Jamming of Deformable Polygons

We introduce the deformable particle (DP) model for cells, foams, emulsions, and other soft particulate materials, which adds to the benefits and eliminates deficiencies of existing models. The DP model combines the ability to model individual soft particles with the shape-energy function of the vertex model, and adds arbitrary particle deformations. We focus on 2D deformable polygons with a shape-energy function that is minimized for area $a_0$ and perimeter $p_0$ and repulsive interparticle forces. We study the onset of jamming versus particle asphericity, $\mathcal{A}=p_0^2/4\pi a_0$, and find that the packing fraction grows with $\mathcal{A}$ until reaching ${\mathcal{A}}^{*}=1.16$ of the underlying Voronoi cells at confluence. We find that DP packings above and below ${\mathcal{A}}^{*}$ are solidlike, which helps explain the solid-to-fluid transition at ${\mathcal{A}}^{*}$ in the vertex model as a transition from tension- to compression-dominated regimes.

Figure 1

Schematic of deformable polygons with $N_v=34$ vertices (with the position of the $j$th vertex in the $m$th polygon given by ${\overrightarrow{v}}_{m,j}$), area $a_m$, and perimeter $p_m$. $l_{m,j}=p_m/N_v$ is the line segment between vertices $j$ and $j+1$ in polygon $m$. Two methods were used to model edges of deformable polygons. In (a) and (b), we show the RS method, which fixes centers of disks with diameter $\delta$ at polygon vertices. In (c) and (d), we show the SS method, which models polygon edges as circulo-lines with width $\delta$. $d_{\min }$ is the minimum distance between line segments $l_{m,j}$ and $l_{n,k}$.

Figure 2

(a) Packing fraction at jamming onset ${\varphi }_J$ (normalized by ${\varphi }_{\max }$), (b) deviation of ${\varphi }_J$ from the confluent value, $1-{\varphi }_J$, (c) coordination number $z_J$, and (d) average friction coefficient $\mu$ (for the RS model) for DP packings versus asphericity $\mathcal{A}$. In (a) and (c), $\mathcal{A}$ is normalized by ${\mathcal{A}}_v$ of a regular polygon with $N_v$ vertices. The dashed lines in (a) and (b) indicate $\mathcal{A}={\mathcal{A}}^{*}\approx 1.16$. In (a), we also show ${\varphi }_J/{\varphi }_{\max }\approx 0.81$ (with ${\varphi }_{\max }=1$) for $N=64$ monodisperse, frictional discs using the Cundall-Strack (CS) model with $\mu =0.65$ (filled diamond). In (c), the dashed line indicates $z_J(\mathcal{A}/{\mathcal{A}}_v)=z_J(1)+z_0(\mathcal{A}/{\mathcal{A}}_v-1{)}^{\beta }$, where $z_J(1)\approx 3.3$, $\mu =0.65$, $z_0\approx 3.9$, and $\beta \approx 0.25$.

Figure 3

DP packings for the RS model with $N_v=34$ and (a) $\mathcal{A}=1.03$, (b) 1.08, and (c) 1.16, near ${\mathcal{A}}^{*}$. Polygonal cells (solid lines) surrounding each deformable particle are obtained from surface-Voronoi tessellation.

Figure 4

(a) Distribution of areas $a_t$ of surface-Voronoi tessellated polygons for $\mathcal{A}=1.03$ (squares), 1.12 (circles), and 1.22 (triangles) for $N=1000$ monodisperse deformable particles with $N_v=12$ and the RS model. The distributions $\mathcal{P}(x)$ are plotted against $x=(a_t-a_{\min })/(⟨a_t⟩-a_{\min })$, where $a_{\min }$ is the minimum tessellated area for each packing. Fits to the $k$-gamma distribution [Eq. (4)] are shown as solid, dot-dashed, and dashed lines for $\mathcal{A}=1.03$, 1.12, and 1.22, respectively. Inset: shape parameter $k$ versus $\mathcal{A}$ from fits of $\mathcal{P}(x)$ to Eq. (4). (b) Bulk $\mathcal{B}$ and (c) shear $\mathcal{G}$ moduli for DP packings using the model in (a) versus $\mathcal{A}$ for $N=32$ (upward triangles), 64 (circles), 200 (squares), 512 (stars), and 1000 (leftward triangles). The inset to (c) shows system-size scaling of $\mathcal{G}$. The dashed lines have slope $-1$.