Figure 1

Dynamic shear moduli of a coarsening foam, in both the (a) frequency and (b) time domains; symbols in (a) denote the imposed strain amplitude. The dashed curves in (a) and (b) are a fit to $G^{*}(\omega )=G_o(1+\sqrt{i\omega /{\omega }_n})$ and $G(t)=G_{\circ }(1+1/\sqrt{\pi {\omega }_{n}t})$, predicted for nonaffine bubble motion in the absence of any time evolution. The solid curve in (b) is an empirical fit to $G(t)$; appropriately transformed, it gives the solid curves matching the storage and loss moduli in (a).Reuse & Permissions

Figure 2

(a) Stress relaxation moduli for foams sheared at various rates, as listed. This is given from the stress $\sigma (t)$ following the superposition of a step strain $\Delta \gamma$ on steady shear: $G(t,\dot{\gamma })=[\sigma (t)-\sigma (0)]/\Delta \gamma$. (b) The corresponding relaxation spectra, $H(t,\dot{\gamma })\approx dG(t,\dot{\gamma })/d\ln t$.Reuse & Permissions

Figure 3

(a) Transient shear modulus, (b) transient shear relaxation times, and (c) viscosity; symbols represent data as a function of shear strain rate; parallel horizontal lines in (a) and (b) represent value and uncertainty in the zero strain rate results. The vertical dotted lines spanning (a)–(c) denote the characteristic shear rates set by the yield strain divided, respectively, by the time between rearrangements and the duration of rearrangements in a quiescent foam. In (a), the solid curve is $G_0\exp [-\dot{\gamma }/(0.1{\mathrm{s}}^{-1})]$ where $G_0=2300\mathrm{d}\mathrm{y}\mathrm{n}/{\mathrm{c}\mathrm{m}}^2$ is the shear modulus. In (b), the open diamonds indicate where $H(t,\dot{\gamma })$ is a global maximum and plusses indicate where $G(t,\dot{\gamma })$ falls to $1/e$ of its initial value; the solid line is a power law with exponent of $-1/2$; the dashed curves indicate DWS time scales for rearrangements (${\tau }_o$) and shear (${\tau }_s$). In (c), power-law fits are shown by solid lines. The picture of “soft-glassy rheology” [19] predicts that power laws in $\eta (\dot{\gamma })$ and $G^{*}(\omega )$ be related [20].Reuse & Permissions