Figure 4.31 shows the distribution of stresses through the system. It was calculated by summing up the elastic and capillary forces transferred over a set of vertical and horizontal cuts through the simulation box. This enabled us to measure the average values of the stress tensor and their spatial distribution through the system.
As can be observed here, the distributions widen when the capillary strength β is increased, meaning that absolute stresses grow larger locally. While the shear stress remains low on average, the average normal stress becomes more negative when the capillary action becomes stronger. This means that the capillary forces drive network to shrink, although actual shrinkage is prevented by the periodic boundary conditions.
0 5 10 15 20 25
-0.8 -0.6 -0.4 -0.2 0 0.2
relative frequency
(a)
0 5 10 15 20 25
-0.4 -0.2 0 0.2 0.4
relative frequency
(b)
Figure 4.31: Histograms of the normal (a) and shear (b) components of the stress tensor in a network of density q = 9, for varying capillary strengths β.
Conclusions and Outlook
5.1 Conclusions
We set out in this work to develop a model that enables us to analyze the effect that a wetting liquid has on the morphology and structure of a disordered network of elastic fibers. As a starting point, we considered the problem of finding the interfacial shape of a liquid condensing at the crossing point of two smooth and straight cylinders, in chapter 2. In an analytical approach to the problem, we considered two different approximations. The first is based on the theory of Princen [Pri69a, Pri69b, Pri70] and is valid for small crossing angles of the cylinders. The second is valid for arbitrary crossing angles, but only for large negative Laplace pressures. We verified that the asymptotic case of small Laplace pressures is reproduced by the Princen theory, and derived the scaling exponents for the interfacial free energy, the liquid volume, the Laplace pressure, the wetted length and the torque as a function of the crossing angle.
We verified these scaling behaviors in numerical minimizations of the free energy, and compared the scaling coefficients derived from numerics and from analytical theory.
As a result, we are able to characterize the capillary torque that arises between the two fibers as a power law with an exponent that depends solely on the thermodynamic ensemble of the wetting problem, and that is valid for a very wide range of crossing angles.
In order to describe the elastic network, we amended the well-known Mikado model to the case of diluted crosslinks. In this way, we are able to allow for rearrangements and transport of liquid through the network. Since this amendment introduces an additional parameter (the crosslinking probability pl), we investigated the influence of this parameter on the elastic properties of a diluted Mikado network. To do this, we implemented the Pebble Game
algorithm by Jacobs and Thorpe [JT95] to investigate the topological rigidity of diluted Mikado networks. In this way, we were able to identify rigidly connected clusters. We could reproduce the literature results for the undiluted case, and found a shift in the rigidity percolation with pl, accompanied by a qualitative change in the topological properties of rigid clusters in the network.
With these ingredients, we were able to develop a new model suitable for computer simulations of wet elastic networks. In this model, fibers are elasti-cally deformed by liquid bridges at the contact points, which leads to capillary cohesion of the fibers. This is a very effective way to simulate wet fiber sys-tems, since the effect of the wetting liquid is accounted for by only a small number of discrete liquid bridge objects, obviating the need to simulate the fluid dynamic problem of the liquid in parallel.
We had to invest some effort in order to arrive at such a model in which the translational forces and the torques behave consistently. Since the capillary action is governed by the crossing angles, both aligning torques and transla-tional forces arise, the latter being induced by the curvatures of the fibers.
We therefore introduced an interpolation scheme to smear out the curvature over the whole of the discretized fiber length, and calculated the resulting forces. Additional force terms arise when liquid bridges lose contact and be-come stretched. It is of great importance to ensure proper tracking of the loci of liquid bridges, and to account for formation and rupture events. Doing this in a computationally efficient way is the key point to applying this kind of model. While models for wet systems of spheres that take a wetting liquid into account by means of liquid bridge objects are already widely used, the equivalent description for elongated elastic fibers is, to our knowledge, new, and is the main achievement of this work.
Applying this model to Mikado networks, we found that for low interfa-cial tensions and high elastic moduli of the fibers, the distribution of crossing angles in the system is changed slightly. Once the capillary interaction be-comes stronger and crosses a threshold, cooperative effects set in, in which the aligning action of the liquid bridges leads to new fiber crossings, and new liquid bridges appear as a consequence. This leads to bundling of the fibers mediated by large accumulations of liquid bridges, and consequently to the growth of pores in the network. The latter effect is interesting for many appli-cations such as aerosol, where the pore structure influences the efficient, and composite materials, where it influences the mechanical properties.
The threshold value of the capillary interaction strength depends on the
concentration of fibers in the network. Our investigations show that even below the rigidity percolation point, the presence of liquid bridges organizes the fibers into larger cell structures that give the network a higher rigidity, as evidenced by the fact that the point where vanishingly small capillary forces do not start the cooperative process of bundling anymore lies in the interval between the point of connectivity percolation and that of rigidity percolation.
For the limit case of large interfacial tensions and low elastic moduli, we found that the process of bundling and pore growth is limited by the internal structure of the network, and identified an upper limit of the pore size given by the mesh of connected loops.