Compressive elasticity of polydisperse biopolymer gels

We theoretically predict the nonlinear elastic responses of polydisperse biopolymer gels to uniaxial compression. We analyze the competition between compressive stiffening due to polymer densification by out-going solvent flow and compressive softening due to continuous polymer buckling. We point out that the polydispersity in polymer lengths can result in an intrinsic, equilibrium mode of nonaffine compression: nonuniform strain but with uniform force distribution, which is found to be more energetically preferable than affine deformation. In this case, the gel softens significantly after the onset of polymer buckling at small compression, but as compression increases, densification-induced stiffening becomes important and a modulus plateau should be observed for a large range of strain. We also relate our results to recent compression experiments on collagen gels and fibrin gels.

Figure 1

Schematic illustrations of a polydisperse biopolymer gel of zero Poisson ratio under uniaxial compression: reference state (a) and deformed state (b). ${\ell }_c$ and ${\ell }_c^{\prime }$ denote the undeformed (contour) length and the deformed length of polymer segments between cross-linkers, respectively. ${\xi }_{\mathrm{eff}}$ and ${\xi }_{\mathrm{eff}}^{\prime }$ denote the effective mesh size of the undeformed and deformed gel, respectively. ${\sigma }^g$ ($<0$ for compression) is the external stress applied on the gel and ${\epsilon }^g$ ($<0$ for compression) is the global strain of the gel due to the externally applied force.

Figure 2

Rod gels and semiflexible polymer gels undergoing affine deformation. The differential Young's modulus $E$ (normalized by the linear shear modulus $G_0^A$) as a function of compressive strain $|{\epsilon }^g|$ for monodisperse gels (dotted line) and for polydisperse gels (solid line, with $\mathcal{R}=0.6$). Two reference cases are also plotted: gels without buckling-induced softening (dash-dotted line) and gels without densification-induced stiffening (dashed line). The polymer buckling occur at $|{\epsilon }_c^g|$ (here chosen to be 0.1). For polydisperse gels, all the polymers buckle at $|{{\epsilon }_c^g}^{\prime }|$. Inset: The fraction $\alpha$ of buckled polymer is plotted as a function of $|{\epsilon }^g|$.

Figure 3

Rod gels and semiflexible polymer gels undergoing nonaffine deformation with uniform force distribution. The differential Young's modulus $E$ (normalized by the linear shear modulus $G_0^F$) as a function of compressive strain $|{\epsilon }^g|$ for monodisperse gels (dotted line) and for polydisperse gels (solid line, with $\mathcal{R}=0.6$). Note that the modulus-strain curve of polydisperse rod gels almost overlap completely with polydisperse semiflexible polymer gels. Two reference cases are also plotted: gels without buckling-induced softening (dash-dotted line) and gels without densification-induced stiffening (dashed line). The polymer buckling occur at $|{\epsilon }_c^g|$ (chosen to be 0.1). $|{{\epsilon }_c^g}^{\prime }|$ is large than 1, i.e., complete buckling does not occur in this case. Inset: The fraction $\alpha$ of buckled polymer is plotted as a function of $|{\epsilon }^g|$.

Figure 4

Comparison of the strain energy of three deformation modes for bidisperse (semiflexible polymer) gels before the presence of polymer buckling (with $\mathcal{R}=0.6$ and $\mathcal{P}=0.5$). Inset: Energy comparisons for bidisperse rod gels in the absent of polymer buckling. The affine deformation and the nonaffine deformation with uniform compressive force (NA-Force) are found to be identical for rod gels at unbuckled states. In both rod gels and semiflexible polymer gels, NA-Force is more energetically preferable than affine deformation.

Figure 5

Bidisperse rod gels and bidisperse semiflexible polymer gels undergoing affine and nonaffine deformation with uniform force distribution. The differential Young's modulus $E$ (normalized by the linear shear modulus $G_0$) as a function of compressive strain $|{\epsilon }^g|$. The polymer buckling occur at $|{\epsilon }_c^g|$ (chosen to be 0.1).

Figure 6

Energy distribution over polymer length for partially buckled gels: (a) energy distribution function (per length) $f_{\mathrm{rod}}$ of rod gels and (b) $f_{\mathrm{polymer}}$ of semiflexible polymer gels. Strain energy is found to be concentrated in buckled polymers in all cases considered here. Here we have taken $\mathcal{R}=0.6$, ${\epsilon }_c^g=-0.1$, and ${\epsilon }^g=-0.3$.