Colloidal Particle Dynamics, Mean Square Displacement, and Drag Force Dynamics (lecture 2)

Physical Properties and Trajectory of Colloidal Particles

• Colloidal Particle Definition and Geometry:

  • An isolated colloidal particle is suspended inside a molecular solvent.
  • Although particle shapes can vary, colloidal particles are commonly modeled and conceptualized as spheres.
  • The particle size is characterized by its diameter $d$, which ranges across three orders of magnitude from $10\,\text{nm}$ to $10\,\mu\text{m}$ ($10\,\text{microns}$ or micrometers).

• Particle Trajectory and Unbiased Random Walk:

  • When observed under an optical microscope starting at initial time $t = 0$, the particle exhibits stochastic jiggling motion.
  • Over time, the particle drifts away from its starting location along a non-deterministic path.
  • The particle has no isotropic bias or preferred direction of movement; it moves randomly up, down, left, right, in-plane, and out-of-plane.
  • Despite the lack of a directional preference, net translational displacement occurs over time. For instance, a particle may travel a cumulative distance equivalent to roughly $5$ times its own diameter over a given time interval $\Delta t$.

Quantification of Motion via Vector Displacement and Mean Square Displacement

• Self-Diffusion Constant ($D^s$):

  • The self-diffusion constant (denoted as $D^s$ or $D_S$) quantifies the rate or speed of the stochastic random walk.
  • It defines how far on average a colloidal particle is expected to translate within a specified time duration.

• Position Vectors and Displacement Vector Definitions:

  • The position of a particle is defined relative to its geometric center.
  • At initial time $t = t_0$, the particle position is defined by the 3D position vector $\mathbf{r}(t_0)$, which encapsulates its Cartesian coordinates $(x, y, z)$.
  • After an elapsed time interval $\Delta t$, the position vector becomes $\mathbf{r}(t_0 + \Delta t)$.
  • The absolute spatial position is arbitrary; the fundamental dynamic variable is the spatial difference, defined as the displacement vector $\Delta\mathbf{r}(\Delta t)$:

$\Delta\mathbf{r}(\Delta t) = \mathbf{r}(t_0 + \Delta t) - \mathbf{r}(t_0)$

• Components of the Displacement Vector:
  • In three-dimensional space, the displacement vector is decomposed into three orthogonal scalar functions of $\Delta t$:

$\Delta\mathbf{r}(\Delta t) = \begin{pmatrix} \Delta x(\Delta t) \\ \Delta y(\Delta t) \\ \Delta z(\Delta t) \end{pmatrix}$

*   In standard optical microscopy experiments, observation is typically limited to two-dimensional diffusion in the focal plane ($x\text{--}y$ plane). Motion along the third spatial dimension ($z$-axis) causes the particle to move out of focus, making 3D tracking experimental resolution more complex.

• Ensemble and Temporal Averaging:
  • Averages taken across multiple experimental particle trajectories or repeated time observations are denoted using angle brackets $\langle \dots \rangle$.
  • Because the motion is completely unbiased in space, the mean displacement vector evaluates to the zero vector:

$\langle \Delta\mathbf{r}(\Delta t) \rangle = \mathbf{0}$

*   A mean displacement vector of zero does **not** signify that the particle is stationary; it strictly indicates the absence of a preferred directional bias in the random walk.

• Vector Magnitude and Dot Product:
  • To quantify the true spatial extent of movement, vector length (magnitude) is evaluated using the vector dot product with itself:

$\|\mathbf{a}\| = (a_x^2 + a_y^2 + a_z^2)^{1/2} = (\mathbf{a} \cdot \mathbf{a})^{1/2}$

*   The square of a vector dotted into itself is conventionally written as $\mathbf{a} \cdot \mathbf{a} = a^2$.

• Mean Square Displacement (MSD):
  • The scalar quantity measuring spatial dispersion over time is the Mean Square Displacement (MSD), defined as:

$\text{MSD}(\Delta t) = \langle \Delta\mathbf{r}(\Delta t) \cdot \Delta\mathbf{r}(\Delta t) \rangle = \langle \Delta\mathbf{r}(\Delta t)^2 \rangle$

*   If $\text{MSD}(\Delta t) = 0$, the particle is strictly frozen and motionless.
*   Because the dot product reduces vector components to a single scalar number, MSD eliminates spatial directional dependence.
*   In an unbounded medium (e.g., a particle dropped in an ocean), there is no theoretical upper bound on how far a particle can diffuse; the MSD increases indefinitely with time $\Delta t$. Tapering off or plateauing of MSD occurs exclusively when particles encounter physical confinement bounds (such as container or beaker walls).

Long-Time Scaling of MSD and the Self-Diffusion Constant

• Brownian Motion:

  • Stochastic colloidal motion is termed Brownian motion, named after Robert Brown, who first recorded this phenomena while observing pollen grains suspended in water.
  • The terms Brownian motion, diffusive motion, and random walk are functionally interchangeable.

• Temporal Scaling Regimes of MSD:

  • At very short initial times, particle trajectories can exhibit non-linear transient behavior.
  • At intermediate and long time intervals, the MSD of all unconstrained random walks transitions into a strictly linear function of $\Delta t$.

• Mathematical Extraction of $D^s$:

  • The long-time slope of the MSD plotted against $\Delta t$ is directly proportional to the self-diffusion constant $D^s$:

$\text{Slope} \propto 2 d_{\text{spatial}} D^s$

*   The rigorous mathematical definition of the self-diffusion constant in arbitrary spatial dimensions $d_{\text{spatial}}$ is:

$D^s = \frac{1}{2 d_{\text{spatial}}} \lim_{\Delta t \rightarrow \infty} \frac{d}{d(\Delta t)} \text{MSD}(\Delta t)$

*   In three dimensions ($d_{\text{spatial}} = 3$), the prefactor becomes $\frac{1}{2 \times 3} = \frac{1}{6}$, giving:

$D^s = \frac{1}{6} \lim_{\Delta t \rightarrow \infty} \frac{d}{d(\Delta t)} \text{MSD}(\Delta t)$

*   A larger self-diffusion constant $D^s$ corresponds to a faster rate of spatial spreading for a colloidal population.

Kinematic Framework and Solvent-Particle Force Dynamics

• Hierarchical Kinematic Chain:
  • Connecting microscopic mechanics to macroscopically measured diffusion constants follows classical Newtonian kinematics:

$\text{Forces } \mathbf{F}(t) \longrightarrow \text{Acceleration } \mathbf{a}(t) \longrightarrow \text{Velocity } \mathbf{v}(t) \longrightarrow \text{Displacement } \Delta\mathbf{r}(\Delta t) \longrightarrow \text{MSD}(\Delta t) \longrightarrow D^s$

• Newton's Second Law of Motion:
  • Classical mechanics governs both astronomical bodies and microscopic particles in fluid media:

$\mathbf{F}_{\text{total}} = m \mathbf{a}(t) = m \frac{d\mathbf{v}(t)}{dt}$

• Microscopic Origin of Forces:
  • Colloidal particles are surrounded by fluid solvent molecules that lie below the optical resolution limit of microscopes.
  • Because particle surface area scales with diameter squared ($\propto d^2$), the number of neighboring solvent collisions scales as $d^2$. For a $10\,\mu\text{m}$ sphere immersed in solvent molecules measuring a few angstroms, tracking every individual molecular collision is computationally prohibitive and analytical overkill.

Implicit Modeling of Solvent Effects: The Drag Force and Stokes' Law

• Langevin and Einstein Implicit Force Formalism:

  • Paul Langevin and Albert Einstein proposed replacing explicit solvent molecular tracking with an implicit continuum model using two effective force terms:
    1. A systematic drag force that resists macroscale particle movement and dissipates kinetic energy.
    2. A fluctuating random Brownian force that imparts stochastic momentum kicks.

• The Drag Force ($\mathbf{F}^d$):

  • Moving a particle through a stagnant fluid creates velocity gradients in the surrounding fluid, requiring mechanical work and inducing an opposing drag force $\mathbf{F}^d(t)$.
  • The simplest linear velocity-dependent formulation for drag is:

$\mathbf{F}^d(t) = -\zeta \mathbf{v}(t)$

*   The drag coefficient $\zeta$ must be a strictly positive scalar ($\zeta > 0$) so that the force vector strictly opposes the instantaneous direction of motion ($-\mathbf{v}(t)$).

• Stokes' Law for Colloidal Spheres:
  • For an isolated, dilute spherical colloidal particle translating in a fluid of viscosity $\eta$ at rest, the drag coefficient is given by Stokes' law:

$\zeta = 6 \pi \eta r = 3 \pi \eta d$

*   Where $r = \frac{d}{2}$ is the sphere radius, $d$ is the sphere diameter, and $\eta$ is the dynamic fluid viscosity of the solvent.
*   Solvents with higher viscosity (e.g., honey relative to water) possess significantly higher $\zeta$ values, resulting in stronger resistive forces.

Hydrostatic Effects: Gravitational and Buoyancy Forces in Sedimentation

• Sedimentation Phenomenon:

  • Sedimentation describes the net directional translation of particles vertically under the influence of gravity.

• Gravitational Force ($\mathbf{F}^g$):

  • Direct gravity acts downwards along the negative $z$ coordinate axis:

$\mathbf{F}^g = -m_c g \mathbf{\hat{z}} = -\rho_c V g \mathbf{\hat{z}}$

*   Where $\rho_c$ is the mass density of the colloidal material, $V$ is the particle volume, $g$ is the gravitational acceleration constant, and $\mathbf{\hat{z}} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$ is the vertical unit vector.

• Buoyancy Force ($\mathbf{F}^b$) and Archimedes' Principle:
  • Gravity creates a depth-dependent hydrostatic pressure gradient within the surrounding solvent ($P_{\text{bottom}} > P_{\text{top}}$).
  • This hydrostatic pressure imbalance yields an upward buoyant force $\mathbf{F}^b$ acting on the submerged particle volume according to Archimedes' principle:

$\mathbf{F}^b = + m_s g \mathbf{\hat{z}} = + \rho_s V g \mathbf{\hat{z}}$

*   Where $\rho_s$ is the mass density of the displaced solvent medium.

Mathematical Modeling of Particle Sedimentation via Differential Equations

• One-Dimensional Equation of Motion:
  • Restricting vertical translational dynamics to the $z$-axis (ignoring transverse $x, y$ dynamics and fluctuating Brownian forces temporarily), Newton's second law gives:

$m \frac{d v_z(t)}{dt} = F_z^d + F_z^b + F_z^g$

$m \frac{d v_z(t)}{dt} = -\zeta v_z(t) + \rho_s V g - \rho_c V g$

$m \frac{d v_z(t)}{dt} = -\zeta v_z(t) + (\rho_s - \rho_c) V g$

• Density Difference Dynamics:

  • If $\rho_c > \rho_s$ (particle denser than fluid), the term $(\rho_s - \rho_c) < 0$, causing heavy particles to sink downwards.
  • If $\rho_c < \rho_s$ (particle lighter than fluid), the term $(\rho_s - \rho_c) > 0$, causing light particles to float upwards.
  • The drag force always opposes the motion regardless of whether the particle sinks or floats, guaranteeing that both rising and falling particles reach a finite terminal velocity.

• Classification of the Sedimentation Equation:

  • The governing dynamic equation:

$m \frac{d v_z(t)}{dt} + \zeta v_z(t) = (\rho_s - \rho_c) V g$

*   This equation is classified mathematically as a **first-order, inhomogeneous ordinary differential equation (ODE)**:
    *   *First-order:* Contains only a first time derivative of velocity ($\frac{d v_z}{dt}$).
    *   *Inhomogeneous:* Contains a non-zero constant driving term $(\rho_s - \rho_c) V g$ uncoupled from velocity $v_z$
    *   *Ordinary:* Velocity depends strictly on a single independent variable (time $t$).

• Characteristic Momentum Relaxation Time Scale ($\tau_m$):
  • Dividing the differential equation by mass $m$ yields:

$\frac{d v_z(t)}{dt} + \frac{\zeta}{m} v_z(t) = \frac{(\rho_s - \rho_c) V g}{m}$

*   Dimensional analysis confirms that the ratio $\frac{m}{\zeta}$ carries units of time. This defines the momentum relaxation time scale $\tau_m$:

$\tau_m = \frac{m}{\zeta}$

*   Rewriting the ODE using $\tau_m$:

$\frac{d v_z(t)}{dt} + \frac{1}{\tau_m} v_z(t) = \frac{(\rho_s - \rho_c) V g}{m}$

*   The time constant $\tau_m$ governs the exponential transient period required for a particle to settle from rest into its steady-state terminal velocity.

• Solution Strategy via Integrating Factor:
  • To solve this first-order inhomogeneous ODE, both sides are multiplied by an integrating factor defined by the exponential of the time coefficient integral:

$\text{Integrating Factor} = \exp\left(\int \frac{1}{\tau_m} dt\right) = e^{t / \tau_m}$

Questions & Discussion

• Question: Does the displacement of a random walk eventually taper off or plateau over long times?

  • Answer: In an unbounded fluid (such as an infinitely deep ocean), there is no spatial limit; the mean square displacement grows linearly indefinitely without tapering off. Tapering off occurs only if the particle motion is physically bounded by confinement walls, such as the sides of a beaker.

• Question: How is the mathematical magnitude or length of a spatial vector calculated?

  • Answer: Vector magnitude is calculated by taking the square root of the sum of the squared individual components, which corresponds to the square root of the vector's dot product with itself: $\sqrt{\mathbf{a} \cdot \mathbf{a}}$.

• Question: In Newton's second law $m \mathbf{a} = \sum \mathbf{F}$, what physical mechanism generates the acceleration?

  • Answer: Forces generate acceleration.

• Question: Why does honey exert greater drag on a moving particle than water?

  • Answer: Honey possesses a much higher dynamic viscosity ($\eta$) than water, directly increasing the drag coefficient $\zeta = 6 \pi \eta r$.

• Question: What fluid force opposes the downward gravitational force acting on a submerged colloid?

  • Answer: The buoyancy force, which originates from hydrostatic pressure gradients within the solvent acting on the particle volume.