Density Profiles and Direct Correlation Functions from Density Functional Theory in Binary Hard-Sphere Crystals: Substitutional Solid and Interstitial Solid Solution

Using classical density functional theory with the White Bear II functional, this study determines the equilibrium density profiles and species-resolved direct correlation functions for binary hard-sphere crystals, revealing that substitutional alloys retain narrow Gaussian-like density peaks similar to single-component systems while interstitial solutions exhibit delocalized small species and distinct correlation behaviors characterized by a geometric picture dependent on vacancy concentration.

The Gist

Imagine a giant, invisible dance floor made of perfectly round, hard balls. Some balls are big (like bowling balls), and some are small (like marbles). In physics, we call these "hard spheres." When you pack them tightly together, they don't just float randomly; they snap into a rigid, organized grid, like a crystal.

This paper is a deep dive into what happens when you mix two sizes of these balls together on that dance floor. The researchers used a powerful mathematical tool called Density Functional Theory (DFT)—think of it as a super-accurate crystal ball that can predict exactly where every single ball is likely to be, without needing to build a physical model.

They looked at two specific ways these mixed balls can arrange themselves:

1. The "Substitutional" Crystal (The Swap Meet)

Imagine a dance floor where everyone is supposed to stand on a specific spot in a grid.

• The Setup: Most spots are taken by the big bowling balls. But occasionally, a big ball is swapped out for a slightly smaller marble.
• The Result: It's a bit like a game of musical chairs where the chairs are slightly different sizes. The marbles fit in the big chairs, but they wiggle a little more because they have extra room.
• The Finding: The researchers found that in this scenario, the marbles stay pretty close to their assigned spots. Their "density profile" (a map of where they are most likely to be found) looks just like the big balls: tight, neat, Gaussian-shaped peaks. It's a very orderly, predictable crowd.

2. The "Interstitial" Solid Solution (The Hiding Game)

Now, imagine a different dance floor.

• The Setup: The big bowling balls form a perfect, rigid grid. But now, we have tiny marbles (much smaller than the bowling balls).
• The Result: The marbles don't try to stand on the same spots as the bowling balls. Instead, they squeeze into the empty gaps (holes) between the big balls.
• The Finding: This is where it gets wild. The tiny marbles aren't stuck in one specific hole. They are delocalized. Imagine a ghost that can be in the kitchen, the living room, and the hallway all at once. The tiny marbles are "smeared out" across the whole unit cell. They spend the most time in the octahedral holes (the big gaps), but they also drift into the tetrahedral holes (smaller gaps) and the tunnels connecting them. They are essentially a fluid trapped inside a solid cage.

The "Direct Correlation Function" (The Social Network of Balls)

The paper also calculates something called the Direct Correlation Function (DCF).

• The Analogy: Think of this as a "social network" map. If I nudge one ball, how does that affect the feeling (or probability) of another ball being somewhere else?
• The Big Discovery:
  • In the Substitutional case (swapping sizes), the social network looks normal. Everyone is connected to their neighbors in a standard way.
  • In the Interstitial case (big balls + tiny gaps), the social network gets weird.
    • The Big Balls: They are very sensitive to empty spots (vacancies). If a big ball is missing, it creates a huge "ripple" in the system. The math shows a massive spike in correlation, roughly proportional to 1 / (number of empty spots). It's like a crowd screaming when a seat is empty.
    • The Tiny Balls: Because they are so free to roam (delocalized), their "social network" looks very different from the big balls. They don't care as much about specific empty spots because they are already floating everywhere.

The "Diffusion Pathway" (The Tunnel)

For the tiny marbles in the Interstitial crystal, the researchers mapped out the energy cost of moving from one hole to another.

• The Metaphor: Imagine the tiny marbles are hikers trying to cross a mountain range. The "octahedral hole" is a cozy valley, and the "tetrahedral hole" is another valley. To get from one to the other, they have to cross a pass.
• The Finding: The pass is low! It only takes a tiny bit of energy (about 2 units of thermal energy) to hop over. This means the tiny marbles are very mobile. They can zip around the crystal lattice easily, almost like they are liquid, even though they are trapped inside a solid crystal.

Why Does This Matter?

This isn't just about math; it helps us understand:

1. Colloids: Tiny particles used in paints, inks, and medicines.
2. Alloys: How metals mix at the atomic level.
3. Elasticity: How hard it is to squish or stretch these materials.

In a nutshell: The paper shows that when you mix big and small hard spheres, the small ones can either behave like shy neighbors staying in their own spots (Substitutional) or like energetic ghosts roaming the entire house (Interstitial). The math proves that the "ghosts" are surprisingly free to move, which changes how the whole crystal reacts to stress and pressure.

Technical Summary

1. Problem Statement

The paper addresses the challenge of understanding the microscopic structure and thermodynamic properties of binary hard-sphere (HS) crystals, specifically focusing on two distinct phases:

• Substitutional Crystals (SC): Where large and small spheres randomly occupy the same face-centered cubic (fcc) lattice sites.
• Interstitial Solid Solutions (ISS): Where large spheres form an fcc lattice, and smaller spheres occupy the octahedral interstitial voids, potentially delocalizing within the unit cell.

While the density profiles ($\rho(r)$) and direct correlation functions (DCF, $c^{(2)}$) for single-component HS crystals have been studied, the behavior of binary systems remains less understood. Specifically, there is a lack of resolved data on:

1. The equilibrium density profiles of binary crystals, particularly how the small species distributes in ISS phases.
2. The species-resolved inhomogeneous two-body direct correlation functions $c^{(2)}_{ij}(r, r')$ in these complex crystal environments.
3. The relationship between these crystal DCFs and their liquid-phase counterparts (e.g., Percus-Yevick solutions).

2. Methodology

The authors employ Classical Density Functional Theory (DFT) using the highly accurate White Bear II (WBII) functional derived from Fundamental Measure Theory (FMT).

• System Definition: A binary mixture of hard spheres with diameters $\sigma_L$ (large) and $\sigma_S = q\sigma_L$ (small).
  • SC Case: $q=0.9$, composition $x_S=0.3$.
  • ISS Case: $q=0.3$, composition $x_S=0.3$, with large spheres at high packing fraction ($\eta_L=0.55$).
• Minimization Strategy:
  • The grand potential $\Omega[\rho]$ is minimized to find equilibrium density profiles $\rho_L(r)$ and $\rho_S(r)$.
  • A two-stage minimization is used to handle the complexity of the crystal lattice and vacancy concentration ($n_{vac}$):
    1. Fix $n_{vac}$ (and thus the unit cell volume) to minimize free energy with respect to the density profiles $\rho(r)$.
    2. Minimize the resulting free energy per particle with respect to $n_{vac}$.
  • The density field is discretized on a $128^3$ grid within a cubic unit cell. Initial guesses use Gaussian peaks at lattice sites (fcc sites for large spheres, octahedral/tetrahedral sites for small spheres).
• Calculation of DCF:
  • The species-resolved DCF, $c^{(2)}_{ij}(r, r')$, is computed via the second functional derivative of the excess free energy functional (Eq. 3).
  • Numerical implementation utilizes automatic differentiation (machine learning frameworks) to compute derivatives of the FMT functional.
  • The DCF is analyzed as a 6-dimensional function, visualized by fixing a reference point $r$ and plotting the dependence on the displacement vector $\Delta r = r' - r$ along crystallographic directions ([100], [110], [111]).

3. Key Contributions

1. Fully Resolved Density Profiles: The authors provide the first fully resolved 3D equilibrium density profiles for binary HS crystals using WBII-FMT, distinguishing between the localized behavior in SC and the delocalized behavior in ISS.
2. Species-Resolved DCFs: They compute and visualize the full matrix of direct correlation functions ($LL, LS, SS$) for both crystal types, a significant step beyond previous single-component studies.
3. Geometric Interpretation of Crystal DCF: The paper proposes a geometric picture for the $LL$ component of the DCF, interpreting it as a superposition of short-ranged, nearly isotropic "base functions" centered at fcc lattice sites, scaled by the vacancy concentration.
4. Vacancy Concentration Analysis: The study quantifies the equilibrium vacancy concentration ($n_{vac}$) for both phases and links the magnitude of the DCF directly to $1/n_{vac}$.

4. Key Results

A. Density Profiles and Vacancy Concentrations

• Substitutional Crystal (SC): The density profiles for both large and small species are narrow, isotropic Gaussian peaks centered at fcc lattice sites, closely resembling the single-component crystal. The small species is slightly more delocalized due to larger free volume.
  • Equilibrium $n_{vac} \approx 1 \times 10^{-5}$.
• Interstitial Solid Solution (ISS):
  • Large Species: Forms sharp Gaussian peaks at fcc sites.
  • Small Species: Highly delocalized. While peaks exist at octahedral and tetrahedral holes, the density is non-zero throughout the unit cell, forming "tunnels" between sites.
  • Vacancy Concentration: $n_{vac} \approx 3.7 \times 10^{-4}$, significantly higher than in the SC or single-component crystals.
  • Diffusion Pathway: The first-order DCF ($c^{(1)}$) along the path from octahedral to tetrahedral holes reveals a moderate energy barrier ($\approx 2 k_B T$), suggesting high mobility for small spheres in the ISS phase.

B. Direct Correlation Functions (DCF)

• Magnitude Scaling: The $LL$ component of the DCF in both SC and ISS phases exhibits a characteristic magnitude of $\sim 1/n_{vac}$. This divergence as $n_{vac} \to 0$ is a hallmark of crystalline DCFs, distinct from the liquid phase ($O(10)$).
• Substitutional Crystal: The species-resolved components ($LL, LS, SS$) are very similar to each other and to the single-component crystal DCF, showing a pronounced minimum near $d \approx \sigma_L/2$.
• Interstitial Solid Solution:
  • Anisotropy: The DCFs are highly anisotropic and lack the characteristic dip seen in SC or single-component crystals, reflecting the complex unit cell structure.
  • Reference Point Dependence:
    • At the fcc lattice site (X): The DCF is relatively isotropic, and all components ($LL, LS, SS$) scale with $1/n_{vac}$.
    • At the interstitial site (Y): The $LL$ component still scales with $1/n_{vac}$ (due to nearby vacant lattice sites), but components involving small spheres ($LS, SS$) have significantly smaller magnitudes because the insertion probability for small spheres at interstitial sites is not limited by $n_{vac}$ in the same way.
  • Liquid Limit: When artificially increasing $n_{vac}$ to 0.1 (simulating a "liquid-like" crystal), the $LS$ and $SS$ components of the ISS DCF converge to the Percus-Yevick (PY) liquid solution, suggesting the small species behaves effectively as a liquid within the lattice, whereas the large species retains solid-like correlations.

5. Significance

• Theoretical Advancement: This work extends the application of FMT to complex binary crystal phases, providing a rigorous framework for calculating inhomogeneous correlation functions that were previously inaccessible via simulation (due to the difficulty of sampling $g(r, r')$ in 3D crystals).
• Physical Insight: The results clarify the nature of "solid solutions" in colloidal systems. The ISS phase is shown to be a hybrid state where the large species forms a rigid lattice while the small species exhibits liquid-like mobility and correlation properties.
• Elastic Constants: The computed DCFs are essential inputs for calculating generalized elastic constants, which describe the free energy response to unit cell deformations. This paves the way for a deeper understanding of the mechanical stability of binary colloidal crystals.
• Validation of Models: The study validates the White Bear II functional's ability to reproduce equilibrium vacancy concentrations and complex density distributions, reinforcing its utility for studying hard-sphere mixtures.