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Distinct elastic properties and their origins in glasses and gels


Glasses and gels, common amorphous solids with diverse applications, share intriguing similarities, including rigidity without translational order and dynamic slowing during ageing. However, their various underlying differences have not yet been explained. Here, through simulations, we elucidate distinct elastic properties related to temperature, observation times and ageing in glasses and gels, uncovering the underlying mechanisms. Configurational constraints, characterized by vibrational mean-squared displacements, similarly impact shear and bulk moduli in gels, but uniquely affect the shear modulus in glasses. As glasses age, a persistent trend of stiffening emerges, in contrast to gels, which initially stiffen and subsequently soften. We attribute these differences to mechanisms minimizing free energy: structural ordering in glasses and interface reduction in gels. Our findings not only reveal distinct behaviours but also shed light on the origin and evolution of elasticity in non-equilibrium disordered solids, with implications for amorphous material application and design.

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Fig. 1: Ageing dynamics and observation-time-dependent elasticity in glasses and gels.
Fig. 2: Age-dependent elasticity in glasses and gels.
Fig. 3: Role of inherent elasticity and MSD on thermal elasticity.
Fig. 4: Age-dependent potential energy and structure in glasses and gels.
Fig. 5: Illustration of the origin of elasticity and distinct ageing mechanisms in glasses and gels.

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The data that support the findings of this study are available from the corresponding author upon reasonable request.

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The codes used in this study are available from the corresponding author upon reasonable request.


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H.T. acknowledges the support by the Grant-in-Aid for Specially Promoted Research (JSPS KAKENHI Grant No. JP20H05619) from the Japan Society for the Promotion of Science (JSPS). M.T. acknowledges the support from JSPS KAKENHI (Grant No. JP20K14424). Y.W. acknowledges the support from Shanghai Jiao Tong University via the scholarship for outstanding PhD graduates. We thank the Supercomputer Center, the Institute for Solid State Physics, the University of Tokyo, for the use of the facilities.

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H.T. designed and supervised the project. Y.W. performed research. Y.W. and M.T. analysed data. All authors discussed the results and wrote the paper.

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Correspondence to Hajime Tanaka.

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Nature Physics thanks Raffaele Pastore, Lei Xu and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Snapshots of the gelation process.

Particles are colour-coded to differentiate those in the foreground from those in the background.

Extended Data Fig. 2 Self-intermediate scattering functions of glasses and gels.

a, Self-intermediate scattering function Fs(k, Δt, tw) for glasses at different ages (tw/τB = 9.6 × 101, 9.6 × 102, 9.6 × 103, 9.6 × 104, 9.6 × 105), depicted from left to right. Here, k is the peak wavenumber of the structural factor. b, Fs(k, Δt, tw) of all particles (dashed lines) and only isostatic particles (solid lines) for gels at different ages (tw/τB = 102, 103, 104, 105, 106), shown from left to right. The vertical dashed lines in a and b indicate the period of oscillatory deformation used in measuring the moduli of glasses and gels, respectively.

Extended Data Fig. 3 Ageing dynamics and observation-time-dependent elasticity in glasses with the Lennard-Jones potential (LJ glasses) and gels with the Wang-Frenkel potential (WF gels).

a, Illustration of the typical structure of an LJ glass. Particles are colour-coded based on their sizes. b, MSD \({\langle {\Delta} r^2 \rangle}\) versus observation time Δt for LJ glasses at different ageing time, tw. Here, the time unit is scaled by the Brownian time, τB. c, Observation-time-dependent shear modulus G and bulk modulus K of LJ glasses. Short bars indicate the affine moduli, GA and KA. d, Illustration of the typical structure of a WF gel. Particles are colour-coded based on their sizes. e, MSD of all particles \(\langle \Delta r^2 \rangle\) (dashed curves) and only isostatic particles \({\langle \Delta r^2 \rangle}_{\rm{iso}}\) (solid curves) versus Δt for WF gels at various tw. f, Observation-time-dependent shear modulus G and bulk modulus K of WF gels. Inset: the same data as the main panel with the vertical axis zoomed out to show the affine moduli GA and KA, as indicated by the short bars. The same plot as the Fig. 1 in the main text.

Extended Data Fig. 4 Amplitude and frequency sweep moduli in gels.

a, Shear stress σxy versus time tt under oscillatory shear strain ϵxy (solid blue curve), where Δt is the period time. The black points represent individual σxy, and the red curve represents the fitting results using \({\sigma }_{xy}={\sigma }_{0}\sin (\omega t+\delta )\). b, Strain amplitude γ sweep of storage shear modulus \({G}^{{\prime} }\) and loss shear modulus G. c, Frequency ω = 2π/(Δt/τB) sweep of \({G}^{{\prime} }\) and G. d, Pressure p and volumetric strain ϵb versus tt under oscillatory uniform compression ϵb (blue solid curve). The red curve represents the fitting results using \(p={p}_{0}\sin (\omega t+\delta )+{p}_{{{{\rm{c}}}}}\), where pc is the initial pressure of the system. e, Volume strain amplitude γb sweep of storage bulk modulus \({K}^{{\prime} }\) and loss bulk modulus K. f, Frequency ω = 2π/(Δt/τB) sweep of \({K}^{{\prime} }\) and K. The vertical dashed lines mark the amplitude and frequency used to measure the moduli in Fig. 2d–f. In this work, G and K refer specifically to the storage moduli \({G}^{{\prime} }\) and \({K}^{{\prime} }\).

Extended Data Fig. 5 Schematic representation of the observation-time (Δt) or frequency (ω)-dependent shear modulus.

The levels of affine (GA) and plateau shear moduli (Gp) are indicated on the left axis.

Extended Data Fig. 6 Age-dependent elasticity in LJ glasses and WF gels.

The age-dependent thermal elasticity G (a,d), K (b,e), modulus ratio G/K and Poisson ratio ν (c,f) (red circles) and corresponding inherent elasticity at zero temperature (blue squares) in LJ glasses (ac) and WF gels (d–f). The observation time Δt used to measure these moduli are marked by the vertical dashed lines in Extended Data Fig. 3. The same plot as the Fig. 2 in the main text.

Extended Data Fig. 7 Scaled moduli versus mean-squared displacement (MSD) in glasses and gels.

a, Shear modulus G scaled by the affine modulus GA (infinite-frequency modulus) versus the inverse of MSD \({\langle \Delta {r}^{2}\rangle }^{-1}\) in glasses. b, Shear modulus G scaled by GA versus the inverse of isostatic MSD \({\langle \Delta {r}^{2}\rangle }_{{{{\rm{iso}}}}}^{-1}\) in gels. c, Scaled shear modulus G/GIS versus MSD \({\langle \Delta {r}^{2}\rangle }_{{{{\rm{rel}}}}}^{-1}\) in glasses. d, Scaled shear modulus G/GIS and bulk modulus K/KIS versus MSD of isostatic particles \({\langle \Delta {r}^{2}\rangle }_{{{{\rm{iso}}}},{{{\rm{rel}}}}}^{-1}\) in gels. e, Scaled shear modulus G/GIS and bulk modulus K/KIS versus original MSD of all particles \({\langle \Delta {r}^{2}\rangle }^{-1}\) in gels.

Extended Data Fig. 8 Role of inherent elasticity and MSD on thermal elasticity in the LJ glasses and WF gels.

a, Inverse MSD \(\langle \Delta r^2\rangle^{-1}\) of LJ glasses at different observation times Δt plotted against scaled waiting time twB. b, Ratios between thermal elasticity and inherent elasticity M/MIS of LJ glasses versus twB for different Δt. c, Scaled shear modulus G/GIS of LJ glasses versus \(\langle \Delta r^2\rangle^{-1}\) for different Δt. d, Inverse MSD of isostatic particles \((Z\geq6)\) \({\langle \Delta r^2\rangle}^{-1}_{\rm{iso}}\) of WF gels versus twB for different Δt. e,f, G/GIS and K/KIS of WF gels versus twB (e) and \({\langle \Delta r^2\rangle}^{-1}_{\rm{iso}}\) (f) for different Δt. The same plot as the Fig. 3 in the main text.

Extended Data Fig. 9 Age-dependent structure of glasses in inherent states.

a, Orientational order parameter Ω and cage anisotropy parameter η. b, Pressure p. Note that the scales of all vertical axes are the same as those shown for the corresponding structural parameters in Fig. 4 in the main text.

Extended Data Fig. 10 Age-dependent potential energy and structure in the LJ glasses and WF gels.

a, Age-dependent potential energy per particle (red open circles), E, and per contact (blue open squares), 2E/Z, in LJ glasses. b, Age-dependent orientational order parameter and Ω anisotropy parameter η of Voronoi cells in LJ glasses, where Ω = 0 indicates perfect order and η = 1 indicates isotropic cells. c, Age-dependent pressure p in LJ glasses. d, The same as in a, for WF gels. e, Age-dependent contact number Z in WF gels. f, Age-dependent loop number Nloop in WF gels. The same plot as the Fig. 4 in the main text.

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Wang, Y., Tateno, M. & Tanaka, H. Distinct elastic properties and their origins in glasses and gels. Nat. Phys. (2024).

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