Surface Tension Controls the Hydraulic Fracture of Adhesive Interfaces Bridged by Molecular Bonds

Biological function requires cell-cell adhesions to tune their cohesiveness; for instance, during the opening of new fluid-filled cavities under hydraulic pressure. To understand the physical mechanisms supporting this adaptability, we develop a stochastic model for the hydraulic fracture of adhesive interfaces bridged by molecular bonds. We find that surface tension strongly enhances the stability of these interfaces by controlling flaw sensitivity, lifetime, and optimal architecture in terms of bond clustering. We also show that bond mobility embrittles adhesions and changes the mechanism of decohesion. Our study provides a mechanistic background to understand the biological regulation of cell-cell cohesion and fracture.

Figure 1

Adhesive interfaces formed by a tense membrane (here, a line with tension $T$) bridged by molecular bonds and enclosing a cavity at pressure $P$. Open bonds can stochastically attach depending on the separation $\delta$ to form a stochastic slip bond, whose off rate depends exponentially on force according to Bell’s model [24]. The system is periodic. Following Laplace’s law, each line segment between two neighboring closed bonds is an arc of circle or radius $R=T/P$. This allows us to compute the force on each closed bond $F$ and the distance between opposite potential partners $\delta$, which in turn determine the unbinding and binding rates $k_{\mathrm{off}}$ and $k_{\mathrm{on}}$. We denote by $b$ the spacing between bonds and by $W$ the size of a selected crack. In this model, increasing tension flattens the membrane over the crack, favoring rebinding, but does not affect the force on the closed bonds (top-right inset).

Figure 2

(a) Kymograph of a representative stochastic simulation, where closed bonds are marked in black and open bonds in white, for $\tilde{T}=4$, $\tilde{P}=0.1$. (b) Corresponding time evolution of the number of closed bonds $N_b$. (c) Critical crack size ${\tilde{W}}_{\text{crit}}$ as a function of the inverse pressure $1/\tilde{P}$ for different tensions $\tilde{T}$. Error bars throughout the Letter indicate the standard error of the mean (SEM), here for 2000 realizations. By plotting $\tilde{P}{\tilde{W}}_{\text{crit}}$ as a function of $\tilde{T}$, each of the curves collapses to a point (inset), where the dotted line is a theoretical prediction. (d) Kymographs describing trajectories of cracks at pressure $\tilde{P}=1/8$ and initial crack width ${\tilde{W}}_i$ indicated by dots in the ($1/\tilde{P}$,$\tilde{W}$) plane, where the red line indicates the boundary of stability ($\tilde{W}={\tilde{W}}_{\text{crit}}$) for $\tilde{T}=4$ and the purple line for $\tilde{T}=8$. (i) $\tilde{T}=4$, ${\tilde{W}}_i=7<{\tilde{W}}_{\text{crit}}$. (ii) $\tilde{T}=4$, ${\tilde{W}}_i=12>{\tilde{W}}_{\text{crit}}$, (iii) $\tilde{T}=8$, ${\tilde{W}}_i=12<{\tilde{W}}_{\text{crit}}$. (e) Logarithm of the average lifetime $\ln ({\tilde{t}}_f)$ as a function of $\tilde{P}$ for different tensions, same color code as in (c). The inset shows the relation between ${\tilde{t}}_f$ and $\tilde{P}$ in linear scale for $\tilde{T}=4$ and $\tilde{T}=8$. Strength is defined operationally as the critical pressure for which ${\tilde{t}}_f=100$. The inset shows that, for a given $\tilde{P}$, doubling tension can change the lifetime by orders of magnitude.

Figure 3

Influence of mobility of molecular bonds: Stochastic dynamics of a system with 50 potential bonds arranged initially in a regular lattice of spacing $b$, subjected to $\tilde{T}=8$ and $\tilde{P}=0.1$, with a precrack of width $W_i=16b$ (smaller than $W_{\text{crit}}\approx 19b$), where open bonds diffuse with diffusivity $D$ and closed bonds diffuse with $D/2$ biased by the lateral force they experience. Bonds cannot move past each other and cannot move closer than $b/8$. (a) Fraction of the 500 simulations leading to failure (red) or healing (green) after a period of 10 $t_0$ for each diffusion constant $\tilde{D}$. (b) Examples of trajectories leading to failure for each $\tilde{D}$, movies 1 and 2 [23]. (c) Relation between crack length $W$ and the number of open bonds at the crack $N_{\text{crack}}$, obtained by averaging 50 failure trajectories, for different diffusivities. For fixed molecules ($\tilde{D}=0$) the relation is linear with slope $b$, whereas for large diffusivity, crack length initially grows by motion of bonds at nearly constant $N_{\text{crack}}$ leading to a concentrated adhesion patch, and then by progressive bond breaking with slope $b/8$ (black line), indicative of a maximal bond packing.

Figure 4

(a) Sequence of architectures of an adhesive interface with increasing cluster size and cluster separation, ${\tilde{W}}_i$, but constant bond separation and average density. (b) Logarithm of lifetime ${\tilde{t}}_f$ of a cluster as a function of ${\tilde{W}}_i$ for different tensions $\tilde{T}$ and for $\tilde{P}=0.1$ (average lifetimes obtained from 500 simulations). Optimal cluster size is well approximated by ${\tilde{W}}_{\text{crit}}/2$. The inset shows the results for $\tilde{T}=8$ where regimes I, II, and III are highlighted. Filled stars, circles, and open stars denote regimes I, II, and III.