1. Brownian motion

Random displacements of a small particlein a fluid are controlled by kinetic energydissipation (1). The mean displacement iszero but the average mean squared displacement(MSD) is finite, growing linearly with time lag $\tau$

Reference

Effect of Collective MolecularReorientations on Brownian Motion ofColloids in Nematic Liquid Crystal

1.1 Theory and Concept:

1. Basic Brownian Motion:

• A particle suspended in a fluid, due to the constant bombardment by fluid molecules, undergoes random motion known as Brownian motion.
• The average movement or displacement (Δr) over time is zero (since the motion is random). But the mean squared displacement (MSD) isn't. It grows linearly with the time lag τ.

2. Diffusion Coefficient:

The self-diffusion coefficient (D) of a random walker in a fluid is described by the Stokes-Einstein relation. It is given by:

$$D = \frac{k_B T}{6 \pi \eta R}$$

Where:

• $k_B$ is the Boltzmann constant.
• $T$ is the temperature.
• $\eta$ is the viscosity of the fluid.
• $R$ is the radius of the random walker (or particle).

This relation arises from the balance between the thermal energy (given by $k_B T$) and the viscous drag (related to $\eta$ and $R$) that the particle experiences in the fluid.

• The equation ⟨Δr^2(τ)⟩ = 6Dτ relates the mean squared displacement with the time lag using the diffusion coefficient D.
• D gives a measure of how quickly a particle spreads out in the fluid.

3. Anomalous Behavior:

• In some fluids, the growth isn't linear. This deviation is termed as "anomalous behavior".
• When α < 1, it's called subdiffusion. This is observed in polymer networks, F-actin networks, and surfactant dispersions.
• When α > 1, it's called superdiffusion. Seen in concentrated bacterial suspensions and polymer-like micelles.

4. Orientational Order & Nematic Fluids:

• Some fluids have molecules with a predominant orientation, described using a "director".
• A uniaxial nematic fluid is a simple fluid where molecules have a dominant orientation.
• Different viscosities for motions parallel and perpendicular to the director make the diffusion anisotropic, resulting in different diffusion coefficients D_|| and D_⊥.

5. Anisotropic Diffusion:

• Definition: Diffusion that differs in different directions.
• Context: In this study, diffusion is anisotropic in the nematic phase due to the alignment of molecules.

6. Velocity Auto-Correlation Function (VACF):

• Definition: Represents how the velocity of a particle at one time is related to its velocity at another time.
• Interpretation: If VACF is close to zero, it signals normal diffusion. Positive and negative values indicate superdiffusion and subdiffusion, respectively.

7.Nematic Liquid Crystals:

• Definition: A type of liquid crystal where molecules align in parallel but are not arranged in well-defined planes.
• Anchoring: Determines how molecules near the boundary align relative to the boundary. Types:
  • Perpendicular: Molecules align perpendicular to the surface.
  • Tangential: Molecules align tangentially to the surface.

1.2 Data, Equations, and Properties in Paper:

Effect of Collective MolecularReorientations on Brownian Motion ofColloids in Nematic Liquid Crystal

1. Diffusion Coefficient and MSD:

• ⟨Δr^2(τ)⟩ = 6Dτ

2. Anomalous Behavior Equation:

• ⟨Δr^2(τ)⟩ ∝ τ^α

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3. Measured MSD Values:

• Isotropic phase (at 60°C): Diffusion coefficient $$ D = 9.2 \times 10^{-16} m^2/s $$.
• Nematic phase (at 50°C):
  • Perpendicular anchoring: $$ D_{||} = 1.9 \times 10^{-16} m^2/s$$ and $$ D_{⊥} = 1.4 \times 10^{-16} m^2/s$$.
  • Tangential anchoring: $$ D_{||} = 2.2 \times 10^{-16} m^2/s $$ and $$ D_{⊥} = 1.3 \times 10^{-16} m^2/s $$.

4. Equations:

• velocity autocorrelation functions :
  • $$ C_{v_{||}}(t) = \langle v_x(t) v_x(0) \rangle$$
  • $$ C_{v_{⊥}}(t) = \langle v_y(t) v_y(0) \rangle$$

5. Power Law Representation of MSD:

• Oversimplified model for understanding anomalous diffusion.
• Equation: $$MSD \propto t^a $$.
• Determined values: For subdiffusive domain:
  • $a_{||} = 0.35 \pm 0.01 $ and $a_{⊥} = 0.30 \pm 0.01$.
  • For superdiffusion: $a_{||} = 1.32 \pm 0.01$ and $a_{⊥} = 1.20 \pm 0.01$.

6. Probability Distribution of Particle Displacements:

• Varies with time lags.
• Normal Diffusion: Coincides with a Gaussian fit at long time lags (e.g., 40s).
• Subdiffusion: Central portion (small displacements) is Gaussian at intermediate time lags (e.g., 10s), but large displacements are less likely.
• Superdiffusion: Large displacements more probable than in normal diffusion at short time lags (e.g., 1s).

1.3 Case study

Further study

Colloidal matter: Packing, geometry, and entropy

Command of active matter by topological defects and patterns Directional self-locomotion of active droplets enabled by nematic environment

Mean first-passage times of non-Markovian random walkers in confinement

Chiral liquid crystal colloids

Control of colloidal placement by modulated molecular orientation in nematic cells