Supplementary Figure 4: Local elasticity measurements of polyacrylamide gels using ferrofluid droplets.

(a) Measurements of the apparent elasticity E_A for three polyacrylamide gels. The apparent elasticity, given by E_A= (5/12) (k_d+k), combines the material's elastic modulus and the capillary stress (Supplementary Note). Its value is measured by applying a uniform and constant magnetic field to ferrofluid droplets (Supplementary Movie 2) injected into polyacrylamide gels and fitting their equilibrium strain the theoretical predictions from the 1D description (Supplementary Note). Unlike the elastic modulus of the gel, the capillary stress depends on the droplet size (∼γ/R). Therefore, injecting droplets with different radius R and measuring the apparent elastic modulus E_A allows the decoupling of their contributions. To do so, we measured the apparent elasticity of gels with three different elastic moduli (magenta triangles, blue circles and black squares) using droplets with different radius R, thereby changing the scale of capillary stress γ/R. The elastic modulus of the gel surrounding the droplet was obtained by fitting the data using a linear relation (magenta, blue and black lines) and extrapolating this relation for vanishing capillary stress (vanishing 1/R; large radius). The data points in the limit of vanishing $1/R$ (magenta pentagon, blue inverse triangle and black diamond) are measurements using a parallel plate rheometer, which were not included in the fit. (b) Comparison of the values of the elastic modulus measured using ferrofluid droplets and those obtained using a parallel plate rheometer. The local measurement of the elasticity using the ferrofluid droplets agrees well with the bulk measurement technique. The error bars in the elastic modulus E measured using the rheometer represent one standard deviation (s.d.), and the error bars in the elastic modulus E measured using ferrofluid drops correspond to the error associated to the linear fit as 1/R approaches zero.