Question

Consider particles that are oriented at an angle \(\theta\) with respect to the \(x\) axis. The distribution of particles at different angles is \(c(\theta)\). The flux \(J(\theta)\) of particles through different angles depends on the gradient as in Fick's law: $$ J(\theta)=-\Theta\left(\frac{\partial c}{\partial \theta}\right), $$ where \(\Theta\) is the orientational diffusion coefficient. (a) Write an expression for the rotational diffusion equation. (b) If the orientational friction coefficient for a sphere of radius \(r\) is \(f_{\text {or }}=8 \pi \eta r^{3}\), write an expression for \(\Theta(r)\).

Short answer

(a) \( \frac{\partial^2 c}{\partial \theta^2} = 0 \). (b) \( \Theta(r) = \frac{k_B T}{8\pi \eta r^3} \).

Step-by-step solution

Understand Fick's Law for Rotational Diffusion

Given the flux equation for particles at an angle θ: \[ J(\theta) = -\theta \left(\frac{\partial c}{\partial \theta}\right) \]This equation is analogous to Fick's law of diffusion, but applied to angles instead of spatial coordinates.

Express Conservation of Particles

The rotational diffusion equation can be derived from the conservation of particles. For steady-state conditions, the flux of particles must satisfy the continuity equation: \[ \frac{\partial c}{\partial t} = -\frac{\partial J}{\partial \theta} \]Since we are considering steady-state, we set the time derivative to zero.

Substitute Flux into Continuity Equation

Using the given flux expression, substitute it into the continuity equation: \[ 0 = -\frac{\partial J}{\partial \theta} = -\frac{\partial }{\partial \theta} \left( -\Theta \frac{\partial c}{\partial \theta} \right) \]Simplify this to get: \[ 0 = \Theta \frac{\partial^2 c}{\partial \theta^2} \]

Write the Rotational Diffusion Equation

Rearrange the equation to get the rotational diffusion equation: \[ \frac{\partial^2 c}{\partial \theta^2} = 0 \]

Determine the Orientational Diffusion Coefficient \(\Theta\)

Given the orientational friction coefficient for a sphere of radius r: \[ f_{\text{or}} = 8\pi \eta r^3 \]and Einstein's relation for diffusion, \(\Theta = \frac{k_B T}{f_{\text{or}}}\), where \(k_B\) is Boltzmann’s constant and T is the temperature, we have: \[ \Theta(r) = \frac{k_B T}{8\pi \eta r^3} \]

Key concepts

Fick's Law

Fick's Law describes the flux of particles due to diffusion. In the context of rotational diffusion, this law is applied to particles oriented at an angle, \( \theta \), with respect to the x-axis. The flux, \( J(\theta) \), represents the flow of particles through different angles. The equation:
\[ J(\theta) = -\Theta\left(\frac{\partial c}{\partial \theta}\right) \]
indicates that the flux is proportional to the gradient of the particle concentration, \( c(\theta) \), with respect to \( \theta \). Here, \( \Theta \) is the orientational diffusion coefficient. The negative sign shows that particles move from regions of high concentration to low concentration. This concept is an extension of linear diffusion (found in spatial coordinates) to rotational or angular diffusion.

Orientational Diffusion Coefficient

The orientational diffusion coefficient, denoted as \( \Theta \), is a measure of how quickly particles diffuse in terms of their orientation. In the context of rotational motion, the coefficient plays a critical role in the flux equation:
\[ J(\theta) = -\Theta\left(\frac{\partial c}{\partial \theta}\right) \]
It is analogous to the diffusion coefficient in linear diffusion but here, it describes the rotational or orientational aspect. For a sphere of radius \( r \), the orientational friction coefficient \( f_{\text{or}} \) is given by:
\[ f_{\text{or}} = 8 \pi \eta r^3 \]
where \( \eta \) is the dynamic viscosity of the medium. The larger the friction coefficient, the more resistance the particle faces while rotating.

Einstein Relation

The Einstein relation connects the diffusion coefficient to thermal energy and friction. For rotational diffusion, this relation is applied to the orientational diffusion coefficient \( \Theta \). According to the Einstein relation:
\[ \Theta = \frac{k_B T}{f_{\text{or}}} \]
where:\

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• \( k_B \) is Boltzmann’s constant.
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• \( T \) is the temperature in kelvin.
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• \( f_{\text{or}} \) is the orientational friction coefficient, given by \( 8 \pi \eta r^3 \) for a sphere of radius \( r \).
\

This equation highlights that \( \Theta \) is inversely proportional to \( f_{\text{or}} \), which means that increased friction results in lower diffusion. Combining these, the orientational diffusion coefficient for a sphere is:
\[ \Theta(r) = \frac{k_B T}{8 \pi \eta r^3} \]
Understanding this relation helps in determining how different factors such as temperature, particle size, and medium viscosity affect the rotational diffusion of particles.