Abstract
Soft solids with tunable mechanical response are at the core of new material technologies, but a crucial limit for applications is their progressive aging over time, which dramatically affects their functionalities. The generally accepted paradigm is that such aging is gradual and its origin is in slower than exponential microscopic dynamics, akin to the ones in supercooled liquids or glasses. Nevertheless, time- and space-resolved measurements have provided contrasting evidence: dynamics faster than exponential, intermittency and abrupt structural changes. Here we use 3D computer simulations of a microscopic model to reveal that the timescales governing stress relaxation, respectively, through thermal fluctuations and elastic recovery are key for the aging dynamics. When thermal fluctuations are too weak, stress heterogeneities frozen-in upon solidification can still partially relax through elastically driven fluctuations. Such fluctuations are intermittent, because of strong correlations that persist over the timescale of experiments or simulations, leading to faster than exponential dynamics.
Similar content being viewed by others
Introduction
Soft condensed matter (proteins, colloids or polymers) easily self-assembles into gels or amorphous soft solids with diverse structure and mechanics1,2,3,4,5. The nanoscale size of particles or aggregates makes these solids sensitive to thermal fluctuations, with spontaneous and thermally activated processes leading to rich microscopic dynamics. Besides affecting the time evolution, or aging, of the material properties at rest, such dynamics interplay with an imposed deformation and are hence crucial also for the mechanical response6,7,8,9,10. In the material at rest, thermal fluctuations can trigger ruptures of parts of the microscopic structure that are under tension or can promote coarsening and compaction of initially loosely packed domains, depending on the conditions under which the material initially solidified. The diffusion of particles or aggregates following such local reorganizations (often referred to as ‘micro-collapses’) is governed by a wide distribution of relaxation times, due to the disordered tortuous environment in which it takes place, that is, the microstructure of a soft solid, be it a thin fractal gel or a densely packed emulsion11,12. Therefore, localization, caging and larger-scale cooperative rearrangements in such complex environment will lead to slow, subdiffusive dynamics, similar to the microscopic dynamics close to a glass transition13,14.
Several quasi-elastic scattering experiments or numerical simulations that can access micro- and nanoscale rearrangements in soft matter have indeed confirmed that the dynamics in aging soft solids are slower than exponential. Measuring the time decay of the correlations of the density fluctuations or of the local displacements of particles or aggregates, the experiments find that time correlations follow a stretched exponential decay 
with β<1, akin to the slow cooperative dynamics of supercooled liquids and glasses, in a wide range of gels and other soft solids15,16,17,18,19. Nevertheless, in the past few years there has been emerging evidence, through a growing body of similar experiments, that microscopic dynamics in the aging of soft materials can be instead faster than exponential. In a wide range of soft amorphous solids including colloidal gels, biopolymer networks, foams and densely packed microgels, time correlations measured via quasi-elastic scattering or other time-resolved spectroscopy techniques appear to decay as 
with β>1 (and in most cases 1.3≤β≤1.5)20,21,22,23,24,25,26,27,28,29,33. Therefore, we expect our general picture to have a much wider relevance. Second, the physical mechanisms we propose help rationalize several experimental observations in very different materials, ranging from biologically relevant soft solids to metallic glasses31,45,46,50. In a quasi-equilibrium scenario, enthalpic and thermal degrees of freedom may still couple and stress correlations decay relatively fast. When the material is deeply quenched and jammed, instead, recovering the coupling between the distinct degrees of freedom and restoring equilibrium will require timescales well beyond the ones accessible in typical experiments or simulations. The result will be intermittent dynamics and compressed exponential relaxations. The competition between Brownian motion and elastic effects through the relaxation of internal stresses illustrated in this work suggests different scenarios for the energy landscape underlying the aging of soft jammed materials. When thermal fluctuations screen the long-range elastic strain transmission, microscopic rearrangements may open paths to deeper and deeper local minima in a rugged energy landscape. Compressed exponential dynamics, instead, evoke the presence of flat regions and huge barriers, with the possibility of intermittent dynamics, abrupt rearrangements and avalanches34,35. Investigating how such different dynamical processes couple with imposed deformations will provide a new rationale, and have important implications, for designing mechanics, rheology and material performances.
Methods
Numerical model and viscoelastic parameters
The particles in the model gel interact through a potential composed of two terms. The first force contribution derives from a Lennard–Jones-like potential of the form:
The second contribution confers an angular rigidity to the inter particles bonds and takes the form:
The strength of the interaction is controlled by Λ(r) and vanishes over two particles diameters:
where Θ denotes the Heaviside function. The evolution of the gel over time is obtained by solving the following Langevin equation for each particle:
where σ is the particle diameter, ξ(t) is a random white noise that models the thermal fluctuations and is related to the friction coefficient ηf by means of its variance 
. To be in the overdamped limit of the dynamics ηf is set to 10, and the timestep δt used for the numerical integration is δt=0.005. The parameters of the potential are chosen such that the disordered thin percolating network starts to self assemble at kBT/∈ = 0.05. One convenient choice to achieve this configuration is given by this set of parameters: A=6.27, a=0.85, B=67.27, θ=65° and w=0.3. The system is composed of N=62,500 particles in a cubic simulation box of a size L=76.43σ with periodic boundary conditions, the number density N/L3 is fixed at 0.14, which corresponds approximatively to a volume fraction of 7%. All initial gel configurations are the same, prepared with the protocol described in ref. 40, which consists in starting from a gas configuration (kBT/∈ = 0.5) and letting the gel self-assemble upon slow cooling down to kBT/∈ = 0.05. We then quench this gel configuration by running a simulation with the dissipative dynamics 
until the kinetic energy drops to zero(10−24). All simulations have been performed using a version of LAMMPS suitably modified by us52.
Stress calculation and cutting strategy
We let the initial gel configuration evolve with equation (4) for each of the different values of kBT/∈ considered here, while using the following procedure to cut network connections. At each timestep, we characterize the state of stress of a gel configuration by computing the virial stresses as 
, where the Greek subscripts stand for the Cartesian components x, y, z and 
represents the contribution to the stress tensor of all the interactions involving the particle i and V is the total volume of the simulation box52. 
contains, for each particle, the contributions of the two-body and the three-body forces evenly distributed among the particles that participate in them:
The first term on the RHS denotes the contribution of the two-body interaction, where the sum runs over all the N2 pair of interactions that involve the particle i. (ri, F i) and (r′, F′) denote, respectively, the position and the forces on the two interacting particles. The second term indicates the three-body interactions involving the particle i. We consider a coarse-graining volume Ωcg centered around the point of interest r and containing around 9–10 particles on average, and define the local coarse-grained stress based on the per-particle virial contribution as 
. For a typical starting configuration of the gel, the local normal stress 
reflect the heterogeneity of the structure and tend to be higher around the nodes, due to the topological frustration of the network. We consider that breaking of network connections underlying the aging of the gel is more prone to happen in the regions where local stresses tend to be higher, as found also in refs 40, 41. Hence, to mimic the aging in the molecular dynamics simulations we scan the whole structure of the gel and remove one of the bonds (by turning off the well in 
) whose contribution to the local normal stress 
is the largest (prevalently bonds between particles belonging to the network nodes). As the simulation proceeds, local internal stresses redistribute in the aging structure of the gel and the locations of more probable connection rupture (as well as their number) change over time. All simulations discussed here have been performed with a rate 
, corresponding to removing only ∼5% of the total network connections over the whole simulation time window. We have run several tests varying the removal rate in order to verify that changing the rate (having kept all other parameters constant) does not modify our outcomes and the emerging physical picture. Overall, varying Γ over nearly two orders of magnitudes, we recover the same results, as long as the τr=1/Γ between two rupture events allows for at least partial stress relaxation (see Fig. 1).
Stress autocorrelation function
The autocorrelation function of the stress fluctuations 
is computed as follows: 
, where fδσ is the auto-covariance and takes the form 
and 
. We have computed the autocorrelation function of the stress fluctuations over partial time series and over the whole duration of the simulations (see Fig. 5 and Supplementary Fig. 2).
Fitting procedure for the intermediate scattering functions
To extract the β exponent and the structural relaxation time τq, we fit the last decay of the coherent scattering function F(q,t)tw=0 simultaneously with the incoherent scattering function 
, defined as 
. This observable is less noisy and has the same exponent as the coherent scattering function shown in Fig. 3 (see also Supplementary Fig. 3). Having fitted Fs(q,t) with a stretched exponential type of the form 
and having extracted the exponent β, such exponent is used to initiate the fit of F(q,t) by locking this parameter and letting all the others free. This procedure helps us to improve the quality of the fit used to extract the relaxation time τq from the coherent scattering.
Data availability
The authors declare that all data supporting the findings of this study are available within the article and its Supplementary Information Files. All raw data can be accessed on request to the authors.
Additional information
How to cite this article: Bouzid, M. et al. Elastically driven intermittent microscopic dynamics in soft solids. Nat. Commun. 8, 15846 doi: 10.1038/ncomms15846 (2017).
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
References
Trappe, V. et al. Jamming phase diagram for attractive particles. Nature 411, 772 (2001).
Storm, C. et al. Nonlinear elasticity in biological gels. Nature 435, 191–194 (2005).
Helgeson, M. E., Moran, S. E., An, H. Z. & Doyle, P. S. Mesoporous organohydrogels from thermogelling photocrosslinkable nanoemulsions. Nat. Mater. 11, 344–352 (2012).
Peter, J. et al. Gelation of particles with short-range attraction. Nature 453, 499 (2008).
Gibaud, T. et al. Unexpected decoupling of stretching and bending modes in protein gels. Phys. Rev. Lett. 110, 058303 (2013).
Perge, C., Taberlet, N., Gibaud, T. & Manneville, S. Time dependence in large amplitude oscillatory shear: a rheo-ultrasonic study of fatigue dynamics in a colloidal gel. J. Rheol. 58, 1331–1357 (2014).
Hsiao, L. C., Newman, R. S., Glotzer, S. C. & Solomon, M. J. Role of isostaticity and load-bearing microstructure in the elasticity of yielded colloidal gels. Proc. Natl Acad. Sci. 109, 16029–16034 (2012).
Conrad, J. C. et al. Arrested fluid-fluid phase separation in depletion systems: Implications of the characteristic length on gel formation and rheology. J. Rheol. 54, 421–438 (2010).
Rajaram, B. & Mohraz, A. Microstructural response of dilute colloidal gels to nonlinear shear deformation. Soft Matter 6, 2246–2259 (2010).
Manley, S. et al. Time-dependent strength of colloidal gels. Phys. Rev. Lett. 95, 048302 (2005).
de Gennes, P. G. La percolation: un concept unificateur. Recherche 7, 919–927 (1976).
Bouchaud, J.-P. & Georges, A. Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys. Rep. 195, 127–293 (1990).
Binder, K. & Kob., W. Glassy Materials and Disordered Solids: An Introduction to Their Statistical Mechanics World Scientific (2005).
Berthier, L. et al. Dynamical Heterogeneities in Glasses, Colloids, and Granular Media Oxford University Press (2011).
Segrè, P. N., Prasad, V., Schofield, A. B. & Weitz, D. A. Glasslike kinetic arrest at the colloidal-gelation transition. Phys. Rev. Lett. 86, 6042–6045 (2001).
Manley, S. et al. Glasslike arrest in spinodal decomposition as a route to colloidal gelation. Phys. Rev. Lett. 95, 238302 (2005).
Bandyopadhyay, R. et al. Evolution of particle-scale dynamics in an aging clay suspension. Phys. Rev. Lett. 93, 228302 (2004).
Jabbari-Farouji, S., Wegdam, G. H. & Bonn, D. Gels and glasses in a single system: evidence for an intricate free-energy landscape of glassy materials. Phys. Rev. Lett. 99, 065701 (2007).
Krall, A. H. & Weitz, D. A. Internal dynamics and elasticity of fractal colloidal gels. Phys. Rev. Lett. 80, 778 (1998).
Cipelletti, L., Manley, S., Ball, R. C. & Weitz, D. A. Universal aging features in the restructuring of fractal colloidal gels. Phys. Rev. Lett. 84, 2275–2278 (2000).
Ramos, L. & Cipelletti, L. Ultraslow dynamics and stress relaxation in the aging of a soft glassy system. Phys. Rev. Lett. 87, 245503 (2001).
Bellour, M. et al. Aging processes and scale dependence in soft glassy colloidal suspensions. Phys. Rev. E 67, 031405 (2003).
Angelini, R. et al. Dichotomic aging behaviour in a colloidal glass. Soft Matter 9, 10955–10959 (2013).
Mansel, B. W. & Williams, M. A. K. Internal stress drives slow glassy dynamics and quake-like behaviour in ionotropic pectin gels. Soft Matter 11, 7016–7023 (2015).
Conrad, H. et al. Correlated heterogeneous dynamics in glass-forming polymers. Phys. Rev. E 91, 042309 (2015).
Ruta, B. et al. Silica nanoparticles as tracers of the gelation dynamics of a natural biopolymer physical gel. Soft Matter 10, 4547–4554 (2014).
Schosseler, F., Kaloun, S., Skouri, M. & Munch, J. P. Diagram of the aging dynamics in laponite suspensions at low ionic strength. Phys. Rev. E 73, 021401 (2006).
Guo, H., Ramakrishnan, S., Harden, J. L. & Leheny, R. L. Gel formation and aging in weakly attractive nanocolloid suspensions at intermediate concentrations. J. Chem. Phys. 135, 154903 (2011).
Chung, B. et al. Microscopic dynamics of recovery in sheared depletion gels. Phys. Rev. Lett. 96, 228301 (2006).
Chaudhuri, P. & Berthier, L. Ultra-long-range dynamic correlations in a microscopic model for aging gels. ar**v:1605.09770.
Duri, A. & Cipelletti, L. Length scale dependence of dynamical heterogeneity in a colloidal fractal gel. Europhys. Lett. 76, 972 (2006).
Cipelletti, L. et al. Universal non-diffusive slow dynamics in aging soft matter. Faraday Discuss. 123, 237 (2003).
Bouchaud, J. P. & Pitard, E. Anomalous dynamical light scattering in soft glassy gels. Eur. Phys. J. E 6, 231–236 (2001).
Bouchaud, J.-P. Anomalous relaxation in complex systems: from stretched to compressed exponentials anomalous transport: foundations and applications, 327–345 (2008).
Ferrero, E. E., Martens, K. & Barrat, J.-L. Relaxation in yield stress systems through elastically interacting activated events. Phys. Rev. Lett. 113, 248301 (2014).
Godec, A., Bauer, M. & Metzler, R. Collective dynamics effect transient subdiffusion of inert tracers in flexible gel networks. New J. Phys. 16, 092002 (2014).
Dinsmore, A. D., Prasad, V., Wong, I. Y. & Weitz, D. A. Microscopic structure and elasticity of weakly aggregated colloidal gels. Phys. Rev. Lett. 96, 185502 (2006).
Dibble, C. J., Kogan, M. & Solomon, M. J. Structural origins of dynamical heterogeneity in colloidal gels. Phys. Rev. E 77, 050401 (2008).
Colombo, J. & Del Gado, E. Self-assembly and cooperative dynamics of a model colloidal gel network. Soft Matter 10, 4003–4015 (2014).
Colombo, J. & Del Gado, E. Stress localization, stiffening, and yielding in a model colloidal gel. J. Rheol. 58, 1089–1116 (2014).
Colombo, J., Widmer-Cooper, A. & Del Gado, E. Microscopic picture of cooperative processes in restructuring gel networks. Phys. Rev. Lett. 110, 198301 (2013).
Irving, J. H. & Kirkwood, J. G. The statistical mechanical theory of transport processes. iv. the equations of hydrodynamics. J. Chem. Phys. 18, 817–829 (1950).
Del Gado, E. & Kob, W. Length-scale-dependent relaxation in colloidal gels. Phys. Rev. Lett. 98, 28303 (2007).
Saw, S., Ellegaard, N. L., Kob, W. & Sastry, S. Structural relaxation of a gel modeled by three body interactions. Phys. Rev. Lett. 103, 248305 (2009).
Lieleg, O. et al. Slow dynamics and internal stress relaxation in bundled cytoskeletal networks. Nat. Mater. 10, 236 (2011).
Maccarrone, S. et al. Ultra-long range correlations of the dynamics of jammed soft matter. Soft Matter 6, 5514–5522 (2010).
Chaudhuri, P., Berthier, L. & Kob, W. Universal nature of particle displacements close to glass and jamming transitions. Phys. Rev. Lett. 99, 060604 (2007).
Ciamarra, M. P., Pastore, R. & Coniglio, A. Particle jumps in structural glasses. Soft Matter 12, 358–366 (2015).
Landau, Lev D & Lifshitz, E. M. in Theory of Elasticity (Course of Theoretical Physics) 3rd edn, Vol. 7, 109 (Butterworth-Heinemann, 1986).
Ruta, B. et al. Atomic-scale relaxation dynamics and aging in a metallic glass probed by x-ray photon correlation spectroscopy. Phys. Rev. Lett. 109, 165701 (2012).
Plimpton, S. Fast parallel algorithms for short-range molecular dynamics. J. Comp. Phys. 117, 1–19 (1995).
Thompson, A. P., Plimpton, S. J. & Mattson, W. General formulation of pressure and stress tensor for arbitrary many-body interaction potentials under periodic boundary conditions. J. Chem. Phys. 131, 154107 (2009).
Acknowledgements
The authors thank Jean-Louis Barrat, Ludovic Berthier, Luca Cipelletti, Thibaut Divoux, Ezequiel Ferrero, Kirsten Martens, Peter D. Olmsted and Veronique Trappe for insightful discussions. L.V.B. was supported from CAPES Foundation, Ministry of Education of Brazil (Proc. N 88888.059093/2013-00). E.D.G. acknowledges support from the Swiss National Science Foundation (Grant No. PP00P2 150738), the National Science Foundation (Grant No. NSF PHY11-25915). All authors thank Georgetown University.
Author information
Authors and Affiliations
Contributions
M.B., J.C. and E.D.G. designed the research; M.B. performed the research and analysed the data; J.C. and M.B. developed analytical tools; L.V.B. developed visualization tools. M.B. and E.D.G. wrote the paper.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
About this article
Cite this article
Bouzid, M., Colombo, J., Barbosa, L. et al. Elastically driven intermittent microscopic dynamics in soft solids. Nat Commun 8, 15846 (2017). https://doi.org/10.1038/ncomms15846
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/ncomms15846
- Springer Nature Limited