2.1. Equations of motion

Note that in the following analysis, we notate ${\boldsymbol A}\boldsymbol {\cdot }{\boldsymbol a}$ and ${\boldsymbol A}\boldsymbol {\cdot }{\boldsymbol B}$ as ${\boldsymbol A}{\boldsymbol a}$ and ${\boldsymbol A}{\boldsymbol B}$, respectively, where ${\boldsymbol a}\in \mathbb {R}^{3},\;{\boldsymbol A},{\boldsymbol B}\in \mathbb {R}^{3\times 3}$, since a second-order tensor can be seen as an operator denoting linear transformations. We also apply the Einstein summation convention to any repeated indices in the subscripts within a term. Vectors or tensors are denoted in bold. To study the diffusion of an anisotropic particle in a medium, we consider the motions of the particle in space under the application of external force and torque and thermal agitations. Let $\{{\boldsymbol e}_i: i=1,2,3\}$ be a global orthonormal frame and $\{{\boldsymbol f}_i: i=1,2,3\}$ be an orthonormal body frame fixed on the particle. The position and orientation of the particle are described by kinematic variables: ${\boldsymbol x}={\boldsymbol x}(t)$ for the position of the centre of mass and rigid rotation matrix ${\boldsymbol Q}(t)={\boldsymbol e}_i\otimes {\boldsymbol f}_i(t)$ for the orientation. Associated with the skew-symmetric matrix $\dot {{\boldsymbol Q}}^{T}{\boldsymbol Q}$, the angular velocity ${{\boldsymbol {\omega }}}$ is introduced such that for any vector ${\boldsymbol a}\in {\mathbb {R}}^{3}$, $\dot {{\boldsymbol Q}}^{T}{\boldsymbol Q} {\boldsymbol a}={{\boldsymbol {\omega }}}\times {\boldsymbol a}$. In particular, we have

(2.1)\begin{equation} \dot{{\boldsymbol f}}_i=\dot{{\boldsymbol Q}}^{T}{\boldsymbol Q}{\boldsymbol f}_i={{\boldsymbol{\omega}}}\times {\boldsymbol f}_i. \end{equation}

Let ${\boldsymbol v} ={\dot {{\boldsymbol x}}}(t)$ be the velocity of the centre of mass, $m$ the mass and ${\boldsymbol I}\in {\mathbb {R}}^{3\times 3}_{sym}$ (respectively, ${{\boldsymbol \tau }}^{e}$ and ${\boldsymbol g}^{e}$) the moment of inertia (respectively, external torque and force) with respect to the centre of mass. The equations of motions for the particle can then be written as

(2.2)\begin{equation} \left. \begin{gathered} \frac{\textrm{d}}{\textrm{d}t} (m{{\boldsymbol v}})={-}{\boldsymbol R}^{tt} {\boldsymbol v} -{\boldsymbol R}^{tr}{{\boldsymbol{\omega}}}+{\boldsymbol g}^{e} +{{\boldsymbol{\sigma}}}_t\boldsymbol{\xi},\\ \frac{\textrm{d}}{\textrm{d}t}(\boldsymbol I\boldsymbol{\omega})={-}({\boldsymbol R}^{tr})^{T}{\boldsymbol v} -{\boldsymbol R}^{rr} \boldsymbol{\omega}+{{\boldsymbol \tau}}^{e}+{{\boldsymbol{\sigma}}}_r \boldsymbol{\xi}, \end{gathered} \right\} \end{equation}

where ${{\boldsymbol {\xi }}}(t)\in \mathbb {R}^{3}$ denotes the uncorrelated white noises satisfying that

(2.3a,b)\begin{equation} \langle \xi(t)\rangle =0,\quad \langle \xi_i(t) \xi_j(t')\rangle=\delta_{ij}\delta(t-t'), \end{equation}

${{\boldsymbol {\sigma }}}_t,{{\boldsymbol {\sigma }}}_r \in \mathbb {R}^{3\times 3}$ are the fluctuation coefficients that will be determined by the fluctuation–dissipation theorem (Kubo Reference Kubo1966), and ${\boldsymbol R}^{({tt}, {tr}, {rr})}\in {\mathbb {R}} ^{3\times 3}$ represent the friction coefficient tensors or the inverse of the mobility tensor. In other words, the drag force ${\boldsymbol g}$ and torque ${{\boldsymbol \tau }}$ on the particle from the ambient fluid are related to the particle linear and angular velocity by

(2.4)\begin{equation} \begin{bmatrix}{\boldsymbol g}\\ {{\boldsymbol \tau}} \end{bmatrix}={-} \begin{bmatrix} {\boldsymbol R}^{tt} & {\boldsymbol R}^{tr} \\ ( {\boldsymbol R}^{tr})^{T} & {\boldsymbol R}^{rr} \\ \end{bmatrix}\begin{bmatrix} {\boldsymbol v} \\ {{\boldsymbol{\omega}}} \end{bmatrix}. \end{equation}

Let ${\hat {\boldsymbol R}}^{({tt}, {tr}, {rr})}$, ${\hat {\boldsymbol I}}$, $\hat {{{\boldsymbol {\omega }}}},{\hat {\boldsymbol {\sigma }}}_r, \hat {{{\boldsymbol \tau }}}^{e}$ be the representation of quantities ${\boldsymbol R}^{({tt}, {tr}, {rr})}, {\boldsymbol I}$, ${{{\boldsymbol {\omega }}}},{{\boldsymbol {\sigma }}}_r, {{{\boldsymbol \tau }}}^{e}$ with respect to the body frame $\{{\boldsymbol f}_i: i=1,2,3\}$. It is standard to show that they are related by the following transformations:

(2.5)\begin{equation} \left. \begin{gathered} ({\boldsymbol R}^{({tt}, {tr}, {rr})}, {\boldsymbol I}, {{\boldsymbol{\sigma}}}_r)={\boldsymbol Q}^{T}({\hat{\boldsymbol R}}^{({tt}, {tr}, {rr})}, {\hat{\boldsymbol I}}, {\hat{\boldsymbol{\sigma}}}_r){\boldsymbol Q},\\ ({{{\boldsymbol{\omega}}}}, {{{\boldsymbol \tau}}}^{e})={\boldsymbol Q}^{T}(\hat{{{\boldsymbol{\omega}}}}, \hat{{{\boldsymbol \tau}}}^{e}). \end{gathered} \right\} \end{equation}

In particular, the friction tensors ${\hat {\boldsymbol R}}^{({tt}, {tr}, {rr})}$ and moment of inertia tensor ${\hat {\boldsymbol I}}$ depend only on the shape of the particle and their properties and explicit formulae have been studied in details at, e.g., Bernal & De La Torre (Reference Bernal and De La Torre1980). The coupled system (2.2) concerning translational and rotational degrees of freedom could be simplified on account of the geometric symmetry of the particle and the energy scale $k_BT$ of thermal agitations. (Here $k_B$ is the Boltzmann constant and $T$ is the absolute temperature.) First, for an axisymmetric particle, $\hat {\boldsymbol {R}}^{tr}=0$. In addition, with respect to the body frame (2.2) can be rewritten as

(2.6)\begin{equation} {\hat{\boldsymbol I}} \dot{{\hat{\boldsymbol{\omega}}}}+{\hat{\boldsymbol \omega}}\times ({\hat{\boldsymbol I}} {{\hat{\boldsymbol \omega}}})={-} \hat{{\boldsymbol R}}^{rr} {\hat{\boldsymbol \omega}}+\hat{{{\boldsymbol \tau}}}^{e}+\hat{{{\boldsymbol{\sigma}}}}_r \hat{{{\boldsymbol{\xi}}}}. \end{equation}

Since the rotational kinetic energy $\frac {1}{2} {\hat {\boldsymbol \omega }}\boldsymbol {\cdot }{\hat {\boldsymbol I}}{\hat {\boldsymbol \omega }}\sim k_BT$ and inertia torque ${\hat {\boldsymbol I}} \dot {{\hat {\boldsymbol \omega }}}\sim \eta \sqrt {k_BTa/\rho }$, we have

(2.7)\begin{equation} \frac{|{\hat{\boldsymbol{\omega}}}\times {\hat{\boldsymbol I}}{\hat{\boldsymbol{\omega}}}|}{|{\hat{\boldsymbol I}} \dot{{\hat{\boldsymbol \omega}}}|}\sim\frac{ |\boldsymbol{\omega}|^{2}}{|\dot{{{\boldsymbol{\omega}}}}|}\sim \sqrt{\frac{k_B T \rho}{\eta^{2} a}}\ll 1, \end{equation}

where $\rho$ (respectively, $a$) is the density (respectively, characteristic size) of the particle and $\eta$ is the viscosity of the ambient fluid. Upon neglecting the term ${\hat {\boldsymbol \omega }}\times {\hat {\boldsymbol I}}{\hat {\boldsymbol \omega }}$, we can rewrite (2.2) as

(2.8)\begin{equation} \left. \begin{gathered} m\dot{{\boldsymbol v}}={-}{\boldsymbol Q}^{T}\hat{\boldsymbol{R}}^{tt} {\boldsymbol Q} {\boldsymbol v} +{\boldsymbol g}^{e} +{{\boldsymbol{\sigma}}}_t\boldsymbol{\xi},\\ {\hat{\boldsymbol I}} \dot{{\hat{\boldsymbol \omega}}}={-} \hat{{\boldsymbol R}}^{rr} {\hat{\boldsymbol{\omega}}}+\hat{{{\boldsymbol \tau}}}^{e}+\hat{{{\boldsymbol{\sigma}}}}_r \hat{{{\boldsymbol{\xi}}}}. \end{gathered} \right\} \end{equation}

The stochastic differential equations (2.8) describe the microscopic motions of the particle under the application of external forces and thermal agitations. To fix the unknown fluctuation coefficients ${{\boldsymbol {\sigma }}}_t$ and ${{\boldsymbol {\sigma }}}_r$ and relate (2.8) with macroscopic observables, we introduce a p.d.f. $P({\boldsymbol r}, t)$ for a generic stochastic process ${\boldsymbol r}={\boldsymbol r}(t)$. In other words, $P({\boldsymbol r}_0, t){\rm d}{\boldsymbol r}$ represents the probability of random variables ${\boldsymbol r}(t)$ taking values from the infinitesimal volume element ${\rm d}{\boldsymbol r}$ centred at ${\boldsymbol r}_0$. From the master equation and neglecting higher-order moments, we find that the p.d.f. satisfies the Fokker–Planck equation (Van Kampen Reference Van Kampen2007)

(2.9)\begin{equation} \frac{\partial P({{\boldsymbol r}}, t)}{\partial t}=\sum_{i,j}\frac{\partial}{\partial r_i}\left\{ -\alpha_i P({\boldsymbol r}, t)+\frac{\beta_{ij}}{2}\frac{\partial }{ \partial r_j}P({\boldsymbol r}, t)\right\}, \end{equation}

where the coefficients are given by

(2.10)\begin{equation} \left. \begin{gathered} {{\boldsymbol \alpha}}(t) =\lim_{{\Delta t} \rightarrow 0} \frac{\langle {\boldsymbol r}(t+{\Delta t} )-{\boldsymbol r}(t)\rangle }{{\Delta t} }, \\ {{\boldsymbol \beta}}(t) =\lim_{{\Delta t} \rightarrow 0} \frac{\langle {[{\boldsymbol r}(t+{\Delta t} )- {\boldsymbol r}(t)] \otimes [{\boldsymbol r}(t+{\Delta t} )- {\boldsymbol r}(t)]}\rangle }{{\Delta t} }. \end{gathered} \right\} \end{equation}

Here we emphasize that the physical meaning of variables ${\boldsymbol r}$ depend on the context. In particular, variables ${\boldsymbol r}$ may represent velocity, position, orientation, or all of them in the subsequent sections.

2.2. Fluctuation coefficients

To see the implications of the stochastic differential system (2.8), we first determine the fluctuation coefficients ${{\boldsymbol {\sigma }}}_t$ and ${{\boldsymbol {\sigma }}}_r$ and how they depend on the orientation matrix ${\boldsymbol Q}$. To this end, we consider a process with ${\boldsymbol Q}$ being essentially constant by, e.g. exerting an external torque ${{\boldsymbol \tau }}^{e}$ that penalizes deviations from the prescribed orientation. In the absence of external force (${\boldsymbol g}^{e}=0$), the stochastic differential equation (2.8₁) is recognized as a vectorial Langevin equation,

(2.11)\begin{equation} m \,\textrm{d}{\boldsymbol v}={-}{\boldsymbol R}^{tt}{\boldsymbol v}\,\textrm{d}t + {{\boldsymbol{\sigma}}}_t {{\boldsymbol{\xi}}}\,\textrm{d}t. \end{equation}

It is straightforward to verify that the solution to (2.11) satisfies (see e.g. Evans Reference Evans2012)

(2.12)$$\begin{gather} {\boldsymbol v}(t)=\exp\left(- \frac{t}{m} {\boldsymbol R}^{tt}\right){\boldsymbol v}(0)\nonumber\\ +\frac{1}{m} \int_0^{t} \exp\left(-\frac{t-s}{m}{\boldsymbol R}^{tt} \right) {{\boldsymbol{\sigma}}}_t {{\boldsymbol{\xi}}}(s)\,\textrm{d}s . \end{gather}$$

Inserting (2.12) into (2.10) (with ${\boldsymbol r}$ replaced by ${\boldsymbol v}$), by (2.3a,b) we find the coefficients associated with the Fokker–Planck equation (2.9) for the probability distribution $P=P({\boldsymbol v}, t)$ of particle in the velocity space,

(2.13)\begin{equation} {{\boldsymbol \alpha}}={-}\frac{1}{m}{\boldsymbol R}^{tt}{\boldsymbol v}\quad {\textrm{and }}\quad {{\boldsymbol \beta}}=\frac{1}{m^{2}}{{\boldsymbol{\sigma}}}_t{{\boldsymbol{\sigma}}}_t^{T}. \end{equation}

Therefore, the Fokker–Planck equation governing the probability distribution of the particle in the velocity space can be written as

(2.14)\begin{equation} \frac{\partial P({\boldsymbol v}, t)}{\partial t}=\sum_{i,k}\frac{\partial }{\partial v_i}\left[ -\alpha_i P({\boldsymbol v}, t) +\frac{\beta_{ik}}{2} \frac{\partial }{\partial v_k} P({\boldsymbol v}, t) \right]. \end{equation}

Meanwhile, from the classical statistical physics the stationary equilibrium p.d.f. $P^{s}({\boldsymbol v})$ should be given by the Maxwell–Boltzmann distribution,

(2.15)\begin{equation} P^{s}({{\boldsymbol v}})\propto \exp{\left[-\frac{m|{\boldsymbol v}|^{2}}{2k_BT}\right]}. \end{equation}

For consistency, we require (2.15) to be a stationary solution to (2.14) and conclude that ${{\boldsymbol \beta }} m/2k_BT={\boldsymbol R}^{tt}/m$, i.e.

(2.16)\begin{equation} {{\boldsymbol{\sigma}}}_t{{\boldsymbol{\sigma}}}_t^{T}=2k_BT {\boldsymbol R}^{tt}. \end{equation}

We remark that the above relation is a ramification of the fluctuation–dissipation theorem.

Similarly, by the second equation of (2.8) we have a Langevin equation for angular velocity in the absence of an external torque (${{\boldsymbol \tau }}^{e}=0$),

(2.17)\begin{equation} \textrm{d}{\hat{\boldsymbol \omega}}={-} {\hat{\boldsymbol I}}^{{-}1} \hat{{\boldsymbol R}}^{rr} {\hat{\boldsymbol{\omega}}}\,\textrm{d}t +{\hat{\boldsymbol I}}^{{-}1}\hat{{{\boldsymbol{\sigma}}}}_r \hat{{{\boldsymbol{\xi}}}}\,\textrm{d}t. \end{equation}

Therefore, we find that for consistency,

(2.18)\begin{equation} {\hat{\boldsymbol{\sigma}}}_r{\hat{\boldsymbol{\sigma}}}_r^{T}=2k_BT \hat{{\boldsymbol R}}^{rr}, \end{equation}

which gives rise to the stationary Maxwell–Boltzmann distribution to the Fokker–Planck equation (2.9) (with ${\boldsymbol r}$ replaced by ${\hat {\boldsymbol \omega }}$) in the angular-velocity space,

(2.19)\begin{equation} P^{s}({\hat{\boldsymbol \omega}})\propto \exp\left[-\frac{{\hat{\boldsymbol \omega}}\boldsymbol{\cdot} {\hat{\boldsymbol I}}{\hat{\boldsymbol \omega}}}{2k_BT}\right]. \end{equation}

2.3. Diffusivity

We are interested in the macroscopic diffusivity of an anisotropic particle in space under an alignment field in the long-time limit. For this purpose, we introduce a certain parameterization of the rotation matrix ${\boldsymbol Q}$, for example the Euler angles ${{\boldsymbol {\varTheta }}}=(\varTheta _1, \varTheta _2, \varTheta _3)$, to fix the orientation of the particle with respect to the global frame $\{{\boldsymbol e}_i: i=1,2, 3\}$. The angular velocity ${{\boldsymbol {\omega }}}$ and rate of change of parameters ${\boldsymbol {\varTheta }}$ are in general related by a linear transformation,

(2.20)\begin{equation} {{\boldsymbol{\omega}}}={\boldsymbol T} \dot{{\boldsymbol{\varTheta}}}. \end{equation}

Note that for anisotropic particles, the mobility tensors ${\boldsymbol M}^{{tt}, {rr}}$ and transformation matrix ${\boldsymbol T}$ in general depend on parameters ${\boldsymbol {\varTheta }}$ (but not on the position ${\boldsymbol x}$), which is sometimes omitted in the notation for brevity.

Neglecting the effect of inertia, we can write the equations of motion (2.2) as

(2.21)\begin{equation} \left. \begin{gathered} {\dot{{\boldsymbol x}}} ={\boldsymbol M}^{tt} {\boldsymbol g}^{e} +{\boldsymbol M}^{tt} {{\boldsymbol{\sigma}}}_t\boldsymbol{\xi}_t,\\ {\boldsymbol T} \dot{{\boldsymbol{\varTheta}}}= {\boldsymbol M}^{rr}{{\boldsymbol \tau}}^{e}+{\boldsymbol M}^{rr}{{\boldsymbol{\sigma}}}_r \boldsymbol{\xi}_r, \end{gathered} \right\} \end{equation}

where $({\boldsymbol M}^{tt} , {\boldsymbol M}^{rr})=(({\boldsymbol R}^{tt} )^{-1}, ({\boldsymbol R}^{rr})^{-1} )$ are the translational and rotational mobility tensors of the particle. Furthermore, both the external force ${\boldsymbol g}^{e}$ and torque ${{\boldsymbol \tau }}^{e}$ are assumed to be conservative and hence are related to the gradient of potential $V^{tt}({\boldsymbol x})$ for force and $V^{rr}({\boldsymbol {\varTheta }})$ for torque. To find the exact relation between ${{\boldsymbol \tau }}^{e}$ and $V^{rr}({\boldsymbol {\varTheta }})$, we notice that the rate of work done by this external torque at any angular velocity is given by $-({\textrm {d}}/{\textrm {d}t}) V^{rr}({\boldsymbol {\varTheta }})={{\boldsymbol {\omega }}}\boldsymbol {\cdot } {{\boldsymbol \tau }}^{e}$ and hence, by (2.20), we obtain

(2.22)\begin{equation} {{\boldsymbol \tau}}^{e}={-} {\boldsymbol T}^{{-}T} \boldsymbol{\nabla}_\varTheta V^{rr}({\boldsymbol{\varTheta}}) . \end{equation}

From the stochastic equations (2.21), by (2.10), (2.16) and (2.18) we find the coefficients associated with translational variables ${\boldsymbol x}$ and rotational variables ${\boldsymbol {\varTheta }}$ for the Fokker–Planck equations,

(2.23)\begin{equation} \left. \begin{gathered} {{\boldsymbol \alpha}}^{tt}={-{\boldsymbol M}^{tt}} \boldsymbol{\nabla}_{\boldsymbol x} V^{tt}({\boldsymbol x}), \quad {{\boldsymbol \beta}}^{tt}=2k_BT {\boldsymbol M}^{tt},\\ {{\boldsymbol \alpha}}^{rr}={-}{\boldsymbol T}^{{-}1}{\boldsymbol M}^{rr}{\boldsymbol T}^{{-}T}\boldsymbol{\nabla}_\varTheta V^{rr}(\boldsymbol{\varTheta}), \\ {{\boldsymbol \beta}}^{rr}=2k_BT {\boldsymbol T}^{{-}1}{\boldsymbol M}^{rr}{\boldsymbol T}^{{-}T}. \end{gathered} \right\} \end{equation}

Therefore, the Fokker–Planck equation for the p.d.f. $P=P({\boldsymbol x}, {\boldsymbol {\varTheta }}, t)$ in the position– orientation space $({\boldsymbol x}, {\boldsymbol {\varTheta }})$ can be written as

(2.24)$$\begin{gather} \frac{\partial }{\partial t} P({\boldsymbol x}, {\boldsymbol{\varTheta}}, t)=\sum_{i,j}\left\{\frac{\partial }{\partial x_i} \left[\left(-\alpha^{tt}_iP({\boldsymbol x}, {\boldsymbol{\varTheta}}, t)+\frac{\beta^{tt}_{ij}}{2}\frac{\partial }{\partial x_j}P({\boldsymbol x}, {\boldsymbol{\varTheta}}, t)\right)\right]\right.\nonumber\\ \left.+ \frac{\partial }{\partial \varTheta_i}\left[\left(-\alpha^{rr}_iP({\boldsymbol x}, {\boldsymbol{\varTheta}}, t) + \frac{\beta^{rr}_{ij}}{2}\frac{\partial }{\partial \varTheta_j}P({\boldsymbol x}, {\boldsymbol{\varTheta}}, t)\right)\right]\right\} . \end{gather}$$

It is straightforward to verify that

(2.25)\begin{equation} \left. \begin{gathered} P^{s}({\boldsymbol x}, {\boldsymbol{\varTheta}})=P^{s}({\boldsymbol x})P^{s}({\boldsymbol{\varTheta}}), \\ P^{s}({\boldsymbol x})\propto \exp\left[{-\frac{V^{tt}({\boldsymbol x})}{k_BT}}\right], \quad P^{s}({\boldsymbol{\varTheta}})\propto \exp\left[{-\frac{V^{rr}({\boldsymbol{\varTheta}})}{k_BT}}\right], \end{gathered} \right\} \end{equation}

is a stationary solution to (2.24), which is consistent with the classical statistical mechanics.

It appears to be reasonable to interpret the tensors ${{\boldsymbol \beta }}^{tt}/2$ and ${{\boldsymbol \beta }}^{rr}/2$ as the macroscopic translational and rotational diffusivities of the particle, respectively. The caveat lies in that both tensors in general depend on the orientation variables ${\boldsymbol {\varTheta }}$. For rotational diffusion, (2.24) implies a time scale

(2.26)\begin{equation} T_{ rot}\sim \frac{(2{\rm \pi})^{2}}{|{{\boldsymbol \beta}}^{rr}|}. \end{equation}

If we are only interested in the translational diffusion in space at a time scale that is much larger than $T_{ rot}$, the rotational motion may be assumed to be statistically stationary in the sense that the p.d.f. $P=P({\boldsymbol x}, {\boldsymbol {\varTheta }}, t)$ is of the following form:

(2.27)\begin{equation} P({\boldsymbol x}, {\boldsymbol{\varTheta}}, t) =P_\ast({\boldsymbol x}, t)P^{s}({\boldsymbol{\varTheta}})\propto P_\ast({\boldsymbol x}, t)\exp\left[{-\frac{V^{rr}({\boldsymbol{\varTheta}})}{k_BT}}\right]. \end{equation}

Inserting the above equation into (2.24) and integrating over ${\boldsymbol {\varTheta }}$-space, we obtain the standard diffusion equation for the probability of distribution of the particle $P=P_\ast ({\boldsymbol x}, t)$ in physical space in the absence of external force $(V^{tt}\equiv 0)$, as follows:

(2.28)\begin{equation} \frac{\partial }{\partial t} P_\ast({\boldsymbol x}, t)= \textrm{div} ({\boldsymbol D}^{eff} \boldsymbol{\nabla} P_\ast({\boldsymbol x}, t)), \end{equation}

where the effective diffusion tensor is given by

(2.29)\begin{equation} {\boldsymbol D}^{eff} = k_BT\int {\boldsymbol M}^{tt}({\boldsymbol{\varTheta}}) P^{s}({\boldsymbol{\varTheta}})\,\textrm{d}{\boldsymbol{\varTheta}}. \end{equation}

The above relation (2.29) between diffusivity and mobility can be regarded as a generalization of the classical Stoke–Einstein's relation. It should be noted that even if the physical conditions are different, (2.28), which governs the probability of distribution, possesses the same terms associated with the effective diffusivity as that derived in Aurell et al. (Reference Aurell, Bo, Dias, Eichhorn and Marino2016). Differently, in our work, the drift velocity due to the non-zero external force and its impact on the probability distribution over the orientation space vanish, and the probability distribution in the orientation space is dominated by the external alignment field (torque).

For an anisotropic particle, the diffusivity along a particular direction, e.g. ${\boldsymbol e}_1$-direction, in general depends on the orientation p.d.f. $P^{s}({\boldsymbol {\varTheta }})$ and hence the external alignment field. Nevertheless, noticing the transformation (2.5), we have

(2.30)\begin{align} {{Tr}}({\boldsymbol D}^{eff} ) &= k_BT\int {{Tr}} [{\boldsymbol M}^{tt}({\boldsymbol{\varTheta}})] P^{s}({\boldsymbol{\varTheta}})\,\textrm{d}{\boldsymbol{\varTheta}}\nonumber\\ &=k_BT{{Tr}}(\hat{{\boldsymbol M}}^{tt}), \end{align}

where ${\hat {{\boldsymbol M}}}^{tt}=({\hat {\boldsymbol R}}^{tt})^{-1}$ is the mobility tensor with respect to the body frame and hence independent of the orientation of the particle for isotropic ambient fluids. In particular, if there is no external alignment field, the effective diffusivity tensor will be isotropic and satisfies

(2.31)\begin{equation} D^{{eff}}_{11}=D^{{eff}}_{22}=D^{{eff}}_{33}=\tfrac{1}{3}k_BT{{Tr}}(\hat{{\boldsymbol M}}^{tt}). \end{equation}

2.4. Effective translational diffusion coefficient of CNT aligned by electric field

We now apply (2.29) to calculate the diffusion coefficients of a suspended CNT aligned by an AC electric field in an isotropic fluid. The shape of a CNT will be approximated as a prolate spheroid with aspect ratio $e$ ($\gg 1$) and major semiaxis length $a$. Because of axisymmetry, it suffices to describe the orientation of the CNT by specifying two angles $({\theta }, {\varphi })$. As illustrated in figure 1, let ${\theta }$ be the angle between the symmetry axis ${\boldsymbol f}_1$ of the body frame and ${\boldsymbol e}_1$ of the global frame, and ${\varphi }$ be the angle between ${\boldsymbol e}_2$ and the projected ray of ${\boldsymbol f}_1$ on the ${\boldsymbol e}_2$–${\boldsymbol e}_3$-plane. In terms of $({\theta }, {\varphi })$, the rigid rotation matrix ${\boldsymbol Q}$ can be explicitly written as

(2.32)\begin{equation} {\boldsymbol Q}({\varphi} ,{\theta} ) =\begin{bmatrix} \cos\theta & \sin\theta \cos\varphi & \sin\theta \sin\varphi \\ \sin\theta & -\cos\theta \cos\varphi & -\cos\theta \sin\varphi \\ 0 & \sin\varphi & -\cos\varphi \end{bmatrix}. \end{equation}

According to Kim & Karrila (Reference Kim and Karrila1991), the off-diagonal components of $\hat {{\boldsymbol M}}^{tt}$ vanish in account of the axisymmetry of a spheroid whereas the diagonal components of $\hat {{\boldsymbol M}}^{tt}$ are given by

(2.33)\begin{equation} \left. \begin{gathered} \hat{M}_{11}=\frac{e{\left[ {\left( 2e^{2}-1\right)\ln \left(e+\sqrt{e^{2}-1}\right) - e\sqrt{e^{2}-1}} \right]}}{{8{\rm \pi} \eta a{\left(e^{2}-1\right)^{{3}/{2}}}}},\\ \hat{M}_{22}=\hat{M}_{33}=\frac{e{\left[ {\left( 2e^{2}-3 \right)\ln \left(e+\sqrt{e^{2}-1}\right) +e\sqrt{e^{2}-1}} \right]}}{{16{\rm \pi} \eta a{\left(e^{2}-1\right)^{{3}/{2}}}}}, \end{gathered} \right\} \end{equation}

where $\eta$ is the viscosity of the ambient fluid. For $e\gg 1$, we have approximations

(2.34)\begin{equation} \left. \begin{gathered} \left( 2e^{2}-1\right) \approx 2e^{2},\quad \ln \left(e+\sqrt{e^{2}-1}\right)\approx \ln 2e\gg 1,\quad e\sqrt{e^{2}-1}\approx e^{2},\\ \left(e^{2}-1\right)^{{3}/{2}}\approx e^{3},\quad \left( 2e^{2}-3 \right)\approx 2e^{2}. \end{gathered} \right\} \end{equation}

Then for fixed minor semiaxis length $b=\frac {a}{e}$ of the spheroid, (2.34) and (2.33) lead to

(2.35)\begin{equation} \hat{M}_{11},\ \hat{M}_{22},\ \hat{M}_{33} \,\propto\, \frac{\ln e}{e}. \end{equation}

By (2.5) and (2.32), we find the diagonal components of the mobility tensor with respect to the global frame are given by

(2.36)\begin{equation} \left. \begin{gathered} M^{tt}_{11}({\varphi} ,{\theta} )=\cos^{2}{\theta} \hat{M}^{tt}_{11}+\sin^{2}{\theta} \hat{M}^{tt}_{22},\\ M^{tt}_{22}({\varphi} ,{\theta} )=\cos^{2}{\varphi} (\sin^{2}{\theta} \hat{M}^{tt}_{11}+\cos^{2}{\theta} \hat{M}^{tt}_{22})+\sin^{2}{\varphi} \hat{M}^{tt}_{22},\\ M^{tt}_{33}({\varphi} ,{\theta} )=\sin^{2}{\varphi} (\sin^{2}{\theta} \hat{M}^{tt}_{11}+\cos^{2}{\theta} \hat{M}^{tt}_{22})+\cos^{2}{\varphi} \hat{M}^{tt}_{22}. \end{gathered} \right\} \end{equation}

Figure 1.

Representation of the current configuration.

Furthermore, the applied AC electric field aligns the CNT along the field direction because of the induced dipole. The time-averaged alignment potential can be written as (Jones Reference Jones1995)

(2.37)\begin{equation} V^{rr}({\theta} )= \epsilon_m |{\boldsymbol E}|^{2} a^{3} \zeta\sin^{2}{\theta} , \end{equation}

where $|{\boldsymbol E}|$ denotes the amplitude of the external AC electric field, and $\epsilon _m$ is the dielectric constant of the isotropic ambient fluid. In addition, the dimensionless factor $\zeta$ is given by (Jones Reference Jones1995)

(2.38)\begin{equation} \zeta=\dfrac{2{\rm \pi} \left(L_2-L_1\right)}{3e^{2}}\text{Re} \left[\dfrac{\left(1-\dfrac{\bar{\epsilon}_m}{\bar{\epsilon}_p}\right)^{2}} {\left[\dfrac{\bar{\epsilon}_m}{\bar{\epsilon}_p}+L_1\left(1-\dfrac{\bar{\epsilon}_m}{\bar{\epsilon}_p}\right)\right] \left[\dfrac{\bar{\epsilon}_m}{\bar{\epsilon}_p}+L_2\left(1-\dfrac{\bar{\epsilon}_m}{\bar{\epsilon}_p}\right)\right]}\right], \end{equation}

where $\text {Re}$ denotes the real part, $L_1$ and $L_2$ are geometrical parameters given by

(2.39a,b)\begin{equation} L_1=\frac{e\ln (e+\sqrt{e^{2}-1})-\sqrt{e^{2}-1}}{(e^{2}-1)^{{3}/{2}}},\quad L_2=\tfrac{1}{2}(1-L_1), \end{equation}

and $\bar {\epsilon }_p$ (respectively, $\bar {\epsilon }_m$) is the complex permittivity of the particle (respectively, medium) and given by ($\epsilon$, permittivity; $\sigma$, conductivity; $\omega$, AC frequency)

(2.40)\begin{equation} \bar{\epsilon}_{p,m}={\epsilon}_{p,m}-{{i}} \frac{\sigma_{p,m}}{\omega} . \end{equation}

For parameters consistent with our experiment (CNT and mineral oil, and frequency up to 1 kHz), we notice that

(2.41)\begin{equation} \frac{|\bar{\epsilon}_{m}|}{|\bar{\epsilon}_{p}|}\ll L_1\ll 1, \end{equation}

so the factor $\zeta$ can be simplified as

(2.42)\begin{equation} \zeta\approx\frac{2{\rm \pi} }{3}\left[\frac{1}{\ln{2e}-1}\right]. \end{equation}

Inserting (2.37) into (2.25), we find the stationary p.d.f. of the CNT orientation,

(2.43)\begin{equation} P^{s}(\theta)=\frac{{\sin \theta}\exp\left[-\dfrac{\epsilon_ma^{3}\zeta |{\boldsymbol E}|^{2}\sin^{2}{\theta} }{k_BT}\right]}{\int_0^{\rm \pi} \exp\left[-\dfrac{\epsilon_m a^{3}\zeta |{\boldsymbol E}|^{2}\sin^{2}{\theta} }{k_BT}\right]{\sin \theta}\,\textrm{d}\theta}. \end{equation}

The derivation of (2.43) is given in the Appendix.

Consequently, by (2.29) and (2.36) we find that the effective diffusion coefficients along three axes of the global frame are given by

(2.44)\begin{equation} \left. \begin{gathered} D^{{eff}}_{11}=\hat{D}^{{tt}}_{11}- (\hat{D}^{{tt}}_{11}-\hat{D}^{{tt}}_{22})\int_0^{\rm \pi} P^{s}(\theta)\sin^{2}{\theta} \,\textrm{d}{\theta} ,\\ D^{{eff}}_{22}=D^{{eff}}_{33} =\hat{D}^{{tt}}_{22}+ \tfrac{1}{2}(\hat{D}^{{tt}}_{11}-\hat{D}^{{tt}}_{22})\int_0^{\rm \pi} P^{s}(\theta)\sin^{2}{\theta} \,\textrm{d}{\theta}, \end{gathered} \right\} \end{equation}

where $\hat {{\boldsymbol D}}=k_BT{\hat {{\boldsymbol M}}}$ is the diffusivity with respect to the body frame. In addition, all the off-diagonal components of ${\boldsymbol D}^{eff}$ will vanish by symmetry.

We remark that if $|{\boldsymbol E}|=0$, the p.d.f. (2.43) is clearly a uniform distribution, and by directly integrating (2.44) we find ${\boldsymbol D}^{eff}$ is isotropic with diffusivity $\frac {1}{3}{{Tr}}(\hat {{\boldsymbol D}})$, as noticed before in (2.31). When $|{\boldsymbol E}|\rightarrow +\infty$, the p.d.f. (2.43) would be a delta function at $\theta =0$, implying complete alignment with electric field, and hence

(2.45a,b)\begin{equation} D^{{eff}}_{11}=\hat{D}^{tt}_{11},\quad D^{{eff}}_{22}=D^{{eff}}_{33}=\hat{D}^{{tt}}_{22}=\hat{D}^{{tt}}_{33}. \end{equation}

When $\epsilon _m a^{3}\zeta |{\boldsymbol E}|^{2} /k_BT\gg 1$, the p.d.f. (2.43) may be approximated by a Gaussian distribution which can be used to fix the factor $\zeta$ or the electrical properties of nanoparticles by experimentally measuring the p.d.f. (Guo, Su & Guo Reference Guo, Su and Guo2012; Castellano et al. Reference Castellano, Akin, Giraldo, Kim, Fornasiero and Shan2015). For general electric field strength, the integrals in (2.44) can be numerically evaluated, with sample results shown in the Results and discussion section (§ 4).

3.1. Experimental set-up

Multiwall CNTs (110–170 nm in diameter and ranging between 5 and 9 $\mathrm {\mu }\textrm {m}$ in length, Sigma Aldrich $\#$659258) are first dispersed in light mineral oil (Drakeol 7 LT Mineral Oil, Calumet Specialty Products and Partners, L.P.) at a volume fraction ${\sim }0.001\,\%$ using a bath sonicator (Fisher Scientific – FS60 Ultrasonic Cleaner), before being placed between a pair of parallel electrodes. Since the suspension is extremely dilute, interactions between particles can be safely neglected in our experiments. The AC electric field in the experiments is provided by an arbitrary function generator (Tektronix AFG3200C) connected to a high-frequency amplifier (TREK 2100HF). Field strengths of 0 $\textrm {V}\,\textrm {mm}^{-1}$, 3.3 $\textrm {V}\,\textrm {mm}^{-1}$, 6.7 $\textrm {V}\,\textrm {mm}^{-1}$, 13.3 $\textrm {V}\,\textrm {mm}^{-1}$, 20 $\textrm {V}\,\textrm {mm}^{-1}$, 26.7 $\textrm {V}\,\textrm {mm}^{-1}$ and 33.3 $\textrm {V}\,\textrm {mm}^{-1}$ are used while the AC frequency is fixed at 1 kHz and the field is always in the ${\boldsymbol e}_1$-direction. We measure approximately 10–30 different CNTs at each field strength. The Brownian motion of individual particles is recorded by a high-speed monochrome CCD camera (pco.edge sCMOS, PCO AG) mounted on an inverted optical microscope (Olympus IX71, Olympus Corp.) with a ${\times }40$ objective lens (Olympus LUCPLFN ${\times}40$, N.A. 0.6, Olympus Corp). The sampling frequency is chosen to be 10 Hz ($\Delta T=0.1\ \textrm {s}$) and the time window of each measurement is 400 s. We combine at least 10 time windows to calculate one averaged translational diffusion coefficient at each field strength. The overall observation time, as the sum of these windows, is much longer than the rotational diffusion time scale. Therefore, our measurement of translational diffusion is unlikely to be affected by rotational diffusion (see supplementary material). The trajectory of the centre of mass of the particles is then extracted from the recorded images using a custom-written MATLAB program (see supplementary material). A schematic of the experimental set-up is shown in figure 2.

Figure 2.

Experimental set-up for measurement of CNT diffusion under an aligning electric field.

3.2. Data processing

In order to determine the translational diffusion coefficient, we first calculate the MSD of the particle for consecutive lag times in the $x$- (parallel to the field ) and $y$- (perpendicular to the field) directions,

(3.1)\begin{equation} \left. \begin{gathered} MSD_x(\tau)=\frac{1}{N-k}\sum_{n=0}^{N-k-1}\left[x(n\Delta t)-x((n+k)\Delta t)\right]^{2}, \\ MSD_y(\tau)=\frac{1}{N-k}\sum_{n=0}^{N-k-1}\left[y(n\Delta t)-y((n+k)\Delta t)\right]^{2}, \end{gathered} \right\} \end{equation}

where $N$ is the total number of points in each measurement and $\tau =k\Delta T$ ($k=1,2,\ldots ,N-1$) is the lag time. The MSD generally grows linearly with the lag time $\tau$ for ideal Brownian motion,

(3.2)\begin{equation} \left. \begin{gathered} MSD_x(\tau)=2D_x\tau, \\ MSD_y(\tau)=2D_y\tau, \end{gathered} \right\} \end{equation}

where $D_x$ and $D_y$ are the translational diffusion coefficient in x and y direction, respectively. However, the average velocity of the CNTs may not vanish for the considered time interval because of the inevitable background motion (see supplementary material) of the fluid due to the non-uniformity of the field or other sources; a better approximation of MSD versus $\tau$ would be of the following form:

(3.3)\begin{equation} \left. \begin{gathered} MSD_x(\tau)=2D_x\tau+v_x^{2}\tau^{2}, \\ MSD_y(\tau)=2D_y\tau+v_y^{2}\tau^{2}, \end{gathered} \right\} \end{equation}

where $v_x$ and $v_y$ are the field-induced translational velocities in the $x$- and $y$-directions, respectively. Therefore, we use a quadratic fitting algorithm to process the experimental MSD data and extract the coefficients associated with the linear term to determine the diffusion coefficients. Only the first 50 MSD data points are used in our fitting to ensure the accuracy of the results. The calculated diffusion coefficients of different CNTs are normalized and combined into an averaged value at each field strength based on the CNT dimensions using the formulae

(3.4)\begin{equation} \left. \begin{gathered} \langle D_x \rangle=\left\langle\frac{2a D_x}{ln(2e)} \right\rangle \cdot \left\langle\frac{ln(2e)}{2a} \right\rangle,\\ \langle D_y \rangle=\left\langle\frac{2a D_y}{ln(2e)} \right\rangle \cdot \left\langle\frac{ln(2e)}{2a} \right\rangle. \end{gathered} \right\} \end{equation}

Because the CNTs could be broken into shorter pieces by ultrasonication, the actual lengths of the CNTs were measured optically with the microscope during the experiments. An average length of $3.5\ \mathrm {\mu }\textrm {m}$ was observed. The diameter of the CNTs was taken to be 140 nm.