The vibrational density of states of a disordered gel model

Structure and elasticity of model disordered, polydisperse, and defect-free polymer networks

The elasticity of disordered and polydisperse polymer networks is a fundamental problem of soft matter physics that is still open. Here, we self-assemble polymer networks via simulations of a mixture of bivalent and tri- or tetravalent patchy particles, which result in an exponential strand length distribution analogous to that of experimental randomly cross-linked systems. After assembly, the network connectivity and topology are frozen and the resulting system is characterized. We find that the fractal structure of the network depends on the number density at which the assembly has been carried out, but that systems with the same mean valence and same assembly density have the same structural properties. Moreover, we compute the long-time limit of the mean-squared displacement, also known as the (squared) localization length, of the cross-links and of the middle monomers of the strands, showing that the dynamics of long strands is well described by the tube model. Finally, we find a relation connecting these two localization lengths at high density and connect the cross-link localization length to the shear modulus of the system.

Structural, mechanical, and vibrational properties of particulate physical gels

Our lives are surrounded by a rich assortment of disordered materials. In particular, glasses are well known as dense, amorphous materials, whereas gels exist in low-density, disordered states. Recent progress has provided a significant step forward in understanding the material properties of glasses, such as mechanical, vibrational, and transport properties. In contrast, our understanding of particulate physical gels is still highly limited. Here, using molecular dynamics simulations, we study a simple model of particulate physical gels, the Lennard-Jones (LJ) gels, and provide a comprehensive understanding of their structural, mechanical, and vibrational properties, all of which are markedly different from those of LJ glasses. First, the LJ gels show sparse, heterogeneous structures, and the length scale ξ_s of the structures grows as the density is lowered. Second, the LJ gels are extremely soft, with both shear G and bulk K moduli being orders of magnitude smaller than those of LJ glasses. Third, many low-frequency vibrational modes are excited, which form a characteristic plateau with the onset frequency ω_* in the vibrational density of states. Structural, mechanical, and vibrational properties, characterized by ξ_s, G, K, and ω_*, respectively, show power-law scaling behaviors with the density, which establishes a close relationship between them. Throughout this work, we also reveal that LJ gels are multiscale, solid-state materials: (i) homogeneous elastic bodies at long lengths, (ii) heterogeneous elastic bodies with fractal structures at intermediate lengths, and (iii) amorphous structural bodies at short lengths.

Normal modes of weak colloidal gels

The normal modes and relaxation rates of weak colloidal gels are investigated in calculations using different models of the hydrodynamic interactions between suspended particles. The relaxation spectrum is computed for freely draining, Rotne-Prager-Yamakawa, and accelerated Stokesian dynamics approximations of the hydrodynamic mobility in a normal mode analysis of a harmonic network representing several colloidal gels. We find that the density of states and spatial structure of the normal modes are fundamentally altered by long-ranged hydrodynamic coupling among the particles. Short-ranged coupling due to hydrodynamic lubrication affects only the relaxation rates of short-wavelength modes. Hydrodynamic models accounting for long-ranged coupling exhibit a microscopic relaxation rate for each normal mode, λ that scales as l^{-2}, where l is the spatial correlation length of the normal mode. For the freely draining approximation, which neglects long-ranged coupling, the microscopic relaxation rate scales as l^{-γ}, where γ varies between three and two with increasing particle volume fraction. A simple phenomenological model of the internal elastic response to normal mode fluctuations is developed, which shows that long-ranged hydrodynamic interactions play a central role in the viscoelasticity of the gel network. Dynamic simulations of hard spheres that gel in response to short-ranged depletion attractions are used to test the applicability of the density of states predictions. For particle concentrations up to 30% by volume, the power law decay of the relaxation modulus in simulations accounting for long-ranged hydrodynamic interactions agrees with predictions generated by the density of states of the corresponding harmonic networks as well as experimental measurements. For higher volume fractions, excluded volume interactions dominate the stress response, and the prediction from the harmonic network density of states fails. Analogous to the Zimm model in polymer physics, our results indicate that long-ranged hydrodynamic interactions play a crucial role in determining the microscopic dynamics and macroscopic properties of weak colloidal gels.

Vibrational and structural signatures of the crossover between dense glassy and sparse gel-like attractive colloidal packings

We investigate the vibrational modes of quasi-two-dimensional disordered colloidal packings of hard colloidal spheres with short-range attractions as a function of packing fraction. Certain properties of the vibrational density of states (vDOS) are shown to correlate with the density and structure of the samples (i.e., in sparsely versus densely packed samples). Specifically, a crossover from dense glassy to sparse gel-like states is suggested by an excess of phonon modes at low frequency and by a variation in the slope of the vDOS with frequency at low frequency. This change in phonon mode distribution is demonstrated to arise largely from localized vibrations that involve individual and/or small clusters of particles with few local bonds. Conventional order parameters and void statistics did not exhibit obvious gel-glass signatures as a function of volume fraction. These mode behaviors and accompanying structural insights offer a potentially new set of indicators for identification of glass-gel transitions and for assignment of gel-like versus glass-like character to a disordered solid material.

Gels of DNA nanostars never crystallize

Using state-of-the-art numerical techniques, we show that, upon lowering the temperature, tetravalent DNA nanostars form a thermodynamically stable, fully bonded equilibrium gel. In contrast to atomic and molecular network formers, in which the disordered liquid is always metastable with respect to some crystalline phase, we find that the DNA nanostar gel has a lower free energy than the diamond crystal structure in a wide range of concentrations. This unconventional behavior, here verified for the first time in a realistic model, arises from the large arm flexibility of the DNA nanostars, a property that can be tuned by design. Our results confirm the thermodynamic stability of the recently experimentally realized DNA hydrogels.