Markov Models of Coarsening in Two-Dimensional Foams with Edge Rupture

Abstract

We construct Markov processes for modeling the rupture of edges in a two-dimensional foam. We first describe a network model for tracking topological information of foam networks with a state space of combinatorial embeddings. Through a mean-field rule for randomly selecting neighboring cells of a rupturing edge, we consider a simplified version of the network model in the sequence space $\ell _1({\mathbb {N}})$ which counts total numbers of cells with $n\ge 3$ sides (n-gons). Under a large cell limit, we show that number densities of n-gons in the mean field model are solutions of an infinite system of nonlinear kinetic equations. This system is comparable to the Smoluchowski coagulation equation for coalescing particles under a multiplicative collision kernel, suggesting gelation behavior. Numerical simulations reveal gelation in the mean-field model, and also comparable statistical behavior between the network and mean-field models.

Typical and Atypical Reactions

Fig. 12

Typical and atypical reactions. The rupturable edge in each diagram is the horizontal edge in the center of the diagram. An edge with a $\circ$ symbol in its center denotes an isthmus. A vertical edge with W denotes that a vertex of the rupturable edge is contained on a wall. For the 3-isthmus, 3-isthmus+wall, and 5-isthmus, a square with $L_i$ for $i = 1,2$ denotes a left arc connected to an isthmus with $L_i$ sides. For isthmus and isthmus+wall, the square with j denotes a j-gon connected to an isthmus, and the square with L denotes a left arc containing L sides (including the two contained in the k-gon and connected to the isthmus)

By removing the condition in Definition 6 that a rupturing edge must be typical, we can consider the broader collection of atypical configurations and their corresponding reactions. A diagram of the thirteen different local configurations and the twelve different reactions for typical and atypical edges are given in Fig. 12. For some of these reactions, there are cells which undergo both edge and face merging, so for simplicity the collection of reacting cells and their products are listed as a single reaction. For each reaction listed, we assume a sufficient number of sides in each reactant cell so that all products have at least three sides. The set of atypical edges includes isthmuses, whose rupture disconnects the foam. If we wished to continue rupturing after rupturing an isthmus, it would be necessary to relax the requirement of connectivity in a simple foam, which in turn would further increase possible reactions. Even more reactions are possible by permitting foams to include loops (1-gons) and multiedges (2-gons). For now, we withhold from enumerating this rather complicated set of reactions.

We now give an informal derivation for how the enumeration in Fig. 12 is obtained. This is done by counting reactions in configurations arising from whether a rupturable edge $e = \{u_0, v_0\}$ or its neighbors are isthmuses. We begin by considering configurations with no isthmuses. We have already discussed the three typical reactions (10)–(12). There is also the possibility that an interior edge e contains two edge neighbors and a single vertex neighbor containing both $u_0$ and $v_0$. This cell wraps around several other cells to contain both vertices, so we call such a configuration a halo.

If e is not an isthmus, it is possible for either one or two incident edges to be isthmuses, but no more. This follows from the fact that if two isthmuses are incident to a vertex, then the third incident edge must be an isthmus as well. This creates four possible configurations: two containing one isthmus neighbor with or without a vertex contained on the boundary, and another containing two isthmus neighbors (both of which producing the same reaction $C_i+C_j \rightharpoonup C_{i+j-6})$. Since the original edge is not an isthmus, each of these configurations after rupture remains connected.

We finally consider the set of configurations for when e is an isthmus. If no other edges are isthmuses, then e can be in the interior of S or have a single vertex in $\partial S$ (two such vertices on $\partial S$ would imply that e is not an isthmus). One or both of $u_0$ or $v_0$ can have all of its incident edges as isthmuses. If one vertex of e has three incident isthmuses, then the other vertex can either be on $\partial S$, or have one or three incident isthmuses. In total, there are five different reactions with e as an isthmus.

Some care is needed when counting the products for reactions with isthmuses. Under the left path interpretation for face sides, isthmuses count for two sides. Additionally, the rupture of an isthmus will disconnect the network. This results in the creation of a new ‘island’ cell with a left path of exterior edges around the island, which are also removed from the cell originally contained ruptured isthmus e. In all reactions, the change in total number of sides is given by the number of boundary vertices in e minus six.

In each atypical reaction, the process of edge removal and insertion is indeed the same as typical reactions. Updates for left loops in the combinatorial foam are more complicated, and will depend on the local configuration. As an example, let us consider the isthmus neighbor configuration, which has a single vertex neighbor $f_1$ and two edge neighbors $f_2, f_3$. We write the left loops of these neighbors as

$$\begin{aligned} f_1 =&[ v_1, v_0,v_2, A_2], \quad f_2 = [ v_2, v_0, u_0, u_2, A_1], \end{aligned}$$

(46)

$$\begin{aligned} f_{3} =&[ u_1, u_0, v_0,v_1, A_3, u_2, u_0, u_1, A_4], \end{aligned}$$

(47)

where $\{u_0, u_1\}$ is an isthmus, and $A_1, \ldots , A_4$ are left arcs. After rupture, there are two cells remaining, with left loops

$$\begin{aligned} f_2' =[ v_1, v_2, A_2] , \quad f_{2,3}' = [ u_1, u_2, A_1, v_2, v_1,A_3, u_2, u_1, A_4]. \end{aligned}$$

(48)