Visualizing the Strain Field in Semiflexible Polymer Networks: Strain Fluctuations and Nonlinear Rheology of $F$-Actin Gels

We image semiflexible polymer networks under shear at the micrometer scale. By tracking embedded probe particles, we determine the local strain field, and directly measure its uniformity, or degree of affineness, on scales of $2–100\mu \mathrm{m}$. The degree of nonaffine strain depends upon the polymer length and cross-link density, consistent with theoretical predictions. We also find a direct correspondence between the uniformity of the microscale strain and the nonlinear elasticity of the networks in the bulk.

Figure 1

Particle position $y$ at height $z$, showing displacement in the shear direction for a typical sample. Different symbols represent different strain amplitudes. The inset shows a schematic of the setup. A shear cell consisting of a fixed top plate and a movable bottom plate is mounted on an inverted microscope.

Figure 2

(a) For a strongly cross-linked network ($c_A=1\mathrm{mg}/\mathrm{ml}$, $R=0.06$), the $K^{\prime }$ measured by 2P microrheology (lines) at different $\gamma$ (solid line: 0%; dashed line: 15%; dash-dotted line: 24%) is compared with that measured by bulk rheology (symbols) at different ${\sigma }_0$ (squares: 0.3 Pa; circles: 0.5 Pa; triangles: 0.75 Pa). (b) For a weakly cross-linked network ($c_A=0.5\mathrm{mg}/\mathrm{ml}$, $R=0.002$), the $K^{\prime }$ measured by 2P microrheology (lines) at different $\gamma$ (solid line: 0%; dashed line: 36%) is compared with that measured by bulk rheology (symbols) at different ${\sigma }_0$ (squares: 0; circles: 0.02 Pa). All data are taken at $25°\mathrm{C}$.

Figure 3

(a) Schematic summary of the physical picture described in the text. At $r<l_0$, the dense network and weak network exhibit two extreme behaviors of $N(r)$, scaling as $r^2$ and $r^0$, respectively. At $r>l_0$, the behavior is intermediate between the two extremes. The solid lines indicate the experimentally accessible region. The inset shows schematically the relative position of two particles after $\gamma$ is applied. The solid line is the actual separation while the dashed line is the separation if the strain were strictly affine. The deviation in angle is $\Delta \theta$. (b) $N(r)$ for $F$-actin networks with $L=10\mu \mathrm{m}$, and different cross-link density, $c_A=1\mathrm{mg}/\mathrm{ml}$, $R=0.06$ (squares, strain 18%–30%); $c_A=0.75\mathrm{mg}/\mathrm{ml}$, $R=0.01$ (circles, strain 24%–30%); $c_A=0.5\mathrm{mg}/\mathrm{ml}$, $R=0.002$ (triangles, strain 23%–28%). The solid line shows $\sim r^{0.6}$ and the dashed line shows $\sim r^{0.3}$. (c) $N(r)$ for $F$-actin networks with fixed cross-link density ($c_A=1\mathrm{mg}/\mathrm{ml}$, $R=0.06$), and different $L$, $10\mu \mathrm{m}$ (squares, strain 18%–30%), $1.6\mu \mathrm{m}$ (circles, 10%–18%) and $1.1\mu \mathrm{m}$ (triangles, 11%–20%). All data are taken at $25°\mathrm{C}$.

Figure 4

Relation of (a) scalar nonaffinity parameter $S$, and (b) exponent of $r$-dependence of $N(r)$ to nonlinear rheology for different relative microstructure parameters, $L/\lambda$. The stiffening (circles) and weakening (triangles) samples, distinguished by microrheology experiments shown in Fig. 2, are separated by the dotted line. For all samples, $\gamma$ ranges from 10%–30%. In the inset of (a), $\Delta \overrightarrow{r}$ is defined as the vector connecting the actual position of a single particle after shear (circle) and its position after a strictly affine shear (dotted circle). All data are taken at $25°\mathrm{C}$.