Random close packing fraction of bidisperse discs: Theoretical derivation and exact bounds

This paper presents a theoretical framework using cell order distribution to derive the highest mathematically possible random close packing fraction for bidisperse discs, along with exact upper and lower bounds for this value as a function of size ratio and concentration.

The Gist

Imagine you are trying to fill a large, flat box with coins. You want to pack as many coins as possible into the box without them forming a perfect, shiny, ordered pattern like a honeycomb. You just want them jumbled up as tightly as possible. This is the "Random Close Packing" problem.

For a long time, scientists have struggled to predict exactly how full this box can get before it's impossible to add more coins without them accidentally lining up into a perfect crystal.

This paper by Raphael Blumenfeld tackles a more complex version of this puzzle: What if you have two different sizes of coins? (Maybe some are pennies and some are quarters).

Here is the breakdown of the paper using simple analogies:

1. The Problem: The "Infinite Maze" of Packing

Usually, when you try to pack things, the result depends on how you do it. Do you shake the box? Do you pour the coins in slowly? Do you drop them from a height?

• The Analogy: Imagine trying to find the shortest path through a maze that has infinite doors and infinite paths. If you try every single way to pack the coins, you'll never finish.
• The Solution: Instead of testing every possible way to shake the box, this paper looks at the shape of the empty spaces left between the coins.

2. The Tool: The "Cell Order Distribution" (The Neighborhood Map)

The author introduces a clever way to look at the packing. Imagine drawing a line connecting the centers of any two coins that are touching. This creates a web of triangles and squares between the coins.

• The "Cells": The empty spaces enclosed by these lines are called "cells."
• The "Order": A cell surrounded by 3 coins is a "triangle" (Order 3). A cell surrounded by 4 coins is a "square" (Order 4).
• The Insight: The paper argues that if you have too many "triangles" (3-cells) all clumped together, the coins will accidentally form a perfect crystal (order). To keep the packing "random" (disordered), you need to limit how many triangles are neighbors with each other.

3. The "Disorder Rule": The "No-Party" Policy

To ensure the coins stay jumbled and don't turn into a crystal, the author proposes a strict rule:

• The Metaphor: Imagine the coins are people at a party. If a group of identical people (same-size coins) sits together in a perfect triangle, they might start organizing a dance routine (crystallizing).
• The Rule: The paper calculates that for the party to remain chaotic and fun, a "triangle group" should, on average, have only one identical neighbor. If they have two or three identical neighbors, they start forming a clique, and the whole system turns into an ordered crystal.
• The Result: This rule tells us exactly what ratio of small coins to large coins ($p$) we need to use to guarantee the packing stays messy. If you have too many small coins or too few, they will inevitably line up.

4. The Big Discovery: The "Theoretical Ceiling"

Once the author established the "Disorder Rule," they calculated the absolute maximum amount of space the coins could fill while still obeying that rule.

• The Analogy: Think of it like a ceiling on a room. You can't build a house higher than the ceiling, no matter how good your bricks are.
• The Finding: The paper derives an exact mathematical formula for this "ceiling" (the maximum packing fraction, $\phi_{RCP}$).
  • It depends on the size difference between the coins ($D$).
  • It depends on the percentage of small coins ($p$).
• The Surprise: The paper found that for any specific size difference, there is a "sweet spot" percentage of small coins where you can pack the most material possible without creating a crystal.

5. Why This Matters

• For Engineers: If you are designing concrete, soil, or pharmaceutical powders (which are often mixtures of different sizes), this paper tells you the theoretical limit of how dense you can make them. If your experiment goes higher than this number, you know you accidentally created a crystal, even if you didn't see it.
• For Scientists: It solves a problem that has been debated for decades. Instead of guessing or running millions of computer simulations, we now have a precise mathematical map.
• The "Upper Bound": The paper proves that you can never pack disordered coins tighter than this calculated number. It's a hard limit.

Summary in One Sentence

By treating the empty spaces between mixed-size coins like a neighborhood map and setting a strict rule to prevent "cliques" (crystals) from forming, this paper calculates the absolute maximum density possible for jumbled coins, giving scientists a precise target for how tightly they can pack materials without them organizing themselves.

Technical Summary

1. Problem Statement

The paper addresses the long-standing challenge of theoretically predicting the Random Close Packing (RCP) fraction ($\phi_{RCP}$) for bidisperse discs in two dimensions ($d=2$).

• Context: While the RCP of monodisperse discs has been recently resolved, many experimental and numerical studies utilize bidisperse systems (two distinct disc sizes) specifically to suppress crystallization and achieve denser disordered states.
• The Challenge: Predicting $\phi_{RCP}$ is difficult because:
  1. The parameter space of packing processes is infinite-dimensional, making trial-and-error approaches insufficient.
  2. Ensuring "disorder" is difficult; most packing processes naturally lead to ordered crystalline states (e.g., hexagonal/trigonal lattices) which are denser than disordered states.
  3. Existing disorder criteria are often a-posteriori (requiring a structure to be built before testing for order), which does not help in predicting the maximum density a-priori.
• Goal: To derive the highest mathematically possible packing fraction, $\phi_{RCP}(p, D)$, for bidisperse discs as a function of the size ratio $D$ and the concentration of smaller discs $p$, while guaranteeing the state remains disordered.

2. Methodology

The author employs the Cell Order Distribution (COD) framework, previously developed for monodisperse systems, to bypass the infinite-parameter-space problem.

• Cell Order Distribution (COD):

  • A packing is modeled as a graph where disc centers are nodes and contacts are edges.
  • The smallest voids enclosed by edges are "cells."
  • The "order" ($k$) of a cell is the number of edges surrounding it.
  • $Q_k$ represents the fraction of cells of order $k$.
  • Key Insight: The packing fraction $\phi$ is directly determined by the COD. For bidisperse discs, $\phi$ is a function of the mean cell order $\bar{k}$ and the mean cell area.

• A-Priori Disorder Criterion:

  • To ensure disorder without generating a specific packing, the author proposes a criterion based on the probability of forming large crystalline clusters.
  • Crystallization in 2D bidisperse systems often manifests as large clusters of identical 3-cells (triangular arrangements).
  • The Criterion: A packing is guaranteed to be disordered if the probability of a 3-cell having two or three identical neighbors is less than $1/3$. This leads to a mathematical bound on the fraction of 3-cells ($Q_3$) relative to the concentration parameter $u$ (an auxiliary variable related to $p$).

• Theoretical Derivation:

  • The analysis assumes isotropic, infinite packings with no "rattlers" (small discs trapped in voids of three large discs), limiting the size ratio to $1 < D < D_{max} = \sqrt{3}/(2-\sqrt{3})$.
  • The author derives exact expressions for the mean area of cells ($\bar{S}_k$) and the area occupied by discs within those cells ($\bar{S}^{disc}_k$) for 3-cells and 4-cells.

3. Key Contributions

A. Exact Upper Bound ($\phi_3$)

The author derives the theoretical maximum packing fraction assuming the packing consists only of 3-cells ($Q_3 = 1$).

• While a packing of only 3-cells is likely topologically impossible to maintain in a disordered state for all parameters, this configuration provides a strict mathematical upper bound for any disordered packing.
• The derivation accounts for four possible 3-cell configurations (combinations of small and large discs) and calculates their weighted mean areas based on an auxiliary probability parameter $u$.
• Result: $\phi_3(p, D)$ represents the absolute ceiling for disordered packing density.

B. Exact Lower Bound ($\phi_{low}$)

The author derives a lower bound by considering the maximum fraction of 3-cells ($Q_{3max}$) that can exist without triggering crystallization, as determined by the disorder criterion.

• The remaining fraction of cells must be 4-cells (which have lower packing density than 3-cells).
• Result: $\phi_{low}$ is the minimum density achievable for a disordered state at the limit of stability against crystallization.

C. The Disorder Criterion and Valid Range

The paper establishes a rigorous a-priori disorder criterion that defines the valid range of concentration $p$ for a given size ratio $D$.

• It identifies a "safe zone" where disorder is mathematically guaranteed.
• Key Finding: For any size ratio $1 < D < D_{max}$, disorder is assured if the concentration of small discs satisfies $0.312 < p < 0.825$.
• Warning on Common Practices: The paper highlights that a common experimental choice—setting the two disc types to occupy equal area ($p = D^2/(D^2+1)$)—increases the risk of crystallization for size ratios $D \lesssim 3$, as this often pushes $p$ outside the safe disorder range.

D. The Exact $\phi_{RCP}(p, D)$ Function

By minimizing the fraction of 4-cells (maximizing $Q_3$) while staying within the disorder bounds, the author calculates the exact $\phi_{RCP}(p, D)$.

• Crucial Observation: For every size ratio $D$, the maximum possible packing fraction coincides exactly with the upper bound ($\phi_3$) within the range of $p$ where disorder is guaranteed.
• This implies that the densest disordered packings are those that are "as close to 3-cell only as possible" without forming large crystalline domains.

4. Results

• Quantitative Bounds: The paper provides exact analytical formulas and numerical plots for $\phi_{RCP}$, $\phi_{upper}$, and $\phi_{lower}$ across the range $1 \le D \le 6.0$.
• Peak Densities: The maximum $\phi_{RCP}$ varies with $D$. For example, at $D=3.0$, the peak density occurs at a specific $p$ and reaches values significantly higher than typical monodisperse RCP ($\approx 0.82-0.86$), approaching the theoretical limit of ordered 3-cell packings ($\approx 0.905$ for specific configurations, though the disordered limit is slightly lower).
• Validation: The derived bounds improve upon previous literature (e.g., Florian's bounds). The author notes that existing simulations often yield lower densities because they fail to optimize the concentration $p$ to the theoretical $p_{max}$ or inadvertently form crystalline clusters.

5. Significance

• Theoretical Breakthrough: This is the first analytical derivation of the exact upper and lower bounds for the RCP of bidisperse discs, moving beyond empirical ranges to mathematical certainty.
• Experimental Guidance: The derived disorder criterion provides a "recipe" for experimentalists and simulationists. By choosing $p$ and $D$ within the identified safe ranges (specifically avoiding the equal-area condition for small $D$), researchers can maximize packing density while avoiding crystallization.
• Process Independence: By using the COD, the results are independent of the specific packing protocol (e.g., compression rate, vibration method), solving the "infinite parameter space" problem.
• Future Extensions: The methodology is presented as a framework that can be extended to polydisperse systems (three or more sizes) and potentially to 3D spheres, although the combinatorial complexity increases significantly.

In summary, the paper resolves the bidisperse RCP problem by defining a rigorous disorder criterion based on cell topology, proving that the densest disordered state is bounded by a 3-cell-only configuration, and providing exact formulas to guide the design of high-density disordered materials.