MATE 664 Lecture 09

Atomic Models for Diffusion (II): Gas & Liquid Phases

Author

Dr. Tian Tian

Published

February 2, 2026

Note

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Recap of Lecture 08

Key ideas from last lecture:

Analytical solution to diffusion problems (infinite domain)
- Semi-infinite (half-half) solution
- Line / point source solution
- Superimposition method
Separation of variables method (finite / bounded domain)
Laplace transform (will not be covered in exam)

Learning outcomes

After this lecture, you will be able to:

Recall the conditions behind the Einstein diffusion equation
Understand why the Einstein relation is universal across states of matter
Connect gas, liquid, and solid diffusion within a unified framework
Derive Arrhenius-type diffusivity from a 1D potential well model
Identify assumptions and missing physics in simple diffusion models

Recap: linking atomistic motion to diffusion

From last lecture:

Diffusion emerges from random microscopic motion
Mean squared displacement (MSD) is the key observable
Einstein equation links MSD to macroscopic diffusivity

Einstein diffusion equation

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:

3D: $\langle R^2\rangle = 6Dt$
2D: $\langle R^2\rangle = 4Dt$
1D: $\langle R^2\rangle = 2Dt$

Random-walk model solution to MSD

MSD in a 1D random-walk model is the expectance value of $\text{Distance}^2$:

\[\begin{align} \langle R^2(t)\rangle &= N_{\tau} \langle r^2\rangle \\ &= \Gamma \tau \langle r^2\rangle \end{align}\]

Linking Einstein equation to random walk model

Two methods should give the same result for $\langle R^2(t)\rangle$
$t$ in Einstein’s method is just $\tau$ in random walk model

Compare (1D case):

\[\begin{align} \langle R^2(\tau)\rangle &= 2 D \tau \\ &= \Gamma \tau \langle r^2\rangle \end{align}\]

Results for diffusivity $D$

\[\begin{align} D &= \frac{\Gamma \langle r^2\rangle}{6};\quad \text{3D} \\ &= \frac{\Gamma \langle r^2\rangle}{4};\quad \text{2D}\\ &= \frac{\Gamma \langle r^2\rangle}{2};\quad \text{1D} \end{align}\]

In-depth analysis of microscopic $D$ formula

Units:

Diffusivity $D$: $\text{m}^2/\text{s}$
Mean jump length $\langle r^2\rangle$: $\text{m}$
Frequency $\Gamma$: $\text{s}^{-1}$
The macroscopic diffusivity is directly linked to microscopic movements!

Correction for correlated jumps

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:

Movement in tangled polymer chains
Vacancy-mediated diffusion in solids
Push-out mechanism of interstalcy diffusion

3D Diffusivity correction:

\[\begin{align} D &= \frac{\Gamma \langle r^2\rangle}{6} f \\ f &\leq 1\;\quad\text{for correlated-jump} \end{align}\]

Application of microscopic diffusivity equation

The master equation for microscopic diffusivity becomes

\[\begin{align} \boxed{ D = \frac{\Gamma \langle r^2\rangle}{6} f } \end{align}\]

Applicable fields?

Gas? ✅ – kinetic theory of gases
Liquid? ✅ – Stokes-Einstein equation
Amorphous material / polymers? ✅ – Modified Stokes-Einstein
Solid? ✅ – various mechanism

Case 1: gas-phase diffusivity

$\Gamma$ and $r^2$ in gas phase can be solved using statistical mechanics treatment (kinetic theory). Some important results:

Average velocity $\langle u \rangle$ ($m$: molecular weight)

\[ \langle u \rangle = \sqrt{\frac{8 k_B T}{\pi m}} \]

Mean free path $\lambda$ ($n$: number density; $d$: molecular diameter)

\[ \lambda = \frac{1}{\sqrt{2} \pi d n} \]

Relation to microscopic parameters:

\[\begin{align} \Gamma &= \frac{\langle u \rangle}{\lambda} \\ \langle r^2 \rangle &= 2 \lambda^2 \\ D &= \frac{1}{3} \langle u \rangle \lambda \end{align}\]

Gas-phase diffusivity: $T, p$ relations

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.

Case 2: liquid-phase diffusivity

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 $D \propto \Gamma \langle r^2 \rangle$.

Newton’s equation for transport in viscous liquid

Goal: from dynamic equation of the system –> expressions for $\Gamma$ and $\langle r^2 \rangle$
Governing Equation

\[\begin{align} m a &= F_{d} - k_f v \\ m \frac{\partial^2 x}{\partial t^2} &= F_{d} - k_f \frac{\partial x}{\partial t} \\ \vdots & \vdots \\ \frac{m}{2} \frac{\partial}{\partial t}\left[ \frac{\partial (x^2)}{\partial t} \right] - m \left( \frac{\partial x}{\partial t} \right)^2 &= x F_{d,x} - \frac{k_f}{2} \frac{\partial (x^2)}{\partial t} \end{align}\]

Transport in viscous liquid: results

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!

Linking diffusivity results

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 $M = f_k^{-1}$.

Diffusivity in liquid: Stokes-Einstein equation

When the mobility $M$ uses the Stokes equation $f_{drag} = 6 \pi \eta R v$, we arrive at the Stokes-Einstein equation:

\[\begin{align} D &= M k_B T\\ &= \frac{\cancel{v}}{6 \pi \eta R \cancel{v}} k_B T \\ &= \frac{k_B T}{6 \pi \eta R} \end{align}\]

Viscosity $\eta$ (unit $\text{Pa}\cdot \text{s}$) typically follows thermal activation $\eta \propto \exp(A/T)$
What about the radius $R$ for non-spherical objects?

Polymer diffusivity models

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

\[\begin{align} R_h &\sim \begin{cases} N^{3/5} b & \text{good solvent} \\ N^{1/2} b & \theta \text{theta solvent} \end{cases} \end{align}\]

Good solvent: polymer–solvent interaction is favorable
Theta solvent: polymer–solvent interaction is neutral (like ideal chain)

Case 3: diffusivity in solids

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:

Mechanism: an abstraction of atomic motion in crystals
Sites: available spaces for atoms to move in / out
- Substitutional site: an atom replaces the position of another
- Interstitial site: an atom is inserted into the spaces between two atomic sites
Barrier: when moving atoms in either substitional or interstitial mechanisms, the inter-atomic potential causes moving atoms to “feel” higher energy states

Key of diffusion theory: barrier ⇔ diffusivity

Solid diffusion: potential well model

The easiest model to illustrate the effect of an energy barrier is the potential well model. Our goal is still to find:

Expression for $\Gamma$, $r^2$ (and optionally $f$)
Derive $D = \frac{\Gamma r^2}{6} f$

Step-function potential well model

Key results:

Attempted frequency $\nu$ (due to kinetic energy)

\[ \nu = \sqrt{\frac{k_B T}{2 \pi m}} \frac{1}{L_w} \]

Cross-barrier frequency $\Gamma'$

\[ \Gamma' = \nu e^{-\frac{E^a}{k_B T}} \]

Diffusivity

\[\begin{align} D &= \sqrt{\dfrac{k_B T}{2 \pi m}} \frac{(L_w + L_a)^2}{L_w} e^{-\dfrac{E^a}{k_B T}} \\ &= D_0 e^{-\dfrac{E^a}{k_B T}} \end{align}\]

More complex landscape: parabolic well model

Near the bottom of well, $E(x) = E_{min} + \frac{\beta}{2} (x - x_{min})^2$
More realistic for interatomic potential (similar to springs)
Modifications from the step-function well:

\[\begin{align} \Gamma' &= \frac{1}{2 \pi} \sqrt{\frac{\beta}{m}} e^{-\frac{E^a}{k_B T}} \end{align}\]

Again, the jumping frequency is related to thermal activation.

What is missing in this picture?

One-particle energy landscape –> many-body landscape
Static potential –> configuration-dependent $E(\vec{x})$
Energy only –> entropic effect in many-body system
Uncorrelated jump –> correlated movement
Single jump event –> symmetry multiplication in lattice

Case 3: diffusion in many-body potential landscape

Different modes of movement co-exist
General solution still follows the Boltzmann-distribution-like equation (see KOM eq. 7.25 )

\[\begin{align} \Gamma' &= \nu e^{-\frac{G^m}{k_B T}} \\ &= \nu e^{\frac{S^m}{k_B}} e^{-\frac{H^m}{k_B T}} \\ \end{align}\]

Attempt frequency related to jumping angular momentum: $\nu = \omega_J / (2 \pi)$
Activation entropy $S^m$ related to all angular momenta in the system.

In-depth discussion about $D$

Arrhenius-type equation:

\[ D = D_0 \exp(- \frac{H^m}{k_B T}) \]

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

Summary

In this lecture we covered the following topics:

Link between micro- and macroscopic diffusivity descriptions through the Einstein equation
Master equation $D = \dfrac{\Gamma r^2}{6}f$
Case studies of diffusivities in gas and liquid phases
Derivation of Arrhenius diffusivity equation using the potential well model

--- title: "MATE 664 Lecture 09" subtitle: "Atomic Models for Diffusion (II): Gas & Liquid Phases" author: "Dr. Tian Tian" date: "2026-02-02" format: html: {} revealjs: output-file: slides.html pdf: output-file: L09.pdf --- ::: {.content-visible when-format="html" unless-format="revealjs"} ::: {.callout-note} - Slides 👉 [Open presentation🗒️](./slides.html) - PDF version of course note 👉 [Open in pdf](./L09.pdf) - Handwritten notes 👉 [Open in pdf](./public/L09_annotated.pdf) ::: ::: ## Recap of Lecture 08 {.center} Key ideas from last lecture: - Analytical solution to diffusion problems (infinite domain) - Semi-infinite (half-half) solution - Line / point source solution - Superimposition method - Separation of variables method (finite / bounded domain) - Laplace transform (will not be covered in exam) ## Learning outcomes {.center} After this lecture, you will be able to: - Recall the conditions behind the Einstein diffusion equation - Understand why the Einstein relation is universal across states of matter - Connect gas, liquid, and solid diffusion within a unified framework - Derive Arrhenius-type diffusivity from a 1D potential well model - Identify assumptions and missing physics in simple diffusion models ## Recap: linking atomistic motion to diffusion {.center} From last lecture: - Diffusion emerges from random microscopic motion - Mean squared displacement (MSD) is the key observable - Einstein equation links MSD to macroscopic diffusivity ## Einstein diffusion equation Define MSD as the normalized second moment of concentration: ```{=tex} \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: - 3D: $\langle R^2\rangle = 6Dt$ - 2D: $\langle R^2\rangle = 4Dt$ - 1D: $\langle R^2\rangle = 2Dt$ ## Random-walk model solution to MSD MSD in a 1D random-walk model is the expectance value of $\text{Distance}^2$: ```{=tex} \begin{align} \langle R^2(t)\rangle &= N_{\tau} \langle r^2\rangle \\ &= \Gamma \tau \langle r^2\rangle \end{align} ``` ## Linking Einstein equation to random walk model - Two methods should give the same result for $\langle R^2(t)\rangle$ - $t$ in Einstein's method is just $\tau$ in random walk model :::{.columns} :::{.column width="50%"} Compare (1D case): ```{=tex} \begin{align} \langle R^2(\tau)\rangle &= 2 D \tau \\ &= \Gamma \tau \langle r^2\rangle \end{align} ``` ::: :::{.column width="50%"} Results for diffusivity $D$ ```{=tex} \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} ``` ::: ::: ## In-depth analysis of microscopic $D$ formula ```{=tex} \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: - Diffusivity $D$: $\text{m}^2/\text{s}$ - Mean jump length $\langle r^2\rangle$: $\text{m}$ - Frequency $\Gamma$: $\text{s}^{-1}$ - The macroscopic diffusivity is directly linked to microscopic movements! ## Correction for correlated jumps 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: - Movement in tangled polymer chains - Vacancy-mediated diffusion in solids - Push-out mechanism of interstalcy diffusion 3D Diffusivity correction: ```{=tex} \begin{align} D &= \frac{\Gamma \langle r^2\rangle}{6} f \\ f &\leq 1\;\quad\text{for correlated-jump} \end{align} ``` ## Application of microscopic diffusivity equation The master equation for microscopic diffusivity becomes ```{=tex} \begin{align} \boxed{ D = \frac{\Gamma \langle r^2\rangle}{6} f } \end{align} ``` Applicable fields? - Gas? ✅ -- kinetic theory of gases - Liquid? ✅ -- Stokes-Einstein equation - Amorphous material / polymers? ✅ -- Modified Stokes-Einstein - Solid? ✅ -- various mechanism ## Case 1: gas-phase diffusivity $\Gamma$ and $r^2$ in gas phase can be solved using statistical mechanics treatment (kinetic theory). Some important results: - Average velocity $\langle u \rangle$ ($m$: molecular weight) $$ \langle u \rangle = \sqrt{\frac{8 k_B T}{\pi m}} $$ - Mean free path $\lambda$ ($n$: number density; $d$: molecular diameter) $$ \lambda = \frac{1}{\sqrt{2} \pi d n} $$ - Relation to microscopic parameters: ```{=tex} \begin{align} \Gamma &= \frac{\langle u \rangle}{\lambda} \\ \langle r^2 \rangle &= 2 \lambda^2 \\ D &= \frac{1}{3} \langle u \rangle \lambda \end{align} ``` ## Gas-phase diffusivity: $T, p$ relations Using ideal gas law $p = n k_B T$, it is possible to show ```{=tex} \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. ## Case 2: liquid-phase diffusivity 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: ```{=tex} \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 $D \propto \Gamma \langle r^2 \rangle$. ## Newton's equation for transport in viscous liquid - Goal: from dynamic equation of the system --> expressions for $\Gamma$ and $\langle r^2 \rangle$ - Governing Equation ```{=tex} \begin{align} m a &= F_{d} - k_f v \\ m \frac{\partial^2 x}{\partial t^2} &= F_{d} - k_f \frac{\partial x}{\partial t} \\ \vdots & \vdots \\ \frac{m}{2} \frac{\partial}{\partial t}\left[ \frac{\partial (x^2)}{\partial t} \right] - m \left( \frac{\partial x}{\partial t} \right)^2 &= x F_{d,x} - \frac{k_f}{2} \frac{\partial (x^2)}{\partial t} \end{align} ``` ## Transport in viscous liquid: results We're more interested in the average displacement $x^2$. In 1D system it follows: ```{=tex} \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! ## Linking diffusivity results 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 $\langle R^2 \rangle = 6 Dt$: ```{=tex} \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 $M = f_k^{-1}$. ## Diffusivity in liquid: Stokes-Einstein equation When the mobility $M$ uses the Stokes equation $f_{drag} = 6 \pi \eta R v$, we arrive at the Stokes-Einstein equation: ```{=tex} \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} ``` - Viscosity $\eta$ (unit $\text{Pa}\cdot \text{s}$) typically follows thermal activation $\eta \propto \exp(A/T)$ - What about the radius $R$ for _non-spherical_ objects? ## Polymer diffusivity models In linear polymer with $N$ repeating units, each having characteristic lenth $b$, the self-diffusivity follows: ```{=tex} \begin{align} D^* &= \frac{k_B T}{6 \pi \eta R_h} \end{align} ``` and the hydrodynamic radius $R_h$ scales as ```{=tex} \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} ``` - Good solvent: polymer–solvent interaction is favorable - Theta solvent: polymer–solvent interaction is neutral (like ideal chain) ## Case 3: diffusivity in solids 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: - *Mechanism*: an abstraction of atomic motion in crystals - *Sites*: available spaces for atoms to move in / out - Substitutional site: an atom replaces the position of another - Interstitial site: an atom is inserted into the spaces between two atomic sites - *Barrier*: when moving atoms in either substitional or interstitial mechanisms, the inter-atomic potential causes moving atoms to "feel" higher energy states Key of diffusion theory: barrier ⇔ diffusivity ## Solid diffusion: potential well model The easiest model to illustrate the effect of an energy barrier is the potential well model. Our goal is still to find: 1) Expression for $\Gamma$, $r^2$ (and optionally $f$) 2) Derive $D = \frac{\Gamma r^2}{6} f$ ## Step-function potential well model Key results: - Attempted frequency $\nu$ (due to kinetic energy) $$ \nu = \sqrt{\frac{k_B T}{2 \pi m}} \frac{1}{L_w} $$ - Cross-barrier frequency $\Gamma'$ $$ \Gamma' = \nu e^{-\frac{E^a}{k_B T}} $$ - Diffusivity ```{=tex} \begin{align} D &= \sqrt{\dfrac{k_B T}{2 \pi m}} \frac{(L_w + L_a)^2}{L_w} e^{-\dfrac{E^a}{k_B T}} \\ &= D_0 e^{-\dfrac{E^a}{k_B T}} \end{align} ``` ## More complex landscape: parabolic well model - Near the bottom of well, $E(x) = E_{min} + \frac{\beta}{2} (x - x_{min})^2$ - More realistic for interatomic potential (similar to springs) - Modifications from the step-function well: ```{=tex} \begin{align} \Gamma' &= \frac{1}{2 \pi} \sqrt{\frac{\beta}{m}} e^{-\frac{E^a}{k_B T}} \end{align} ``` Again, the jumping frequency is related to thermal activation. ## What is missing in this picture? - One-particle energy landscape --> many-body landscape - Static potential --> configuration-dependent $E(\vec{x})$ - Energy only --> entropic effect in many-body system - Uncorrelated jump --> correlated movement - Single jump event --> symmetry multiplication in lattice ## Case 3: diffusion in many-body potential landscape - Different modes of movement co-exist - General solution still follows the Boltzmann-distribution-like equation (see _KOM eq. 7.25_ ) ```{=tex} \begin{align} \Gamma' &= \nu e^{-\frac{G^m}{k_B T}} \\ &= \nu e^{\frac{S^m}{k_B}} e^{-\frac{H^m}{k_B T}} \\ \end{align} ``` - Attempt frequency related to jumping angular momentum: $\nu = \omega_J / (2 \pi)$ - Activation entropy $S^m$ related to all angular momenta in the system. ## In-depth discussion about $D$ - Arrhenius-type equation: $$ D = D_0 \exp(- \frac{H^m}{k_B T}) $$ - 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 ## Summary In this lecture we covered the following topics: - Link between micro- and macroscopic diffusivity descriptions through the Einstein equation - Master equation $D = \dfrac{\Gamma r^2}{6}f$ - Case studies of diffusivities in gas and liquid phases - Derivation of Arrhenius diffusivity equation using the potential well model