Polymer networks and gels


Michael LangToni Müller, Kiran Suresh-Kumar, Reinhard Scholz, and Jens-Uwe Sommer


Cross-linking of polymers leads to a combination of liquid (viscosity) and solid-state features (shape stability, no dissolution). In modern polymer research, cross-linking processes are applied to control materials properties in combination with structure-formation and self-organization processes. Even though rubber elasticity has been investigated since more than one century, convincing theoretical models that address all the possible structure-property relationships are not at hand. This is due the randomness of the cross-linking process that defines the particular local disorder and heterogeneity of the material. As an example, the Figure shows the frozen in concentration fluctuations of a gel that is swollen to equilibrium.

In our work, we focus on two particular classes of problems. The first class concerns general aspects of polymer networks and gels such as rubber elasticity, equilibrium degree of swelling, relaxation phenomena, residual bond orientations, network structure, etc … . Here, a development of fundamental concepts from simplified models to more realistic scenarios is of particular interest.

The second class of problems is based upon model systems that have a distinct molecular structure that leaves its signature in the physical response of the networks:

  • Reversible gels start to flow beyond the lifetime of the reversible bonds.
  • Slide-ring gels exhibit an unusual deformation dependence that is not yet fully understood.
  • Olympic gels are held together by a mutual concatenation of strands, which imposes an unusual swelling behavior.
  • Star-polymer networks are model systems that allow to suppress some of the most abundant cyclic defects.
  • Cross-linked polymer brushes are two-dimensional networks with unexpected resistance towards a collapse when cross-linked in the swollen state.

The rubber elasticity of non-entangled [1,2] and entangled [3] networks and was in the focus of our most recent research. Our results indicate that some more theoretical work is needed for an improved modeling of rubber elasticity. In particular, the classical way of approximating modulus by the sum of an unperturbed melt-entanglement and the phantom model cross-link contribution needs to be questioned. The former underestimates the entanglement contributions as chains are stretched by fluctuations that freeze in while cross-linking [3]. The latter is an over-estimate of the cross-link contribution as finite cyclic structures contribute less to elasticity as compared to an infinite tree structure that is assumed in theory [1,2]. Future work will have to combine these corrections with other, more detailed consideration of the network structure [4].

Tendomers are a special class of cross-linked rotaxanes where the cross-linked rings are concentrated at one end of the rotaxane [5]. This special molecular design causes a three stage deformation behavior. Below a stall force, there is a first linear regime with a low deformation of the molecule. For applied forces beyond the stall fore, a super-elastic regime is observed where the molecule deforms even softer than a linear chain of same degree of polymerization. Finally, the super-elastic regime crosses over to a saturation regime that is governed by the finite extensibility of the polymer backbone. Thus, tendomers are promising building blocks to design force-sensitive elastomers.

Equilibrium swelling and network structure were analyzed for randomly linked networks that result from a cross-linking co-polymerization reaction [6] and Olympic gels [7]. Both systems showed an unexpected scaling of the equilibrium degree of swelling that was understood after developing a detailed model for the network structure including network defects. For the randomly linked networks of Ref. [6], an explicit consideration of entanglements and the elastically inactive material is necessary to understand the swelling data. The swelling of Olympic gels [7], on the other hand, is a combination of the des-interspersion of overlapping non-concatenated strands with an additional swelling process of the “des-interspersed” gel.

Recently, the research network FOR2811 has started that investigates amphi-philic co-networks with a particular focus on the relation between local phase separation, possible block-copolymer mesophases, network structure and material properties. Future work may cover the interpenetrating networks, the formation of micro-gels, or the kinetics of swelling and de-swelling of polymer gels.



  1. Lang, M.,
    On the resistor analogy for estimating the elasticity of polymer networks,
    Macromolecules 52 (2019) 6266-6273.
  2. Lang, M.,
    Elasticity of phantom model networks with cyclic defects,
    ACS Macro Letters 7 (2018) 536-539.
  3. Lang, M.,
    Relation between Cross-Link Fluctuations and Elasticity in Entangled Polymer Networks,
    Macromolecules 50 (2017) 2547-2555.
  4. Lang, M.,
    Cyclic structures in polymer model networks,
    Macromolecular Symposia 285 (2019) 1800168.
  5. Müller, T.;  Sommer, J.-U. ; Lang M.,
    Tendomers – force sensitive bis-rotaxanes with jump-like deformation behavior
    Soft Matter 15 (2019) 3671-3679.
  6. Lang, M. ; John, A. ; Sommer, J.-U.,
    Model simulations on network formation and swelling as obtained from cross-linking co-polymerization reactions,
    Polymer 82 (2016) 138-155.
  7. Lang, M. ; Fischer, J. ; Werner, M. ; Sommer, J.-U.,
    Swelling of Olympic gels,
    Physical Review Letters 112 (2014) 238001(5).