Atomic Models for Diffusion (II): Gas & Liquid Phases
2026-02-02
Key ideas from last lecture:
After this lecture, you will be able to:
From last lecture:
Define MSD as the normalized second moment of concentration:
\[\begin{align} \langle R^2(t)\rangle &= \frac{\int_0^\infty r^2 c(r,t)\,4\pi r^2 dr} {\int_0^\infty c(r,t)\,4\pi r^2 dr} \end{align}\]Using solution to point source diffusion, Einstein showed:
MSD in a 1D random-walk model is the expectance value of \(\text{Distance}^2\):
\[\begin{align} \langle R^2(t)\rangle &= N_{\tau} \langle r^2\rangle \\ &= \Gamma \tau \langle r^2\rangle \end{align}\]Compare (1D case):
\[\begin{align} \langle R^2(\tau)\rangle &= 2 D \tau \\ &= \Gamma \tau \langle r^2\rangle \end{align}\]Results for diffusivity \(D\)
\[\begin{align} D &= \frac{\Gamma \langle r^2\rangle}{6};\quad \text{3D} \\ &= \frac{\Gamma \langle r^2\rangle}{4};\quad \text{2D}\\ &= \frac{\Gamma \langle r^2\rangle}{2};\quad \text{1D} \end{align}\]Units:
The random-walk model considers “uncorrelated” jumps, where subsequent events are independent of previous steps, where \(\langle R^2\rangle = N_{\tau} \langle r^2\rangle\).
In reality, some atomic motions are “correlated”, so that the direction / probability of the next move are linked to previous events. Examples include:
3D Diffusivity correction:
\[\begin{align} D &= \frac{\Gamma \langle r^2\rangle}{6} f \\ f &\leq 1\;\quad\text{for correlated-jump} \end{align}\]The master equation for microscopic diffusivity becomes
\[\begin{align} \boxed{ D = \frac{\Gamma \langle r^2\rangle}{6} f } \end{align}\]Applicable fields?
\(\Gamma\) and \(r^2\) in gas phase can be solved using statistical mechanics treatment (kinetic theory). Some important results:
\[ \langle u \rangle = \sqrt{\frac{8 k_B T}{\pi m}} \]
\[ \lambda = \frac{1}{\sqrt{2} \pi d n} \]
Using ideal gas law \(p = n k_B T\), it is possible to show
\[\begin{align} D &\propto \sqrt{T} (\frac{p}{k_B T})^{-1} \\ &= \frac{T^{3/2}}{p} \end{align}\]Methods like the Chapman-Enskog theory has such power laws with \(T\) and \(p\), and can predict \(D\) in gas phase within 10% error compared to experimental results.
Recall the macroscopic model for diffusivity (Einstein relation) that links mobility \(M\) to diffusivity \(D\) using steady-state motion of a particle in a frictous fluid:
\[\begin{align} D &= M k_B T \\ M &= \frac{\text{[velocity]}}{\text{[driving force]}} \end{align}\]At steady-state, the driving force \(F_{d}\) equals the opposite of dragging force \(f_{drag}\). In the linear regime, we can often model \(f_{drag} = k_{f} v\), where \(k_{f}\) is the friction factor.
We will see the mobility treatment is essentially the same as \(D \propto \Gamma \langle r^2 \rangle\).
Goal: from dynamic equation of the system –> expressions for \(\Gamma\) and \(\langle r^2 \rangle\)
Governing Equation
We’re more interested in the average displacement \(x^2\). In 1D system it follows:
\[\begin{align} \langle \frac{\partial (x^2)}{\partial t} \rangle &= \frac{2 k_B T}{k_f} \\ \langle x^2 \rangle &= \frac{2 k_B T}{k_f} t \end{align}\]Which means \(\langle x^2 \rangle\) linearly increases with time!
For 3D system where \(x, y, z\) motions are independent, \(\langle R^2 \rangle = \langle x^2 \rangle + \langle y^2 \rangle + \langle z^2 \rangle = \frac{6 k_B T}{k_f} t\). We can reuse the result \(\langle R^2 \rangle = 6 Dt\):
\[\begin{align} D &= \frac{\langle R^2 \rangle}{6 t} \\ &= \frac{\cancel{6} k_B T \cancel{t}}{k_f \cancel{6} \cancel{t}} \\ &= \frac{k_B T}{k_f} \\ &= M k_B T \end{align}\]We’ve arrived at the macroscopic conclusion! The last expression uses the relation \(M = f_k^{-1}\).
When the mobility \(M\) uses the Stokes equation \(f_{drag} = 6 \pi \eta R v\), we arrive at the Stokes-Einstein equation:
\[\begin{align} D &= M k_B T\\ &= \frac{\cancel{v}}{6 \pi \eta R \cancel{v}} k_B T \\ &= \frac{k_B T}{6 \pi \eta R} \end{align}\]In linear polymer with \(N\) repeating units, each having characteristic lenth \(b\), the self-diffusivity follows:
\[\begin{align} D^* &= \frac{k_B T}{6 \pi \eta R_h} \end{align}\]and the hydrodynamic radius \(R_h\) scales as
\[\begin{align} R_h &\sim \begin{cases} N^{3/5} b & \text{good solvent} \\ N^{1/2} b & \theta \text{theta solvent} \end{cases} \end{align}\]The relation \(D = \frac{\Gamma r^2}{6} f\) can be applied to diffusion in solids, if we know the type of mechanism
Some terminologies:
Key of diffusion theory: barrier ⇔ diffusivity
The easiest model to illustrate the effect of an energy barrier is the potential well model. Our goal is still to find:
Key results:
\[ \nu = \sqrt{\frac{k_B T}{2 \pi m}} \frac{1}{L_w} \]
\[ \Gamma' = \nu e^{-\frac{E^a}{k_B T}} \]
Again, the jumping frequency is related to thermal activation.
\[ D = D_0 \exp(- \frac{H^m}{k_B T}) \]
In the plot only activation enthalpy relevant, entropic effect largely ignored
Activation enthalpy can be readily obtained from modern quantum chemistry / density functional theory calculation
Theoretical calculation for \(D\) covered in later part of the course
In this lecture we covered the following topics: