Discontinuous bundling transition in semiflexible polymer networks induced by Casimir interactions

Fluctuation-induced interactions are an important organizing principle in a variety of soft matter systems. We investigate the role of fluctuation-based or thermal Casimir interactions between cross linkers in a semiflexible network. One finds that, by integrating out the polymer degrees of freedom, there is an attractive logarithmic potential between nearest-neighbor cross linkers in a bundle, with a significantly weaker next-nearest-neighbor interaction. Here we show that a one-dimensional gas of these strongly interacting linkers in equilibrium with a source of unbound ones admits a discontinuous phase transition between a sparsely and a densely bound bundle. This discontinuous transition induced by the long-ranged nature of the Casimir interaction allows for a similarly abrupt structural transition in semiflexible filament networks between a low cross linker density isotropic phase and a higher cross link density bundle network. We support these calculations with the results of finite element Brownian dynamics simulations of semiflexible filaments and transient cross linkers.

Figure 1

(a) A typical realization of a semiflexible gel. The fluctuations of a given filament (blue) are reduced by the cross linking (orange linkers) between it and the network, which is itself a cross linked gel (red linkers). (b) Finite element simulation of bundle formation via the Casimir-induced linker condensation: doubly (singly) bound linkers shown in red (blue). (c) Finite element-based test of Casimir interactions between three linkers (triangles) on one filament.

Figure 2

Comparison of theoretical ($▴$) Fourier displacement mode amplitudes for a filament hinged at both ends to those computed ($★$) from finite element simulations for time discretizations $\mathrm{\Delta }t={10}^{-2}\mathrm{s}$, ${10}^{-4}\mathrm{s}$, and ${10}^{-6}\mathrm{s}$, shown in (a), (b), and (c), respectively. Large (small) time steps are necessary to sample slow (fast) modes, and collectively the chosen time steps allow one to accurately examine the first 30 modes.

Figure 3

The Fourier amplitude covariance matrix $\langle A_n{A}_m\rangle$ for the first 10 displacement modes $A_n$ from simulation of a filament pinned at $D=0.4L$. The area and shape of each element represents its log-normalized magnitude and sign (square $+$ vs circle -), respectively. The color bar indicates the percent error relative to Eq. (6). The error magnitude is consistent across different pinning locations.

Figure 4

The equation of state of the 1D Casimir gas for different values of the interaction strength $d_{\perp }\alpha$. The dashed lines representing solutions to Eq. (17) agree with Monte Carlo simulations (points). At high densities, the hard-core repulsion dominates forcing the equation of state to approach that of the Tonk's gas. Inset: The Casimir interaction dramatically reduces the pressure at low densities. For $d_{\perp }\alpha >2$ it vanishes for $\rho <{\rho }_{\mathrm{crit}}$.

Figure 5

Langmuir isotherms of a Casimir gas in equilibrium with an ideal gas of cross linkers at fixed chemical potential. The points represent results from grand canonical Monte Carlo simulations of $N=10000$ particles adsorbing onto a line and interacting with the hard-core Casimir potential. For $d_{\perp }\alpha >2$ the system undergoes a condensation transition in the thermodynamic limit: The cross-linkers spontaneously condense to the critical density ${\rho }_{\mathrm{crit}}=\frac{d_{\perp }\alpha -2}{d_{\perp }\alpha -1}$ at ${\mu }_{\mathrm{crit}}=k_{\mathrm{B}}Tlog\left[\frac{{\lambda }_c}{{\lambda }_t}\left(d_{\perp }\alpha -1\right)\right]$. Note the softening of the transition for $d_{\perp }\alpha =4$ due to finite size effects.

Figure 6

Theoretical and simulated Langmuir isotherms of linkers adsorbed onto a filament bundle. ($★$) Simulation data for a $5\mu \mathrm{m}$ bundle with persistence length $l_p=9.2\mu \mathrm{m}$. ($▴$) Both the system size and the persistence length doubled to study finite size scaling effects. The transition is noticeable sharper and is approaching the expected thermodynamic limit.