Unit 3: Diffusion Phenomena & Brownian Motion

Diffusion is fundamentally a thermodynamic process driven by the minimization of Gibbs free energy, rather than a purely mechanical concentration effect.

Chemical Potential Gradient as Generalized Force

For a solute species $i$ in solution, the chemical potential $\mu_i$ is given by:

where $a_i$ is activity, $\gamma_i$ is the activity coefficient, and $c_i$ is concentration.

The thermodynamic force $F_{\text{therm}}$ acting on a single molecule is the negative spatial gradient of chemical potential per particle:

For an ideal solution ($\gamma_i = 1$, $a_i = c_i$):

Terminal Drift Velocity and Flux

Under this thermodynamic force, solute molecules acquire a terminal drift velocity $v_{\text{drift}}$ balanced by the Stokes hydrodynamic friction coefficient $f$:

The macroscopic molar diffusion flux $J$ (moles passing through unit area per unit time) is the product of concentration and velocity:

Comparing with Fick's First Law $J = -D \frac{\partial c_i}{\partial x}$ yields the Stokes-Einstein Relation:

This links macroscopic diffusion directly to molecular thermal agitation $k_B T$ and solvent viscosity $\eta$.

Master Table: Diffusion Coefficients Across States of Matter & Molecular Sizes

Diffusion coefficients span more than twelve orders of magnitude from light gases to solid-state crystals:

| Medium / Phase | Diffusing Species | Temperature ($T$) | Diffusion Coefficient $D$ ($\text{m}^2/\text{s}$) | Characteristic Time $\tau = L^2 / (2D)$ for $L = 10\;\mu\text{m}$ | |---|---|---|---|---| | Gas ($1\text{ atm}$) | $H_2$ in Air | $298\text{ K}$ | $7.8 \times 10^{-5}$ | $0.64\;\mu\text{s}$ | | Gas ($1\text{ atm}$) | $CO_2$ in Air | $298\text{ K}$ | $1.6 \times 10^{-5}$ | $3.1\;\mu\text{s}$ | | Gas ($1\text{ atm}$) | Benzene in Air | $298\text{ K}$ | $8.8 \times 10^{-6}$ | $5.7\;\mu\text{s}$ | | Liquid (Aqueous) | Water self-diffusion | $298\text{ K}$ | $2.299 \times 10^{-9}$ | $21.7\text{ ms}$ | | Liquid (Aqueous) | $NaCl$ (mutual) | $298\text{ K}$ | $1.61 \times 10^{-9}$ | $31.1\text{ ms}$ | | Liquid (Aqueous) | Glucose ($M = 180$) | $298\text{ K}$ | $6.73 \times 10^{-10}$ | $74.3\text{ ms}$ | | Liquid (Aqueous) | Bovine Serum Albumin ($66\text{ kDa}$)| $298\text{ K}$ | $6.07 \times 10^{-11}$ | $824\text{ ms}$ | | Liquid (Aqueous) | Tobacco Mosaic Virus ($40\text{ MDa}$)| $298\text{ K}$ | $4.6 \times 10^{-12}$ | $10.9\text{ s}$ | | Viscous Liquid | Glycerol self-diffusion | $298\text{ K}$ | $1.7 \times 10^{-12}$ | $29.4\text{ s}$ | | Polymer Melt | Polystyrene in melt | $450\text{ K}$ | $1.0 \times 10^{-15}$ | $14\text{ hours}$ | | Solid Crystal | $C$ in $\alpha\text{-Fe}$ (interstitial) | $1200\text{ K}$ | $1.0 \times 10^{-10}$ | $500\text{ ms}$ | | Solid Crystal | $Au$ in $Cu$ (substitutional) | $1200\text{ K}$ | $5.0 \times 10^{-14}$ | $1000\text{ s}$ | | Solid Crystal | $Cu$ in $Cu$ self-diffusion | $300\text{ K}$ | $10^{-34}$ | $> 10^{18}\text{ years}$ |

Adolf Fick (1855) formulated the phenomenological laws of diffusion by drawing a direct mathematical analogy to Fourier's law of heat conduction and Ohm's law of electrical conduction.

Statement of Fick's First Law

In a one-dimensional isotropic medium, the diffusion flux $J_x$ (amount of substance crossing a unit area perpendicular to the $x$-axis per unit time) is proportional to the negative spatial concentration gradient:

where:

• $J_x$ is diffusion flux (SI: $\text{mol}/(\text{m}^2\cdot\text{s})$ or $\text{kg}/(\text{m}^2\cdot\text{s})$).
• $D$ is the diffusion coefficient (SI: $\text{m}^2\text{/s}$ or $\text{cm}^2\text{/s}$).
• $\frac{\partial c}{\partial x}$ is the concentration gradient (SI: $\text{mol/m}^4$).

The negative sign signifies that mass transport proceeds spontaneously from regions of higher chemical potential/concentration to regions of lower concentration.

General Vector Formulation

In three dimensions:

For anisotropic media (e.g., layered crystals, stretched polymers), $D$ is a second-rank tensor:

Steady-State Diffusion Through a Membrane

Under steady-state conditions, $\frac{\partial c}{\partial t} = 0$, meaning the flux $J$ is uniform across a planar membrane of thickness $L$ separated by constant concentrations $c_1$ and $c_2$:

Introducing the membrane partition coefficient $K = c_{\text{mem}} / c_{\text{aq}}$ and membrane permeability $P_{\text{perm}} = \frac{K D}{L}$:

When concentration varies with both position and time, mass conservation dictates non-steady-state diffusion.

Derivation from the Continuity Equation

Consider a volume element $\Delta V = A \Delta x$ between planes at $x$ and $x + \Delta x$. The rate of accumulation of solute in this volume element is:

Mass conservation requires that accumulation equals the net inflow minus outflow across the boundary planes:

Dividing by $A \Delta x$ and taking the limit $\Delta x \to 0$:

This is the Continuity Equation for diffusion.

Substituting Fick's first law $J_x = -D \frac{\partial c}{\partial x}$:

If the diffusion coefficient $D$ is independent of concentration (dilute regime):

This is Fick's Second Law of Diffusion (the classic parabolic diffusion equation).

Multi-Dimensional Geometries

1. Three-Dimensional Cartesian:

2. Cylindrical Coordinates (radial symmetry):

3. Spherical Coordinates (radial symmetry):

Analytical solutions to Fick's second law depend on initial and boundary conditions.

1. Instantaneous Planar Source (Gaussian Spreading)

Consider an infinitesimal sheet at $x = 0$ containing $N_0$ moles of solute per unit area injected at $t = 0$ into an infinite medium ($-\infty < x < \infty$):

The fundamental solution (Green's function) of $\frac{\partial c}{\partial t} = D \frac{\partial^2 c}{\partial x^2}$ is a spreading Gaussian:

Key features:

• Standard deviation of the spreading profile is $\sigma(t) = \sqrt{2 D t}$.
• Peak concentration at the center $x = 0$ decays as $c(0, t) = \frac{N_0}{\sqrt{4 \pi D t}} \propto t^{-1/2}$.

2. Semi-Infinite Medium with Constant Surface Concentration (Error Function)

Consider a semi-infinite medium ($x \ge 0$) initially at uniform concentration $c_0$, whose surface at $x = 0$ is held at fixed concentration $c_s$ for all $t > 0$:

Using the similarity variable $\eta = \frac{x}{\sqrt{4 D t}}$, the partial differential equation reduces to an ordinary differential equation:

Integrating yields the solution in terms of the Gauss error function ($\text{erf}$):

where:

The penetration depth where concentration reaches halfway ($c = \frac{c_s + c_0}{2}$) occurs at $z \approx 0.4769$:

In 1905, Albert Einstein and Marian Smoluchowski established that macroscopic diffusion is the statistical ensemble manifestation of microscopic Brownian motion.

Discrete One-Dimensional Random Walk

Consider a particle starting at the origin $x = 0$ at $t = 0$. In each discrete time step $\tau$, the particle steps a distance $\pm \lambda$ with equal probability $p = 1/2$. After $N$ steps (time $t = N \tau$), the particle's position is:

The average displacement is zero due to symmetry:

The mean squared displacement (MSD) is:

Because successive steps are statistically uncorrelated, $\langle \Delta x_i \Delta x_j \rangle = 0$ for $i \neq j$, and $\langle \Delta x_i^2 \rangle = \lambda^2$:

Connection to Fick's Second Law

In the continuum limit ($\lambda \to 0, \tau \to 0$ such that $\frac{\lambda^2}{2\tau} = D$):

In three dimensions, with independent motion along $x, y, z$:

This is the celebrated Einstein-Smoluchowski equation:

Jean Perrin used this equation in 1908 to track the Brownian motion of colloidal mastic particles, experimentally calculating Avogadro's number $N_A$ and confirming the physical reality of atoms.

University Honors Research Monograph: Anomalous Subdiffusion & Macromolecular Crowding in Cell Biology

Inside living biological cells, the aqueous cytoplasm is not a dilute Newtonian solvent, but a densely crowded viscoelastic gel packed with proteins, RNA, cytoskeletal filaments, and organelles occupying $20 - 40\%$ of total cell volume ($200 - 400\text{ g/L}$ macromolecular density):

• Breakdown of Fickian Scaling: Instead of the classical linear Einstein Brownian mean-squared displacement $\langle r^2(t) \rangle = 6 D t$, fluorescent correlation spectroscopy and single-particle tracking (SPT) reveal power-law subdiffusive scaling:

where $\alpha \approx 0.70 - 0.85$ in mammalian cytoplasm, and $\Gamma_\alpha$ is the anomalous transport coefficient ($\text{m}^2/\text{s}^\alpha$).

• Physical Mechanisms of Subdiffusion:

1. Steric Obstruction & Fractal Percolation: Dense networks of actin microfilaments and microtubules impose geometric tortuosity, trapping molecules in transient dead-ends.

2. Continuous-Time Random Walks (CTRW): Non-specific transient binding interactions between diffusing enzymes and cytoplasmic macromolecules generate heavy-tailed power-law waiting time distributions ($\psi(t) \propto t^{-(1+\alpha)}$).

3. Viscoelastic Hydrodynamics (Fractional Brownian Motion): Cytoplasmic polymer relaxation times span multiple orders of magnitude, generating long-range temporal memory in drag forces.

• Consequences for In Vivo Reaction Kinetics: Subdiffusion dramatically slows down large macromolecular search times while accelerating local geminate radical and enzyme-substrate re-encounters.

Paul Langevin (1908) formulated the first dynamical equation for Brownian motion by separating the total force on a particle into a systematic friction term and a fluctuating stochastic force.

The Langevin Equation

For a Brownian particle of mass $m$ and velocity $v(t)$:

where:

• $-\gamma v(t)$ is the macroscopic frictional drag ($\gamma = 6 \pi \eta r$).
• $R(t)$ is the fluctuating, zero-mean stochastic Gaussian white noise representing instantaneous molecular impacts from the solvent.

Properties of $R(t)$:

1. $\langle R(t) \rangle = 0$.
2. $\langle R(t) R(t') \rangle = 2 \gamma k_B T \delta(t - t')$ (Fluctuation-Dissipation Theorem).

Velocity Autocorrelation Function (VACF)

Multiplying by $v(0)$ and ensemble averaging:

Integrating yields an exponential decay with momentum relaxation time $\tau_m = m / \gamma$:

Green-Kubo Formula for Diffusion

The diffusion coefficient is the time integral of the velocity autocorrelation function:

recovering the Stokes-Einstein relation from microscopic stochastic dynamics.

Transport across biological and synthetic semipermeable membranes couples concentration diffusion with electrical potential gradients and osmotic pressure.

Osmotic Flux & Kedem-Katchalsky Formalism

When a membrane separates a pure solvent from a solution containing non-permeating solute, chemical potential equality across the membrane establishes an osmotic pressure $\Pi$:

(van 't Hoff equation).

Under combined hydrostatic pressure difference $\Delta P$ and osmotic pressure difference $\Delta \Pi$, the total volume flux $J_v$ is:

where:

• $L_p$ is the hydraulic permeability coefficient.
• $\sigma_{\text{ref}}$ is the Staverman reflection coefficient ($0 \le \sigma_{\text{ref}} \le 1$; $\sigma_{\text{ref}} = 1$ for a completely impermeable solute).

The solute flux $J_s$ across the membrane is:

where $\omega$ is the solute permeability coefficient and $\bar{c}_s$ is the mean intra-membrane solute concentration.

Donnan Equilibrium

When a semipermeable membrane separates an electrolyte solution containing an impermeable macromolecular poly-ion (e.g., protein $P^{z-}$ with $Na^+$ counterions) on side 1 from diffusible $NaCl$ on side 2, electrochemical equilibrium for diffusible $Na^+$ and $Cl^-$ requires:

Assuming unit activity coefficients:

Electroneutrality demands:

• Side 1: $[Na^+]_1 = [Cl^-]_1 + z [P^{z-}]_1$
• Side 2: $[Na^+]_2 = [Cl^-]_2 = c$

Because $[Na^+]_1 > [Cl^-]_1$, substitution shows:

The presence of non-diffusible poly-ions forces an asymmetric distribution of small mobile ions, producing a permanent transmembrane electric potential: the Donnan Potential:

Measuring self-diffusion coefficients $D$ non-invasively at molecular and macromolecular length scales is accomplished using nuclear spin-phase tagging via Pulsed-Field-Gradient NMR (PFG-NMR) and intensity autocorrelation analysis via Dynamic Light Scattering (DLS).

1. Pulsed-Field-Gradient NMR (Stejskal-Tanner Diffusion Metrology)

Pulsed-Field-Gradient Spin-Echo (PGSE) NMR, introduced by E.O. Stejskal and J.E. Tanner (1965), measures self-diffusion by applying pulsed spatial magnetic field gradients $g(z)$ along the static magnetic field axis $B_0$.

The PGSE Pulse Sequence

1. A $90^\circ$ radiofrequency pulse flips macroscopic magnetization into the transverse $x-y$ plane.
2. A pulsed gradient of amplitude $g$ and duration $\delta$ is applied. Nuclei at spatial position $z_1$ precess at local Larmor frequency $\omega(z_1) = \gamma (B_0 + g z_1)$, acquiring a spatially dependent phase angle:

1. A $180^\circ$ inversion pulse at time $\tau$ inverts spin phases: $\phi \to -\phi$.
2. After a diffusion delay $\Delta$, a second gradient pulse of identical magnitude $g$ and duration $\delta$ is applied. If molecules moved to new position $z_2$ via Brownian diffusion during interval $\Delta$, the re-phasing is incomplete:

The Stejskal-Tanner Attenuation Formula

Averaging over the 1D Gaussian displacement probability distribution $P(z_2 - z_1, \Delta) = \frac{1}{\sqrt{4 \pi D \Delta}} \exp\left(-\frac{(z_2 - z_1)^2}{4 D \Delta}\right)$, the attenuated NMR echo intensity $I(g)$ is:

where:

• $\gamma$ is the nuclear gyromagnetic ratio ($2.675 \times 10^8\text{ rad}/(\text{s}\cdot\text{T})$ for $^1H$).
• The correction $-\delta/3$ accounts for Brownian diffusion occurring during the finite duration of the gradient pulses.

Plotting $\ln(I/I_0)$ against the Stejskal-Tanner parameter $b = \gamma^2 g^2 \delta^2 (\Delta - \delta/3)$ produces a linear slope equal to $-D$. PFG-NMR measures diffusion coefficients from $10^{-9}\text{ m}^2/\text{s}$ (small molecules) down to $10^{-14}\text{ m}^2/\text{s}$ (polymers, lipid bilayers, confined porous media).

2. Dynamic Light Scattering (Photon Correlation Spectroscopy)

Dynamic Light Scattering (DLS) measures the Brownian translational diffusion coefficient of colloidal nanoparticles, micelles, and globular proteins suspended in liquid.

Temporal Intensity Autocorrelation

A monochromatic laser beam ($\lambda_0$) illuminates a dilute colloidal suspension. The scattered light intensity $I(t)$ collected at angle $\theta$ fluctuates randomly due to constructive and destructive interference caused by Brownian motion of scattering centers. The normalized second-order temporal intensity autocorrelation function $g^{(2)}(\tau)$ is:

By the Siegert relation for Gaussian optical fields:

where $\beta \le 1$ is an optical coherence factor, and $g^{(1)}(\tau)$ is the normalized electric field autocorrelation function. For monodisperse spheres:

where the decay rate $\Gamma$ is directly proportional to the translational diffusion coefficient $D$:

and $q$ is the magnitude of the scattering wave vector:

Hydrodynamic Size Extraction

Measuring $\Gamma$ as a function of $q^2$ yields the mutual diffusion coefficient $D = \Gamma / q^2$. Using the Stokes-Einstein equation, the hydrodynamic radius $R_h$ is directly obtained: