Session P07: Origin of Rigidity and the Nature of the Yielding Transition in Solids

Live

It is known that if local atomic rearrangements leading to exchange of neighbours are allowed, no crystalline solid can be rigid in the thermodynamic limit. We show that crystals loose rigidity as a consequence of a symmetry breaking, first order, phase transition at zero deformation [1]. The usual rigid, elastic, response to shape changes, a defining character of a crystalline solid, occurs because the time needed for such rearrangements diverge as the magnitude of deformation approaches zero. This picture gives us a fundamentally new viewpoint on the phenomenon of yielding, i.e. the loss of rigidity of a crystal when deformed beyond a limit, viz. the yield point. The phenomenon of yielding is now simply the nucleation of bubbles of the thermodynamically stable phase within the metastable, rigid crystal. An outcome of this theory is that the yield point is always a weak function of the rate of deformation and vanishes in the quasistatic and thermodynamic limits. The analytic form derived by us for the yield point as a function of the rate of deformation is able to explain experimental data over 15 orders of magnitudes in time [2]. Finite size effects, that are substantial, are also shown to be responsible for creating the illusion of a non-zero quasistatic yield point in small crystals. Finally these results can be understood within a Ginzburg Landau theory where lattice dislocations arise naturally at interfaces between rigid and stress free states.

1] P. Nath et al., Proc. Natl. Acad. Sci. USA 115, E4322 (2018).
[2] V. S. Reddy et al. Phys. Rev. Lett. 124, 025503 (2020).   

Live

Simulations of dense, athermal assemblies of self-propelled soft particles with infinite persistence time display intriguing mechanical properties as a function of the strength of the active propulsion force and the packing fraction of the system. Applying active propulsion to an initial, passively-jammed state results in unjamming and subsequent flow of the system. We find that there is a threshold force above which the system fluidizes indefinitely but below which the system experiences dynamical arrest, and that this threshold force increases as the density of the system is increased.

We analyze various mechanical and network properties in the final jammed states as well as over time series of such systems to illuminate the differences between packings formed with passive and active constituents, as well as understand the underlying mechanisms of dynamical arrest in this extreme active limit. We also investigate the scaling behavior of system quantities such as the contact force distribution and pair correlation function in the jammed states, appropriately accounting for the active force.   

Live

Colloidal suspensions can undergo ergodic to nonergodic transitions that depend on the colloidal concentration, temperature, or other conditions. The transitions are typically characterized by localization of the colloids and the emergence of macroscopic elasticity. However, the relationship between these phenomena at very different length scales is not fully clear. We report a study employing x-ray photon correlation spectroscopy (XPCS) and in situ rheometry to investigate the microscopic dynamics and rheology of Laponite suspensions, composed of nanoscale discoidal colloids with concentrations from 3.25 to 3.75 wt%, which evolve over time from a fluid to a soft glass that displays aging behavior. The XPCS characterizes the particle localization during formation and aging of the soft-glass state. The fraction of localized particles f₀ increases rapidly during the early formation stage and more slowly during subsequent aging, while the localization length r_loc steadily decreases. Despite the strongly varying rates of aging at different concentrations, both f₀ and r_loc scale with the elastic shear modulus G’ in a manner independent of concentration. In the later aging stage, the scaling between r_loc and G’ agrees quantitatively with a prediction of naive mode coupling theory.   

Live

We have developed a method to quantify the dissipation and stiffness (viscoelasticity) of nanoscale interactions in aqueous environment. Experiments were performed using two cantilever-deflection detection schemes and displacement-detection using a fibre interferometer-based home-built setup. Cantilever is excited using two mechanisms: base and tip excitation. A single protein molecule (Titin-I27), tethered between cantilever-tip and substrate, is pulled with constant speed (~ 20 nm/s). Cantilever-tip is driven with low frequency (~ 133 Hz), amplitude (~ 1Å) in the experiment, and phase and amplitude are recorded. We analyzed the data using existing theoretical models and also a model proposed by us. Pairing different excitation and detection schemes; and analyzing the data with different models, a robust study of mechanical properties of I27 molecule was made. Our observed results show inconsistencies with previous studies. We believe it is due to the inappropriate selection of experimental parameters and theoretical models. This leads artefacts in the results. We propose the appropriate ways of performing measurements and theoretical model for data analysis which accurately quantify the dissipation and stiffness of nanoscale interactions in aqueous environment.   

Live

Clay is one the most important materials in earth's crust and has wide applications in geotechnical, environmental, and biomedical engineering. It has complex mechanical properties due to its particulate nature and complex physico-chemical interactions between primary particles. In this presentation, a multiscale framework will be presented, linking the mechanical properties of illite from atomistic up to macroscopic scales. Illite is a typical type of clay with flexible plate-like particles. The framework includes the study of inter-particle interaction at atomistic scale through free energy perturbation calculations with molecular dynamics simulations, from which the potential of mean force will be used to calibrate the coarse-grained force field to be used in meso-scale simulations. The modes of deformation and mechanical response of the mesoscopic systems are incorporated into a thermodynamically consistent constitutive model that can be used to simulate clay behaviour at macroscopic level. The simulation results will be compared with experimental results on the same material. The framework can provide good guidance on similar multiscale study on physico-chemical properties of complex materials.   

Live

From jammed packings to biological tissues, rigidity plays a crucial role in the integrity and functionality of systems. Typically, Maxwell constraint counting (CC) is used to predict whether a system will be rigid or not based on how constrained its individual components are; however, there are cases where CC fails. For instance, epithelial tissues have been known to exhibit fluid-solid transitions in development or due to underlying disease. Similarly, biopolymer networks can show rigidity transitions under applied strain. In both cases, the transition can happen without any changes to the number of constraints, demonstrating the limitations of CC. Using energy expansion, we show that rigidity is an inherently nonlinear phenomenon and that CC is only guaranteed to work for specific systems. We introduce the framework of second order rigidity to classify systems based on their internal prestresses and make general predictions about their rigidity. We show for example that prestresses can rigidify under-constrained systems such as spring networks and vertex models of epithelia. This formalism unifies our understanding of the origin of rigidity and can be useful in material design by predicting the stability of rigid systems under external loads.   

Live

Periodic boundary conditions are used in studies of jamming to lessen finite size effects. However, the use of these boundaries can lead to strange and unintuitive results. We show that when critically jammed sphere packings are replicated they unjam through the creation of new zero modes. In fact, systems with fewer than the dimension, d, states of self-stress will always unjam upon replication. We draw links between this result and the constraints needed for shear stability, as well as the Guest-Hutchinson modes found in crystals.   

Live

The nonaffine lattice dynamics framework was developed only for small deformations and T=0. The introduction of the instantaneous normal modes (INM) extended the theory to finite T. We propose to extend the theory to large deformations by constructing the INM of deformed states (γINM) using the instantaneous Affine Transform (AT) from the non-deformed state (γ = 0) to the deformed state as a proxy for non-equilibrated protocol that generates the γINMs. We compare our theoretical results to those obtained from athermal quasistatic simulations (AQS). The increase of γ has a similar effect on VDOS of γINM as the increase of the temperature on the INM, thus leading to a similar softening of the material. To directly test if the softening occurs, we calculate the shear modulus and use it to reconstruct the stress-strain, which for our choice of (γINM) shows a significant drop of the stress around 10% deformation, in agreement with AQS.
This may open up the possibility to provide predictions of stress-strain behaviour and yielding using only snapshots of underformed samples as input, since the construction of γINM is a purely analytical procedure.   

Live

The amount of energy transferred from a throwing device to a homogeneous elastic projectile can reach 250 % the kinetic energy of a rigid projectile for the right tuning between the timescale of the thrower acceleration and the eigenperiod of the projectile [1].
Here we show that the kinetic energy of a rigid projectile is greatly increased thanks to a geometrical change: the addition of an intermediate soft layer between the throwing device and the projectile [2]. Experiments have been performed with a centimeter long rigid polymer and a soft hydrogel whose length has been varied. The throwing device has a harmonic motion and an initial acceleration of about 10 times standard gravity.
Simulations and analytical models based on the resolution of the wave equation in the soft part show that the optimal energy transfer emerges for the right time tuning between the throwing device and the projectile as well as for the right proportion of soft material. From the measurements we show that the rigid projectile can obtain 400 % the kinetic energy it gets without a soft layer.

[1] Raufaste et al. , Phys. Rev. Lett. 2017
[2] Celestini et al. , Phys. Rev. Appl. 2020   

Live

Recently, a Monte Carlo simulation study [1] has shown that yielding in crystalline solids is associated with an underlying quasi-static first-order phase transition. As a consequence, in the limit of a deformation with sufficiently low strain rate, the rigid response of a crystal due to a shape change of its boundaries corresponds to a metastable state that transforms to a stable state where internal stresses are eliminated, maintaining the crystalline order. A nucleation theory based on these findings predicts the yield point as a function of strain rate and shows agreement with data from experiment and molecular dynamics (MD) simulations over 15 orders of magnitude [2]. In the case of amorphous solids (glasses), a MD simulation study, using an athermal quasi-static (aqs) deformation protocol [3], have found a sharp stress drop in the stress-strain relation that marks the transition from an elastic response of the glass to plastic flow. In Ref. [3], this stress drop has been interpreted as a non equilibrium first-order transition, leading to the occurrence of shear localization, i.e. shear banding, after the drop. Using MD simulations of glassforming model systems under shear, we study the conditions for the occurrence of shear bands at finite temperatures [4] and discuss the response of glasses to an external shear in the limit of zero strain rate. We argue that there is no quasi-static first-order transition as in the case of crystalline solids.

[1] P. Nath et al., Proc. Natl. Acad. Sci. USA 115, E4322 (2018).
[2] V. S. Reddy et al., Phys. Rev. Lett. 124, 025503 (2020).
[3] M. Ozawa et al., Proc. Natl. Acad. Sci. USA 115, 6656 (2018).
[4] M. Golkia et al., Phys. Rev. E 102, 023002 (2020).   

Not Participating

We report the results of an experimental investigation of the motion of a solid particle through a collection of hydrogel particles in water. The volume of fluid is kept fixed in each run of the experiment and the number of particles is changed to set the volume fraction. Fluid-like behaviour is found at small volume fractions and visco-elastic properties are found at higher particle concentrations. Slowing of the motion, including divergence in the propagation speed is found at intermediate volume fractions. This is accompanied by an intriguing linear speed dependence over a small range of volume fractions.