Power-law creep and residual stresses in a carbopol gel

Abstract

We report on the interplay between creep and residual stresses in a carbopol microgel. When a constant shear stress σ is applied below the yield stress σ _y, the strain is shown to increase as a power law of time, γ(t) = γ ₀ + (t/τ)^α, with an exponent α = 0.39 ± 0.04 that is strongly reminiscent of Andrade creep in hard solids. For applied shear stresses lower than some typical value σ _c ≃ 0.2σ _y, the microgel experiences a more complex, anomalous creep behavior, characterized by an initial decrease of the strain, that we attribute to the existence of residual stresses of the order of σ _c that persist after a rest time under a zero shear rate following preshear. The influence of gel concentration on creep and residual stresses are investigated as well as possible aging effects. We discuss our results in light of previous works on colloidal glasses and other soft glassy systems.

Appendices

Appendix A: Rheological characterization of carbopol ETD 2050 samples

Flow curve measurement

The flow curve of our 1 % wt carbopol microgel recorded through a downward shear-rate sweep is shown in Fig. 10 together with the best HB fit, $\sigma =\sigma _{\text {y}}+k\dot {\gamma }^{n}$, yielding σ _y = 10.2 Pa, n = 0.60, and k = 3.3 Pa.s ⁿ. Let us emphasize here our method for estimating the yield stress σ _y. Instead of fitting the flow curve with the full HB model that involves three free parameters, which can raise some convergence issues due to nonlinearity depending on the fitting algorithm and on the initial guess for the parameters (Mullineux 2008), we first set σ _y to some arbitrary value and compute the best linear fit of ln(σ−σ _y) vs. $\ln \dot {\gamma }$. Fitting a straight line in logarithmic coordinates rather than fitting σ−σ _y vs. $\dot {\gamma }$ as a power law allows us to give the same weight to small shear rates and to larger ones. We compute the residuals χ ² of this fit, defined as the sum of the squared distances from the experimental data to the fit. We then vary σ _y systematically over a realistic range and look for a minimum of χ ² vs. σ _y. As shown in the upper right inset of Fig. 10, χ ² goes through a well-defined minimum: the stress σ _y = 10.2 Pa corresponding to this minimum is thus taken as the yield stress of our microgel. It can be checked in the upper left inset of Fig. 10 that with this value of σ _y, the difference σ−σ _y follows a strict power law of $\dot {\gamma }$ with no significant deviation over the whole range of shear rates. This method allows us to confidently estimate the yield stress of our microgels to within 1 % for a given flow curve measurement. In the case of Fig. 10, this procedure yields σ _y = 10.2 Pa, n = 0.60, and k = 3.3 Pa.s ⁿ. We note that a direct HB fit of the same data with three free parameters leads to significantly different estimates (σ _y = 9.0 Pa, n = 0.55, and k = 4.5 Pa.s ⁿ), which illustrates the importance of using a careful fitting procedure as described here.

Fig. 10

Flow curve, shear stress σ vs. shear rate $\dot {\gamma }$, of a 1 % wt carbopol microgel measured in a sand-blasted cone-and-plate geometry by decreasing the shear rate in logrithmically-spaced steps of 5 s each. The red solid line is the best Herschel-Bulkley fit, $\sigma =\sigma _{\text {y}}+k\dot {\gamma }^{n}$, with σ _y = 10.2 Pa, n = 0.60, and k = 3.3 Pa.s ⁿ, inferred from the minimization procedure described in the text. The upper left inset shows σ−σ _y with σ _y = 10.2 Pa as a function of $\dot {\gamma }$ in logarithmic scales. The upper right inset shows the residuals χ ² of the best power-law fit of σ−σ _y vs $\dot {\gamma }$ when varying the value of the yield stress σ _y (see text)

Still, it should be noted that we observe a reproducibility of the best HB fit parameters of only about 10 %. For instance, for the same preparation batch, we found values of σ _y ranging from 9.5 to 10.5 Pa from one loading of the cone-and-plate geometry to the other. Since the flow curves are measured through decreasing shear rate sweeps, the loading history is efficiently erased at high shear rates and such variations can only be explained by small differences in the sample volume from one loading to the other. We checked that the various flow curves are simply shifted along the stress axis from one measurement to the other and that the scatter is typically 1 Pa, which confirms that variations in HB parameters essentially stem from variations in the loaded sample volume.

Linear and nonlinear viscoelastic measurements

The linear viscoelastic moduli of our microgels are shown as a function of oscillation frequency f in Fig. 11a. The storage modulus G ^′(f→0) = 37 Pa increases weakly over the whole frequency range. Here again, we observe variations of G ^′ by about 10 % from one loading to the other. The loss modulus G ^″ remains always smaller than G ^′ and can be well fitted by a power law G ^″∼f ^0.44 at high frequencies. The oscillatory strain sweep of Fig. 11b shows that the linear regime, characterized by an elastic modulus G ₀ = 34 Pa, extends up to strain amplitudes of about 100 %. The nonlinear regime involves a sharp drop of the storage modulus and a local maximum in the loss modulus. This corresponds to a case of “weak strain overshoot” as classified by Huyn et al. in their review on large-amplitude oscillatory shear (Hyun et al. 2011) and appears as a distinctive feature of soft glassy materials, including microgels (de Souza Mendes et al. 2014). In systems like emulsions and microgels, the maximum in G ^″ is generally attributed to enhanced dissipation due to local irreversible particle rearrangements that progressively invade the whole sample before yielding and complete fluidization (Mason et al. 1995; Knowlton et al. 2014). We note that the point at which G ^′ and G ^″ cross corresponds to a stress of about 36 Pa, significantly above the yield stress measured from the flow curve in Fig. 10. Since this article is mostly devoted to creep experiments performed below the yield stress, where the strain shall not increase beyond 100 %, we do not expand more on the nonlinear behavior of our carbopol ETD 2050 samples and refer the reader to the cited literature for more details.

Fig. 11

a Elastic modulus $G^{\prime }$ (red) and viscous modulus $G^{\prime \prime }$ (blue) vs. frequency f after a rest time t _w = 300 s following preshear under a constant shear rate $\dot {\gamma }_{\text {p}}=100~\mathrm {s}^{-1}$ for a duration t _p = 60 s. The strain amplitude is fixed to γ = 1 % and the frequency is logarithmically swept down. The orange solid line shows that $G^{\prime \prime }\sim f^{0.44}$ for $f\gtrsim {0.2}~{\text {Hz}}$. b $G^{\prime }$ (red, left axis), $G^{\prime \prime }$ (blue, left axis) and stress amplitude σ (black, right axis) vs strain amplitude γ after the same preparation protocol as in a. The frequency is fixed to f = 1 Hz and the strain amplitude γ is logarithmically swept up with a duration of 16 s per point. One has $G^{\prime }(\gamma ^{*})=G^{\prime \prime }(\gamma ^{*})$ for γ ^∗≃200 % which corresponds to σ ^∗≃36 Pa. The red solid line is the best linear fit, σ = G ₀ γ for γ < 80 %, leading to G ₀ = 34 Pa. The dotted line indicates the end of the linear regime at γ ≃ 80 %

Appendix B: Influence of the carbopol concentration

The robustness of our findings has been tested by considering two other concentrations of carbopol ETD 2050, namely C = 0.6 % wt and 2 % wt. As shown in Fig. 12a for similar normalized stresses, σ/σ _y≃0.4, power-law creep is observed in all three samples. Table 1 gathers the rheological parameters of the various samples as well as the results of the analysis of strain responses in the Andrade-like regime as described in “Creep dependence on the applied stress”. We find a remarkably robust mean Andrade exponent of α ≃ 0.4 for all concentrations. Here again, the prefactor $G^{\prime }_{0}$ deduced from the initial elastic deformation γ ₀ is in good agreement with the elastic modulus G ^′. The parameter σ ₀ is non-zero and increases with C in the same fashion as the elastic modulus and the yield stress. This suggests that residual stresses also come into play for C = 0.6 % wt and 2 % wt.

Fig. 12

Creep and residual stresses in carbopol microgels with concentrations C = 0.6 % wt (blue), 1 % wt (black), and 2 % wt (red). a Strain responses γ(t)−γ ₀ for σ ≃ 0.4σ _y plotted in logarithmic scales together with Andrade fits (yellow solid lines): [C, σ/σ _y, γ ₀, α] = [0.6 % wt, 0.38, 9.9 %, 0.40], [1 % wt, 0.36, 8.9 %, 0.37], [2 % wt, 0.35, 16.2 %, 0.40]. b Strain responses γ(t) for σ/σ _y≤0.1 plotted in semilogarithmic scales for [C, σ/σ _y] = [0.6 % wt, 0.04], [1 % wt, 0.04], [2 % wt, 0.1]. c Residual stress σ _r (after a relaxation over t _w = 600 s) as a function of the preshear rate $\dot {\gamma }_{\text {p}}$ for t _p = 60 s. Solid lines are power laws with exponent −0.2

Table 1 Elastic modulus $G^{\prime }$ and yield stress σ _y (see Appendix A), average Andrade exponent 〈α〉 and fitting parameters $G^{\prime }_{0}$ and σ ₀ (see “Analysis of a typical creep test”), and residual stress σ _r measured 600 s after preshear at $\dot {\gamma }_{\text {p}}=100~\mathrm {s}^{-1}$ (see “Evidence for residual stresses”) for carbopol microgels with different concentrations C

Indeed, for applied stresses such that σ/σ _y < 0.1, the same decreasing trend is observed in the strain response for all three concentrations (see Fig. 12b). The presence of residual stresses for C = 0.6 % wt and 2 % wt is further confirmed through flow cessation experiments performed as in “Evidence for residual stresses”. Figure 12c shows that σ _r decreases roughly as $\dot {\gamma }_{\text {p}}^{-0.2}$ for all concentrations with an amplitude that scales with C like G ^′ and σ _y (see Table 1). We conclude that a similar interplay between creep and residual stress occurs at low stresses whatever the carbopol concentration.

Appendix C: Influence of the rest protocol

In order to test the influence of the rest protocol, Fig. 13 compares creep experiments following a rest period performed either under a zero shear stress or under a zero shear rate (or more precisely under small strain oscillations with amplitude 1 % and frequency 1 Hz, which were checked to effectively correspond to a zero shear rate). As shown in Fig. 13a for σ = 5 Pa, the Andrade exponent is robust to a change of rest protocol but the initial deformation γ ₀ is slightly smaller when rest is imposed under a zero shear rate. Moreover, when the imposed stress is decreased to σ = 0.4 Pa, anomalous creep characterized by a decreasing strain is recovered only in the case of rest under a zero shear rate (see Fig. 13b). In the case of rest performed under a zero shear stress, Andrade-like response persists down to the lowest imposed stresses as shown by the fit in Fig. 13b.

Fig. 13

Influence of the rest protocol on creep in a 1 % wt carbopol microgel. Following preshear at $\dot {\gamma }_{\text {p}}=100~\mathrm {s}^{-1}$ for t _p = 10 s, a rest time of t _w = 300 s is imposed either under a zero shear rate (black circle) or under a zero shear stress (white square) on the same loading of the cone-and-plate geometry. a Strain responses γ(t)−γ ₀ for σ = 5 Pa plotted in logarithmic scales together with Andrade fits (red solid lines) yielding respectively γ ₀ = 11.4 % and α = 0.38 for rest under a zero shear rate and γ ₀ = 13.4 % and α = 0.35 for rest under a zero shear stress. b Strain responses γ(t) for σ = 0.4 Pa plotted in semilogarithmic scales. The red solid line is the Andrade fit in the case of rest under a zero shear stress with γ ₀ = 1.2 % and α = 0.33

Andrade fit parameters are displayed in Fig. 14 which confirms the good collapse of the exponents α independent of the rest protocol, provided the imposed stress is large enough and anomalous creep is avoided. The linear fits of γ ₀ vs. σ show that the difference noted above in the initial deformation does not stem from the slope $G^{\prime }_{0}$ which remains close to the elastic modulus of the microgel but from the intercept σ ₀ which is significantly larger in the case of rest under zero shear rate (0.7 Pa) than in the case of rest under zero shear stress (0.2 Pa). These results indicate that imposing a zero shear stress during the rest time essentially cancels out residual stresses so that power-law creep is observed even at very low applied stresses.

Fig. 14

Andrade fit parameters for the two rest protocols used in Fig. 13 on the same loading (black circle zero shear rate, white square zero shear stress): a exponent α and b initial elastic deformation γ ₀, as a function of the applied stress σ. The vertical dashed line shows the typical stress of 2 Pa below which anomalous creep is observed when rest is imposed at a zero shear rate. Solid lines in b are linear fits $\sigma =G^{\prime }_{0}\gamma _{0}+\sigma _{0}$ with $G^{\prime }_{0}={26.3}~\text {Pa}$ and σ ₀ = 0.7 Pa for rest under a zero shear rate (red line) and $G^{\prime }_{0}={27.0}~\text {Pa}$ and σ ₀ = 0.2 Pa for rest under a zero shear stress (blue line)