Wouter G. Ellenbroek

Critical scaling in linear response of frictionless granular packings near jamming.

We study the origin of the scaling behavior in frictionless granular media above the jamming transition by analyzing their linear response. The response to local forcing is non-self-averaging and fluctuates over a length scale that diverges at the jamming transition. The response to global forcing becomes increasingly nonaffine near the jamming transition. This is due to the proximity of floppy modes, the influence of which we characterize by the local linear response. We show that the local response also governs the anomalous scaling of elastic constants and contact number.

Stability of jammed packings I: the rigidity length scale

In 2005, Wyart et al. [Europhys. Lett., 2005, 72, 486] showed that the low frequency vibrational properties of jammed amorphous sphere packings can be understood in terms of a length scale, called *, that diverges as the system becomes marginally unstable. Despite the tremendous success of this theory, it has been difficult to connect the counting argument that defines * to other length scales that diverge near the jamming transition. We present an alternate derivation of * based on the onset of rigidity. This phenomenological approach reveals the physical mechanism underlying the length scale and is relevant to a range of systems for which the original argument breaks down. It also allows us to present the first direct numerical measurement of *.

Rigidity loss in disordered systems : three scenarios

We reveal significant qualitative differences in the rigidity transition of three types of disordered network materials: randomly diluted spring networks, jammed sphere packings, and stress-relieved networks that are diluted using a protocol that avoids the appearance of floppy regions. The marginal state of jammed and stress-relieved networks are globally isostatic, while marginal randomly diluted networks show both overconstrained and underconstrained regions. When a single bond is added to or removed from these isostatic systems, jammed networks become globally overconstrained or floppy, whereas the effect on stress-relieved networks is more local and limited. These differences are also reflected in the linear elastic properties and point to the highly effective and unusual role of global self-organization in jammed sphere packings.

Geometry and the onset of rigidity in a disordered network

Disordered spring networks that are undercoordinated may abruptly rigidify when sufficient strain is applied. Since the deformation in response to applied strain does not change the generic quantifiers of network architecture, the number of nodes and the number of bonds between them, this rigidity transition must have a geometric origin. Naive, degree-of-freedom-based mechanical analyses such as the Maxwell-Calladine count or the pebble game algorithm overlook such geometric rigidity transitions and offer no means of predicting or characterizing them. We apply tools that were developed for the topological analysis of zero modes and states of self-stress on regular lattices to two-dimensional random spring networks and demonstrate that the onset of rigidity, at a finite simple shear strain γ^{★}, coincides with the appearance of a single state of self-stress, accompanied by a single floppy mode. The process conserves the topologically invariant difference between the number of zero modes and the number of states of self-stress but imparts a finite shear modulus to the spring network. Beyond the critical shear, the network acquires a highly anisotropic elastic modulus, resisting further deformation most strongly in the direction of the rigidifying shear. We confirm previously reported critical scaling of the corresponding differential shear modulus. In the subcritical regime, a singular value decomposition of the networks compatibility matrix foreshadows the onset of rigidity by way of a continuously vanishing singular value corresponding to the nascent state of self-stress.

Mechanics from Calorimetry: Probing the Elasticity of Responsive Hydrogels

Temperature-sensitive hydrogels based on polymers such as poly(N-isopropylacrylamide) (PNIPAM) undergo a volume phase transition in response to changes in temperature. During this transition, distinct changes in both thermal and mechanical properties are observed. Here, we illustrate and exploit the inherent thermodynamic link between thermal and mechanical properties by showing that the compressive elastic modulus of PNIPAM hydrogels can be probed using differential scanning calorimetry. We validate our approach by using conventional osmotic compression tests. Our method could be particularly valuable for determining the mechanical response of thermosensitive submicron-sized and/or oddly shaped particles, to which standard methods are not readily applicable.

Self-consistent field lattice model for polymer networks

A lattice model based on polymer self-consistent field theory is developed to predict the equilibrium statistics of arbitrary polymer networks. For a given network topology, our approach uses moment propagators on a lattice to self-consistently construct the ensemble of polymer conformations and cross-link spatial probability distributions. Remarkably, the calculation can be performed “in the dark”, without any prior knowledge on preferred chain conformations or cross-link positions. Numerical results from the model for a test network exhibit close agreement with molecular dynamics simulations, including when the network is strongly sheared. Our model captures nonaffine deformation, mean-field monomer interactions, cross-link fluctuations, and finite extensibility of chains, yielding predictions that differ markedly from classical rubber elasticity theory for polymer networks. By examining polymer networks with different degrees of interconnectivity, we gain insight into cross-link entropy, an important quantity in the macroscopic behavior of gels and self-healing materials as they are deformed.

Mechanical properties of single supramolecular polymers from correlative AFM and fluorescence microscopy

We characterize the structure and mechanical properties of 1,3,5-benzenetricarboxamide (BTA) supramolecular polymers using correlative AFM and fluorescence imaging. AFM allows for nanoscale structural investigation but we found that statistical analysis is difficult because these structures are easily disrupted by the AFM tip. We therefore correlate AFM and fluorescence microscopy to couple nanoscale morphological information to far-field optical images. A fraction of the immobilized polymers are in a clustered or entangled state, which we identify based on diffraction limited fluorescence images. We find that clustered and entangled polymers exhibit a significantly longer persistence length that is broader distributed than single unentangled polymers. By comparison with numerical simulations we find significant heterogeneity in the persistence length of single unentangled polymers, which we attribute to polymer–substrate interactions and the presence of structural diversity within the polymer.