Dynamical Density Functional Theory for Orientable Colloids Including Inertia and Hydrodynamic Interactions

Abstract

Over the last few decades, classical density-functional theory (DFT) and its dynamic extensions (DDFTs) have become powerful tools in the study of colloidal fluids. Recently, previous DDFTs for spherically-symmetric particles have been generalised to take into account both inertia and hydrodynamic interactions, two effects which strongly influence non-equilibrium properties. The present work further generalises this framework to systems of anisotropic particles. Starting from the Liouville equation and utilising Zwanzig’s projection-operator techniques, we derive the kinetic equation for the Brownian particle distribution function, and by averaging over all but one particle, a DDFT equation is obtained. Whilst this equation has some similarities with DDFTs for spherically-symmetric colloids, it involves a translational-rotational coupling which affects the diffusivity of the (asymmetric) particles. We further show that, in the overdamped (high friction) limit, the DDFT is considerably simplified and is in agreement with a previous DDFT for colloids with arbitrary-shape particles.

Appendices

Appendix 1: Derivation of Generalised Langevin Equations

1.1 The Fokker–Planck Equation

Here we outline the derivation of the time-evolution equation for the probability distribution function of a system of arbitrary-shape particles. This derivation can be thought as an application of the works of Kirkwood [49], Murphy and Aguirre [63], Deutch and Oppenheim [20] and Wilemski [77] in conjunction with the results of Dahler and Sather [18], Condiff and Dahler [16], Evans [29] and Condiff and Brenner [15], to consider arbitrary-shape particles. The derivation presented below can be also understood as a generalisation of the one carried by Archer [1] within the context of point-like particles.

As in Sect. 2, consider N identical, asymmetric colloidal particles with mass m immersed in a fluid of $n\gg N$ identical bath particles with mass $m_b$. Throughout this section upper case letters will refer to colloidal particles while lower case ones indicate bath particles. The phase-space coordinates of the i-th bath particle are $\mathbf {x}_j\doteq (\mathbf {r}_j,\,\mathbf {p}_j)$, with “$\doteq$” denoting “by definition”, $\mathbf {r}_j$ being the position vector and $\mathbf {p}_j=m_b\dot{\mathbf {r}}_j$ the canonical momentum. In addition, the dynamical state of the j-th colloidal particle is determined by $\mathbf {X}_j\doteq (\mathbf {R}_j,\mathbf {P}_j)$, where $\mathbf {R}_j$ is its centre-mass position vector and $\mathbf {P}_j=m\dot{\mathbf {R}}_j$ is its conjugate momentum, along with the pair $\varvec{\Omega }_j\doteq (\varvec{\alpha }_j,\varvec{\pi }_j)$ which comprises the rotational degrees of freedom, with $\varvec{\alpha }_j$ being the Eulerian angles as in Sect. 2 and $\varvec{\pi }_j$ their conjugate momenta [18]. The angular velocity of a particle, $\varvec{\omega }_j$, is determined by the time derivative of the Euler angles. In the principal-axes frame $\mathfrak {B}$, we have the relation $\varvec{\omega }_j=\varvec{\Lambda }_j^\top \dot{\varvec{\alpha }}_j$, where [16, 47]

$$\begin{aligned} \varvec{\Lambda }_j^\top = \begin{pmatrix} \cos \chi _j &{} \sin \theta _j\sin \chi _j &{} 0\\ -\sin \chi _j &{} \sin \theta _j\cos \chi _j &{} 0\\ 0 &{} \cos \theta _j &{} 1 \end{pmatrix}\ \Leftrightarrow \varvec{\Lambda }_j^{-1} = \begin{pmatrix} \cos \chi _j &{} \csc \theta _j\cos \chi _j &{} -\cot \theta _j\sin \chi _j\\ -\sin \chi _j &{} \csc \theta _j\sin \chi _j &{} -\cot \theta _j\cos \chi _j\\ 0 &{} 0 &{} 1 \end{pmatrix}. \end{aligned}$$

(43)

Accordingly, $\varvec{\omega }_j'=\mathcal {R}_j^\top \varvec{\omega }_j= \varvec{\Xi }_j\,\dot{\varvec{\alpha }}_j$ under the space-fixed frame $\mathfrak {S}$, with $\varvec{\Xi }_j\doteq \mathcal {R}_j^\top \varvec{\Lambda }_j^\top$. These can be related with their corresponding angular momenta by, $\mathbf {L}_j=\mathbb {I}\,\varvec{\omega }_j$ and $\mathbf {L}'_j=\mathbb {I}'_j\varvec{\omega }'_j$ respectively.

Thus, the dynamical state of the system at any given instant represents a single point in a $6(N+n)-$dimensional space, $\Gamma$ [52],

$$\begin{aligned} \mathfrak {s}(t)\doteq & {} (\mathbf {x}_1(t),\dots ,\mathbf {x}_n(t),\mathbf {X}_1(t),\dots ,\mathbf {X}_N(t),\varvec{\Omega }_1(t),\dots ,\varvec{\Omega }_N(t))\nonumber \\\equiv & {} (\mathbf {x}^n(t),\mathbf {X}^N(t),\varvec{\Omega }^N(t))\in \Gamma \end{aligned}$$

(44)

where we made use of the notation $\mathbf {x}^n= \mathbf {x}_1\dots \mathbf {x}_n$, $\mathbf {X}^N=\mathbf {X}_1\dots \mathbf {X}_N$ and $\varvec{\Omega }^N=\varvec{\Omega }_1,\dots ,\varvec{\Omega }_N$. From classical mechanics, the evolution of the system is completely determined by the initial conditions for positions and momenta of all particles. This time evolution, which prescribes a unique trajectory $\mathfrak {s}(t;t_0)$, is fully described by the Lagrangian and the Hamiltonian of the system. In the following we use the former to get the relation between $\varvec{\pi }_j$ and $\dot{\varvec{\alpha }}_j$ in order to construct the Hamiltonian. Then Hamilton’s equations along with Liouville’s theorem will be employed to get the time-evolution equation for the PDF of the system so that its phases lie in a differential region of $\Gamma$ with centre placed at $\mathfrak {s}(t)$. First, the Lagrangian of the system can be written as,¹

$$\begin{aligned} \mathcal {L}=&\mathcal {L}_b+\mathcal {L}_B,\\ \mathcal {L}_b=&\sum _{i=1}^n\frac{1}{2}m_b\dot{\mathbf {r}}_j\cdot \dot{\mathbf {r}}_j-\mathcal {U}(\mathbf {r}^N,\mathbf {R}^n,\varvec{\alpha }^n),\nonumber \\ \mathcal {L}_B=&\sum _{i=1}^N\frac{1}{2} \left( m\dot{\mathbf {R}}_j\cdot \dot{\mathbf {R}}_j+\dot{\varvec{\alpha }}_j\cdot (\varvec{\Xi }_j^\top \mathbb {I}'_j\varvec{\Xi }_j) \dot{\varvec{\alpha }}_j\right) -V(\mathbf {R}^N,\varvec{\alpha }^N)\nonumber , \end{aligned}$$

(45)

with $V(\mathbf {R}^N,\varvec{\alpha }^N)$ the potential energy due to short-range interactions exclusively between colloidal particles, and

$$\begin{aligned} \mathcal {U}(\mathbf {r}^n,\mathbf {R}^N,\varvec{\alpha }^N) = U(\mathbf {r}^n)+\sum _{\mu =1}^N\mathfrak {u}_\mu (\mathbf {r}^N,\mathbf {R}_\mu ,\varvec{\alpha }_\mu ), \end{aligned}$$

(46)

the short-range intermolecular potential energy coming from the interaction between bath particles, $U(\mathbf {r}^n)$, and the interaction of each colloidal particle with the whole bath, $\mathfrak {u}_\mu (\mathbf {r}^N,\mathbf {R}_\mu ), \forall \mu =1,\dots ,N$. From these equations we obtain

$$\begin{aligned} \mathbf {p}_j=\frac{\partial \mathcal {L}}{\partial \dot{\mathbf {r}}_j}=m_b\dot{\mathbf {r}_j},\quad \mathbf {P}_j=\frac{\partial \mathcal {L}}{\partial \dot{\mathbf {R}}_j}=m\dot{\mathbf {R}_j},\quad \varvec{\pi }_j=\frac{\partial \mathcal {L}}{\partial \dot{\varvec{\alpha }}_j}= (\varvec{\Xi }_j^\top \mathbb {I}'_j\varvec{\Xi }_j)\dot{\varvec{\alpha }_j}, \end{aligned}$$

(47)

Therefore, $\varvec{\pi }_j$ can be easily related with the angular momentum $\mathbf {L}_j$ via [15, 16]

$$\begin{aligned} \varvec{\pi }_j = (\varvec{\Xi }_j^\top \mathbb {I}'_j\varvec{\Xi }_j)\dot{\varvec{\alpha }}_j \equiv \varvec{\Lambda }_j{\mathbf {L}}_j, \end{aligned}$$

(48)

which will be used to perform the appropriate change of variables later on. We now write the Hamiltonian function of the system as

$$\begin{aligned} \mathcal {H}= & {} \mathcal {H}_b + \mathcal {H}_B \\ \mathcal {H}_b= & {} \sum _{i=1}^n\frac{\mathbf {p}_j\cdot \mathbf {p}_j}{2m_b}+\mathcal {U}(\mathbf {r}^N,\mathbf {R}^n,\varvec{\alpha }^n)\nonumber \\ \mathcal {H}_B= & {} \sum _{i=1}^N\left( \frac{\mathbf {P}_j\cdot \mathbf {P}_j}{2m}+\frac{1}{2}\varvec{\pi }_j\cdot (\varvec{\Xi }_j^\top \mathbb {I}'_j\varvec{\Xi }_j)^{-1}\varvec{\pi }_j \right) +V(\mathbf {R}^N,\varvec{\alpha }^n).\nonumber \end{aligned}$$

(49)

According to Liouville’s theorem, the $n+N$ particle distribution function, $\varrho ^{(n+N)}(\mathbf {x}^n,\mathbf {X}^N;t)$, will evolve according to

$$\begin{aligned} \partial _t \varrho ^{(n+N)}(\mathbf {x}^n,\mathbf {X}^N,\varvec{\Omega }^N;t)+\left( i\mathfrak {L}_b+i\mathfrak {L}_B^T+i\mathfrak {L}_B^R\right) \,\varrho ^{(n+N)}(\mathbf {x}^n,\mathbf {X}^N;t)=0 \end{aligned}$$

(50)

with the Liouvillian, $\mathfrak {L}\doteq \mathfrak {L}_b+\mathfrak {L}_B^T+\mathfrak {L}_B^R$,

$$\begin{aligned} i\mathfrak {L}_b =&\sum _{j=1}^n\left( \frac{\mathbf {p}_j}{m_b}\cdot \frac{\partial }{\partial \mathbf {r}_j}+ \mathbf {f}_j\cdot \frac{\partial }{\partial \mathbf {p}_j}\right) \\ i\mathfrak {L}_B^T =&\sum _{j=1}^N\left( \frac{\mathbf {P}_j}{m}\cdot \frac{\partial }{\partial \mathbf {R}_j}+ \mathbf {F}_j\cdot \frac{\partial }{\partial \mathbf {P}_j}\right) \nonumber \\ i\mathfrak {L}_B^R =&\sum _{j=1}^N\left( \dot{\varvec{\alpha }}_j\cdot \frac{\partial }{\partial \varvec{\alpha }_j}+ \dot{\varvec{\pi }}_j\cdot \frac{\partial }{\partial \varvec{\pi }_j}\right) \nonumber \end{aligned}$$

(51)

where $\mathbf {f}_j=-\frac{\partial }{\partial \mathbf {r}_j}(U+\sum _{\mu =1}^N\mathfrak {u}_\mu )$ and $\mathbf {F}_j=-\frac{\partial }{\partial \mathbf {R}_j}(V+\mathfrak {u}_j)$ are the instantaneous forces acting on bath and colloidal particles, respectively. From a conceptual point of view [18], the quantities ${\varvec{\alpha }}_j$ and ${\varvec{\pi }}_j$ are less convenient than the angular velocities and momenta, $\varvec{\omega }_j$ and $\mathbf {L}_j$. This can be easily checked by considering the rotational kinetic energy

$$\begin{aligned} K_{R}= \sum _{j=1}^N \frac{\dot{\varvec{\alpha }}_j}{2}\cdot (\varvec{\Xi }_j^\top \mathbb {I}'_j\varvec{\Xi }_j) \dot{\varvec{\alpha }}_j \equiv&\ \sum _{j=1}^N \frac{\varvec{\omega }_j}{2}\cdot \mathbb {I}\varvec{\omega }_j\\ K_{R}= \sum _{j=1}^N \frac{\varvec{\pi }_j}{2}\cdot (\varvec{\Xi }_j^\top \mathbb {I}'_j\varvec{\Xi }_j)^{-1}\varvec{\pi }_j \equiv&\ \sum _{i=1}^N\frac{\mathbf {L}_j}{2}\cdot (\mathbb {I}^{-1}\mathbf {L}_j).\nonumber \end{aligned}$$

(52)

Transforming now Eq. (50) into an equivalent equation for

$$\begin{aligned} F^{(n+N)}(\mathbf {x}^n,\mathbf {X}^N,\varvec{\alpha }^N,\mathbf {L}^N;t)=&\left| \frac{\partial (\mathbf {x}^n,\mathbf {X}^N,\varvec{\alpha }^N,\varvec{\pi }^N)}{\partial (\mathbf {x}^n,\mathbf {X}^N,\varvec{\alpha }^N,\mathbf {L}^N)}\right| \varrho ^{(n+N)}(\mathbf {x}^n,\mathbf {X}^N,\varvec{\Omega }^N;t)\\ =&\prod _{j}\sin \theta _j\,\varrho ^{(n+N)}(\mathbf {x}^n,\mathbf {X}^N,\varvec{\Omega }^N;t)\nonumber \end{aligned}$$

(53)

gives

$$\begin{aligned} \partial _t F^{(n+N)}(\mathbf {x}^n,\mathbf {X}^N,\varvec{\Omega }^N;t)+\left( i\mathfrak {L}_b+i\mathfrak {L}_B^T+i\widetilde{\mathfrak {L}}_B^R\right) \,F^{(n+N)}(\mathbf {x}^n,\mathbf {X}^N,\varvec{\Omega }^N;t)=0 \end{aligned}$$

(54)

with [15, 16, 18]

$$\begin{aligned} i\widetilde{\mathfrak {L}}_B^R = \sum _{j=1}^N\dot{\varvec{\alpha }}_j \cdot \hat{\partial }_{\varvec{\alpha }_j}+\mathbf {T}_j\cdot \frac{\partial }{\partial \mathbf {L}_j} \end{aligned}$$

(55)

where the operator [16] $\hat{\partial }_{\varvec{\alpha _j}}\doteq (\csc \theta _j\frac{\partial }{\partial \theta _j}\sin \theta _j, \frac{\partial }{\partial \phi _j}, \frac{\partial }{\partial \chi _j})^\top$, deduced by using the chain rule [72], has been introduced. In the latter equation, $\mathbf {T}_j = \mathbf {N}_j - \varvec{\omega }_j\times \mathbf {L}_j$ denotes the net torque acting on the j-th colloidal particle, with [16]

$$\begin{aligned} \mathbf {N}_j&= - \varvec{\Lambda }_j^{-1}\frac{\partial }{\partial \varvec{\alpha }_j}\left[ V(\mathbf {R}^N,\varvec{\alpha }^n)+\mathfrak {u}_j(\mathbf {r}^N,\mathbf {R}_j,\varvec{\alpha }_j)\right] \end{aligned}$$

(56)

the torque due to intermolecular interactions along the principal axes of inertia. At this point, it is necessary to introduce the angular-gradient operator (also known as orientational gradient [68])

$$\begin{aligned} \frac{\partial }{\partial \varvec{\Phi }_j}&\doteq \varvec{\Lambda }_j^{-1}\frac{\partial }{\partial \varvec{\alpha }_j}\equiv \hat{\partial }_{\varvec{\alpha }_j}\cdot (\varvec{\Lambda }^\top )^{-1} \end{aligned}$$

(57)

such that $\frac{\partial }{\partial \varvec{\Phi }}= \mathbf {e}_x\frac{\partial }{\partial \Phi _{x}}+\mathbf {e}_y\frac{\partial }{\partial \Phi _{y}}+\mathbf {e}_z\frac{\partial }{\partial \Phi _{z}}$, where $\mathbf {e}_i$ is the unitary vector along axis $i\in \{x,y,z\}$ of the Cartesian frame $\mathfrak {B}$, and

$$\begin{aligned} \frac{\partial }{\partial \Phi _{x}^j}&= \cos \chi _j\frac{\partial }{\partial \theta _j} +\csc \theta _j\cos \chi _j\frac{\partial }{\partial \phi _j} -\cot \theta _j\sin \chi _j\frac{\partial }{\partial \chi _j}\\ \frac{\partial }{\partial \Phi _y^j}&= -\sin \chi _j\frac{\partial }{\partial \theta _j} +\csc \theta _j\sin \chi _j\frac{\partial }{\partial \phi _j} -\cot \theta _j\cos \chi _j\frac{\partial }{\partial \chi _j} \nonumber \\ \frac{\partial }{\partial \Phi _z^j}&=\frac{\partial }{\partial \chi _j}.\nonumber \end{aligned}$$

(58)

It is worth mentioning here that the derivative operators $(\frac{\partial }{\partial \Phi _{x}},\frac{\partial }{\partial \Phi _{y}},\frac{\partial }{\partial \Phi _{z}})$ are the generators of rotations of a rigid body about the body-fixed Cartesian frame [40]. This results in

$$\begin{aligned} i\widetilde{\mathfrak {L}}_B^R=\sum _{j=1}^N \frac{\partial }{\partial \varvec{\Phi }_j}\cdot \varvec{\omega }_j + \left( \mathbf {N}_j+\mathbf {L}_j\times \varvec{\omega }_j\right) \cdot \frac{\partial }{\partial \mathbf {L}_j} \end{aligned}$$

(59)

Now we can define $\varvec{\mathfrak {m}}\doteq \text {diag}(m\mathbf {1},\mathbb {I})$ along with the vectors $\varvec{\mathfrak {r}}_j\doteq (\mathbf {R}_j,\varvec{\Phi }_j)$, $\varvec{\mathfrak {p}}_j\doteq (\mathbf {P}_j,\mathbf {L}_j)$ and $\varvec{\mathfrak {f}}_j\doteq (\mathbf {F}_j,\mathbf {T}_j)$, and the operators $\varvec{\nabla }_{\mathfrak {r}_j}\doteq (\frac{\partial }{\partial \mathbf {R}_j},\frac{\partial }{\partial \varvec{\Phi }_j})^\top$ and $\varvec{\nabla }_{\varvec{\mathfrak {p}}_j}\doteq (\frac{\partial }{\partial \mathbf {P}_j},\frac{\partial }{\partial \mathbf {L}_j})^\top$, enabling us to rewrite Eq. (54) in a more compact and convenient way,

$$\begin{aligned}&\partial _t F^{(n+N)}(t)+ \sum _{j=1}^n\left( \frac{\mathbf {p}_j}{m_b}\cdot \frac{\partial }{\partial \mathbf {r}_j}+ \mathbf {f}_j\cdot \frac{\partial }{\partial \mathbf {p}_j}\right) F^{(n+N)}(t) \nonumber \\&\quad + \sum _{j=1}^N\left( \varvec{\nabla }_{\varvec{\mathfrak {r}}_j}\cdot \varvec{\mathfrak {m}}^{-1}\varvec{\mathfrak {p}}_j + \varvec{\mathfrak {f}}_j\cdot \varvec{\nabla }_{\varvec{\mathfrak {p}}_j} \right) F^{(n+N)}(t)=0, \end{aligned}$$

(60)

where explicit dependence on the phase-space coordinates was omitted, but recalled through the superscript $(n+N)$. Following Murphy and Aguirre [63], we introduce the scaling quantity, $\varvec{\lambda }\doteq \varvec{\mathfrak {m}}^{-1/2}$, so that $\widetilde{\varvec{\mathfrak {p}}}_j=\varvec{\lambda }\varvec{\mathfrak {p}}_j$ and hence, the last term of the previous equation becomes

$$\begin{aligned} \sum _{j=1}^N\varvec{\lambda }\left( \varvec{\nabla }_{\varvec{\mathfrak {r}}_j}\cdot \widetilde{\varvec{\mathfrak {p}}}_j + \varvec{\mathfrak {f}}_j\cdot \varvec{\nabla }_{\widetilde{\varvec{\mathfrak {p}}}_j} \right) F^{(n+N)}(t). \end{aligned}$$

(61)

Substitution of Eq. (61) into (60) the results in an equation resembling Liouville’s equation for spherical colloidal particles. Such a result provides a description of the time evolution of the full system. However, our interest rests exclusively on colloidal particles. Thus, our aim is to get the time-evolution equation for the N-particle distribution,

$$\begin{aligned} f^{(N)}(t) \doteq \int d\mathbf {x}^n F^{(n+N)}(t). \end{aligned}$$

(62)

For this purpose, Zwanzig’s projection technique can be applied as in the case of spherical colloidal particles [20, 27, 63, 77]. Following the work of Murphy and Aguirre [63], for the arbitrary initial state at $t=-t_I$ we choose one where the bath particles are in equilibrium with the instantaneous positions of the colloidal particles. This means that

$$\begin{aligned} F^{(n+N)}(\mathbf {x}^n,\mathbf {X}^N,\varvec{\Omega }^N;-t_I)=\rho _n^\dag (\mathbf {x}^n)\,f^{(N)}(\mathbf {X}^N,\varvec{\Omega }^N;-t_I), \end{aligned}$$

(63)

with $\rho _n^\dag$ the canonical distribution of the n bath particles in the instantaneous potential created by the colloidal particles. The last step before integrating Eq. (60) to remove the dependence on fast variables involves the definition of the projection operator,

$$\begin{aligned} \hat{\mathcal {P}}\doteq \rho _n^\dag (\mathbf {x}^n)\int d\mathbf {x}^n, \end{aligned}$$

(64)

and its complementary operator $\hat{\mathcal {Q}}=1-\hat{\mathcal {P}}$. Thus, the integration of Eq. (60) over the fast variables $\mathbf {x}^n$ is equivalent to applying $(\rho _n^\dag )^{-1}\hat{\mathcal {P}}$ on both sides of such an equation. Although considerable algebraic manipulations are required, we can follow the work of Murphy and Aguirre [63] and Lebowitz and Résibois [51] step by step to finally reach the desired time-evolution equation for the projected distribution function $f^{(N)}$,

$$\begin{aligned} \partial _tf^{(N)}(t)+\sum _{j=1}^N&\varvec{\lambda }\left( \varvec{\nabla }_{\varvec{\mathfrak {r}}_j}\cdot \widetilde{\varvec{\mathfrak {p}}}_j+\varvec{\mathfrak {F}}_j\cdot \varvec{\nabla }_{\widetilde{\varvec{\mathfrak {p}}}_j}\right) f^{(N)}(t) \\ =&\sum _{j,k=1}^N \varvec{\lambda }^2\,\varvec{\nabla }_{\varvec{\mathfrak {p}}_j}\cdot \int _{-t_I}^{t}dt'\varvec{\gamma }_{jk}(\varvec{\lambda };t,t')\left( \varvec{\nabla }_{\widetilde{\varvec{\mathfrak {p}}}_k}+\beta \,\widetilde{\varvec{\mathfrak {p}}}_k\right) f^{(N)}(t')\nonumber \end{aligned}$$

(65)

where $\beta =1/k_BT$, where $k_B$ is the Boltzmann constant, T is the temperature imposed by the bath, and $\varvec{\mathfrak {F}}$ is the equilibrium average force and torque, i.e.

$$\begin{aligned} \varvec{\mathfrak {F}}_j\equiv & {} \left\langle \begin{array}{l} \mathbf {F}_j\\ \mathbf {T}_j \end{array} \right\rangle ^\dag =\int d\mathbf {x}^n \rho _n^\dag (\mathbf {x}^n)\,\varvec{\mathfrak {f}}_j(\mathbf {r}^n,\mathbf {R}^N,\varvec{\alpha }^N)\nonumber \\= & {} -\left[ \begin{array}{l} -\frac{\partial }{\partial \mathbf {R}_j} V(\mathbf {R}^N,\varvec{\alpha }^N)\\ -\frac{\partial }{\partial \varvec{\Phi }_j}V(\mathbf {R}^N,\varvec{\alpha }^N)- \varvec{\omega }_j\times \mathbf {L}_j \end{array} \right] +\left\langle \varvec{\nabla }_{\varvec{\mathfrak {r}}_j}\mathfrak {u}(\mathbf {r}^n,\mathbf {R}_j,\varvec{\alpha }_j)\right\rangle ^\dag \nonumber \\= & {} \left[ \begin{array}{l} -\frac{\partial }{\partial \mathbf {R}_j} (V+\psi )\\ -\frac{\partial }{\partial \varvec{\Phi }_j}(V+\psi )- \varvec{\omega }_j\times \mathbf {L}_j \end{array} \right] , \end{aligned}$$

(66)

with $\langle .\rangle ^\dag$ the equilibrium average over the fast variables. Equation (66) also includes the definition of the potential of mean force $\psi \doteq \langle \mathfrak {u}\rangle ^\dag$, i.e. the potential which gives rise to the average (over all configurations of the n bath molecules) force and torque acting on the jth colloidal particle at any given configuration keeping all the colloidal particles frozen. Thus, the fluid-equilibrium average force and torque includes the contribution of a postulated vacuum colloid-colloid interaction potential, V, and the solvent contribution to the total force and torque, $\psi$. This combination in turn results in a solvent-averaged potential of mean force, $\widetilde{V}\doteq V+\psi$, which can be obtained from a given physical model, e.g. the DLVO theory for the interaction of charged colloidal particles [75] or the ten Wolde-Frenkel potential [79]. Finally, the tensor $\varvec{\gamma }_{jk}$ is given by

$$\begin{aligned} \varvec{\gamma }_{jk}(\varvec{\lambda };t,t')&=\left\langle \varvec{\mathfrak {f}}_j(t)\otimes e^{i(t'-t)\hat{\mathcal {Q}}\mathfrak {L}}(\varvec{\mathfrak {f}}_k(t)-\varvec{\mathfrak {F}}_k)\right\rangle ^\dag \\&\equiv \left\langle \begin{array}{ccc} \mathbf {F}_j(t)\otimes e^{i(t'-t)\hat{\mathcal {Q}}\mathfrak {L}}(\mathbf {F}_k(t)-\varvec{\mathcal {F}}_k) &{}&{} \mathbf {F}_j(t)\otimes e^{i(t'-t)\hat{\mathcal {Q}}\mathfrak {L}}(\mathbf {T}_k(t)-\varvec{\mathcal {T}}_k)\\ \mathbf {T}_j(t)\otimes e^{i(t'-t)\hat{\mathcal {Q}}\mathfrak {L}}(\mathbf {F}_k(t)-\varvec{\mathcal {F}}_k) &{}&{} \mathbf {T}_j(t)\otimes e^{i(t'-t)\hat{\mathcal {Q}}\mathfrak {L}}(\mathbf {T}_k(t)-\varvec{\mathcal {T}}_k) \end{array} \right\rangle ^\dag \nonumber . \end{aligned}$$

(67)

where $\varvec{\mathcal {F}}\doteq \langle \mathbf {F}_k\rangle ^\dag$ and $\varvec{\mathcal {T}}\doteq \langle \mathbf {T}\rangle ^\dag$. The behaviour of Eq. (65) as the product $m_b\,\varvec{\mathfrak {m}}^{-1}$ vanishes can be obtained by simply letting $\varvec{\lambda }\rightarrow \mathbf {0}$. In such a limit, the friction tensor $\varvec{\gamma }$ can be approximated by the first term of a multi-power series in $\varvec{\lambda }$ [63],

$$\begin{aligned} \varvec{\gamma }_{jk}(\varvec{\lambda };t,t')\sim \left\langle \varvec{\mathfrak {f}}_j(t)\otimes (\varvec{\mathfrak {f}}_k(t')-\varvec{\mathfrak {F}}_k) \right\rangle \end{aligned}$$

(68)

which yields

$$\begin{aligned}&\partial _tf^{(N)}(t)+\varvec{\lambda }\sum _{j=1}^N\left( \varvec{\nabla }_{\varvec{\mathfrak {r}}_j}\cdot \widetilde{\varvec{\mathfrak {p}}}_j+\varvec{\mathfrak {F}}_k\cdot \varvec{\nabla }_{\widetilde{\varvec{\mathfrak {p}}}_j}\right) f^{(N)}(t) \nonumber \\&\quad = \sum _{j,k=1}^N \varvec{\lambda }^2\,\varvec{\nabla }_{\varvec{\mathfrak {p}}_j}\cdot \int _{-t_I}^{t}dt'\left\langle \varvec{\mathfrak {f}}_j(t)\otimes (\varvec{\mathfrak {f}}_k(t')-\varvec{\mathfrak {F}}_k) \right\rangle \left( \varvec{\nabla }_{\widetilde{\varvec{\mathfrak {p}}}_k}+\beta \,\widetilde{\varvec{\mathfrak {p}}}_k\right) f^{(N)}(t'). \end{aligned}$$

(69)

The last step, the “Markovianization” of this time-evolution equation with memory, is the most controversial one [9, 59, 60, 75] as it requires the assumption that $f^{(N)}(t')$ is very slowly varying compared to the correlation $\langle \varvec{\mathfrak {f}}_j(t)\otimes \varvec{\mathfrak {f}}_k(t) \rangle ^\dag$, so that [63]

$$\begin{aligned}&\int _{-t_I}^{t}dt'\left\langle \varvec{\mathfrak {f}}_j(t)\otimes (\varvec{\mathfrak {f}}_k(t')-\varvec{\mathfrak {F}}_k) \right\rangle \left( \varvec{\nabla }_{\widetilde{\varvec{\mathfrak {p}}}_k}+\beta \,\widetilde{\varvec{\mathfrak {p}}}_k\right) f^{(N)}(t')\nonumber \\&\quad \sim \int _{-\infty }^{t}dt'\left\langle \varvec{\mathfrak {f}}_j(t)\otimes (\varvec{\mathfrak {f}}_k(t')-\varvec{\mathfrak {F}}_k)\right\rangle \left( \varvec{\nabla }_{\widetilde{\varvec{\mathfrak {p}}}_k}+\beta \,\widetilde{\varvec{\mathfrak {p}}}_k\right) f^{(N)}(t). \end{aligned}$$

(70)

Such an approximation is widely known to produce an incorrect description of the velocity correlation function if the colloidal particles have a similar density to that of the bath particles [9, 59, 60, 75]. Equation (70) gives an exponential decay, $\sim e^{-t}$, for the velocity correlation at large times, while with memory, such a decay is algebraic [75], $\sim t^{-3/2}$. Nevertheless, it has also been pointed out that such long-time tails are very small compared to the exponential component predicted under the Markovianized theory [75]. Although these could be reasons to avoid this critical step, the advantages of getting an FPE are significant. In contrast, the approximation is completely valid when both $m_b/m$ and $N_{\text {Kn}}\doteq r_0/R_0$ (the Knudsen number, with $r_0$ being a characteristic length scale for fluid intermolecular interactions and $R_0$ is a characteristic colloidal particle length scale) are considered very small [67]. Nevertheless, it was argued by Bocquet and Piasecki [9] that under such circumstances sedimentation of colloidal particles could occur. This seems to restrict the applicability of the resultant theory to microgravity scenarios. However, it is not clear at all what the actual significance of these algebraic long-time tails is and each case should be judged on its own merits [59]. Thus Eq. (70) comprises, on the one hand, an uncontrolled approximation. On the other hand, it has been extensively used in statistical mechanics [17, 44, 67, 80]. For instance, the same hypothesis underlies a recent derivation of a unified DDFT to include inertia and HIs [35], and is also involved in modern theories describing nucleation of colloidal systems and macromolecules [26, 54–56]. Moreoever, the results obtained are in perfect agreement with experiments and simulations, corroborating the smallness of the error related to Eq. (70) when it comes to describing systems of interacting colloidal particles. More akin to the problem at hand, the hypothesis is tacitly assumed within the seminal work of Dickinson [21], where a generalised algorithm to simulate protein diffusional problems is proposed. Therefore, while we cannot really justify such an assumption in a rigorous manner it does, nevertheless, represents the state-of-the-art in modelling colloidal systems. With this in mind, we can finally obtain the FPE related to Eq. (69) when Eq. (70) is taken into consideration,

$$\begin{aligned}&\partial _tf^{(N)}(t)+\sum _{j=1}^N\left( \varvec{\nabla }_{\varvec{\mathfrak {r}}_j}\cdot \varvec{\mathfrak {m}}^{-1}\varvec{\mathfrak {p}}_j+\varvec{\mathfrak {F}}_j\cdot \varvec{\nabla }_{\varvec{\mathfrak {p}}_j}\right) f^{(N)}(t) \nonumber \\&\quad = \sum _{j,k=1}^N \,\varvec{\nabla }_{\varvec{\mathfrak {p}}_j}\cdot \varvec{\Gamma }_{jk}(\varvec{\mathfrak {r}}^N)\left( \varvec{\mathfrak {p}}_k+k_BT\varvec{\mathfrak {m}}\,\varvec{\nabla }_{\varvec{\mathfrak {p}}_k}\right) f^{(N)}(t), \end{aligned}$$

(71)

with the translational, rotational and coupled translational-rotational friction tensors,

$$\begin{aligned} \varvec{\Gamma }_{jk}(\mathbf {r}^N) \equiv \left( \begin{array}{cc} \varvec{\Gamma }_{jk}^{TT} &{} \varvec{\Gamma }_{jk}^{TR}\\ \varvec{\Gamma }_{jk}^{RT} &{} \varvec{\Gamma }_{jk}^{RR}\\ \end{array}\right) \doteq \beta \varvec{\mathfrak {m}}^{-1}\int _{0}^{\infty }ds\,\left\langle \varvec{\mathfrak {f}}_j(t)\otimes (\varvec{\mathfrak {f}}_k(t-s)-\varvec{\mathfrak {F}}_k) \right\rangle . \end{aligned}$$

(72)

1.2 Equations of Motion

The FPE previously derived can be rewritten in the less compact but more explicit form,

$$\begin{aligned} \partial _t f^{(N)}(t)+&\sum _{j=1}^N \varvec{\nabla }_{\varvec{\mathfrak {r}}_j}\cdot \left( \varvec{\mathfrak {m}}^{-1}\varvec{\mathfrak {p}}_jf^{(N)}(t)\right) - \varvec{\nabla }_{\varvec{\mathfrak {p}}_j} \cdot \left[ \left( \varvec{\mathfrak {F}}_j-\sum _{k=1}^N\varvec{\Gamma }_{jk}(\varvec{\mathfrak {r}}^N)\varvec{\mathfrak {p}}_k \right) f^{(N)}(t)\right] \\&= \sum _{j,k=1}^N \left[ \varvec{\nabla }_{\varvec{\mathfrak {p}}_j} \otimes \varvec{\nabla }_{\varvec{\mathfrak {p}}_k} \right] :\left( k_BT\varvec{\mathfrak {m}}\,\varvec{\Gamma }_{jk}(\varvec{\mathfrak {r}}^N)f^{(N)}(t)\right) \nonumber \end{aligned}$$

(73)

which is equivalent to the system of SDEs [7, 48, 72]

$$\begin{aligned} \dot{\varvec{\mathfrak {r}}}_j(t)=&\ \varvec{\mathfrak {m}}^{-1}\varvec{\mathfrak {p}}_j(t)\\ \dot{\varvec{\mathfrak {p}}}_j(t)=&\ \varvec{\mathfrak {F}}_j-\sum _{k=1}^N\varvec{\Gamma }_{jk}(\varvec{\mathfrak {r}}^N)\varvec{\mathfrak {p}}_k + \sum _{k=1}^N\mathbf {A}_{jk}\varvec{\xi }_k(t)\nonumber \end{aligned}$$

(74)

where $\varvec{\xi }_j=(\mathbf {f}_j,\mathbf {t}_j)^\top$ is a 6-dimensional Gaussian white noise representing the random forces, $\mathbf {f}_j$, and torques, $\mathbf {t}_j$, acting upon the j-th particle, such that $\langle \xi _j^a(t)\rangle =0$ and $\langle \xi _j^a(t)\xi _k^b(t')\rangle = 2\delta _{jk}\delta ^{ab}\delta (t-t')$, where $\langle .\rangle$ refers to the average over an ensemble of the white-noise realisations. The strength of these random forces and torques is given by the tensor $\mathbf {A}_{jk}$ which obeys the fluctuation-dissipation relation,

$$\begin{aligned} k_BT \varvec{\mathfrak {m}}\,\varvec{\Gamma }_{jk}(\varvec{\mathfrak {r}}^N) = \sum _{l=1}^N\mathbf {A}_{jl}(\varvec{\mathfrak {r}}^N)\mathbf {A}_{kl}(\varvec{\mathfrak {r}}^N), \end{aligned}$$

(75)

and

$$\begin{aligned} \mathbf {A}_{jk}\varvec{\xi }_k\equiv \begin{pmatrix} \mathbf {A}_{jk}^{TT}&{}\mathbf {A}_{jk}^{TR}\\ \mathbf {A}_{jk}^{RT}&{}\mathbf {A}_{jk}^{RR} \end{pmatrix} \begin{pmatrix} \mathbf {f}_{k}\\ \mathbf {t}_{k} \end{pmatrix}. \end{aligned}$$

(76)

Coming back to the expanded notation, the system of equations (74) becomes

$$\begin{aligned} \frac{d\mathbf {R}_j}{dt}=&\,\frac{1}{m}\mathbf {P}_j,\end{aligned}$$

(77)

$$\begin{aligned} \frac{d\varvec{\Phi }_j}{dt}=&\,\mathbb {I}^{-1}\mathbf {L}_j,\end{aligned}$$

(78)

$$\begin{aligned} \frac{d\mathbf {P}_j}{dt}=&\, -\frac{\partial }{\partial \mathbf {R}_j}\widetilde{V}(\mathbf {R}^N,\varvec{\alpha }^N)-\sum _{k=1}^N\left( \varvec{\Gamma }_{jk}^{TT}\mathbf {P}_k+\varvec{\Gamma }_{jk}^{TR}\mathbf {L}_k\right) +\sum _{k=1}^N\mathbf {A}_{jk}^{TT}\mathbf {f}_k(t)\nonumber \\&\,\,\, +\mathbf {A}_{jk}^{TR}\mathbf {t}_k(t) \end{aligned}$$

(79)

$$\begin{aligned} \frac{d\mathbf {L}_j}{dt}\equiv \mathbb {I}\frac{d\varvec{\omega }_j}{dt}=&-\frac{\partial }{\partial \varvec{\Phi }_j}\widetilde{V}(\mathbf {R}^N,\varvec{\alpha }^N)-\varvec{\omega }_j\times \mathbf {L}_j \, -\sum _{k=1}^N\left( \varvec{\Gamma }_{jk}^{RT}\mathbf {P}_k+\varvec{\Gamma }_{jk}^{RR}\mathbf {L}_k\right) \nonumber \\&\,\,\, +\sum _{k=1}^N\mathbf {A}_{jk}^{RT}\mathbf {f}_k(t)+\mathbf {A}_{jk}^{RR}\mathbf {t}_k(t) \end{aligned}$$

(80)

with Eq. (78) equivalent to the relation $\varvec{\omega }_j=\varvec{\Lambda }_j^\top \dot{\varvec{\alpha }_j}$, as can be verified by using Eq. (57). Equations (77–80), a much less convenient representation of the rotational-translational Langevin equation (74), have been introduced in the studies by Wolynes and Deutch [80], Dickinson [21, 22] and Hernández-Contreras and Medina-Noyola [44] but these authors started from postulated equations instead of the detailed microscopic derivation from the full system of bath and colloidal particles offered here.

Appendix 2: On the Rapid Relaxation of the Fluid: Neglecting Inertia in the Bath

Here we address the question of when the inertia of the fluid bath can be neglected while having finite viscous forces. With this aim we will make use of some of the results derived by Peters [67] in his study on the FPE for coupled rotational and translational motions of structured Brownian particles. The main conclusion, the rapid relaxation of the fluid bath, is ultimately connected with the assumption of negligible inertial effects in the fluid bath whilst considering inertial effects of the colloids.

In a very detailed study, Peters [67] applied the multiple time-scale expansion to the derivation of the FPE for arbitrary-shape colloids. With this method Peters showed that Eq. (71) is the formal time-evolution equation for the distribution function, up to $(m/M)^3$, when both m / M and $N_{\text {Kn}}$ are considered small. It was also argued that the rapid relaxation of the fluid depends upon $N_{\text {Kn}}$ for a system with m / M small. If we now make use of the fact that $m \sim r_0^3\rho _b$ and $M\sim R_0^3\rho _B$, the conditions for the FPE would to be a good description of the colloidal system can be reduced to requiring that $\rho _b/\rho _B$ must be small, which is indeed the condition pointed out by Bocquet and Piasecki [9] and many others [43, 45, 58, 61, 73]. In the following we analyse whether or not this is possible while the ratio between inertia in the fluid bath and viscous forces is small. That is, whether is possible to have a low Reynolds number for the fluid bath along with the condition on the ratio between bath and colloidal densities. In such a case, it would be justified to neglect inertial effects in the bath.

To this end, we consider the case of solid and spherical colloids (far from walls) so that the coupling components of the friction and diffusion tensors vanish, and neglect HIs for the moment. If the radius of the colloidal particles is denoted by $R_0$, the friction tensor takes the simple form [21],

$$\begin{aligned} \varvec{\Gamma }_{jk}^{TT} = \gamma _{T}\delta _{jk}\,\mathbf {1},\quad \varvec{\Gamma }_{jk}^{RR}=\gamma _{R}\delta _{jk}\,\mathbf {1},\quad \varvec{\Gamma }_{jk}^{TR}=\varvec{\Gamma }_{jk}^{RT}=0, \end{aligned}$$

(81)

with $\gamma _{T}=6\pi \eta \, R_0/M$ and $\gamma _{R}=8\pi \eta \,R_0^3/I=20\pi \eta \,R_0/M$, and $\eta$ the dynamic viscosity, satisfying the Stokes–Einstein formula [21]. For the sake of generality, the friction components will be considered equally important, i.e. $\gamma _{T}/\gamma _{R}\sim \mathcal {O}(1)$. For this reason, we can define the following two time scales

$$\begin{aligned} t_0^T = \left( \frac{MR_0^2\,\gamma _{TT}}{k_BT}\right) ,\quad t_0^R=\left( \frac{MR_0^2\,\gamma _{RR}}{k_BT}\right) , \end{aligned}$$

(82)

which are indeed of the same order-of-magnitude.

Considering the natural physical scales of the system, the following dimensionless translational and rotational variables (denoted by an asterisk) are pertinent

$$\begin{aligned}&t = t_0^T\,t*,\&\ \mathbf {R}_j = R_0\,\mathbf {R}^*_j,\&\ \mathbf {P}_j = \frac{k_BT}{\gamma _TR_0}\,\mathbf {P}_j^*, \&\ \mathbf {F}_j = \frac{k_BT}{R_0}\,\mathbf {F}_j^*,\&\ \mathbf {F}_{j,\text {noise}}= \frac{k_BT}{R_0}\,\mathbf {F}_{j,\text {noise}}^*,\ \nonumber \\&t = t_0^R\,t*,\&\ \varvec{\omega }_j = \frac{k_BT}{MR_0^2\gamma _R}\varvec{\omega }^*_j,\&\ \mathbf {L}_j = \frac{k_BT}{\gamma _R}\,\mathbf {L}_j^*, \&\ \mathbf {T}_j = k_BT\,\mathbf {T}_j^*,\&\ \mathbf {T}_{j,\text {noise}}= k_BT\,\mathbf {T}_{j,\text {noise}}^*,\ \end{aligned}$$

(83)

which, when applied in Eqs. (77)–(80), yield the dimensionless equations of motion,

$$\begin{aligned} \frac{d\mathbf {R}_j^*}{dt^*}=&\,\mathbf {P}_j^*,\quad \frac{d\varvec{\Phi }_j^*}{dt^*}=\,\varvec{\omega }_j^*\\ \frac{d\mathbf {P}_j^*}{dt^*}=&\,\gamma _T^{*^2}\left( \mathbf {F}_j^*-\mathbf {P}_j^*+\mathbf {F}_{j,\text {noise}}^*(t)\right) \nonumber \\ \frac{d\mathbf {L}_j^*}{dt^*}=&\,\gamma _R^{*^2}\left( \mathbf {T}_j^*-\mathbf {L}_j^*+\mathbf {T}_{j,\text {noise}}^*(t)\right) \nonumber \end{aligned}$$

(84)

along with the definitions

$$\begin{aligned} \gamma _T^{*}=\gamma _T R_0\sqrt{\frac{M}{k_BT}},\quad \gamma _R^*=\gamma _{R} R_0\sqrt{\frac{M}{k_BT}}. \end{aligned}$$

(85)

What we wish to test is whether or not it is possible to have finite viscous forces, i.e. $\gamma _T^{*}\sim \mathcal {O}(1)$, such that at the same time inertia forces in the fluid bath are negligible. For this purpose both translational and rotational Reynolds numbers must be small. They can be defined as [41],

$$\begin{aligned} \text {Re}^T=\frac{U\,R_0\,\rho _b}{\eta },\quad \text {Re}^R=\frac{\Omega \,R_0^2\,\rho _b}{\eta }, \end{aligned}$$

(86)

respectively. The quantities U and $\Omega$ represent typical linear and angular velocity scales, which can be obtained from the nondimensionalised momenta,

$$\begin{aligned} U=\frac{k_BT}{MR_0\gamma _{T}},\quad \Omega =\frac{k_BT}{MR_0^2\gamma _{R}}. \end{aligned}$$

(87)

Making use of (82), (83), (85) and (87) into (86), we finally reach

$$\begin{aligned} \text {Re}^T=\frac{9}{2\gamma _{T}^{*^2}}\frac{\rho _b}{\rho _B},\quad \text {Re}^R=\frac{15}{\gamma _{R}^{*^2}}\frac{\rho _b}{\rho _B}. \end{aligned}$$

(88)

As already noted, the regime we consider involves, $\gamma _T^{*}\sim \gamma _{R}^*\sim \mathcal {O}(1)$ and $\text {Re}^T\sim \text {Re}^R\ll 1$, which is undoubtedly satisfied when $\rho _b/\rho _B\ll 1$. Then, neglecting inertia forces in the fluid bath is consistent with the separation of time scales we already assumed to obtain the FPE. Further details of the physical interpretation and consequences of this limiting condition were given in Appendix 1.