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Nonlinear viscoelasticity and generalized failure criterion for gels

Bavand Keshavarz,1, ∗ Thibaut Divoux,2, 3 S´ebastienManneville,4 and Gareth H. McKinley1 1Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA 2Centre de Recherche Paul Pascal, CNRS UPR 8641 - 115 avenue Schweitzer, 33600 Pessac, France 3MultiScale Science for Energy and Environment, UMI 3466, CNRS-MIT, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA 4Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France (Dated: January 6, 2017) Polymer gels behave as soft viscoelastic and exhibit a generic nonlinear mechanical response characterized by pronounced stiffening prior to irreversible failure, most often through macroscopic fractures. Here, we aim at capturing the latter scenario for a protein gel using a nonlinear integral built upon (i) the linear viscoelastic response of the gel, here well described by a power-law modulus, and (ii) the nonlinear viscoelastic properties of the gel, encoded into a “damping function”. Such formalism predicts quantitatively the gel mechanical response to a shear start-up experiment, up to the onset of macroscopic failure. Moreover, as the gel failure involves the irreversible growth of macroscopic cracks, we couple the latter response with Bailey’s durability criterion for brittle solids in order to predict the critical values of the stress σc and strain γc at the failure point, and how they scale with the applied shear rate. The excellent agreement between theory and experiments suggests that the crack growth in this soft viscoelastic gel is a Markovian process, and that Baileys’ criterion extends well beyond hard such as , , or minerals.

PACS numbers: 62.20.mj, 83.80.Kn, 82.35.Pq, 83.10.Gr

Introduction.- Polymer gels find ubiquitous applica- ing the Boltzmann superposition principle, the damping tions in material science, from biological tissues to man- function is used to construct a time-strain separable con- ufactured goods, among which food stuffs and medical stitutive equation of K-BKZ (Kaye–Bernstein-Kearsley- products are the most widespread [1–3]. These materi- Zapas) form [14, 15] that predicts the gel mechanical als commonly feature a porous microstructure filled with response to steady-shear experiments. This approach water, which results in -like viscoelastic mechani- robustly captures the strain-stiffening of the gel during cal properties. While soft polymer gels share common start up of steady shear tests up to the appearance of features with hard materials, including delayed failure a stress maximum that is accompanied by the onset of [4, 5], crack propagation [6, 7] or work-hardening [8], the first macroscopic crack. Moreover, in order to link their porous microstructure also confers upon them re- the nonlinear viscoelastic response of the gel to its subse- markable nonlinear viscoelastic properties. Indeed, such quent brittle-like rupture, we adopt the Bailey criterion, soft solids strongly stiffen upon increasing , which describes the gel failure as arising from accumu- which stems from the inherent nonlinear elastic behavior lation of irreversible damage [17, 18]. The combination of the polymer chains composing the gel network [9–12]. of the stress response predicted by the K-BKZ consti- Polymer gels hence endure large strains to failure and tutive formulation with the Bailey criterion allows us to dissipate substantial mechanical work, leading to very predict accurately the scaling of the critical stress and tough and [13]. However, to date strain at failure with variations in the magnitude of the no quantitative link has been made between the nonlin- applied shear rate. Our results extend Bailey’s criterion ear viscoelasticity of polymer gels and the failure that is to viscoelastic soft solids and provide a unified consistent

arXiv:1607.08300v2 [cond-mat.soft] 5 Jan 2017 subsequently observed as the strain-loading is increased framework to describe the failure of protein gels under beyond the initial stiffening regime. various shear loading histories.

In the present Letter, we apply the concept of a strain Experimental.- We consider two acid-induced protein damping function, traditionally used for polymeric liq- gels with substantially different mechanical properties: uids and rubber-like materials [16], to quantify the non- the first one shows pronounced strain-hardening, while linear viscoelastic response of a prototypical protein gel. the second does not. They are prepared by dissolving The form of the damping function is constructed exper- caseinate powder (Firmenich) at 4% wt. (resp. 8% wt.) imentally through a series of independent stress relax- in deionized water under gentle mixing at 600 rpm and ation tests that allow us to probe large deformations T = 35◦C. Homogeneous gelation is induced by dissolv- while injecting very little energy into the gel, hence lim- ing 1% wt. (resp. 8% wt.) glucono-δ-lactone (GDL, Fir- iting as much as possible any plastic damage. Follow- menich) in the protein solution [19, 20]. While still , 2

4

10 0.4 0.3 α 0.2

3 0.1 −3 −2 −1 0 1 10 10 10 10γ 10 10 0 0 10 3 2 10 ) 2.5 0 )[Pa]

γ 2 ,t ( ) 0 0 γ 1 h ( 1.5

γ −1 h

( 10 10 1 G 0.5 0 (t) 3 10 0 0 −3 −2 −1 0 1 10 10 10γ 10 10 1 −2 0 −1 t 10 −3 −2 −1 0 1 10 −1 0 1 2 3 4 10 10 10 10 10 10 10 10 10 10 10 γ0 t[s] FIG. 2: (color online) Strain damping function h(γ0) of a 4% FIG. 1: (color online) Nonlinear relaxation function wt. casein gel as defined in the text. Same color code as in G(γ0, t) = σ(t)/γ0 vs time t determined by step strain tests, Fig. 1. The solid black line is the best fit function by power each one performed on a freshly prepared 4% wt. casein gel. 2 series in γ0 as proposed in [27] which captures the stiffening Colors from blue to red represent strain values ranging from behavior, but does not account for the softening part of the γ0 = 0.002 to γ0 = 5. The black line is the best power-law gel response at strains larger than 50%. The red continuous fit of the data in the linear region (γ0 ≤ 0.01). Inset: Stress line is the best fit function h˜(γ) of the data (see text). Inset: relaxation exponent α extracted from the power-law fit of the same data plotted in semilogarithmic scales. data shown in the main graph with the same color code. The horizontal line is the average exponent α = 0.18 ± 0.01. V α = 266 5 Pa.s [22, 23]. For γ0 & 0.01, the stress re- laxation still± exhibits a power-law decrease in time, with the protein solution is poured into the gap of a cylindri- the same exponent α, after t & 0.1 s but the magnitude cal Couette shear cell connected to a strain-controlled of the stress at a given time first stiffens and then softens (ARES, TA Instruments, Delaware) [21]. In as γ0 is increased. situ gelation is achieved within 12 hours after which ei- Since α is insensitive to the strain amplitude, we can ther a step strain or a constant shear rate is imposed on use the concept of strain-time separability [24] to quan- the sample while the resulting stress response is moni- tify the strain dependence of the stress relaxation re- tored. In both cases, images of the gel deformation are sponse by computing the damping function [16], defined recorded simultaneously to in order to monitor as h(γ ) = G(γ , t)/G(t) where ... denotes the time 0 h 0 it h it the nucleation and growth of cracks. average for 1 t 1000 s for each of the step-strain ≤ ≤ Damping function.- To first characterize the viscoelas- experiments. The resulting damping function reported tic properties of the 4% wt. casein gel, we perform a in Fig. 2 thus fully characterizes the strain dependence series of step strain tests. Each experiment is performed of the viscoelastic response in the material. The gel dis- on a freshly prepared gel and consists of two successive plays a linear response (i.e. h = 1) up to γ0 = 0.1, strain steps. The first step is applied within the linear whereas for intermediate strain amplitudes, the gel ex- deformation regime and the stress relaxation is followed hibits a pronounced strain-stiffening, that is character- over the next 4000 s and serves as a reference for the ized by a maximum value h 2.2 reached at γ = 0.5. ' comparison of two independent experiments. This is fol- Finally, for even larger strains, the material softens due lowed by a second step at a strain amplitude chosen be- to network rupture and the damping function decreases −3 tween 10 γ0 5 and the stress is monitored again abruptly as a power-law function of the imposed strain for 4000 s to≤ measure≤ the gel viscoelastic response. The with an exponent of 3. Yet, we emphasize that for ∼ − stress relaxation functions G(γ0, t) = σ(t)/γ0 associated all step strain tests, the gel remains visually intact even with the second step of strain are reported in Fig. 1. at strain amplitudes as large as γ 5 [25]. 0 ' At low applied strains (γ0 . 0.01), the magnitude of As proposed in [26, 27], the strain hardening portion the viscoelastic stress scales linearly with the imposed of the damping function is captured in a power series ex- ? strain and the relaxation modulus exhibits a remarkable pansion of h (γ0) [28], with the fractal dimension db of power-law decrease over four decades of time, which is the stress-bearing network backbone as the only fitting well modeled by a spring-pot (or fractional viscoelastic parameter. Here we find db = 1.3 0.1 in good agreement element) [22], G(t) = Vt−α/Γ(1 α), where V and α are with other measurements for polymer± gels (see black line the only two material properties− required to characterize in Fig. 2)[29]. However, describing the whole damping the gel, and Γ denotes the Gamma function. By fitting function also requires us to take into account the gel soft- the data for γ0 0.01 we find that the relaxation expo- ening that is measured at strains larger than 0.5. Follow- nent α = 0.18 0≤.01 and the prefactor or “quasiproperty” ing analogous approaches in the literature for rubbery ± 3

3 10 z 10 mm (b) 2 r 10 γ = 0.58 γ = 0.86 γ =1.15 γ =1.44 γ 1 −3 −2 −1 0 1 10

10 10 10 10 10 [Pa] −1 γ!0 = 0.001 s σ 0 γ! = 0.02 s−1 (a) 10 0 γ! = 0.2 s−1 0 2 1 γ! = 0.6 s− 10 ri −1 0 H 10 −1 0 1 2 3 4 ro 10 10 10 10 10 10 t[s]

z ↵ 0 ˙

1 r (c) V [Pa] 10 150 / 0 ) 10 t σ 125 (

100 ) ↵

[Pa] 75 −1 σ 10 γ! = 0.001 s−1 γ! = 0.1 s−1 50 0 0 0 (1 γ! = 0.003 s−1 γ! = 0.2 s−1 25 0 0 10 ) −1 −1

0 1 2 3 4 ↵ −2 γ! = 0.02 s γ! = 0.3 s 10 10 10 10 10 0 0 10 −1 −1 t[s] γ! = 0.03 s γ! = 0.6 s 0 0 (1 0 1 2 3 4 −3 −2 −1 0 1 2 10 10 10 10 10 10 10 10 10 10 10 t[s] γ =˙γ0t

FIG. 3: (color online) (a) Stress response σ vs time t (lower axis) and vs strain γ =γ ˙ 0t (upper axis) of a 4% wt. casein gel −3 −1 to a constant shear rateγ ˙ 0 = 10 s initiated at t = 0. The gray dashed line corresponds to linear viscoelastic response [Eq. (2)]. The black line corresponds to the K-BKZ equation constructed using only the strain-hardening part of the damping ∗ function, h (γ0) and reported as the solid black line in Fig. 2. The continuous red line corresponds to the K-BKZ equation built upon h˜, which includes both the hardening and the softening components of the damping function. Lower inset: same data on semilogarithmic scales. Upper inset: sketch and images of the side view of Couette cell at different strains recorded simultaneously to the experiment reported in the main graph. (b) Stress responses to individual constant shear rate experiments ranging from 10−3 s−1 to 0.6 s−1. Dashed lines indicate the linear response [Eq. (2)], and the continuous lines correspond to ˜ α the K-BKZ predictions using h [Eq. (3)]. (c) Normalized stress responses (1 − α)Γ(1 − α)σ(t)/Vγ˙ 0 vs strain γ for all constant shear rate experiments. networks and polymer melts [16] we use the following viscoelastic stress response using a time-strain separable functional form h˜(γ ) = [1 + (γ /γ )2]/[1 + γ /γ 5], equation of the integral K-BKZ type [14, 30]. The stress 0 0 m | 0 M | where γm = 0.34 and γM = 0.57 are fitting parameters is given by (see ref. [31] for a more detailed derivation): that respectively mark the departure from linearity and Z t 0 0 0 the location of the strain maximum. The quadratic de- σ(t) = G(t t )h(γ)γ ˙ (t )dt (1) −∞ − pendence of the numerator is set by symmetry [27] while the exponent in the denominator, which depends on the The first regime in Fig. 3 is fully accounted for by the strength of the individual network bonds in the gel, is linear viscoelastic response based on the power-law be- fixed by a fit of the damping function at large strains havior of G(t) determined in Fig. 1 for γ 1%. Since 0 ≤ γ γM . This form describes the whole data set re- h(γ) = 1 in this regime the stress can be found analyti- markably well (Fig. 2) and can be used to predict the gel cally [32, 33]: viscoelastic response to start up of steady shear all the Z t V 1−α 0 0 0 γ˙ 0t way to sample failure as we will now illustrate. σ(t) = G(t t )γ ˙ (t )dt = . (2) −∞ − (1 α)Γ(1 α) K-BKZ description of shear start-up.- In Fig. 3, we − − show the evolution of stress and onset of cracking when Equation (2) nicely describes the experimental data for a 4% wt. casein gel is submitted to a constant shear γ . 0.2 without any additional fitting parameter [see −3 −1 rateγ ˙ 0 =10 s . The stress growth σ(γ) can be sepa- dashed gray line in Fig. 3(a) and (c)]. To predict the rated into three consecutive regimes: a linear viscoelastic nonlinear behavior, we substitute the power-law form of regime, characterized by a power-law growth of σ(t) up the relaxation modulus G(t) into Eq. (1) [16, 31, 34] and to γ =γ ˙ t 0.2, followed by a strain-stiffening regime in rearrange to give: 0 ' which σ shows a steeper increase up to a critical strain V α Z γ˙ 0t γ˙ 0 −α γc 0.8 at which the stress goes through a maximum σ(t) = h(γ)γ dγ (3) ' Γ(1 α) 0 σc. Finally, in a third regime, the stress exhibits an − abrupt decrease followed by a slower relaxation at larger where γ =γ ˙ 0t is the total accumulated strain at time t. strains. The gel remains visually intact and homogeneous To capture the strain-stiffening in the we substi- initially, and the first macroscopic fracture appears at the tute the strain hardening form of the damping function ∗ end of the second regime when γ γc and σ σc (see for a protein gel, h (γ), into Eq. (3) to obtain the pre- Movie 1 in the Supplemental Material).' We predict' the diction shown by the solid black line in Fig. 3(a) and 4

(c). This captures the nonlinear response of the gel at (a) (b)(b) 3 moderate strains, but leads to an ever-increasing rate of 10 0 stress growth. The softening part of the damping func- 10 c [Pa] γ tion is crucial to account for the stress evolution observed c experimentally during shear start-up. Substituting h˜ in σ Eq. (3) and integrating numerically we get the red line in 2 −1 Fig. 3(a) and (c) which accurately predicts the mechani- 10 −3 −2 −1 0 1 10 −3 −2 −1 0 1 10 10 10 10 10 10 10 10 10 10 −1 −1 cal response of the protein gel up to the stress maximum, γ˙0[s ] γ˙ 0[s ] without any adjustable parameter. The initial stiffening behavior is described by the numerator of h˜, while the FIG. 4: (color online) Critical stress σc (a) and critical strain γ (b) vs applied shear rateγ ˙ for both a 4% (◦) and a 8% denominator is responsible for the plateauing of the pre- c 0 () casein gel. In both graphs, the dashed and continuous dicted stress response. The subsequent decrease of the lines stand for the prediction issued from the combination of stress observed experimentally in Fig. 3 must be associ- the Bailey criterion and the stress response computed either ated with the growth of macroscopic fractures that can- from just the linear response (dashed lines) or the full K-BKZ not be accounted for by an integral formulation such as equation built upon h∗ (solid lines). the K-BKZ equation, for which the value of the integral always increases monotonically in time. Repeating shear cess σ(t), and F (σ) is the dependence of the time to start-up experiments for variousγ ˙ 0 confirms that Eq. (3) rupture for experiments performed at a series of quantitatively predicts the gel response over almost three constant imposed stresses σ [17]. Independent creep decades of shear rate [Fig. 3(b)]. The universal nature of tests have been performed on casein gels [5] and indi- the response is evident by the rescaling of the experimen- cate that F (σ) = Aσ−β where A is a scale parameter, tal data onto a single master curve [Fig. 3(c)] and this with A = (7.6 0.1) 1013 s.Paβ and β = 5.5 0.1 ± × ± rescaling also holds true for the 8% wt gel [see Fig. 2(a) for the 4% wt. gel [5] and A = (5.0 0.1) 1018 s.Paβ ± × in the Supplemental Material]. and β = 6.4 0.1 for the 8% wt. gel [see Fig. 3 in the ± Moreover, the steady-state stress value predicted by Supplemental Material]. We have also independently de- the K-BKZ formulation coincides closely with the value of termined the rheological response to an arbitrary loading the stress maximum observed experimentally [Fig. 3(b)], history [Eq. (2) and (3)]. When combined with Eq. (2), suggesting that there is a deeper connection between the the Bailey criterion leads to the following analytic ex- failure point of the gel characterized by σc and γc, and pressions for the critical strain and stress at failure under the nonlinear viscoelastic response of the material pre- startup of steady shear: ceding macroscopic failure. In Fig. 4 we show the lo- γ (γ ˙ ) = S γ˙ (1−αβ)/[1+(1−α)β] (4) cus of the stress maximum (σc, γc) for different imposed c 0 γ 0 shear rates, for both the 4% wt. and 8% wt. casein gels. 1/[1+(1−α)β] σc(γ ˙ 0) = Sσγ˙ 0 (5) The critical stress σc (above which macroscopic fractures appear) increases as a weak power law ofγ ˙ 0 with an ex- where Sσ and Sγ are analytic functions of α, V and β [39]. ponent ξ = 0.18 0.01 for the 4% wt. casein gel and Whether the critical stress and strain are constant or in- ± ξ = 0.13 0.01 for the 8% wt. respectively [Fig. 4(a)]. crease/decrease withγ ˙ thus depends on both the pa- ± 0 Moreover, the critical strain γc also displays a power-law rameters in the linear viscoelastic kernel G(t) and on the increase withγ ˙0 for the 8% casein gel, whereas the 4% form of the failure law F (σ). For example, in the 8% wt. casein gel shows a yield strain that is rate independent casein gel (for which α = 0.04 0.01 and β = 6.4 0.1) we ± ± [Fig. 4(b)]. The goal is now to predict such nontrivial find that the critical strain increases withγ ˙0 since the ex- power-law dependences. ponent in Eq. (4) is (1 αβ)/ [1 + (1 α)β] 0.10 0.01. Failure criteria and discussion.- To account quanti- For both the 4% wt. and− 8% wt. gels,− the agreement' ± be- tatively for the different scalings of the crack appear- tween theory (dashed lines in Fig. 4) and experiments is ance coordinates σc and γc withγ ˙ 0, we apply the fail- excellent for the two power-law exponents, again without ure criterion introduced by J. Bailey, already successful any adjustable parameter. However, the prefactor Sγ is to describe the rupture of much stiffer samples such as clearly overestimated for the 4% wt. casein gels (dashed glasses [37] and elastomeric-like materials [38]. This cri- line in Fig. 4). Indeed, 4% wt. casein gels display a pro- terion may be applied under the assumption that the nounced stiffening responsible for the early rupture of failure process is irreversible and results from indepen- the gel, which is not captured by the linear viscoelastic dent damage events [18]. This appears to be the case formulation Eq. (2). Instead, when combined with the for the brittle-like failure scenario of casein gels that Bailey criterion, direct numerical integration of Eq. (3) are well modeled by Fiber Bundle Models that verify accounting for strain-stiffening leads to the correct value the former hypothesis [5, 35, 36]. The Bailey criterion of the prefactor for both 4% and 8% casein gels (solid R τf reads 0 dt/F [σ(t)] = 1, where τf denotes the sample lines in Fig. 4). Hence, one should use the complete lifespan under an arbitrarily given active loading pro- damping function h˜ for predicting the failure point of 5 strain-hardening materials. Soft Matter 10, 672 (2014). Conclusion.- We have shown using soft viscoelastic casein [14] R.B. Bird, R.C. Armstrong and O. Hassager, Dynamics gels that the combination of the K-BKZ formalism, to- of Polymeric , 2nd ed. (John Wiley & Sons, New gether with Bailey’s failure criterion sets a self-consistent York, 1987). [15] R.G. Larson, The Structure and Rheology of Complex framework to capture the linear and non-linear response , Topics in chemical engineering. Oxford University of the gel up to the materials’ yield point. This approach Press (1999). also predicts the scaling of both the critical stress and [16] V. Rol´onand M. Wagner, The damping function in rhe- strain at failure. The present results thus extend the va- ology, Rheol. Acta 48, 245 (2009). lidity of Bailey’s criterion to squishy soft solids, paving [17] G. Bartenev and Y. Zuyev, Strength and Failure of visco- the way for deeper analogies between soft and hard ma- elastic materials (Pergamon Press, 1968), chap. 7, Depen- terials [40]. dence of Strength of elastomers on and type of filler, pp. 193–211. The authors thank M. Bouzid and E. Del Gado for en- [18] A.D. Freed and A.I. Leonov, The Bailey criterion: Sta- lightening discussions, and A. Parker for providing the tistical derivation and applications to interpretations of casein and GDL. This work was funded by the MIT- durability tests and chemical kinetics, Z. Angew. Math. Phys. 53, 160 (2002). France seed fund and by the CNRS PICS-USA scheme [19] J. Lucey, T. van Vliet, K. Grolle, T. Geurts, and P. Wal- (#36939). BK acknowledges financial support from Ax- stra, Properties of Acid Casein Gels Made by Acidifica- alta Coating Systems. tion with Glucono-δ-lactone. 1. Rheological Properties, Int. Dairy J. 7, 381 (1997). [20] B. O’Kennedy, J. Mounsey, F. Murphy, E. Duggan, and P. 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Nonlinear viscoelasticity and generalized failure criterion for polymer gels: supplemental material

1 10 3 γ! = 0.006 s−1 0 10 −1 γ!0 = 0.05 s ↵ −1 0 ˙ γ!0 = 0.1 s

0 0.3 s−1 [Pa] γ!0 = V 10 ′′ −1 / γ!0 =1.0 s ) −1 t G γ!0 = 3.0 s ( 2

10 ) −1 ↵

and 10 4 10 ′

3 10 (1 G

2 10 [Pa] ) −2 σ

1 ↵ 10 1 10 10

0 10 −2 −1 0 1 2 −1 0 1 2 10 10 10 10 10

10 10 10 10 (1 t[s] ω[rad/s] −3 10 −3 −2 −1 0 1 2 10 10 10 10 10 10 γ =˙γ0t FIG. 1: (color online) Linear viscoelastic moduli G0 (upper, filled symbols) and G00 (lower, hollow symbols) as a function of pulsation ω for a strain amplitude of γ = 0.01. The symbols FIG. 2: (color online) Normalized stress responses (1−α)Γ(1− V α • and respectively correspond to the 4% wt. and 8% wt. α)σ(t)/ γ˙ 0 vs strain γ =γ ˙ 0t in the start up of steady shear N −1 casein gels. Dashed and continuous lines correspond to the flow experiments for shear rates ranging from 0.006 s to −1 best fit of the data with the spring-pot model. 3 s for a 8% wt. casein gel. The dashed gray line cor- responds to the linear response (Eqn. 2 in the main text) while the red solid line stands for the K-BKZ prediction con- structed from the power-law linear viscoelastic response plus Supplemental movie the nonlinear damping function h˜ determined independently from step strain experiment (see main text). Inset: non- Supplemental Movie 1 shows the failure of a 4% wt. normalized stress responses for a subset of shear start-up ex- casein gel acidified with 1% wt. GDL in a Couette ge- periments shown in the main graph. ometry for an imposed shear rateγ ˙ =10−3 s−1. The rhe- ological response recorded simultaneously corresponds to 6 that shown in Fig. 3 in the main text. The first two cracks 10 nucleate simultaneously at the top and bottom of the in- 5 ner cylinder with slightly different angular positions and 10 grow towards each other, stopping at the center of the 4 cell. Meanwhile, a second pair of cracks nucleates next 10 to the first one and grows in the same fashion before a

[s] 3 third pair develops, and so on. Such a failure front prop- f 10 τ agates along the cell perimeter while the stress decreases 2 in the third regime. 10 1 10

Supplemental figures 0

10 2 3 10 σ 10 As discussed in the main text we performed tests with 0[Pa] two protein gels: 4% casein-1% GDL and the 8% casein- 8% GDL gels which we henceforth refer to as 4% and 8% FIG. 3: (color online) Failure time τf as a function of the constant stress σ0 applied during creep experiments. The casein gels, respectively. −β solid line is the best power-law fit τf = Aσ , with A = Supplemental Figure 1 shows the frequency depen- (5.0 ± 0.1) × 1018 s.Paβ and β = 6.4 ± 0.1 for the 8% wt. gel. dence of the elastic and viscous moduli of both the 4% wt. and 8% wt. gels. Both gels display a power- law linear rheology that can be modeled by a spring- pot (or fractional) element, as reported in the main Supplemental Figure 2 shows the stress responses to text. The elastic and viscous modulus reads respectively shear start-up experiments at different shear rates, which G0(ω) = Vωα cos(πα/2) and G0(ω) = Vωα sin(πα/2), can be rescaled into a single master curve. The scaling with α = 0.18 0.02 and V = 261 5 Pa.s for the 4% of the failure time as a function of the applied stress, gel and α = 0.04± 0.01 and V = 620± 5 for the 8% gel. measured for creep experiments is plotted in Fig. 3. ± ±