Interfacial Fracture between Highly Cross-Linked Polymer Networks and a Solid Surface: Effect of Interfacial Bond Density

Thermal Conductivity of Polymers: A Simple Matter Where Complexity Matters

Thermal conductivity coefficient κ measures the ability of a material to conduct a heat current. In particular, κ is an important property that often dictates the usefulness of a material over a wide range of environmental conditions. For example, while a low κ is desirable for the thermoelectric applications, a large κ is needed when a material is used under the high temperature conditions. These materials range from common crystals to commodity amorphous polymers. The latter is of particular importance because of their use in designing light weight high performance functional materials. In this context, however, one of the major limitations of the amorphous polymers is their low κ, reaching a maximum value of ≈0.4 W/Km that is 2-3 orders of magnitude smaller than the standard crystals. Moreover, when energy is predominantly transferred through the bonded connections, κ ⩾ 100 W/Km. Recently, extensive efforts have been devoted to attain a tunability in κ via macromolecular engineering. In this work, an overview of the recent results on the κ behavior in polymers and polymeric solids is presented. In particular, computational and theoretical results are discussed within the context of complimentary experiments. Future directions are also highlighted.

Characterizing the shear response of polymer-grafted nanoparticles

Grafting polymer chains to the surface of nanoparticles overcomes the challenge of nanoparticle dispersion within nanocomposites and establishes high-volume fractions that are found to enable enhanced material mechanical properties. This study utilizes coarse-grained molecular dynamics simulations to quantify how the shear modulus of polymer-grafted nanoparticle (PGN) systems in their glassy state depends on parameters such as strain rate, nanoparticle size, grafting density, and chain length. The results are interpreted through further analysis of the dynamics of chain conformations and volume fraction arguments. The volume fraction of nanoparticles is found to be the most influential variable in deciding the shear modulus of PGN systems. A simple rule of mixture is utilized to express the monotonic dependence of shear modulus on the volume fraction of nanoparticles. Due to the reinforcing effect of nanoparticles, shortening the grafted chains results in a higher shear modulus in PGNs, which is not seen in linear systems. These results offer timely insight into calibrating molecular design parameters for achieving the desired mechanical properties in PGNs.

Understanding the application of covalent adaptable networks in self-repair materials based on molecular simulation

Covalent adaptable networks (CANs) are widely used in the field of self-repair materials. They are a group of covalently cross-linked associative polymers that undergo reversible chemical reactions, and can be further divided into dissociative CANs (Diss-CANs) and associative CANs (Asso-CANs). Self-repair refers to the ability of a material to repair itself without external intervention, and can be classified into self-adhesion and self-healing according to the utilization of open stickers. Unlike conventional materials, the viscoelastic properties of CANs are influenced by both the molecular structure and reaction kinetics, ultimately affecting their repair performance. To gain deeper insight into the repair mechanism of CANs, we conducted simulations by using the hybrid MC/MD algorithm, as previously proposed in our research. Interestingly, we observed a significant correlation between reaction kinetics and repair behavior. Asso-CANs exhibited strong mechanical strength and high creep resistance, rendering them suitable as self-adhesion materials. On the other hand, Diss-CANs formed open stickers that facilitated local relaxation, aligning perfectly with self-healing processes. Moreover, the introduction of crosslinkers in the form of small molecules enhanced the repair efficiency. Theoretically, it was found that the repair timescale of Asso-CANs is slower than that of Diss-CANs with identical molecular structures. Our study not only clarifies the similarities and differences between Diss-CANs and Asso-CANs in terms of their self-repairing capabilities, but more importantly, it provides valuable insights guiding the effective utilization of CANs in the development of self-repair materials.

Simulation Study of the Effect of Nanoporous Surfaces on the Adhesion Properties of Cross-Linked Polymer Networks

We have investigated the effect of surface nanopores on the adhesion behavior between cross-linked polymer networks and metal substrates by molecular dynamics simulations. By increasing the cross-linking ratio of the polymer network, the fracture behavior in tensile mode changed from cohesive failure to interfacial failure. In the case of polymers without cross-links, the breaking strengths were almost the same for systems with flat and porous metal substrates. Conversely, in the case of cross-linked polymer networks, the tensile behavior for the porous metal substrates depended on the cross-linking ratio and structure of the polymer chains. For polymer networks consisting of long polymer chains, the force curves in extension mode before the yield points were almost the same for the systems regardless of the surface roughness caused by nanopores. Meanwhile, for highly cross-linked resin networks consisting of short rigid molecules, the yielding strength of the porous metal surfaces showed slightly higher values than that of the flat metal surfaces. The simulation results revealed that the adhesion behavior between cross-linked polymer networks and rough metal surfaces is related not only to the interfacial area but also to the detailed networking topology of the polymers.

Direct visualization of cooperative adsorption of a string-like molecule onto a solid

Natural systems, composite materials, and thin-film devices adsorb macromolecules in different phases onto their surfaces. In general, polymer chains form interfacial layers where their aggregation states and thermal molecular motions differ from the bulk. Here, we visualize well-defined double-stranded DNAs (dsDNAs) using atomic force microscopy and molecular dynamics simulations to clarify the adsorption mechanism of polymer chains onto solid surfaces. Initially, short and long dsDNAs are individually and cooperatively adsorbed, respectively. Cooperative adsorption involves intertwining of multiple chains. The dependence of adsorption on the chain affects the formation of the interfacial layer, realizing different mechanical properties of DNA/filler bulk composites. These findings will contribute to the development of light and durable polymer composites and films for various industrial, biomedical, and environmental applications.

Unexpected Ductility in Semiflexible Polymer Glasses with Entanglement Length Equal to Their Kuhn Length

Semiflexible polymer glasses (SPGs), including those formed by the recently synthesized semiflexible conjugated polymers, are expected to be brittle because classical formulas for their craze extension ratio λ_{craze} and fracture stretch λ_{frac} predict that systems with N_{e}=C_{∞} have λ_{craze}=λ_{frac}=1 and hence cannot be deformed to large strains. Using molecular dynamics simulations, we show that in fact such glasses can form stable crazes with λ_{craze}≃N_{e}^{1/4}≃C_{∞}^{1/4}, and that they fracture at λ_{frac}=(3N_{e}^{1/2}-2)^{1/2}≃(3C_{∞}^{1/2}-2)^{1/2}. We argue that the classical formulas for λ_{craze} and λ_{frac} fail to describe SPGs' mechanical response because they do not account for Kuhn segments' ability to stretch during deformation.

Computational indentation in highly cross-linked polymer networks

Indentation is a common experimental technique to study the mechanics of polymeric materials. The main advantage of using indentation is this provides a direct correlation between the microstructure and the small-scale mechanical response, which is otherwise difficult within the standard tensile testing. The majority of studies have investigated hydrogels, microgels, elastomers, and even soft biomaterials. However, a less investigated system is the indentation in highly cross-linked polymer (HCP) networks, where the complex network structure plays a key role in dictating their physical properties. In this work, we investigate the structure-property relationship in HCP networks using the computational indentation of a generic model. We establish a correlation between the local bond breaking, network rearrangement, and small-scale mechanics. The results are compared with the elastic-plastic deformation model. HCPs harden upon indentation.

Superstretchable Elastomer from Cross-linked Ring Polymers

The stretchability of polymeric materials is critical to many applications such as flexible electronics and soft robotics, yet the stretchability of conventional cross-linked linear polymers is limited by the entanglements between polymer chains. We show using molecular dynamics simulations that cross-linked ring polymers are significantly more stretchable than cross-linked linear polymers. Compared to linear polymers, the entanglements between ring polymers do not act as effective cross-links. As a result, the stretchability of cross-linked ring polymers is determined by the maximum extension of polymer strands between cross-links, rather than between trapped entanglements as in cross-linked linear polymers. The more compact conformation of ring polymers before deformation also contributes to the increase in stretchability.

Simulation Study of the Effects of Interfacial Bonds on Adhesion and Fracture Behavior of Epoxy Resin Layers

The adhesion and fracture behavior of tetraglycidyl-4,4'-diaminodiphenylmethane (TGDDM)/4,4'-diaminodiphenyl sulfone (44DDS)-bisphenol A diglycidyl ether (DGEBA)/44DDS layer interfaces were investigated by molecular dynamics (MD) simulation, mainly focusing on the role of covalent and noncovalent interactions. To accurately investigate the bond dissociation processes, the force field parameters of several bond potentials of the epoxy resin polymers were optimized by density functional theory calculations. In the MD simulations under a tensile load, small voids gradually developed without covalent bond dissociation in the plateau region. In the final large strain region, the stress rapidly increased with bond breaking, leading to failure. When the chemical bonds across the interface between the two layers were removed, the stress-strain curve in the initial elastic region was almost the same as that with interfacial bonds. This showed that the nonbonded interactions governed adhesion strength in the initial elastic region. In contrast, the bonded interactions at interfaces played important roles in the hardening regions because the bonded interactions made the major contribution to the fracture energies. We also investigated the effect of the etherification reaction in cross-linking. It was found that the etherification reaction mainly contributed to the behavior in the late region with large strain. These simulation results revealed that the nonbonded interactions, especially, van der Waals interactions, are important factors for adhesion of the different polymer layers in the small strain region up to the yield point.

Simulation of polymerization induced phase separation in model thermosets

Polymerization induced phase separation (PIPS) in a three component thermoset is studied using molecular dynamics simulations of a new coarse-grained thermoset model. The system includes two crosslinker molecules, which differ in their glass transition temperatures (T_g) and chain length and thus have the potential for phase separation. One crosslinker has a high T_g corresponding to a rubbery behavior, and simulations were performed for a short length (4 beads) and a long length (33 beads). The resin and other crosslinker have low T_g. A coarse-grained model is developed with these features and with interaction parameters determined so that for either rubbery crosslinker length, the system is in the liquid state at the cure temperature. For sufficiently slow reaction rates, the long rubbery molecule exhibits PIPS into a bicontinuous array of nanoscale domains, but the short one does not, reproducing recent experimental results. The simulations demonstrate that the reaction rates must be slow enough to allow diffusion to yield phase separation. Particularly, the reaction rate corresponding to the secondary amine must be very slow, else the structure of crosslinked clusters and the substantially increased diffusion time will prevent PIPS.

Response of adhesive polymer interfaces to repeated mechanical loading and the spatial variation of diffusion coefficient and stresses in a deforming polymer film

Comprehensive molecular simulations are conducted to show that polymer crosslinks preserve the strength of solid-polymer (melt) interfaces when they are subjected to repeated mechanical loading. The spatial variation of the diffusion coefficient and local stresses is also investigated along the polymer thickness, during deformation. After each loading cycle, a reduction in entanglement strength is observed at the fracture site. The work of adhesion also decreases over consecutive loading cycles, when fracture is induced at the same site. Reduction in both, the work of adhesion and the entanglement strength, decreases as the crosslink density increases. Diffusion coefficient and stresses vary significantly and in a complex manner along the film thickness during the entire deformation process. These variations were due to peculiar configurations occurring at each instance of separation, which are analyzed and explained in this work. The variation of diffusion coefficient during deformation suggests that other dynamic properties, such as viscosity, also vary spatially during polymer deformation.

Molecular dynamics simulation of thermo-mechanical behaviour of elastomer cross-linked via multifunctional zwitterions

Coarse-grained (CG) molecular dynamics simulations have been employed to study the thermo-mechanical response of a physically cross-linked network composed of zwitterionic moieties and fully flexible elastomeric polymer chains.

Salt concentration dependence of the mechanical properties of LiPF6/poly(propylene glycol) acrylate electrolyte at a graphitic carbon interface: A reactive molecular dynamics study

This reactive molecular dynamics study explores the salt concentration dependence of the viscoelastic and mechanical failure properties of a poly(propylene glycol)/LiPF₆‐based solid polymer electrolyte (SPE) at a graphitic carbon electrode interface. To account for the finite‐size effect of interface‐confined SPE films, the properties of two distinct film thicknesses are compared with the respective bulk properties. Additionally, the effect of uniaxial compression in the interface‐normal direction on free energy profiles of Li‐ion SPE‐desolvation is studied. © 2018 The Authors. Journal of Polymer Science Part B: Polymer Physics Published by Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2018, 56, 718–730

Uncovering the rupture mechanism of carbon nanotube filled cis-1,4-polybutadiene via molecular dynamics simulation

In this work, by employing molecular dynamics simulations in a united atomistic resolution, we explored the rupture mechanism of carbon nanotube (CNT) filled cis-1,4-polybutadiene (PB) nanocomposites. We observed that the rupture resistance capability increases with the interfacial interaction between PB and CNTs, as well as the loading of CNTs, attributed to the enhanced chain orientation along the deformed direction to sustain the external force, particularly those near voids. The number of voids is quantified as a function of the strain, exhibiting a non-monotonic behavior because of the coalescence of small voids into larger ones at high strain. However, the number of voids is greatly reduced by stronger PB–CNT interaction and higher loading of CNTs. During the rupture process, the maximum van der Waals energy change reflects the maximum conformational transition rate and the largest number of voids. Meanwhile, the strain at the maximum orientation degree of bonds is roughly consistent with that at the maximum square radius of gyration of chains. After the failure, the stress gradually decreases with the strain, accompanied by the contraction of the highly orientated polymer bundles. In particular, with weak interfacial interaction, the nucleation of voids occurs in the interface, and in the polymer matrix in the strong case. In general, this work could provide some fundamental understanding of the voids occurring in polymer nanocomposites (PNCs), with the aim to design and fabricate high performance PNCs.

In this work, by employing molecular dynamics simulations in a united atomistic resolution, we explored the rupture mechanism of carbon nanotube (CNT) filled cis-1,4-polybutadiene (PB) nanocomposites.

Simulational insights into the mechanical response of prestretched double network filled elastomers

This paper deals with molecular-dynamics simulations of the mechanical properties of prestretched double network filled elastomers. To this end, we firstly validated the accuracy of this method, and affirmed that the produced stress-strain characteristics were qualitatively consistent with Lesser's experimental results on the prestretched tri-block copolymers with a competitive double network. Secondly, we investigated the effect of the crosslinking network ratio on the mechanical properties of the prestretched double network homopolymers under uniaxial tension. We found that the prestretched double network contributes greatly to the enhanced tensile stress and ultimate strength at fracture, as well as to the lower permanent set (the residual strain) and dynamic hysteresis loss, both parallel and perpendicular to the prestretching direction. Notably, though, an anisotropic behavior was observed: in the parallel direction, both the first and the second crosslinked networks bore the external force; whereas in the perpendicular direction, only the second crosslinked network was relevantly effective. Finally, the polymer nanocomposites with a prestretched double network exhibited tensile mechanical properties similar to those of the studied homopolymers with prestretched double networks. Summing up the results, it can be concluded that the incorporation of prestretched double networks with a specified crosslinking network ratio seems to be a promising method for manipulating the mechanical properties of elastomers and their nanocomposites, as well as for introducing anisotropy in their mechanical response.

Topological structure and mechanics of glassy polymer networks

The influence of chain-level network architecture (i.e., topology) on mechanics was explored for unentangled polymer networks using a blend of coarse-grained molecular simulations and graph-theoretic concepts. A simple extension of the Watts-Strogatz model is proposed to control the graph properties of the network such that the corresponding physical properties can be studied with simulations. The architecture of polymer networks assembled with a dynamic curing approach were compared with the extended Watts-Strogatz model, and found to agree surprisingly well. The final cured structures of the dynamically-assembled networks were nearly an intermediate between lattice and random connections due to restrictions imposed by the finite length of the chains. Further, the uni-axial stress response, character of the bond breaking, and non-affine displacements of fully-cured glassy networks were analyzed as a function of the degree of disorder in the network architecture. It is shown that the architecture strongly affects the network stability, flow stress, onset of bond breaking, and ultimate stress while leaving the modulus and yield point nearly unchanged. The results show that internal restrictions imposed by the network architecture alter the chain-level response through changes to the crosslink dynamics in the flow regime and through the degree of coordinated chain failure at the ultimate stress. The properties considered here are shown to be sensitive to even incremental changes to the architecture and, therefore, the overall network architecture, beyond simple defects, is predicted to be a meaningful physical parameter in the mechanics of glassy polymer networks.

HORIZONS FOR DESIGN OF FILLED RUBBER INFORMED BY MOLECULAR DYNAMICS SIMULATION

Fillers such as carbon black provide a long-standing and essential strategy for the mechanical reinforcement of rubber in tires and other load-bearing applications. Despite their technological importance, however, the microscopic mechanism of this reinforcement remains a matter of considerable debate. A predictive understanding of filler-based reinforcement could catalyze the design of new rubber-filler composites with enhanced performance. Molecular dynamics simulations of rubber mechanical response in the presence of structured fillers offer a new strategy for resolving the origins of filler-based reinforcement and guiding filler design. Results of for ideal rubber-filler dispersions over a range of filler structures suggest that neither hydrodynamic effects nor non-deformable “bound rubber domains” are necessary to achieve high reinforcement. Moreover, simulations show that particle surface area is a poor predictor of reinforcement. Instead, simulated reinforcement correlates strongly with filler structure, with more rarified filler structure predicting much greater reinforcement at fixed loading. Simulation results are consistent with a scenario in which reinforcement at industrially relevant loadings is dominated by formation of a jammed network of filler particles, suggesting that reinforced rubber can be understood as a superposition of two materials: a rubbery solid, and a jammed granular solid. This perspective points to an opportunity to improve filler-reinforced rubber design by leveraging concepts and expertise developed over many decades in the fields of jamming and granular media.

Alkane-Metal Interfacial Structure and Elastic Properties by Molecular Dynamics Simulation

The structure of amorphous materials near the interface with an ordered substrate can be affected by various characteristics of the adjoining phases, such as the lattice spacing of the adherent surface, polymer chain length, and adhesive strength. To discern the influence of each of these factors, four FCC metal lattices are examined for three chain lengths of n-alkane and van der Waals interfacial interactions are controlled by adjusting the Lennard-Jones 12-6 potential parameters. The role of interaction strength is investigated for a single chain length and substrate combination. Four nanoconfined systems are also analyzed in terms of their mechanical strength. A strong layering effect is observed near the interface for all systems. The distinctiveness of polymer layering, i.e., the maximum density and spatial extent, exhibits a logarithmic dependence on the interaction strength between polymer and substrate. Congruency with the substrate lattice parameter further enhances this effect. Moreover, the elastic modulus of the alkane phase as a function of layer thickness indicates that the effects of ordering within the structure extend beyond the immediately obvious interfacial region.

The Influence of Crosslink Density on the Failure Behavior in Amorphous Polymers by Molecular Dynamics Simulations

The crosslink density plays a key role in the mechanical response of the amorphous polymers in previous experiments. However, the mechanism of the influence is still not clear. In this paper, the influence of crosslink density on the failure behavior under tension and shear in amorphous polymers is systematically studied using molecular dynamics simulations. The present results indicate that the ultimate stresses and the broken ratios (the broken bond number to all polymer chain number ratios) increase, as well as the ultimate strains decrease with increasing crosslink density. The strain concentration is clearer with the increase of crosslink density. In other words, a higher crosslink density leads to a higher strain concentration. Hence, the higher strain concentration further reduces the fracture strain. This study implies that the mechanical properties of amorphous polymers can be dominated for different applications by altering the molecular architecture.

Molecular dynamics simulation of the rupture mechanism in nanorod filled polymer nanocomposites

Through coarse-grained molecular dynamics simulation, we aim to uncover the rupture mechanism of polymer-nanorod nanocomposites by characterizing the structural and dynamic changes during the tension process. We find that the strain at failure is corresponding to the coalescence of a single void into larger voids, namely the change of the free volume. And the minimum of the Van der Walls (VDWL) energy reflects the maximum mobility of polymer chains and the largest number of voids of polymer nanocomposites. After the failure, the stress gradually decreases with the strain, accompanied by the contract of the highly orientated polymer bundles. In particular, we observe that the nucleation of voids prefers to occur from where the ends of polymer chains are located. We systematically study the effects of the interfacial interaction, temperature, the length and volume fraction of nanorods, chain length, bulk cross-linking density and interfacial chemical bonds on the rupture behavior, such as the stress at failure, the tensile modulus and the rupture energy. The rupture resistance ability increases with the increase of the interfacial interaction, rod length, and bulk cross-linking density. With an increase in the interfacial interaction, it induces the rupture transition from mode A (no bundles) to B (bundles). The transition point of the stress at failure as a function of the temperature roughly corresponds to the glass transition temperature. At longer chain length, a non-zero stress plateau occurs. And excessive chemical bonds between polymers and nanorods are harmful to the rupture property. We find that an optimal volume fraction of nanorods exists for the stress-strain behavior, which can be rationalized by the formation of the strongest polymer-nanorod network, leading to the slowest mobility of nanorods.

Coarse-grained molecular dynamics simulations of the tensile behavior of a thermosetting polymer

Using a previously developed coarse-grained model, we conducted large-scale (∼ 85 × 85 × 85 nm(3)) molecular dynamics simulations of uniaxial-strain deformation to study the tensile behavior of an epoxy molding compound, epoxy phenol novolacs (EPN) bisphenol A (BPA). Under the uniaxial-strain deformation, the material is found to exhibit cavity nucleation and growth, followed by stretching of the ligaments separated by the cavities, until the ultimate failure through ligament scissions. The nucleation sites of cavities are rather random and the subsequent cavity growth accounts for much (87%) of the volumetric change during the uniaxial-strain deformation. Ultimate failure of the materials occurs when the cavity volume fraction reaches ∼ 60%. During the entire deformation process, polymer strands in the network are continuously extended to their linear states and broken in the postyielding strain hardening stage. When most of the strands are stretched to their taut configurations, rapid scission of a large number of strands occurs within a small strain increment, which eventually leads to fracture. Finally, through extensive numerical simulations of various loading conditions in addition to uniaxial strain, we find that yielding of the EPN-BPA can be described by the pressure-modified von Mises yield criterion.

Healing of polymer interfaces: Interfacial dynamics, entanglements, and strength

Self-healing of polymer films often takes place as the molecules diffuse across a damaged region, above their melting temperature. Using molecular dynamics simulations we probe the healing of polymer films and compare the results with those obtained for thermal welding of homopolymer slabs. These two processes differ from each other in their interfacial structure since damage leads to increased polydispersity and more short chains. A polymer sample was cut into two separate films that were then held together in the melt state. The recovery of the damaged film was followed as time elapsed and polymer molecules diffused across the interface. The mass uptake and formation of entanglements, as obtained from primitive path analysis, are extracted and correlated with the interfacial strength obtained from shear simulations. We find that the diffusion across the interface is significantly faster in the damaged film compared to welding because of the presence of short chains. Though interfacial entanglements increase more rapidly for the damaged films, a large fraction of these entanglements are near chain ends. As a result, the interfacial strength of the healing film increases more slowly than for welding. For both healing and welding, the interfacial strength saturates as the bulk entanglement density is recovered across the interface. However, the saturation strength of the damaged film is below the bulk strength for the polymer sample. At saturation, cut chains remain near the healing interface. They are less entangled and as a result they mechanically weaken the interface. Chain stiffness increases the density of entanglements, which increases the strength of the interface. Our results show that a few entanglements across the interface are sufficient to resist interfacial chain pullout and enhance the mechanical strength.

Structure and Strength at Immiscible Polymer Interfaces

Thermal welding of polymer-polymer interfaces is important for integrating polymeric elements into devices. When two different polymers are joined, the strength of the weld depends critically on the degree of immiscibility. We perform large-scale molecular dynamics simulations of the structure-strength relation at immiscible polymer interfaces. Our simulations show that immiscibility arrests interdiffusion and limits the equilibrium interfacial width. Even for weakly immiscible films, the narrow interface is unable to transfer stress upon deformation as effectively as the bulk material, and chain pullout at the interface becomes the dominant failure mechanism. This greatly reduces the interfacial strength. The weak response of immiscible interfaces is shown to arise from an insufficient density of entanglements across the interface. We demonstrate that there is a threshold interfacial width below which no significant entanglements can form between opposite sides to strengthen the interface.

Molecular dynamics simulations of polymer welding: strength from interfacial entanglements

Large-scale simulations of thermal welding of polymers are performed to investigate the rise of mechanical strength at the polymer-polymer interface with the welding time t(w). The welding process is at the core of integrating polymeric elements into devices as well as in the thermal induced healing of polymers, processes that require the development of interfacial strength equal to that of the bulk. Our simulations show that the interfacial strength saturates at the bulk shear strength long before polymers diffuse by their radius of gyration. Along with the strength increase, the dominant failure mode changes from chain pullout at the interface to chain scission as in the bulk. The formation of sufficient entanglements across the interface, which we track using a primitive path analysis, is required to arrest catastrophic chain pullout at the interface. The bulk response is not fully recovered until the density of entanglements at the interface reaches the bulk value. Moreover, the increase of interfacial strength before saturation is proportional to the number of interfacial entanglements between chains from opposite sides.