Microscopic dynamics and failure precursors of a gel under mechanical load

Significance

The sudden, catastrophic failure of materials under a constant mechanical load is widespread, occurring from geological scales, as in earthquakes, to biological and soft-matter systems, as in protein, polymer, and colloidal gels. Failure is often preceded by a long induction time, with little if any macroscopic signs of weakening, making it unpredictable. Here, we investigate the failure of a model colloidal gel under a constant load, measuring simultaneously its mechanical response and microscopic dynamics. We show that the gel undergoes a burst of microscopic plastic rearrangements that largely precede failure, undetectable by monitoring conventional structural quantities. The notion of a dynamic precursor thus emerges as a powerful tool to understand and predict the sudden failure of solids.

Abstract

Material failure is ubiquitous, with implications from geology to everyday life and material science. It often involves sudden, unpredictable events, with little or no macroscopically detectable precursors. A deeper understanding of the microscopic mechanisms eventually leading to failure is clearly required, but experiments remain scarce. Here, we show that the microscopic dynamics of a colloidal gel, a model network-forming system, exhibit dramatic changes that precede its macroscopic failure by thousands of seconds. Using an original setup coupling light scattering and rheology, we simultaneously measure the macroscopic deformation and the microscopic dynamics of the gel, while applying a constant shear stress. We show that the network failure is preceded by qualitative and quantitative changes of the dynamics, from reversible particle displacements to a burst of irreversible plastic rearrangements.

Material failure is ubiquitous on length scales ranging from a few nanometers, as in fracture of atomic or molecular systems (1, 2), up to geological scales, as in earthquakes (3, 4). While some attempts have been made to harness failure—for example, to produce new materials with a well-controlled patterning (5)—material failure remains in general an unwanted, uncontrolled, and unpredictable process, widely studied since the pioneering experiments on metallic wires by Leonardo da Vinci in the 15th century (6). Indeed, a better control of the conditions under which material failure may or may not occur and the detection of any precursors that may point to incipient failure are the Holy Grail in many disciplines, from material science (7–10) to biology (11, 12), engineering, and geology (13–16). Failure may occur almost instantaneously, as a consequence of an impulsive load. Often, however, it manifests itself in more elusive ways, as in the sudden, catastrophic breakage of a material submitted to a constant load, where failure may be preceded by a long induction time with little if any precursor signs of weakening. Such delayed failure has been reported in a wide spectrum of phenomena, from earthquakes (17), snow avalanches (18), and failure in biomaterials (11, 12) to the sudden yielding of crystalline (1) solids, composite materials (10, 19), and amorphous systems (20), including viscoelastic soft materials (21, 22), such as adhesives (23) and network-forming materials (24–28).

Delayed failure typically involves creep during the induction time, the sublinear (e.g., power law) increase of sample deformation under a constant load. The microscopic origin of creep is well understood for crystalline solids, where it is attributed to defect motion (29, 30). Power-law creep is also widespread in amorphous materials, but its microscopic origin remains controversial: It has been attributed to the accumulation of irreversible, plastic rearrangements (19, 21, 22, 31, 32), to linear viscoelasticity (12, 28, 33), or to a combination of both (34), with different authors holding contrasting views on similar systems (19, 34). Crucially, a detailed understanding of the creep regime holds the promise of unveiling the origin of the sudden failure of the material, potentially revealing any precursor signs of failure, which are difficult to detect by monitoring macroscopic quantities, such as the deformation rate (10), or mesoscopic, coarse-grained shear velocity maps (28). Clearly, investigations of the evolution of the microscopic structure and dynamics under creep are required, which are however very scarce to date and essentially restricted to numerical works (35).

Here, we address these questions by studying the microscopic dynamics of a soft solid submitted to a constant shear stress, using a unique custom-made apparatus (36, 37) that couples stress-controlled rheology to small-angle static and dynamic light scattering (see Supporting Information for details on the sample and setup geometry). We focus on a gel made of attractive colloidal particles, a model system for network-forming soft solids, which are ubiquitous in soft matter (38) and in biological materials (39). Initially, particles in the gel network undergo both affine (as in ideal elastic solids) and nonaffine displacements, but all displacements are fully reversible. Thus, the initial regime of creep is not due to plasticity but rather to the complex viscoelastic response of the gel network. At larger strains, by contrast, the dynamics are due to irreversible plastic rearrangements that progressively weaken the network, eventually leading to the gel failure. Strikingly, this plastic activity does not increase steadily until failure but rather has a nonmonotonic behavior, peaking thousands of seconds before the macroscopic rupture. Our work thus establishes the notion of dynamic precursor as a powerful tool to understand and predict sudden material failure.

The gel is formed in situ by triggering the aggregation of an initially stable suspension of silica nanoparticles via an enzymatic reaction (see Materials and Methods). The nanoparticles have radius $a=26\mathrm{nm}$ and occupy a volume fraction $\phi =5\%$. Gelation occurs within 3 h, resulting in a network formed by fractal clusters with typical size $\xi \sim a{\phi }^{1/(d_f-3)}\sim 0.5\mu \mathrm{m}$ (40, 41) and fractal dimension $d_f=2$. All experiments are performed at least 48 h after gelation, when the gel viscoelastic properties do not evolve significantly with sample age. Under a constant load, the gel exhibits delayed failure, a feature reported for many network-forming systems (24–28). Fig. 1 demonstrates delayed failure for our gel, by showing the time evolution of the shear strain $\gamma$ and of the strain rate $\dot{\gamma }$ upon imposing a constant stress ${\sigma }_0=240$ Pa at time $t=0$. On time scales shorter than those shown in Fig. 1 ($t<1$ s), the gel responds elastically: $\gamma$ jumps to an elastic shear deformation ${\gamma }_e\sim 4.8\%$, corresponding to a shear modulus $G={\sigma }_0/{\gamma }_e=5000$ Pa, consistent with the low-frequency elastic modulus $G\prime$ measured in oscillatory rheology tests (see Materials and Methods). Following the elastic jump, $\gamma$ grows sublinearly: Both the deformation in excess of the elastic response, $\gamma -{\gamma }_e$, and the shear rate follow power laws, well accounted for by a generalized, or fractional, Maxwell viscoelastic model (42), $\gamma (t)-{\gamma }_e={\gamma }_e{\mathrm{\Gamma }}^{-1}(\alpha ){(t/{\tau }_{FM})}^{\alpha }$, with $\alpha =0.43\pm 0.01$, ${\tau }_{FM}\gtrsim {10}^5\mathrm{s}$, and $\mathrm{\Gamma }(x)$ the Gamma function. Remarkably, this creep regime extends over more than four decades in time, until the gel abruptly fails at $t\approx 2.8\times {10}^4$ s, as signaled by the sharp upturn of both $\gamma$ and $\dot{\gamma }$.

Fig. 1.

Mechanical response and structure evolution of a colloidal gel during creep. Main plot: deformation in excess of the elastic jump ${\gamma }_e=4.8\%$ (blue squares, left axis) and shear rate (red circles, right axis) following a step shear stress of amplitude ${\sigma }_0=240$ Pa, applied at time $t=0$. Lines: power law fits to the data in the initial creep regime ($1\mathrm{s}\le t\le {10}^4\mathrm{s}$), yielding an exponent $\alpha =0.43\pm 0.01$ in the generalized Maxwell viscoelastic model. (Inset) Anisotropy $\chi$ of the scattered intensity as a function of $t$, for $q_{\perp }=q_{\parallel }=2.6\mu {\mathrm{m}}^{-1}$. Triangles: data for the creep test. Line: anisotropy as obtained from $\chi =k\gamma (t)$, with the proportionality coefficient $k=0.26$ determined in independent oscillatory experiments in the linear regime. Solid and dashed lines correspond to the linear regime and to an extrapolation in the nonlinear regime, respectively.

To investigate the relationship between the sudden macroscopic failure of the gel and its microscopic evolution, we inspect static and dynamic light-scattering data collected simultaneously to the rheology measurements (see Fig. 1) (36). Light scattering probes density fluctuations as a function of wavevector $𝐪$: $I(𝐪)\propto {\sum }_{j,k}\exp [-i𝐪\cdot ({𝐫}_j-{𝐫}_k)]$, with $I$ the scattered intensity, ${𝐫}_j$ the position of the $j$-th particle, and $𝐪$ the scattering vector (see Materials and Methods). We use a custom-designed small angle setup (37) based on a complementary metal-oxide-semiconductor (CMOS) detector, allowing measurements on length scales $\sim \pi /q$ in the range $0.8\mu \mathrm{m}-10\mu \mathrm{m}$, comparable to or larger than the cluster size $\xi$. At rest, the scattering pattern depends only on the magnitude of $𝐪$, since the gel is isotropic. During creep, the $q$ dependence of the scattered intensity hardly changes, indicating that the gel structure is fundamentally preserved until sample failure. However, a small anisotropy develops in the static structure factor, similar to that observed for other sheared soft solids (43). We quantify this asymmetry by $\chi (q)=\left[I({𝐪}_{\parallel })-I({𝐪}_{\perp })\right]/\left[I({𝐪}_{\parallel })+I({𝐪}_{\perp })\right]$, with $|{𝐪}_{\parallel }|=|{𝐪}_{\perp }|$ and where $\parallel$ and $\perp$ refer to orientations of the scattering vector parallel and perpendicular to the shear direction, respectively. The Inset of Fig. 1 shows the time dependence of $\chi$. We find that the asymmetry follows the same trend as $\gamma (t)$—that is, that $\chi$ is proportional to $\gamma$ throughout the whole experiment, up to failure. Moreover, the proportionality coefficient is the same as that measured in independent oscillatory experiments in the linear, reversible regime. Thus, structural quantities simply reflect the macroscopic shear deformation, without providing additional information on the fate of the gel.

We now show that the microscopic dynamics are a much more sensitive probe of the gel evolution, unveiling dramatic plastic events that weaken the network thousands of seconds before its macroscopic failure. We measure the two-time intensity correlation function $g_2(𝐪,t_1,t_2)-1\propto {|g_1(𝐪,t_1,t_2)|}^2$, with $g_1$ the field correlation, or intermediate scattering, function (44) (see Materials and Methods). The correlation function is measured simultaneously for several $𝐪$ vectors; Fig. 2A shows representative $g_2-1$ measured at various times $t_1$ during the creep, for $q=3.1\mu {\mathrm{m}}^{-1}$. As for the static intensity, we analyze data separately for scattering vectors parallel and perpendicular to the shear direction. The curves for ${𝐪}_{\parallel }$ (green symbols in Fig. 2A) exhibit a full decay on timescales that grow during creep, eventually reaching $\approx {10}^3\mathrm{s}$. The behavior for ${𝐪}_{\perp }$ is more complex: Initially, $g_2-1$ decays to a quasi-plateau, while a two-step relaxation leading to an almost complete decorrelation is seen at later times. Throughout the experiment, the dynamics along ${𝐪}_{\parallel }$ are faster than for ${𝐪}_{\perp }$. This can be understood by decomposing the particle displacement in its affine and nonaffine components: $𝐫(t_2)-𝐫(t_1)={𝐮}_0(t_1,t_2)+{𝐮}_{na}(t_1,t_2)$. For a particle with coordinate $z$ in the direction of the shear gradient, the affine component is ${𝐮}_0=({\gamma }_2-{\gamma }_1)z{\widehat{𝐞}}_{\parallel }$, with ${\gamma }_i=\gamma (t_i)$ and ${\widehat{𝐞}}_{\parallel }$ the unit vector parallel to the shear direction. Because ${\widehat{𝐞}}_{\parallel }\cdot {𝐪}_{\perp }=0$, correlation functions measured for ${𝐪}_{\perp }$ are only sensitive to ${𝐮}_{na}$, the nonaffine component of the displacement, while the decay of $g_2({𝐪}_{\parallel },t_1,t_2)-1$ reflects both affine and nonaffine motions, resulting in a faster relaxation. In principle, additional contributions to $g_2-1$ may also stem from the spontaneous, thermally activated dynamics of the gel (45). However, here this contribution is negligible (see dashed line in Fig. 2A). Thus, the microscopic dynamics is only related to the shear deformation: This suggests analyzing the dynamics as a function of strain increment rather than time delay.

Fig. 2.

Microscopic dynamics of the gel during creep. (A) Time correlation functions, $g_2-1$, measured in the $q_{\parallel }$ (green) and $q_{\perp }$ (blue to red shades) directions, as a function of time lag $\tau$, for $q=3.1\mu {\mathrm{m}}^{-1}$ and for representative times $t$ after applying a stress step. Black dashed line: spontaneous isotropic dynamics measured on the same sample but at rest. (B) Solid symbols: same data as in A, plotted as a function of the strain increment $\mathrm{\Delta }\gamma$. Additional datasets for intermediate $t$ are shown as small symbols. Line: $g_2-1$ calculated assuming a purely affine deformation. $+$, $\times$: data collected in independent oscillatory experiments following the protocol shown in C. $\times$: maximum decorrelation during a shear cycle of amplitude ${\gamma }_0$; $+$: correlation echo after a full cycle. (C) Shear deformation and correlation function during a shear cycle of amplitude ${\gamma }_0=0.4\%$. $\times$ and $+$ indicate the correlation values reported in B. (D) Nonaffine mean square displacement vs. strain increment during creep (solid symbols) and vs. ${\gamma }_0$ in oscillatory experiments. The dashed line corresponds to the squared cluster size. The solid line is a fit to the data using $<u_{na}^2>=u_{\mathrm{\infty }}^2\frac{{\gamma }_0^2}{{\gamma }_0^2+{\gamma }_c^2}$, with $u_{\mathrm{\infty }}^2=0.12\mu {\mathrm{m}}^2$ and ${\gamma }_c=0.12\%$.

Fig. 2B shows the same data as in Fig. 2A, replotted versus the strain increment $\mathrm{\Delta }\gamma ={\gamma }_2-{\gamma }_1$. A remarkable collapse is seen for the ${𝐪}_{\parallel }$ data, independent of ${\gamma }_1$. This indicates that motion in the ${\widehat{𝐞}}_{\parallel }$ direction is dominated by affine displacements, for which ${𝐮}_0$ is proportional to $\mathrm{\Delta }\gamma$, regardless of the cumulated strain. We confirm this interpretation by calculating $g_2({𝐪}_{\parallel })-1$ for a purely affine shear deformation (see Materials and Methods): The result (line in Fig. 2B) is indeed very close to the data, ruling out sample slip or shear banding, which would result in significant deviations of $g_2-1$ with respect to its theoretical form. Note, however, that the data lay slightly below the theoretical curve, showing that a small nonaffine component must also be present. Nonaffine motion is better resolved by inspecting the ${𝐪}_{\perp }$ correlation functions, which are insensitive to the affine component.

In the following, we thus focus on the microscopic dynamics in the direction perpendicular to shear, which probes only nonaffine displacements. We shall first discuss the linear viscoelastic regime, where we will show that microscopic displacements are fully reversible, and then the nonlinear regime, where irreversible, plastic rearrangements come into play, ultimately causing the gel failure. In the initial regime, $\gamma -{\gamma }_e\le 4\%$ ($t\le 2000\mathrm{s}$, blue shades in Fig. 2B), all data collapse onto a master curve, exhibiting a decay to a quasi-plateau. This collapse is remarkable and sheds light on the nature of the nonaffine deformation observed in the initial regime of the creep. For an ideal solid, the displacement under a shear deformation is purely affine. Nonaffine displacements indicate a departure from this ideal behavior, which may stem from two different physical mechanisms: elastic, reversible response, but with spatial fluctuations of the elastic modulus (46–49), or plastic, irreversible rearrangements (50). The fact that $g_2({𝐪}_{\perp },{\gamma }_1,{\gamma }_2)-1$ is independent of the cumulated strain suggests that no plastic events occur in the initial creep regime. We test this hypothesis by measuring reversibility at the microscopic level in separate oscillatory shear experiments, following the “echo” protocol of refs. 51–53. As shown in Fig. 2C, the sample is submitted to a sinusoidal deformation, $\gamma (t)={\gamma }_0\sin (\omega t)$. The overlap between the initial microscopic configuration and a sheared one is quantified by $g_2({𝐪}_{\perp },t=0,t_2)-1$. We focus on the maximum of $g_2-1$ at the correlation echo, after one full cycle ($+$ in Fig. 2C), and on its first minimum ($t_2=10$ s, $\times$ in Fig. 2C). By repeating the measurements for several ${\gamma }_0$, we obtain the plus and cross symbols displayed in Fig. 2B, for the maximum and the minimum level of correlation, respectively. Up to $\gamma =4\%$, we find no loss of correlation upon setting back to zero the macroscopic deformation; furthermore, the crosses follow the same master curve as the creep data. This confirms that for $t\le 2000\mathrm{s}$, the creep is not due to plastic rearrangements but rather to the slow, fully recoverable deformation of the elastically heterogeneous network, at fixed connectivity. To characterize the strain dependence of nonaffinity, we extract the nonaffine mean-squared displacement $<u_{na}^2>$ via $g_2({𝐪}_{\perp },\mathrm{\Delta }\gamma )-1=\exp (-{𝐪}_{\perp }^2⟨u_{na}^2(\mathrm{\Delta }\gamma )⟩/3)$, the analogous in the strain domain of the usual relationship between correlation functions and mean squared displacement in the low $q$ limit (44). As seen in Fig. 2D, data collected at various $q$ collapse on the same curve, confirming the $q^2$ scaling. Initially, $<u_{na}^2>$ grows as $\mathrm{\Delta }{\gamma }^2$, eventually saturating to a plateau $\approx 0.1\mu {\mathrm{m}}^2$. The quadratic dependence of $<u_{na}^2>$ on strain is the analogous of ballistic dynamics in the time domain; it is the signature of elastic response in an heterogeneous medium (47) and was recently reported for a polymer network (46). The nonaffine displacement saturates at a value close to the cluster size (dotted line in Fig. 2D), consistent with the physical picture that the gel structure cannot be remodeled on length scales larger than the network mesh size without changing its connectivity—that is, without any plastic rearrangements.

We now turn to the $\gamma -{\gamma }_e>4\%$ regime ($t>2000\mathrm{s}$, red shades in Fig. 2B), where $g_2-1$ exhibits a two-step relaxation. The initial decay overlaps with that observed at early times, due to reversible nonaffine deformation. The final decay of $g_2({𝐪}_{\perp })-1$ indicates additional dynamics, which lead to the relaxation of density fluctuations on length scales comparable to or larger than $\xi$. In this regime, the macroscopic deformation is not recovered upon releasing the applied stress: We thus attribute the additional dynamics to irreversible plastic rearrangements. To investigate the evolution of plastic dynamics during creep, we show in Fig. 3 the intensity correlation function for several scattering vectors and a fixed strain increment $\mathrm{\Delta }\gamma =5\%$, versus cumulated strain. This quantity is directly related to the amount of plastic rearrangements occurring over $\mathrm{\Delta }\gamma$. Strikingly, all curves exhibit a negative peak, indicating that the gel undergoes a burst of plastic activity for $13\%\lesssim \gamma \lesssim 22\%$. Remarkably, the minimum of $g_2-1$ occurs at $\gamma \approx 17\%$ ($t=1.9\times {10}^4\mathrm{s}$), as much as 9,000 s before the gel fails, for $\gamma \approx 30\%$. Fig. 3 reveals that the minimum is more pronounced for the largest ${𝐪}_{\perp }$ vectors. Thus, the burst of plastic activity is better seen when probing the dynamics on small length scales, which suggests that macroscopic quantities should be less sensitive to such burst. This is indeed the case for the macroscopic strain: As seen in Fig. 1, only a slight deviation from the sublinear creep is seen around $t=1.9\times {10}^4\mathrm{s}$, at the burst maximum. The strain rate is a more sensitive quantity: Fig. 3 shows that the onset of plasticity coincides with the departure of $\dot{\gamma }$ from its power law behavior in the linear regime (dotted line), thus establishing a direct connection between microscopic dynamics and macroscopic creep (see Supporting Information for a detailed comparison between mechanical and dynamical signatures of plasticity).

Fig. 3.

Microscopic dynamics signals the onset of plasticity. $g_2({𝐪}_{\perp },\gamma -\mathrm{\Delta }\gamma /2,\gamma +\mathrm{\Delta }\gamma /2)-1$ vs. $\dot{\gamma }$. Right axis, red line: macroscopic deformation rate $\dot{\gamma }$ vs. cumulated shear deformation during creep. Dashed line: generalized, or fractional, Maxwell model. Left axis, blue curves: microscopic nonaffine dynamics over a fixed strain increment $\mathrm{\Delta }\gamma =5\%$, for various ${𝐪}_{\perp }$ as indicated by the labels.

To quantify the microscopic plastic activity during creep, we develop a simple model for dynamic light scattering under time-varying conditions. Using strain as the relevant variable and focusing on the dynamics along ${𝐪}_{\perp }$, we assume that reversible and plastic displacements are uncorrelated processes, leading to the factorization $g_1({𝐪}_{\perp },{\gamma }_1,{\gamma }_2)=R({𝐪}_{\perp },\mathrm{\Delta }\gamma )P({𝐪}_{\perp },{\gamma }_1,{\gamma }_2)$, with $R$ and $P$ the contributions due to reversible and plastic displacements, respectively, and where $R$ only depends on the strain increment, as indicated by the experiments. For a stationary process, a general form that captures well different kinds of dynamics is $g_1=\exp [-f(q){(A\mathrm{\Delta }\gamma )}^p]$. The initial decay of the correlation function in the linear, reversible regime discussed in reference to Fig. 2 is an example of this functional form, with $f\sim q^2$, $p=2$, and $A$ the constant, nonaffine root-mean-square particle displacement per unit strain increment. We generalize this form by expressing $P({𝐪}_{\perp },{\gamma }_1,{\gamma }_2)$ as a function of a strain-dependent plastic activity per unit strain increment, $A(\gamma )$:

$$g_1({𝐪}_{\perp },{\gamma }_1,{\gamma }_2)=R({𝐪}_{\perp },\mathrm{\Delta }\gamma )\exp \left[-f({𝐪}_{\perp }){\left({\int }_{{\gamma }_1}^{{\gamma }_2}A(\gamma )\mathrm{d}\gamma \right)}^p\right].$$ [1]

We extract $f{({𝐪}_{\perp })}^{1/p}A(\gamma )$ from the experimental data using Eq. 1 and assuming that $R({𝐪}_{\perp },\mathrm{\Delta }\gamma )$ is the same as in the reversible regime of creep (see Materials and Methods for details). The results are shown in Fig. 4A, for several ${𝐪}_{\perp }$ vectors. Consistent with the findings for the correlation function at a specific strain increment, Fig. 3, the plastic activity per unit strain exhibits a nonmonotonic behavior, with a peak centered around $\gamma =17.1\%$. The height of the peak strongly increases with $q$. According to our model, this is due to the $q$ dependence of the prefactor $f$, since the plastic activity $A$ is a quantity intrinsic to the gel and is thus independent of the probed length scale. We test this assumption by plotting in Fig. 4B the plastic activity scaled by its peak value, $A_0$. An excellent collapse is seen for data spanning a factor of 5 in $q$ vectors, thereby confirming the soundness of the model.

Fig. 4.

Plastic activity as revealed by the microscopic dynamics. (A) Plastic activity per unit strain increment vs. cumulated strain, for various ${𝐪}_{\perp }$. The error bars on the left indicate the uncertainty resulting from averaging data collected at different $\mathrm{\Delta }\gamma$. (B) Collapse of the plastic activity measured at various ${𝐪}_{\perp }$, as indicated by the labels. (C) $q$ dependence of the plastic dynamics. Data are normalized by $q^2$, the behavior expected for diffusive dynamics. (D) Exponent $p$ characterizing the plastic dynamics (see Eq. 1), where $p=1$ corresponds to diffusive dynamics.

To gain insight into the nature of the plastic dynamics, we inspect the $q$ dependence of the prefactor $f({𝐪}_{\perp })A_0^p$ and of the exponent $p$. For $q_{\perp }\le 1.5\mu \mathrm{m}$, the prefactor scales as $q^2$ (see Fig. 4C, where the $q^2$ dependence has been factored out) and $p\approx \mathrm{\hspace{0.17em}1}$. Under these conditions, the contribution of plasticity to the decay of the correlation function over a small strain increment reads $P({𝐪}_{\perp },\gamma ,\gamma +\mathrm{\Delta }\gamma )=\exp [-q_{\perp }^{2}D(\gamma )\mathrm{\Delta }\gamma ]$, with $D(\gamma )\sim A(\gamma )$ a strain-dependent, but $q$-independent, diffusion coefficient. Thus, in the low $q$ regime, the plastic dynamics are diffusive, since $P$ is the analogous in the strain domain of the usual diffusive dynamics in the time domain, for which $g_1=\exp [-q^{2}D\tau ]$ (44). A change of the plastic dynamics occurs beyond $q_{\perp }^{\ast }\approx 2\mu {\mathrm{m}}^{-1}$, corresponding to a length scale $\pi /q_{\perp }^{\ast }\approx 1.6\mu \mathrm{m}$, slightly larger than the cluster size. For $q_{\perp }\ge q_{\perp }^{\ast }$, $f{A}_0^p$ grows sharply, increasingly departing from the $q^2$ scaling. Concomitantly, the $p$ exponent grows up to $p\approx 1.75$ at the largest probed $q$ vectors (Fig. 4C), approaching $p=2$, the exponent characterizing ballistic dynamics. The emerging picture is that of plasticity consisting of irreversible rearrangements, most likely due to bond rupture. On small length scales (large $q$), the dynamics are strongly $q$ dependent and are dominated by local motion associated with such rearrangements. On larger length scales (smaller $q$), the contribution of many events adds up, leading to a diffusive decay of density fluctuations. These events progressively weaken the network, eventually leading to its catastrophic failure.

The experiments reported here unveil the complex evolution of the microscopic dynamics during the creep of a colloidal gel, from reversible nonaffine motion due to the heterogeneous gel structure to a burst of plastic rearrangements that irreversibly weaken the network, providing a microscopic signature of the onset of plasticity. Remarkably, this dynamic precursor occurs midway through the creep. While further theoretical work will be needed to fully understand the origin of the precursor and its temporal location, we emphasize that its occurrence allows one to predict the ultimate fate of the network thousands of seconds before its catastrophic rupture. Ongoing experiments in our group reveal that similar dramatic changes of the microscopic dynamics largely precede failure in a variety of mechanically driven materials, from polymer gels to elastomers and semicrystalline polymers. The notion of dynamic precursor therefore emerges as a powerful concept to understand and predict material failure.

Materials and Methods

Enzyme-Induced Aggregation and Gel Formation.

The gel results from the aggregation of a suspension of silica particles (Ludox TM50, from Sigma Aldrich, diameter $a=26$ nm as determined by small angle neutron scattering, SANS), dispersed at a volume fraction $\phi =5$% in an aqueous solvent containing urea at $1$ M. Particle aggregation is triggered by increasing in situ the ionic strength of the solvent, thanks to the hydrolysis of urea into carbon dioxide and ammonia, a reaction catalyzed by an enzyme (Urease U1500-20KU, from Sigma Aldrich, $35$ U/mL) (54), whose activity depends on temperature $T$. The suspension is prepared at $T\approx 4^{\circ }\mathrm{C}$ and brought at room temperature after loading the cell, thereby activating the enzyme and initiating aggregation. The sol–gel transition occurs $\approx 3$ h after loading the sample in the shear cell. The fractal dimension of the gel network, $d_f=2$, has been determined from independent SANS measurements on a gel prepared following the same protocol.

Oscillatory Rheology in the Linear Regime.

We characterize the mechanical properties of the gel by measuring the frequency-dependent elastic and loss moduli [$G\prime (\omega )$ and $G^{″}(\omega )$, respectively] in the linear regime (${\gamma }_0=0.1\%$) using a commercial rheometer (MCR502 by Anton Paar). Over the range ${10}^{-3}\mathrm{rad}{\mathrm{s}}^{-1}\le \omega \le 10\mathrm{rad}{\mathrm{s}}^{-1}$, $G\prime$ is essentially flat and $G^{″}\sim {\omega }^{-0.4}$, a behavior consistent with the fractional Maxwell model that accounts for the gel creep. The gel slowly ages: $G\prime$ increases as $t^{1/3}$, and the characteristic relaxation time ${\tau }_{FM}$ obtained from the fractional Maxwell model increases linearly with $t$. We let the gel age for 48 h before running a creep experiment, such that during the duration of one experiment (typically a few hours) the viscoelastic properties of the gel do not evolve significantly, with $G\prime \sim 5$ kPa and ${\tau }_{FM}\gtrsim {10}^5$ s. The elastic modulus dominates over the loss modulus at all measured frequencies; for $\omega =1\mathrm{Hz}$, $G\prime /G^{″}\approx 125$.

Light Scattering.

The small-angle light-scattering apparatus is described in detail in ref. 37. In brief, the scattered light is collected by a lens system and forwarded to the detector of a CMOS camera, such that each pixel corresponds to a well-defined scattering vector $𝐪$, with $q=4\pi n{\lambda }^{-1}\sin (\theta /2)$, where $n=1.338$ is the solvent refractive index, $\lambda =632.8\mathrm{nm}$ the laser in vacuo wavelength, and $\theta$ the scattering angle. The $𝐪$-dependent intensity is obtained as $I(𝐪)={⟨I_p⟩}_{𝐪}$, where $I_p$ is the CMOS signal of the $p$-th pixel, corrected for the dark background as in ref. 55, and ${⟨\mathrm{\cdots }⟩}_{𝐪}$ is an average over a small region in $q$ space centered around $𝐪$. For the silica particles used here, the form factor $\approx 1$ in the range of $q$ covered by the setup, such that $I(𝐪)$ is proportional to the static structure factor $S(𝐪)$.

The two-time intensity correlation function is calculated as

$$g_2(𝐪,t_1,t_2)-1=\beta \frac{{⟨I_p(t_1)I_p(t_2)⟩}_{𝐪}}{{⟨I_p(t_1)⟩}_{𝐪}{⟨I_p(t_2)⟩}_{𝐪}}-1,$$

where $\beta \gtrsim 1$ is a setup-dependent prefactor chosen such that $g_2(𝐪,t_1,t_2=t_1)-1=1$. The intensity correlation function is related to the field correlation function $g_1$ by $g_2-1={|g_1|}^2$ (44), with $g_1=F(𝐪,t_1,t_2)/F(𝐪,t_1,t_2=t_1)$ and $F(𝐪,t_1,t_2)=N^{-1}{\sum }_{j,k=1}^N{e}^{i𝐪\cdot [{𝐫}_j(t_1)-{𝐫}_k(t_2)]}$, where the sum runs over the $N$ particles in the scattering volume.

Intermediate Scattering Function for a Purely Affine Deformation.

Following ref. 44 with strain, rather than time, as the independent variable, the intermediate scattering function is expressed as

$$g_1(𝐪,\mathrm{\Delta }\gamma )=\int Q(\mathrm{\Delta }𝐫)\exp (-i𝐪\cdot \mathrm{\Delta }𝐫)\mathrm{d}\mathrm{\Delta }𝐫,$$ [MM1]

where $Q(\mathrm{\Delta }𝐫)$ is the probability distribution function of the particle displacement following a strain increment $\mathrm{\Delta }\gamma$. For a purely affine deformation in the direction of ${\widehat{e}}_{\parallel }$, $Q(\mathrm{\Delta }𝐫)=1/(\mathrm{\Delta }\gamma b)$ and $\mathrm{\Delta }𝐫=z\mathrm{\Delta }\gamma {\widehat{e}}_{\parallel }$, with $b$ and $z$ the cell gap and the coordinate in the direction of the shear gradient, respectively. By inserting these expressions in Eq. MM1, one finds $g_1({𝐪}_{\perp },\mathrm{\Delta }\gamma )=1$ and $g_1({𝐪}_{\parallel },\mathrm{\Delta }\gamma )=\text{sinc}\left(q_{\parallel }\mathrm{\Delta }\gamma b/2\right)$. The corresponding $g_2-1$ function is shown as a line in Fig. 2B.

Extracting the Plastic Activity $A(𝜸)$ from the Light-Scattering Data.

To calculate the plastic activity per unit strain, $A(\gamma )$, we invert Eq. 1:

$${\int }_{{\gamma }_1}^{{\gamma }_2}A(\gamma )\mathrm{d}\gamma ={\left[-\frac{1}{f({𝐪}_{\perp })}\ln \frac{g_1({𝐪}_{\perp },{\gamma }_1,{\gamma }_2)}{R({𝐪}_{\perp },{\gamma }_2-{\gamma }_1)}\right]}^{\frac{1}{p}}.$$ [MM2]

Taking the derivative with respect to ${\gamma }_2$ at fixed ${\gamma }_1$ yields

$$A_{{\gamma }_1}(\gamma )={f{({𝐪}_{\perp })}^{-\frac{1}{p}}\frac{\partial }{\partial {\gamma }_2}{\left[-\ln \frac{g_1({𝐪}_{\perp },{\gamma }_1,{\gamma }_2)}{R({𝐪}_{\perp },{\gamma }_2-{\gamma }_1)}\right]}^{\frac{1}{p}}|}_{{\gamma }_2=\gamma },$$ [MM3]

where the ${\gamma }_1$ index in the l.h.s. of Eq. MM3 indicates that here $A$ is evaluated using data for a specific value of the initial strain ${\gamma }_1$. Operationally, we calculate $A_{{\gamma }_1}(\gamma )$ for several values of ${\gamma }_1$, using $g_1=\sqrt{g_2-1}$ and $R({𝐪}_{\perp },\mathrm{\Delta }\gamma )=\exp (-q_{\perp }^2⟨u_{na}^2⟩/3)$, with $<u_{na}^2>=u_{\mathrm{\infty }}^2\frac{\mathrm{\Delta }{\gamma }^2}{\mathrm{\Delta }{\gamma }^2+{\gamma }_c^2}$ and $u_{\mathrm{\infty }}^2=0.12\mu {\mathrm{m}}^2$, ${\gamma }_c=0.12\%$ (line in Fig. 2D). The derivative in the r.h.s. of Eq. MM3 was performed either numerically on the raw data or analytically on a second order polynomial fit of $\ln (g_1/R)$. We find similar results and use the latter method, which is less sensitive to data noise. Finally, $A(\gamma )$ is obtained by averaging $A_{{\gamma }_1}(\gamma )$ over different choices of ${\gamma }_1$, in the range $1\%\le \gamma \le 6\%$. The exponent $p$ is chosen by repeating the calculation of $A(\gamma )$ for several test values, finally retaining the $p$ value that minimizes the rms residuals between the experimental $g_2-1$ and the correlation functions calculated from $A(\gamma )$ using Eq. 1.