The real-space structure of colloidal hard-sphere glasses is a critical aspect of soft matter physics, as it provides insight into how particles are arranged in disordered, dense systems. Colloidal hard-sphere glasses are systems where spherical particles, typically made of materials like polystyrene or silica, are suspended in a solvent at a volume fraction high enough to cause the system to form a glass-like state, characterized by a lack of long-range order.

Here’s a detailed explanation of the real-space structure of colloidal hard-sphere glasses:

1. Characteristics of Colloidal Hard-Sphere Glasses:

• Hard-Sphere Model: In this model, particles are assumed to interact with each other only via hard-core repulsion, meaning they cannot overlap or pass through one another. These systems lack attractive forces or long-range interactions.
• Glass Transition: As the volume fraction of colloidal particles increases, the system eventually reaches a state where the particles are no longer able to move freely, resulting in a glass transition. This transition occurs at a volume fraction near \( \phi_g \), which is typically around 0.58 for hard-sphere colloidal systems. At this point, the system becomes structurally disordered, and the particles are trapped in a local, non-equilibrium configuration.

2. Structural Features in Real Space:

a) Local Order:

• First Neighbors: In the hard-sphere glass, the particles are arranged in locally ordered structures within the first coordination shell. This results in short-range order where each particle is surrounded by a well-defined set of nearest neighbors.
• Hexagonal Close Packing (hcp) and Face-Centered Cubic (fcc): Though the system does not exhibit long-range translational order, clusters of particles often arrange themselves in local packing motifs, such as hcp or fcc structures, due to the hard-sphere interaction.
• Disordered Clusters: As the system reaches higher volume fractions and enters the glassy state, these local structures become more disordered and "frozen" in place, leading to a heterogeneous arrangement of particles.

b) Intermediate Range Order (IRO):

• Pair Distribution Function: The radial distribution function, \( g(r) \), describes how the particle density varies as a function of distance from a reference particle. In hard-sphere glasses, \( g(r) \) exhibits a sharp peak at the particle diameter (representing the first peak in the radial distribution), corresponding to the first coordination shell. As the volume fraction increases, these peaks become sharper and the system begins to develop more significant intermediate-range order (IRO), though still lacking long-range order.
• Decay of Correlations: Beyond the first peak, the decay of correlations is non-exponential. In a glass, this typically means the decay of the pair distribution function becomes slow and doesn't follow the perfect crystal-like behavior seen in crystals.

c) Glass-Like Packing:

• Cage Formation: At high volume fractions, colloidal particles become trapped in "cages" formed by their neighbors. Each particle is surrounded by a group of particles that restrict its motion, contributing to the glassy behavior. This phenomenon is often visualized as particles being unable to escape from their local environment, even though the system is disordered.
• Jamming and Softening: At the glass transition point, the system "jammed" — the particles are unable to move past one another despite thermal or driving forces. In hard-sphere glasses, this is primarily caused by the steric repulsion between the particles, which increases as the volume fraction grows.

3. Methods for Studying Real-Space Structure:

Several experimental and computational methods are employed to investigate the real-space structure of colloidal hard-sphere glasses:

a) Confocal Microscopy:

• Confocal microscopy allows for high-resolution imaging of colloidal suspensions, providing three-dimensional reconstructions of particle arrangements. By tracking individual particles, researchers can directly observe the local structure, including cage formation and particle mobility in dense systems.