during sliding. After sliding the junction is in a well ordered crystalline state with one additional layer [Fig. 4(c)]. Further strain causes a sequence of similar elastic deformation and plastic yield events, as in the thin polymer ®lms. The reason that crystalline junctions fail through this speci®c mechanism is that they are highly anisotropic. The spacing between (111) planes is larger than that between planes with other orientations. Thus they slide over each other at a lower stress. Two invariant scalars can be used to describe the amount of dilational and shear stress on a system. The ®rst is the hydrostatic pressure p which is given [8] by the average of the diagonal elements of the pressure tensor P: p=Tr[P]/3. The second is the von Mises shear stress [9], J1/22 =[Tr(PÿpI)2 /2]1/2. Landman et al. studied the variation of J21/ 2 during rupture. They found that the system yielded to create a new layer when the local value of J21/2 exceeded a threshold value. This is typical of ductile crystalline materials where the easiest mode of internal motion
involves glide of lattice planes over each other, often through motion of dislocations. The oligomer ®lms discussed above are isotropic. There are no lattice planes where sliding would be easier than in other directions. It would be interesting to investigate whether p or J21/2 controls failure in these systems. While the issue has not been addressed for thin ®lms, it has been studied in simulations of cavitation and shear of bulk polymers. 4. Plastic deformations in bulk polymers Mott et al. have studied cavity formation and plastic deformation in bulk polymeric materials [10,11]. Their model for glassy polypropylene treats the backbone carbon and hydrogen atoms explicitly. Bond lengths and bond angles are ®xed and molecular movement can only occur by rotation around the C±C bonds. Thermal ¯uctuations are precluded. A single chain in a glassy state is con®ned in a cubic unit cell with periodic boundary conditions. The system is forced to dilate by expanding the size of the cell. As in the other systems described above, the system responds elastically at small dilations. Then cavities form suddenly when the density has dropped by about 12%. Just as in the simulations of oligomer ®lms, further dilation produces a series of elastic loading stages and sudden yield events at which all cavities grow simultaneously. Yield events produced jumps in both the pressure and the von Mises stress. Unlike the crystalline junctions described above, yield was more closely associated with release of large negative pressures than with relief of accumulated shear stress. In fact, the shear stress rises during some rearrangements. The pressure decreases in magnitude at each yield event, although not all the way to the unstrained value. The average magnitude of the pressure slowly increases during cavity growth and the magnitude at which yield occurs also rises. This indicates that the system strain hardens or toughens. One reason for the dierence in yield mechanism between polymers and crystals may be that random
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A.R.C. Baljon, M.O. Robbins / Computational and Theoretical Polymer Science 9 (1999) 35±40
polymers do not have lattice planes that facilitate shear. However, the polymer and crystal were also stressed in dierent ways. The polymer was under a uniform dilation in all three directions so that most of the stress was associated with the pressure. The crystal was under a uniaxial strain that also produced a shear stress. That this dierence is responsible for the dierence in yield mechanisms is supported by work of Mott et al. on the shear response of the same polymer system [11]. Here too the system ¯ows through a series of plastic loading and relaxation events. However, now the von Mises stress is larger in magnitude than the pressure and is more closely associated with the onset of yield instabilities. It remains to be seen whether p or J21/2 determines the yield threshold of uniaxially strained oligomer ®lms. One of the surprising results of the shear simulations [11] was that the clusters of atoms which rearranged when the system became unstable were up to 10 nm in size. All particles in these large clusters moved simultaneouslyÐat least within the time resolution of the simulation. Considering other studies of yield phenomena [12], one expects that transformation of the cluster starts when the yield stress is exceeded in some part of the cluster. Local rearrangements cause stress to be transferred to other bonds that may yield in turn. The resulting avalanche transforms the whole cluster. Mott et al. found that the early elastic loading stages of their dilation simulations could be described by the Rose±Smith±Guinea±Ferrante model [10]. This model relates the elastic moduli to a characteristic intermolecular spacing and the intermolecular binding energy. Its success implies that intramolecular bonds are not contributing to the modulii. Work on oligomer ®lms also shows that intermolecular bonds dominate the elastic behavior and yield stress. In particular, the values of the yield stress at cavitation and the ®rst few subsequent yield events are nearly independent of chain length. This is surprising because intramolecular features such as chain length are very important in determining the free energy, and one would expect that the free energy change would be directly related to bond strength. The importance of energy rather than free energy is also seen in studies of biological bonds. 5. Biological bonds Recent simulations of the rupture of a bond between a single ligand and receptor [3] have been motivated by atomic-force-microscope studies of streptavidin±biotin bond strengths [13]. Recognition mechanisms and dissociation pathways of this complex were investigated. Examination of atomic con®gurations shows that the ligand and receptor are held together by a network of hydrogen bonds. Fig. 5 shows that the force goes through a sequence of oscillations as the streptavidin±
Fig. 5. Pulling force as a function of displacement during rupture of a biotinstreptavidin bond at a velocity of 0.015 AÊ/ps. The dashed vertical lines mark the ruptures of hydrogen bonds (bold lines) and water bridges (thin lines). (Reproduced, with permission, from Ref. [3]).
biotin bond is stretched. As in the systems described above, the regions where the force rises are periods where the hydrogen bonds deform elastically. Some of the bonds rupture at each force maximum, allowing the atoms to rearrange rapidly to create a new hydrogenbond network with lower stress. The unbinding force, a measure for molecular recognition, is de®ned as the largest force along the unbinding pathway. Its value, as obtained from simulations by extrapolating to low rupture velocities, agrees nicely with experimental data. As yet unexplained is that the unbinding force is proportional to the enthalpy change during rupture, but independent of changes in free energy [13]. It has been suggested that this is because entropic: changes do not occur until after unbinding [14]. This conclusion is consistent with recent simulations of rupture of oligomer ®lms [15]. They also show that the largest force during the rupture process correlates with the enthalpy change. Moreover, failure occurs at the interface with the lowest enthalpy change rather than that with the lowest free energy change. When the wall±¯uid interaction is much weaker than the ¯uid± ¯uid interaction, the bond ruptures at the wall±¯uid interface (adhesive failure). When the ¯uid±¯uid interaction is weaker, the bond breaks within the adhesive (cohesive failure). The location of failure does not depend on chain length or temperature although both produce large changes in the entropic contribution to the free energy change. A lack of correlation between bond strength and free energy change has also been seen in other experimental systems [16]. 6. Outlook Computational capabilities have reached a point where it is feasible to study rupture processes through
A.R.C. Baljon, M.O. Robbins / Computational and Theoretical Polymer Science 9 (1999) 35±40
molecular dynamics simulations. Molecular mechanisms of rupture in materials as diverse as glassy oligomer ®lms, solid junctions, and biological molecules, all share common aspects. In each of them, rupture takes place through alternating stages of stress accumulation and relief. During the loading stages the system deforms elastically. Most of the work done on the system causes a reversible increase in the potential energy stored in intermolecular bonds due to a decrease in density. Yield starts through local imbalance of forces on one segment of a molecule and rapidly spreads through the system. Studies of oligomer ®lms and biological molecules indicate that the yield stress is determined primarily by energy rather than free energy changes. During the rapid yield stages, energy is dissipated, and stress is relieved. The total amount of dissipated energy is strongly aected by the factors that control the importance of entropy in the system, such as chain length and temperature. The separation between the factors that control yield stress and energy dissipation may open up opportunities to design improved adhesives. Intermolecular bond strengths control the yield stress and the location of failure. The macroscopic strength of bonds is usually determined by the force needed to open a crack. This scales with the work required to rupture a unit area of the bond rather than with the yield stress. In most cases the work is dominated by energy dissipation and thus depends most sensitively on chain length and temperature. The systems considered here were all at such low temperatures that atoms were not free to diuse and no motion occured until the yield stress was exceeded. As a result, the amount of energy dissipation and the entire sequence of stress accumulation and relief stages became independent of the rupture velocity at low velocities. Rupture may be very dierent at high temperatures. In the extreme limit, the material becomes a linear viscous ¯uid. There are no longer separate stages of stress accumulation and release, because molecules can ¯ow smoothly to relieve stress. Energy dissipation arises only from viscous eects, and vanishes linearly in the limit of low rupture velocity [2]. More interesting behavior may occur at intermediate temperatures where polymer ®lms exhibit visco-elastic behavior. Gent et al. [17] reported that the dependence of the work required to rupture a macroscopic polymeric bond on rupture speed and that of the shear modulus on shear rate is qualitatively similar in this regime. This is quite remarkable given the dierence in the nature of the two processesÐfor instance one expects much higher density ¯uctuations during rupture than during shear. The natural inference is that viscoelastic dissipation in the bulk dominates the dissipation mechanisms. However, a detailed quantitative analysis shows that the eective length of the fracture zone deduced from this correspondence is much too small. Further studies are required to explain this discrepancy.
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Although all the systems described above undergo periodic cycles of stress buildup and release, they exhibit dierent types of microstructure. Larger simulations and more complex interactions may reveal more intricate spatial structures. For example, simulations in progress show that increasing the length of oligomers to four times the entanglement length produces rich spatial structure that is characteristic of polymer crazes. Cavities grow and merge to create an interconnected network of ®brils. Each ®bril contains several aligned chains and may merge with other ®brils or branch to create several ®brils. The craze grows by drawing more and more material into the ®brils from a thin, strain softened layer at the craze interface. We plan to use the simulations to study the details of this grow mechanism. Finally, experiments [18,19] and simulations [1] on ultrathin lubricant ®lms show that shear produces rich ¯uctuating patterns in the friction force. Analysis of the experimental data reveals chaotic behavior or broad band noise. Similar complex dynamic behavior has been found in rupture experiments. For instance, Gandur et al. [20] characterized stick-slip behavior during rupture of adhesive tape as a chaotic phenomenon. Non-equilibrium phase transitions, similar to those recently predicted to exist in Releigh-Bernard convection [21] might play a role as well. Hence, a complete understanding of dierent types of dissipation mechanisms and the range of processes that can cause stick-slip motion during rupture will require a better understanding of nonequilibrium phenomena. Acknowledgements The authors would like to thank D. Gersappe and E. Siggia for useful discussions and H. GruÈbmuller and U. Landman for allowing us to reproduce their ®gures. Support from the National Science Foundation through grants NSF96-25792, DMR-9634131, and PHY9407194 is gratefully acknowledged.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
Landman U, Luedtke WD, Gao J. Langmuir 1996;12:2514. Baljon ARC, Robbins MO. Science 1996;271:482. GruÈbmuller H, Heymann B, Tavan P. Science 1996;271:997. Kremer K, Grest GS. J Chem Phys 1990;92:5057. Thompson PA, Grest GS, Robbins MO. Phys Rev Lett 1992;68:3448. Lynden-Bell R. Science 1994;263:1704. Selinger RLB, Lynden-Bell RM, Gelbart WM. J Chem Phys 1993;90:9808. Allen M, Tildesley D. Computer simulation of liquids. Oxford: Clarendon Press, 1987. Johnson KL. Contact mechanics. Cambridge: Cambridge University Press, 1985. Mott P, Argon A, Suter U. Philos Mag 1993;68:537.
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A.R.C. Baljon, M.O. Robbins / Computational and Theoretical Polymer Science 9 (1999) 35±40
[11] Mott P, Argon A, Suter U. Philos Mag 1993;67:931. [12] Nato advanced study institute on disorder and fracture. New York: Plenum Press, 1990. [13] Moy VT, Florin E, Gaub HE. Science 1994;266:257. [14] Berendsen HJC. Science 1996;271:954. [15] Gersappe D, Robbins MO. Phys Rev Lett (submitted). [16] Israelachvili JN, Berman A. Isr J Chem 1995;35:85.
[17] [18] [19] [20]
Gent A, Lai SM. J Polym Sci B 1994;32:1543. Demirel AL, Granick S. Phys Rev Lett 1996;77:4330. Israelachvili JN, Drumond G. personal communication. Gandur MC, Kleine MU, Galembeck F. J Adhesion Sci Technol 1997;11:11. [21] Cakmur RV, Egolf DA, Plapp BB, Bodenschatz E. Phys Rev Lett 1997;79:1853.