Polymers, entanglements and cross-links

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Computer simulations detailing polymers at the level of atoms are shedding light on how these synthetic materials deform and bear loads. They are even providing clues to how bespoke polymers with surface and bulk properties targeted for specific applications can be designed from bottom-up.

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The white, flimsy glass in which you were served the dark, flat cola at the last party you attended was made of a material called polystyrene foam, though, understandably, you could not care less. But incidentally, polystyrene is probably the earliest polymer—a thermoplastic—discovered accidentally by a German pharmacist, Johann Eduard Simon, in 1839. The early years of the twentieth century saw the commercial production of a variety of thermoplastics, such as polyvinyl chloride and polyethene, which soon entered our everyday lives and altered engineering practice in ways no one had envisaged.

Thermoplastics are synthesised from monomers with functionalities of 1 or 2, which means they can connect to other monomers of the same type at one or both ends. When external conditions such as purity and temperature are tightly controlled, the polymerisation reaction proceeds along pathways that polymer chemists understand well, producing long, linear polymers. Though not always necessary, even techniques to make all chains grow to almost precisely the same length have also been perfected.

The sophisticated chemistry gives thermoplastics a very distinctive personality --- they are easy to process into complex shapes, tough and ductile, excellent electrical insulators, not prone to corrosion and, most importantly, as synthetic materials, offer a degree of property tunability unmatched by any other known material.

Think of how easy it is to slide a single strand out of a bowl of noodles. While the analogy between a bowl of noodles and an ensemble of macromolecules is apt in many ways, it breaks down when it comes to explaining the toughness and ductility of solid thermoplastics, which are simply the macromolecular ensemble held at a temperature below its characteristic glass transition temperature.

 

Entanglements and cross-links: holding a polymer together

It is not easy to pull a macromolecule out of the ensemble because, in a solid polymer, the chains loop around each other, forming invisible but frozen interconnections that conjure up the image of a badly dishevelled ball of wool rather than a bowl of slippery noodles. These knots are called entanglements and, polymer scientists believe, are the silent, decisive factor that gives these materials toughness and ductility.

This is what a linear polymer will look like when you zoom in. The chains cannot cut through each other and therefore, form entanglements.

There is another way to prevent the chains from sliding easily past each other. A macromolecule can be connected to other macromolecules at multiple points along its length using another short tie-molecule. This creates a network of macromolecules in much the same way the invisible entanglements do, with the important difference that, while entanglements occur at points dictated by the ensemble’s desire to settle into a thermodynamically more stable state, these `crosslinks’ can be introduced at selected locations in a controlled manner. Thus, since the 1940s, crosslinked thermoplastics have emerged as a designer material—controlling the network density opens the possibility of tailoring properties to an extent not possible with any other natural or synthetic material. An early success story is crosslinked polyethylene, XLPE, which has significantly improved dimensional and shape stability, toughness, heat resistance and insulating properties compared to its uncrosslinked counterpart.

Then there is the other extreme. Hydrogels, which are ultra-soft and biocompatible materials used in superabsorbent products such as diapers, contact lenses, and drug-delivery systems, also qualify as crosslinked polymers. Hydrogels are three-dimensional, cross-linked networks of natural or synthetic hydrophilic polymer networks that can swell by absorbing water and transform into soft, stimuli-sensitive materials. Hydrogels and XLPE, on two ends of the stiffness scale, are testaments to the extreme property modulations that are possible with cross-linked polymers.

Understanding how polymers get their strength: atom by atom

Two groups at IIT Kanpur are trying to understand how these two mechanisms of connecting macromolecular chains, namely entanglements and cross-links, allow a polymer to bear mechanical loads. Manoj Kumar Maurya and Pramod Patel have just submitted their PhD theses on two fascinating aspects of the problem, both analysed at scales where continuum assumptions no longer holds, long, interconnected chains become visible and the discreteness of matter manifests.

It turns out that polymers can be modelled on the computer in ways that retain different amounts of detail. At one end of the spectrum are atomistically detailed models that include every atom in the chains and model their interactions using elaborate schemes that employ even finer quantum-mechanical calculations to calibrate the forces that atoms of a type exert on one another. These calibrations are incredibly sophisticated, in that they not only consider the kinds of atoms interacting, but also the surroundings they are in. These are indeed first-principle models that are often thought to be free of parameters the user needs to estimate. But they can be prohibitive, and even the best modern computers can model extremely short events on extremely tiny samples. On the other end of the spectrum are models based on continuum mechanics, which rely on many parameters that need to be estimated and supplied but have relatively looser constraints on sample sizes or event durations.

Hydrogels in dry state and after swelling.

Between the two extremes lie a host of possibilities that are often grouped under mesoscopic models. Manoj and Pramod use models that fall in this category. In these models, a macromolecular chain is a long linear array of balls connected by springs. Each ball is a so-called superatom that can represent many monomers or a part of one, depending on the degree of `coarse-graining’ that is desired. The non-linear springs capture the interaction between adjacent superatoms, while superatoms in different chains exert forces on each other that are, in some ways, like the gravitational forces celestial bodies experience, but with an extremely short range. These forces are surrogates for short-range nuclear forces and are called ‘non-bonded’.

Manoj uses heavily coarse-grained macromolecules that look like a string of springy beads. Pramod opts for a somewhat lower degree of coarse-graining and adds some additional stiffness to the backbone that is not as floppy as Manoj’s. Importantly, to microscopically accommodate demands for change in the overall volume, he allows parts of his macromolecular chains to perform crankshaft-like manoeuvres whenever the crowd of other chains does not hem them in. Both their models include the non-bonded forces.

The two mesoscopic models view the macromolecular ensemble through microscopes with different magnifications, jettisoning many details that are not germane to the problem they wish to understand. Minimalist models resulting from various degrees of coarse-graining allow us to zero in on the essential physics without being distracted by superfluous details and model systems that are way larger than what detailed atomistics allows.

Scratching and denting cross-linked polymers

How do we build a cross-linked polymer model from here? Manoj Maurya, Ajay Kumar and their mentor Dr Manjesh Singh choose beads at random from a macromolecule and allow them to have dangling bonds – four at the most – through which they can connect to similarly chosen random beads in a neighbouring macromolecule. This process of synthetically driven cross-linking continues until no dangling bonds remain. If just a few random beads are chosen from a chain, they obtain a lightly cross-linked polymer. If all beads are selected, they obtain the extreme case of a highly cross-linked polymer. The computer-generated model now begins to resemble a real crosslinked polymeric system.

The read beads have four dangling bonds each (as shown on the right) and can connect to other red beads on neighbouring chains.

A further touch of realism is added. The bonds that connect neighbouring chains are breakable. Like real bonds, they snap when stretched. The sudden removal of the interconnections between chains due to bond snapping reduces the overall load-bearing capacity of the solid. As interconnects break, chains are unshackled, becoming mobile and suddenly, it is easier to deform the solid into shape.

A typical simulation box with the rigid indenter that can push in as well as slide along the top surface.

The ansatz thus created can now be loaded mechanically. Manoj and his co-workers push a rigid sphere into the surface of the solid. This mimics a process called nano-indentation, in which a small volume of matter is deformed intensely to obtain properties local to the indented region on the surface. If the sphere is first indented and then pulled across the surface, it can mimic the scratch test, often used by engineers to determine how resistant a solid surface is to abrasion or wear.

Using the above framework, the authors demonstrate that, between weakly cross-linked polymers with just a few linkable beads per chain to highly cross-linked ones where every bead is linkable, lies a veritable storehouse of designer materials with tunable properties.

Almost all moduli increase with cross-linking, while the coefficient of friction between the rigid sphere and the material's surface decreases. The force of adhesion between the material and the sphere decreases as well. Further, as the indenter drives into the material, the pent-up deformation energy around it releases, like avalanches, in bursts of bond scissions. Every avalanche is manifested by a steep load drop whose magnitude is correlated with the number of concerted scissions contained in the event.

The left figure shows the force on the polymer as the indenter is pushed in. The thin line shows the number of bonds that break during the process of indentation. Notice that the bonds all break in a very short interval and coincides with load drops. The figure on the right shows how the coefficient of friction and force of adhesion between the rigid intender and the surface vary with degree of crosslinking. From [1].

Compressing entangled polymers

Pramod investigates the compression behaviour of long chained polymers that are not cross-linked but are held together by entanglements. Under compression, many of these polymers exhibit a characteristic stress-strain response with a hump at low strains and a sharp upward trend at large strains, indicating that the polymer bears increasing stress as it is compressed. On unloading, these materials retain a significant amount of plastic strain.

Computationally generated stress-strain response of a linear polymer. The loading protocol involves compression with periodic unloadings to zero stress. Note that the stress-carrying capacity is greatly exaggerated in atomistic simulations due to the applied.

To decipher the role entanglements play in shaping this typical response, we first need to understand what entanglements are and devise ways to detect them. Pramod borrows a concept from knot theory, proposed initially by Gauss in 1833 to describe two intertwined closed curves in space. Called the Gauss linking number, it measures how many times and in which direction two closed, disjoint curves wind around each other. If the Gauss linking number is zero, we can separate the curves without cutting through any of them. For two closed curves, it turns out to be an integer that counts the number of times one winds around another.

Polymer chains are obviously not open curves. Given two chains that wind around each other, you can always separate them by sliding one out along the length of the other. But you can still calculate a Gaussian linking number; for winding open curves, it will not be an integer. Pramod divides each chain in his ensemble into segments and calculates the linking number with similar segments in other chains. The larger the number, the more difficult it is to separate the segments apart. And in a simulation, we can track them as the entire polymer is compressed.

The entanglements execute a number of interesting manoeuvres. When strains are small, they almost do nothing. The imposed deformation is accommodated by short sections of the chains, which execute crankshaft flips whenever they find space around them. Sometimes the flips locally create more space or rearrange the existing voids, but beyond some strain, no more free volume can be created by rearranging the chains alone. This is where entanglements come into play.

Entanglements, especially the ones with low linking number, now start to slip along the chains. They also occasionally disappear, either by sliding out through the ends or by simply unravelling. Both slipping and disentanglement are irreversible events that contribute to plastic strain. Unloading sometimes recovers some of the lost entanglements but the slipped ones dissipate energy even when they return to their original configurations.

Researchers have wondered how thermoplastics, even without cross-links, can often exhibit significant ductility. Pramod’s work clearly shows that its ability to adjust its entanglement network by affecting slip and entanglement losses is the thermoplastic’s way of accommodating plastic strain.

Interestingly, though we do not have as much control over the mechanical response of these materials as we do for the cross-linked versions, both the initial rearrangement of free volume and network adjustments can be influenced by engineering the polymer architecture at the molecular level. This means that, to a limited extent, we can have designer materials even when cross-links are absent.

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[1] Maurya, M.K., Singh, M.K. Computational indentation in weakly cross-linked polymer networks. Int J Adv Eng Sci Appl Math 15, 196–206 (2023). https://doi.org/10.1007/s12572-023-00354-3

[2] Manoj Kumar Maurya,C´eline Ruscher, Debashish Mukherji and Manjesh Kumar Singh, Computational indentation in highly cross-linked polymer networks, Phys. Rev. E 106, 014501, DOI: https://doi.org/10.1103/PhysRevE.106.014501

[3] Ajay Kumar, Manoj Kumar Maurya, and Manjesh Kumar Singh, Computational Study of the Adhesion and Friction Behavior of Cross-Linked Polymer Networks, Langmuir 2025 41 (48), 32560-32568, DOI: 10.1021/acs.langmuir.5c04394

[4] Pramod Kumar Patel, Sumit Basu, An Algorithm for Computing Entanglements in an Ensemble of Linear Polymers, Macromolecular Theory and Simulations, 33, 6 2400035, https://doi.org/10.1002/mats.202400035

[5] Pramod Kumar Patel, Sumit Basu, Micromechanical insights into the uniaxial stress-strain behaviour of glassy amorphous polymers through molecular dynamics simulations, 208, 2026, 106496, https://doi.org/10.1016/j.jmps.2025.106496