Porosity Governs Normal Stresses in Polymer Gels

When sheared, most elastic solids including metals, rubbers, and polymer gels dilate perpendicularly to the shear plane. This behavior, known as the Poynting effect, is characterized by a positive normal stress. Surprisingly, fibrous biopolymer gels exhibit a negative normal stress under shear. Here we show that this anomalous behavior originates from the open-network structure of biopolymer gels. Using fibrin networks with a controllable pore size as a model system, we show that the normal-stress response to an applied shear is positive at short times, but decreases to negative values with a characteristic time scale set by pore size. Using a two-fluid model, we develop a quantitative theory that unifies the opposite behaviors encountered in synthetic and biopolymer gels.

Figure 1

Normal stress difference $N_1=2F/\pi R^2$, where $F$ is the normal force (thrust) reported by the rheometer and $R$ is the sample radius, as a function of the amplitude of the applied oscillatory shear strain. (a) $N_1$ for PAAm [9] prepared with various ratios of monomer-to-cross-linker concentrations. The line indicates a quadratic dependence of $N_1\sim {\gamma }^2$, as expected from the Mooney-Rivlin model [4, 5]. (b) $N_1$ shown for fibrin gels polymerized at $22°\mathrm{C}$ at various fibrinogen concentrations (in $\mathrm{mg}/\mathrm{mL}$). The line indicates a $\sim {\gamma }^2$ dependence, but with negative sign.

Figure 2

(a),(b) Fluorescence confocal microscopy images of fibrin networks whose pore size is tuned by polymerizing under different conditions, at (a) $22°\mathrm{C}$ and (b) $27°\mathrm{C}$. The scale bars are $10\mu \mathrm{m}$. Protein content is ($8\mathrm{mg}/\mathrm{mL}$) in both samples. (c) Normal stress ${\sigma }_N$, given by the apparent normal-stress difference $2F/\pi R^2$ obtained from the rheometer thrust $F$, for four fibrin networks differing in pore size as a function of time after the application of a constant shear stress at $t=0$. The stress relaxation curves are fitted to an exponential decay derived from the two-fluid model in [9] (black lines). The viscoelastic time scale is unrelated to the normal-stress transition, as shown in the inset where the storage moduli (filled symbols) and the loss moduli (open symbols) of the gels are plotted. (d) Schematic representation of the two-fluid model showing an inward, radial contraction of the network (black) relative to the solvent (blue) upon shearing.

Figure 3

Normal stress ${\sigma }_N$, given by the apparent normal stress difference $N_1^{(\text{app})}=2F/\pi R^2$ reported by the rheometer, for a fibrin gel polymerized at $27°\mathrm{C}$ in response to an oscillatory shear stress at oscillation frequencies of 0.001 Hz (left), 0.01 Hz (middle), and 1 Hz (right). The top panels [(a)–(c)] show the Lissajous curves of normal stress versus shear stress (symbols) fitted by the predictions of the two-fluid model in Eq. 2 (red lines) assuming a time constant of 12.5 s. The bottom panels [(d)–(f)] show the corresponding time-dependent normal stress (blue open symbols) and applied shear stress (black dotted lines). The data shown at $\nu =1\mathrm{Hz}$ represent averages with standard deviations obtained by averaging over 34 cycles to compensate for the low sampling frequency of the rheometer. The normal stresses are all negative since they correspond to steady-state values, obtained after initial relaxation (Fig. S2 in [9]).