2.1. Hydrodynamic angular velocity

The hydrodynamic angular velocity $\dot {\gamma }\varOmega$ is required to close (2.1). In the dilute limit, $\varOmega$ can be evaluated by examining the motion of an isolated freely suspended platelet in the absence of Brownian fluctuations. For a symmetric platelet, $\varOmega$ can be calculated exactly by using Bretherton's (Reference Bretherton1962) equation of motion (Kim & Karrila Reference Kim and Karrila2013). This equation describes the relationship between the time derivatives of the particle orientation vector $\boldsymbol {d}$, and the rate of strain and vorticity tensors associated with the undisturbed flow field, $\boldsymbol {\varOmega _{\infty }}=(\boldsymbol {\nabla } \boldsymbol {u}_{\infty }+\boldsymbol {\nabla } \boldsymbol {u}^{T}_{\infty })$ and $\boldsymbol {E}_{\infty }=(\boldsymbol {\nabla } \boldsymbol {u}_{\infty }-\boldsymbol {\nabla } \boldsymbol {u}^{T}_{\infty })$, respectively, as follows:

(2.2)\begin{equation} \dot{\boldsymbol{d}}=\dot{\gamma}\boldsymbol{\varOmega}\times\boldsymbol{d} = \boldsymbol{\varOmega_{\infty}}\times\boldsymbol{d}+\left(\frac{k_{e}^2-1}{k_{e}^2+1}\right)\left(\boldsymbol{E}_{\infty}\boldsymbol{\cdot}\boldsymbol{d} -\boldsymbol{E}_{\infty}\,\boldsymbol{d}\boldsymbol{d}\,\boldsymbol{d} \right). \end{equation}

Using spherical polar coordinates, the orientation vector can be expressed as $\boldsymbol {d}=\sin {\theta }\cos {\phi } \boldsymbol {\hat {e}}_{x} +\sin {\theta }\sin {\phi } \boldsymbol {\hat {e}}_y+\cos {\theta }\boldsymbol {\hat {e}}_{z}$, resulting in the following coupled ordinary differential equations (Kim & Karrila Reference Kim and Karrila2013):

(2.3)\begin{equation} \dot{\theta}=\left(\frac{1-k_{e}^2}{1+k_{e}^2}\right)\frac{\dot{\gamma}}{4}\sin{2\theta}\sin{2\phi}, \quad \dot{\phi}\equiv \dot{\gamma}\varOmega={-}\frac{\dot{\gamma}}{k_{e}^2+1}\left(k_{e}^2\cos^2{\phi}+\sin^2{\phi}\right). \end{equation}

Note that $\dot {\phi }$ does not depend on $\theta$, hence an analysis of the particular 2-D case considered here corresponding to $\theta ={\rm \pi} /2$ is representative of the $\phi$-dependence for all values of $\theta$. The derivation of Bretherton's equation of motion does not make assumptions regarding the boundary condition at the solid surface and therefore applies to platelets with slip. This equation depends on the value of $k_{e}$, which in turn depends on the slip length as well as on the platelet's shape (Luo & Pozrikidis Reference Luo and Pozrikidis2008; Kamal et al. Reference Kamal, Gravelle and Botto2020). The parameter $k_{e}$ can be calculated for $\theta ={\rm \pi} /2$ as the ratio between the hydrodynamic torques acting on a particle held fixed with its long axis held parallel to the flow, $T(0)$, or perpendicular to the flow, $T({\rm \pi} /2)$, according to the following expression (Cox Reference Cox1971):

(2.4)\begin{equation} k_{e}=\sqrt{\frac{T(0)}{T({\rm \pi}/2)}}. \end{equation}

For a platelet satisfying the no-slip boundary condition, $T(0)/T({\rm \pi} /2)$ is positive and therefore $k_{e}$ is a real number (Willis Reference Willis1977). In this case, when the geometric aspect ratio $k = b/a \ll 1$, the parameter $k_{e}$ follows a power-law relationship $k_{e} \propto k^n$ where $n$ is an exponent that depends on the specific geometry of the platelet. For this reason, $k_{e}$ is often called the ‘effective aspect ratio’. For example, an ellipsoid with a no-slip surface has $k_{e}=k$ (Jeffery Reference Jeffery1922), and a disk with blunt edges has $k_{e}\propto k^{3/4}$ (Singh et al. Reference Singh, Koch, Subramanian and Stroock2014). For real $k_{e}$ the solution to (2.3) is

(2.5a,b)\begin{equation} \tan{\theta}=\dfrac{C k_e}{\sqrt{k_e^2 \cos^2{\phi}+\sin^2{\phi}}}, \quad \tan{\phi}={-}k_{e}\tan{\left(\frac{\dot{\gamma}t}{k_{e}+k_{e}^{{-}1}}\right)}, \end{equation}

where $C$ is a positive integration constant. These equations describe periodic rotations of the particle called ‘Jeffery orbits’; the time period of such rotations is

(2.6)\begin{equation} P=2{\rm \pi}\dot{\gamma}^{{-}1}(k_{e}+k_{e}^{{-}1}). \end{equation}

The value of $C$ depends on $\theta (0)$; for $\theta (0) = {\rm \pi}/2$, $C\to \infty$ and $\theta (t)={\rm \pi} /2$ for all values of $t$. Thus, a particle initially rotating in the plane of the flow will continue rotating in the plane of the flow. This is the situation we are aiming to model in the current paper. For finite $Pe$ numbers, trajectories out of the $\theta = {\rm \pi}/2$ plane are of course possible. The case $\theta = {\rm \pi}/2$ for finite $Pe$ is tractable analytically and has been shown, in the no-slip case, to be relevant to the full 3-D dynamics (Leahy et al. Reference Leahy, Koch and Cohen2015). In § 3 we shall consider the general slip case at finite $Pe$.

An example of an orbit for $\theta = {\rm \pi}/2$ and real $k_{e}$ and the corresponding $\varOmega (\phi )$ are given in figures 2(a) and 2(b), respectively. Since $k_{e} \ll 1$, the time scale for the particle to flip is much smaller than the time spent by the particle near $\phi =0$ (Singh et al. Reference Singh, Koch, Subramanian and Stroock2014). For a no-slip platelet of vanishing thickness, $k_{e} \to 0$ as $k \to 0$ and thus the particle will remain in a fixed position in the ad hoc case $k=0$ (Bretherton Reference Bretherton1962). Otherwise the platelet will rotate (unless the platelet has a specially designed, slightly non-axisymmetric shape, see Borker, Stroock & Koch (Reference Borker, Stroock and Koch2018)).

Figure 2. (a) Rotation angle $\phi$ for a particle with real effective aspect ratio $k_{e} = 0.16$ and ${Pe} = \infty$. (b) Associated angular velocity $\dot {\gamma }\varOmega$; notice that $\varOmega <0 \ \forall \phi$. (c) Rotation angle $\phi$ for a particle with imaginary effective aspect ratio $k_{e} = 0.28 \text {i}$ and ${Pe} = \infty$, showing stabilisation at $\phi _{c} \approx 0.27$. (d) Corresponding non-dimensional angular velocity, with the two equilibrium points at $\phi = \pm \phi _{c}$. The ‘${\bigodot}$’ symbol marks the stable angle $\phi _{c}$; the angle denoted by ‘${\bigotimes}$’ is unstable.

For platelets presenting a surface with slip, however, such alignment is possible (Kamal et al. Reference Kamal, Gravelle and Botto2020). The effect of surface slip is to reduce the value of $k_{e}^2$, such that the $\varOmega (\phi )$ curve shifts upwards, reducing the frequency of rotation with respect to the no-slip case (Luo & Pozrikidis Reference Luo and Pozrikidis2008; Zhang et al. Reference Zhang, Xu and Qian2015; Kamal et al. Reference Kamal, Gravelle and Botto2020). At or above a critical slip length that scales with the thickness of the platelet, $k_{e}^2$ becomes negative, and so $\varOmega (\phi )$ will be shifted onto or above the zero line (figure 2d). Such a solution corresponds to a negative value of $T(0)/T({\rm \pi} /2)$ and a purely imaginary complex value of $k_{e}$ (Kamal et al. Reference Kamal, Gravelle and Botto2020). It is instructive to compare the structure of the $\phi -\theta$ equations (describing full 3-D trajectories) for pure and imaginary values of $k_{e}$. For a purely imaginary $k_{e}$, the solution of (2.3) is

(2.7a,b)\begin{equation} \tan{\theta}=\frac{C|k_{e}|}{|k_{e}|^2\cos^2{\phi}+\sin^2{\phi}},\quad \tan{\phi}=|k_{e}|\tanh{\left(\frac{\dot{\gamma}t}{|k_{e}|^{{-}1}-|k_{e}|}\right)}. \end{equation}

Comparing with (2.5a,b), whilst the equation for $\theta$ remains unchanged with respect to the case of a real $k_{e}$, the equation for $\phi$ now admits a stable non-periodic solution for $t \rightarrow \infty$. The particle relaxes to the stable orientation angle $\phi _{c}=\arctan (|k_{e}|)$, as shown in figure 2(c). For this angle, $\varOmega$ has a positive zero root, indicated by the open circle in figure 2(d). The relaxation to equilibrium occurs on a time scale

(2.8)\begin{equation} P_{R}=\dot{\gamma}^{{-}1}(|{k}_e|^{{-}1}-|k_{e}|), \end{equation}

to be compared with the rotational time period of (2.6). For $|{k}_e|\ll 1$, $P$ and $P_{R}$ are proportional to each other. We note that a stable orientation has been predicted for blood cells in shear flow in the case of small internal-to-external viscosity ratios (due to the so-called ‘tank-treading’ motion, see e.g. Keller & Skalak (Reference Keller and Skalak1982)). The analogy with our case is, however, only qualitative, because there is no single slip parameter that can model the dynamic coupling between inner and outer fluids seen in blood cells.

Surface slip at infinite Péclet number thus has significant effects on the rotational dynamics of a platelet. When the slip length is smaller than the platelet's thickness, slip reduces the frequency of the rotational orbits through a decrease in the value of $k_{e}$. However, when the slip length is greater than a threshold value comparable to the platelet's thickness, the platelet aligns indefinitely at an angle $\phi _{c}$ with the flow.

3.1. Random walk trajectories

To illustrate the effects of Brownian fluctuations on the platelet's rotational trajectories, we solve numerically the Langevin representation of (2.1) for different $Pe$ and $k_{e}$. In finite difference form, such Langevin representation describes simply connected paths in the particle orientational space. Using a first-order approximation of the time derivative, the resulting stochastic finite difference equation is (Doi & Edwards Reference Doi and Edwards1988)

(3.1)\begin{equation} \Delta \phi= \dot \gamma \varOmega(\phi) \Delta t + X. \end{equation}

Here, we take $X$ as a Gaussian random variable with zero mean and variance $\left \langle X^2\right \rangle = 2 D_{r} \Delta t$. Solving numerically this equation for several sequences of random numbers provides a set of trajectories distributed according to (2.1). Examples of trajectories calculated from (3.1) for different $Pe$ are shown in figure 3, comparing the cases of real and purely imaginary $k_{e}$. Full rotational cycles, corresponding to a platelet travelling through the full range $\phi \in [-{\rm \pi} /2, {\rm \pi}/2]$, can always be observed if $k_{e}$ is real. When $k_{e}$ is real, the average time period (in units of $Pe$ $t$) for the platelet to perform a full rotational cycle decreases as $Pe$ decreases (figure 3a–c). This behaviour has been predicted in previous theoretical studies on the steady-state rotatory dynamics of plate-like particles (Leal & Hinch Reference Leal and Hinch1971; Hinch & Leal Reference Hinch and Leal1972), and is in agreement with experiments where the instantaneous rotations of elongated particles have been visualised directly (Stover et al. Reference Stover, Koch and Cohen1992; Herzhaft & Guazzelli Reference Herzhaft and Guazzelli1999; Leahy et al. Reference Leahy, Cheng, Ong, Liddell-Watson and Cohen2013), or through rheo-optical or neutron diffraction techniques (Frattini & Fuller Reference Frattini and Fuller1986; Jogun & Zukoski Reference Jogun and Zukoski1999; Brown et al. Reference Brown, Clarke, Convert and Rennie2000). In contrast, if $k_{e}$ is purely imaginary, for large $Pe$ (${Pe}=640$), the platelet fluctuates around a stable orientation angle (figure 3d–f). This angle is approximately the same as the equilibrium angle for $Pe\to \infty$ ($\phi _{c}=\arctan {|k_{e}|}$). When $Pe$ decreases below a threshold value, Brownian fluctuations cause the platelet to perform full rotational cycles. However, the average frequency of these full rotational cycles is significantly reduced in comparison with the no slip case, as seen by the reduction of the number of peaks in figure 3(d–f) as compared with figure 3(a–c) for ${Pe}=64$. For ${Pe} \sim O(1)$, Brownian fluctuations induce a chaotic behaviour whereby the platelet rotates frequently and randomly. In this range, the trajectories for a purely imaginary and a real $k_{e}$ are practically indistinguishable (see ${Pe}=6.4$ in figure 3).

Figure 3. Time evolution of the rotation angle for different $Pe$, as calculated from (3.1), comparing (a–c) real $k_{e}$ ($k_{e}=0.11$) to (d–f) purely imaginary $k_{e}$ ($k_{e}=0.21\textrm {i}$). The red dashed line is the value of $\phi _{c}$ obtained for ${Pe} \to \infty$.

Figure 3 suggests that for large $Pe$, the rotational dynamics depends strongly on whether $k_{e}$ is real or purely imaginary, while for smaller $Pe$ such dependency disappears. Quantifying the values of $Pe$ needed for the rotational dynamics to depend on the value of $k_{e}$ requires solving the Fokker–Planck (2.1).

3.1.1. Time evolution of the orientation distribution function's moments

Taking advantage of the periodicity of $p(\phi ,t)$, any solution of (2.1) can be expressed as a spectral series,

(3.2)\begin{equation} p(\phi,t)=a_0(t)+\sum_{n=1}^{\infty} \left[ a_n(t)\cos{(2n\phi)}+b_n(t)\sin{(2n\phi)}\right]. \end{equation}

Normalisation of $p$ requires $a_0=1/{\rm \pi}$, and assuming $\phi$ to be uniformly distributed at time $t=0$ gives an initial uniform distribution $p(\phi ,0)=1/{\rm \pi}$. In spectral coordinates, this is equivalent to having only one non-zero coefficient $a_0=1/{\rm \pi}$. Numerically, we solve (2.1) by truncating the series at a large value $N$, yielding $2N$ first-order differential equations for the spectral coefficients (Leahy et al. Reference Leahy, Koch and Cohen2015). A truncated value $N=1000$ has been chosen from the analysis of the spectral coefficients $a_n$ and $b_n$. For $n>N$, $a_n, b_n \to 0$ for all values of time $t$ (see figure 4 for $t \to \infty$). The $2N$-coupled equations are solved in time by a Runge–Kutta method (Press et al. Reference Press, Teukolsky, Vetterling and Flannery2007). The mean angle, standard deviation and higher moments can be evaluated directly in terms of the spectral distribution. For example, the mean and variance are

(3.3a,b)\begin{equation} \mu=\left\langle \phi\right\rangle=\sum_{i=1}^{\infty}\frac{\rm \pi}{2n}({-}1)^{1+n}b_n, \quad \sigma^2=\left\langle (\phi-\mu)^2\right\rangle=\frac{{\rm \pi}^3}{12}+\sum_{i=1}^{\infty}\frac{\rm \pi}{2n^2}({-}1)^{n}a_n-\mu^2, \end{equation}

respectively. Here, the close brackets $\left \langle \;\right \rangle$ represent an average over the orientation distribution function $p(\phi ,t)$ for fixed $t$.

Figure 4. Spectral coefficients $a_n$ and $b_n$ for $k_e=0.05\text {i}$ and $Pe=10^{6}$ and $t \to \infty$. The spectral coefficients are truncated at $n=1000$, since $a_n, b_n \to 0$ above this threshold.

Many studies discuss solutions to (2.1) for elongated particles with real $k_{e}$ (Burgers Reference Burgers1938; Peterlin Reference Peterlin1938; Sadron Reference Sadron1953; Scheraga Reference Scheraga1955; Leal & Hinch Reference Leal and Hinch1971; Hinch & Leal Reference Hinch and Leal1972, Reference Hinch and Leal1973; Férec et al. Reference Férec, Heniche, Heuzey, Ausias and Carreau2008; Leahy et al. Reference Leahy, Koch and Cohen2015; Leahy, Koch & Cohen Reference Leahy, Koch and Cohen2017). However, the case of purely imaginary $k_{e}$, to our knowledge, has not yet been analysed. The time evolution of the mean, variance and third-order moment $\left \langle (\phi -\mu )^3\right \rangle$ of the orientation distribution for $k_{e}=0.1\text {i}$ and different $Pe$ is shown in figure 5. Similarly to when $k_{e}$ is real (Leahy et al. Reference Leahy, Koch and Cohen2015), the moments relax in time to a steady value, with a relaxation time that decreases as $Pe$ decreases. Therefore, we can assume that the statistical steady state is reached over a time greater than the relaxation time $P_R=\dot {\gamma }^{-1}(|k_{e}|^{-1}-|k_{e}|)$ of a platelet in absence of Brownian motion.

Figure 5. Mean (a), variance (b) and skewness (c) of the angular probability distribution versus time, for a nanoplatelet with $k_{e}=0.1\text {i}$ and different values of $Pe$.

3.2. Steady-state probability distribution

We solve (2.1) with $\dot {p}=0$ to find the steady-state orientation distribution function $p(\phi )$ using the method described in § 3.1.1. The effect of $Pe$ on $p(\phi )$ comparing real and purely imaginary values of $k_{e}$ is shown in figure 6. For ${Pe} = 1000$, there is a noticeable difference in the shape of the orientation distribution function depending on whether $k_{e}$ is real or purely imaginary. In particular, $p(\phi )$ has a much sharper peak if $k_{e}$ is purely imaginary. This difference can be analysed by considering the limit $Pe\to \infty$. For ${Pe}\to \infty$, (2.1) reduces at steady state to

(3.4)\begin{equation} p\varOmega = c, \end{equation}

for some integration constant $c$. For real $k_{e}$, normalisation of $p$ gives $p=(P\dot {\gamma } | \varOmega |)^{-1}$, where $P$ is the time period (see (2.6)) (Anczurowski & Mason Reference Anczurowski and Mason1967a). Such a probability distribution shown by the (red) dotted line in figure 6 has mean orientation angle $\mu =0$ and finite variance $\sigma ^2$ by the symmetry of $p$ about $\phi =0$. If $k_{e}$ is purely imaginary, however, $1/\varOmega$ diverges as $\phi \to \pm \phi _{c}$, and thus the integrand of $c/\varOmega$ will not be finite unless $k_{e}=0$ or $c=0$, as required by the normalisation constraint for $p$. Thus, the equation $p\varOmega =c$ is only satisfied for $c=0$ and $p(\phi )=\delta (\phi -\phi _{c})$, as expected. As seen in figure 6 for the imaginary $k_{e}$, as $Pe$ increases the probability distribution function indeed converges to a Dirac delta, confirming that for sufficiently large $Pe$ the platelet spends most of the time fluctuating about $\phi _{c}$ (as observed in figure 3d–f).

Figure 6. Steady-state orientation distribution function $p(\phi )$ for a real $(k_e = 0.1)$ or a purely imaginary $(k_e = 0.1\textrm {i})$ value of $k_{e}$ for different $Pe$ numbers. The red dotted line marks the orientation distribution function for ${Pe} \to \infty$ and $k_{e}=0.1$ (a) and the location of $\phi _{c}$ for $k_{e}=0.1 i$ (b). The values [$\mu$, $\sigma ^2$, $\left \langle (\phi -\mu )^3\right \rangle$] in increasing order of $Pe$ are $[0.17,0.39,-0.36]$, $[0.069,0.21,-0.40]$ and $[0.019,0.14,-0.30]$, respectively, for $k_{e}=0.1$; and $[0.19,0.38,-0.37]$, $[0.094,0.16,-0.53]$ and $[0.075,0.033,-1.1]$, respectively, for $k_{e}=0.1\text {i}$.

For ${Pe} = 10$ the orientation distribution functions for $k_{e}=0.1$ and $k_{e}=0.1\text {i}$ are almost identical, and slightly skewed towards positive values of $\phi$ (the value of $\mu$, $\sigma ^2$ and $\left \langle (\phi -\mu )^3\right \rangle$ for $k_{e}=0.1$ and $k_{e}=0.1\text {i}$, given in the caption of figure 6, are within 10 % of each other). Since $Pe$ is not small in this example, hydrodynamic stresses must play a role in creating such skewness. However, the effect of changes to the value of $k_{e}$ due to surface slip on the probability distribution is evidently minor.

Balancing the orders of magnitude of the convective and diffusive fluxes appearing in (2.1) yields a characteristic angle $\varPhi$. From this, we can define a ‘local’ $Pe$ number, ${Pe}_{loc} = \dot {\gamma }\varOmega (\phi ) \varPhi /D_{r}$ (Leal & Hinch Reference Leal and Hinch1971). The effects of Brownian fluctuations on the orientation of the platelets are considered to be ‘weak’ compared with the hydrodynamic angular velocity if, for $\phi$ near zero, ${Pe}_{loc} \gg 1$ (Leal & Hinch Reference Leal and Hinch1971). This condition translates to the Brownian angular velocity being smaller than, using (2.3), $|\varOmega |\sim k_{e}^2$. For $|\varOmega |\sim k_{e}^2$ we have that $\phi$ can be at most $\varPhi \sim |k_{e}|$, and so $|{Pe}_{loc}|$ can be at most of the order of $|k_{e}^3 {Pe}|$.

Thus, for ${Pe} \gg |k_{e}|^{-3}$ Brownian diffusion is subdominant, and the rotation of the platelet can be approximated by the orientation distribution obtained for negligible Brownian fluctuations.

To quantify the value of $Pe$ for which the orientation distribution does not depend on $k_{e}$ to leading order, one can proceed as follows. For a geometric aspect ratio $k\to 0$, we will verify in § 4 that $|k_{e}|\ll 1$ whatever the value of $\lambda$. Expanding the right-hand side of the equation for $\dot {\phi }$ in (2.3) to leading order in $|k_{e}|$, the hydrodynamic velocity is seen to become independent of $k_{e}$ if $\sin^{2} {\phi _m} \gg |k_{e}^2|\cos^{2} {\phi _m}$. Thus, because we are interested in a critical $Pe$ valid for both $\lambda =0$ and finite $\lambda$, the condition for $\phi _m$ is $\tan {\phi _m}\approx \phi _m \gg \max (|k_{e}(\lambda =0)|,|k_{e}(\lambda )|)$. For the rotational dynamics to be independent of $k_{e}$ to leading order, the local Péclet number must be $O(1)$ over the characteristic angle $\varPhi \sim \phi _m$. This regime is associated with a rotational velocity $\varOmega (\phi _m)\approx \dot {\gamma }\phi _m^2$, so the local Péclet number ${Pe} \phi _m^3 \sim 1$. It follows from the condition $\phi _m\gg \max (|k_{e}(\lambda =0)|,|k_{e}(\lambda )|)$ that ${Pe} \ll \min (|k_{e}^{-3}(\lambda =0)|,|k_{e}^{-3}(\lambda )|)$. Thus, Brownian diffusion will control the orientation of the platelet when ${Pe} \ll \min (|k_{e}^{-3}(\lambda =0)|,|k_{e}^{-3}(\lambda )|)$.

The value ${Pe}_{m} = \min (|k_{e}^{-3}(\lambda =0)|,|k_{e}^{-3}(\lambda )|)$ is an important threshold. For ${Pe} \ll {Pe}_{m}$ the value of $k_{e}$ does not affect to leading order the orientation distribution $p(\phi ,t)$. When ${Pe} \gg {Pe}_{m}$ significant changes to the orientation distribution for either a real or purely imaginary value of $k_{e}$ are expected. Figure 7 demonstrates the importance of this threshold on the statistical moments $\mu , \sigma ^2$ and $\left \langle (\phi -\mu )^3\right \rangle$. The black line corresponds to ${Pe}_{m} = |k_{e}|^{-3}$. For $Pe$ above this line, the statistical moments approach the values predicted by using the probability distribution functions corresponding to $Pe \to \infty$. In this case the moments depend strongly on the value of $k_{e}$. For example, if $k_{e}$ is purely imaginary, then the variance $\sigma ^2 \to 0$ and the mean $\mu \to k_{e}$ for $Pe$ above the ${Pe}_{m}=|k_{e}^{-3}|$ line. If instead $k_{e}$ is real then $\sigma ^2$ is instead finite and $\mu \to 0$. Likewise, if $Pe$ is much smaller than $|k_{e}^{-3}|$ the statistical moments depend only on $Pe$ and become essentially independent of $k_{e}$ (compare, for example, the colour maps for $|k_{e}| \leq 0.1$ and ${Pe}\leq 100$ in figure 7).

Figure 7. Colour maps of the mean $\mu$, variance $\sigma ^2$ and skewness $\left \langle (\phi - \mu )^3\right \rangle$ of the steady-state orientation distribution in the ($Pe$, $k_{e}$) space. The black straight line is ${Pe}=|k_{e}|^{-3}$. For $Pe$ above this line, the difference in the moments between real and purely imaginary values of $k_{e}$ becomes noticeable.

Finally, for ${Pe} \ll 1$, the platelet's rotation is almost entirely due to Brownian diffusion. As ${Pe}\to 0$, the leading-order orientation distribution can be described at all times by a uniform distribution (Burgers Reference Burgers1938; Peterlin Reference Peterlin1938; Scheraga Reference Scheraga1955). Such distribution describes randomly rotating platelets that complete an average rotational cycle with a variance much larger than in the pure hydrodynamic flow limit, as also shown in figure 3.

3.3. Threshold Péclet number for full rotations

When the effective aspect ratio $k_{e}$ is purely imaginary, we can define a critical Péclet number ${Pe}_{c}$ separating the stable region, in which the platelet fluctuates about $\phi _{c}$, from the unstable region in which the platelet performs full rotations. Because the hydrodynamic angular velocity is zero for $\phi =\phi _{c}$ and small in the neighbourhood of this critical angle, in the stable region weak Brownian fluctuations always affect the orientation of the platelet for any finite value of $Pe$. When the platelet is in the unstable region, full rotation cycles can occur because the variance $\sigma ^{2}$ is large enough for the fluctuating platelet to ‘jump out’ of the region where $\varOmega (\phi )$ is positive. The width of this ‘hydrodynamic potential well’ is given by the range of values of $\phi$ for which $\varOmega (\phi )$ is positive. This width can be approximated as $2 \arctan (\theta _c) \approx 2 |k_{e}|$ for $|k_{e}| \ll 1$. Therefore, we estimate that the probability for a given particle to perform full rotary cycles becomes significant for $\sigma$ larger than a threshold $\sigma \sim 2 |k_{e}|$.

Using our numerical procedure for calculating $\sigma ^2$, as given in (3.3a,b), we calculate ${Pe}_{c}$ by first computing $\sigma ({Pe},k_{e})$ and then calculating the value of $Pe$ that gives $\sigma =2|k_{e}|$. We emphasise that full rotations of a given platelet can still occur for ${Pe}$ significantly larger than ${Pe}_{c}$, but these events, associated with the tails of the probability distribution, are rare and should not affect the occurrence of a stable orientation on account of the approximate symmetry of $p(\phi )$ about its mean value. Our results for ${Pe}_{c}$ are shown in figure 8 for a range of purely imaginary $k_{e}$. As $|k_{e}|$ decreases, the width of the ‘hydrodynamic potential well’ decreases, so ${Pe}_{c}$ increases. The data appear to be well fitted by ${Pe}_{c}\propto |k_{e}|^{-3.4}$ for small values of $|k_{e}|$. The exponent $-3.4$ is quite close to the exponent $-3$ obtained by balancing the order of magnitude of the convection and diffusive terms in (2.1). The data are expected to approach this theoretical value for values of $|k_{e}|$ smaller than we could simulate. For $|k_{e}| \gtrapprox 0.1$ the local power-law exponent is definitely not close to $-3$. In this region, the line ${Pe}_{c}\propto |k_{e}|^{-4.8}$ appears to be a better fit to the data.

Figure 8. Threshold value of the Péclet number ${Pe}_{c}$ beyond which no platelet rotation is expected, as a function of $|k_{e}|$ (with $k_{e}$ purely imaginary). Black disks are numerical values, lines are fits for $|k_{e}| \gtrapprox 0.1$ (dot–dashed blue line) and $|k_{e}| \lessapprox 0.1$ (dashed red line), respectively.

Figure 8 does not enable us by itself to draw a conclusion about whether a stable orientation occurs for given values of $\lambda$ and $a$, as both $k_{e}$ and $Pe$ depend on these variables. To draw such conclusion we need to map $k_{e}$ and $Pe$ to specific values of $\lambda$ and $a$. This will be done in the next section, by considering particles of a specific geometry and slip properties.

4.1. Effective nanoplatelet's geometry

Nanoplatelets of 2D materials, made for example of carbon (C), boron nitride (BN), or molybdenum disulfide ($\textrm {MoS}_2$), are essentially stacks of atomic crystal layers. To achieve a continuum description of the nanoplatelets in flow, one must define the reference surface that best approximates such platelets from a hydrodynamic standpoint. Molecular dynamics calculations reveal that, because of the smoothing of the molecular flow field by the finite-interaction potential near the edges, the reference surface is a cuboid with rounded edges, as sketched in figure 9. The cuboid has a half-thickness $b = \xi + d_{gg}(n-1)/2$, where $\xi$ is the effective radius of a single atom of the nanoplatelet, $d_{gg}$ is the interlayer spacing and $n$ is the number of stacks. For multilayer graphene, $d_{gg} \approx 3.35$ Å (Chung Reference Chung2002). The effective radius $\xi$ depends on the equilibrium distance between the atoms of the solid and the liquid molecules. For graphene in water, $\xi \sim 1.8\text {--}2.5$ Å (Gravelle et al. Reference Gravelle, Joly, Ybert and Bocquet2014), and comparable values can be expected for BN and MoS$_2$ due to their similar atomic structure (Radisavljevic et al. Reference Radisavljevic, Radenovic, Brivio, Giacometti and Kis2011; Tocci et al. Reference Tocci, Joly and Michaelides2014; Luan & Zhou Reference Luan and Zhou2016). The edges form a semicircle shape for $n=1$ (figure 9a) and a flat face with blunt edges as $n$ increases (figure 9b–d). The shape of the edge slightly affects the value of $k_{e}$: the blunter the edge, the larger the value of $k_{e}$ for a fixed aspect ratio $a/b$ (cf. Singh et al. (Reference Singh, Koch, Subramanian and Stroock2014) for $\lambda =0$).

Figure 9. (a–c) Water density profiles as extracted from MD simulations for a number of layer $n=1$ (a), $2$ (b) and $3$ (c), respectively. The centres of the carbon atoms are located along the grey lines. Details of the molecules dynamics simulations are given in Kamal et al. (Reference Kamal, Gravelle and Botto2020). The colour field is the water density, from white (low density) to blue (high density), and the red dashed lines show the reference surfaces. (d) Sketch of the reference surface for $n=3$, showing the slip length $\lambda$, the effective radius $\xi$ of the carbon atoms and the interlayer spacing $d_{gg}$.

4.2. Range of slip lengths

For atomically smooth surfaces of 2-D nanomaterials such as graphene, the hydrodynamics slip length measured at the planar surface of the particle ranges from a few nanometres to tens of nanometres. As summarised in table 1, experiments performed on graphite (Maali et al. Reference Maali, Cohen-Bouhacina and Kellay2008; Ortiz-Young et al. Reference Ortiz-Young, Chiu, Kim, Voïtchovsky and Riedo2013), and ab initio calculations (Tocci et al. Reference Tocci, Joly and Michaelides2014), suggest $\lambda \approx 10\ \textrm {nm}$ for water and graphene. In the case of NMP or ethanol, the slip length of graphene is also relatively large (${>}10\ \textrm {nm}$) (Gravelle et al. Reference Gravelle, Kamal and Botto2020). Ionic liquid (IL) can also give large slip lengths, but in this case $\lambda$ has been found to depend strongly on the shear rate (Voeltzel et al. Reference Voeltzel, Fillot, Vergne and Joly2018).

Table 1. Literature values of the slip length for different solid material/solvent combinations. The materials and liquids are: diamond-like carbon (DIC); graphene oxide (GO); boron nitride (BN); molybdenum disulfide (MoS$_2$); ionic liquid (IL); n-methyl-2-pyrrolidone (NMP); octamethylcyclotetrasiloxane (OMCTS); cyclopentanone (CPO). Ab initio MD refers to a method allowing for the calculation of electronic behaviour from first principles by using a quantum mechanical method.

The slip length $\lambda$ can also be significant for atomically smooth and chemically homogeneous materials other than graphene. The interaction of BN and water gives $\lambda \sim 3\ \textrm {nm}$ (Tocci et al. Reference Tocci, Joly and Michaelides2014). While this value may seem small, it is still larger than the thickness of single-layer BN. Despite the large scatter in the literature data for $\lambda$, consequence of the dependence of MD simulation results on empirical force fields, table 1 suggests that relatively large slip lengths are not uncommon in 2-D nanomaterials, which is essentially a consequence of the fact than many 2-D nanomaterials have atomically smooth surfaces. To provide theoretical guidelines on a range of realistic values, characteristic of those in table 1, we will evaluate the rotational dynamics of nanoplatelets for $\lambda = 0$, $2$, $20$ and 200 nm.

4.3. Calculation of $k_{e}$ and $D_{r}$

Using a boundary integral method which will be described shortly, we have calculated the effective aspect ratio $k_{e}$ and the rotational diffusion coefficient $D_{r}$ for typical nanoplatelets by solving the incompressible Stokes equations

(4.1a,b)\begin{equation} \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{\sigma}=0,\quad \boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{u}=0, \end{equation}

where $\sigma _{ij}=-\delta _{ij}p + \eta (\partial u_i/\partial x_j +\partial u_j/\partial x_i)$ is the hydrodynamic stress with $p$ the pressure and $\eta$ the viscosity; $\boldsymbol {u}$ is the velocity field. For the reference surface, we use a cuboid with rounded edges (§ 4.1). At the reference surface of the platelet we prescribed the Navier slip boundary condition

(4.2)\begin{equation} \boldsymbol{u}_{{sl}}= \frac{\lambda}{\eta} \boldsymbol{n} \times \boldsymbol{f}\times\boldsymbol{n}, \end{equation}

where $\boldsymbol {f}=\boldsymbol {\sigma }\boldsymbol {\cdot } \boldsymbol {n}$ is the hydrodynamic surface traction and $\boldsymbol {n}$ is the unit normal vector (Lauga, Brenner & Stone Reference Lauga, Brenner and Stone2008). In the present case, the macroscopic time scale is given by the rotational time period $P_{R} = \dot \gamma ^{-1} ( |k_{e}|^{-1} + |k_{e}|)$, which is controlled by the particle aspect ratio and by the shear rate. For realistic parameters, $P_{R}$ is always much larger than the time scale of the microscopic motion of the fluid molecules, which is typically of the order of picoseconds. A boundary integral equation is used to calculate the distribution of $\boldsymbol {f}$ corresponding to each given inclination angle $\phi$ (Pozrikidis Reference Pozrikidis1992). Denoting the reference surface by $S$, the boundary integral equation for a point $\boldsymbol {x}\in S$ is (Luo & Pozrikidis Reference Luo and Pozrikidis2008)

(4.3)\begin{equation} \int_{S} \boldsymbol{n} \boldsymbol{\cdot}\boldsymbol{K}(x',y') \boldsymbol{\cdot} \boldsymbol{u}_{{sl}} \,\mathrm{d} S -\frac{1}{\eta}\int_{S} \boldsymbol{G}(x',y') \boldsymbol{\cdot} \boldsymbol{f}\, \mathrm{d} S = \frac{\boldsymbol{u}_{{sl}}(\boldsymbol{x})}{2}-\boldsymbol{u}_{\infty}(\boldsymbol{x}), \end{equation}

where the integral is over the surface parameterised by the coordinate point $\boldsymbol {x}_1=(x_1,y_1)$. In (4.3), $x'=x_1-x$, $y'=y_1-y$; $\boldsymbol {G}$ and $\boldsymbol {K}$ are Green's functions corresponding to the 2-D ‘Stokeslet’ and ‘Stresslet’, respectively; $\boldsymbol {u}_{\infty }$ is the undisturbed simple shear flow field. The traction is used to calculate the hydrodynamic torque $T(\phi )= \boldsymbol {\hat {e}}_{z}\boldsymbol {\cdot } \int [{\boldsymbol {x}}\times \boldsymbol {f}]\,\mathrm {d}S$ acting on the fixed platelet. In order to achieve high accuracy with the boundary integral algorithm, a non-uniform grid has been implemented with the surface of the particle being divided into two regions, see figure 10(a). Namely, the edge region, made of the edges of arclength $S_{E}$ plus the neighbouring regions of lengths $S_{E}/4$, and the planar region. The total number of grid points, $N$, is distributed between the two regions, with the number of points of the edge region being $N/2$ up to a maximum of $32$ points for each end. Following a convergence analysis (figure 10b,c), we set $N=288$ for platelets with $a/b\leq 100$ and $N=384$ for platelets with $a/b> 100$; the value of $k_{e}$ is then calculated from (2.4).

Figure 10. (a) Example of the discretisation of a $n=1$ surface with $N=96$. (b,c) Effective aspect ratio $|k_e|$ versus the number of discretisation points $N$ for platelets with $\lambda =20\ \textrm {nm}$, $a/b=10$ or $a/b=100$, and either $n=1$ (b) or $n=10$ (c).

The (4.3) is evaluated numerically by using the following method. First, the traction and slip velocity are discretised as $N$ piecewise constant functions $\{\boldsymbol{f}[1]\ldots \boldsymbol {f}[N]\}$ and $\{\boldsymbol {u}^{{sl}}[1]\cdots \boldsymbol {u}^{{sl}}[N]\}$, respectively, each associated with a surface element $S_i=\int _{s_i}^{s_{i+1}}\,\mathrm {d}S$. With this discretisation, for each point $s_j$ for $j =\{1 \cdots N\}$, the discretised form of (4.3) becomes (Kamal et al. Reference Kamal, Gravelle and Botto2020)

(4.4)\begin{align} \sum_{i=1}^{N}\left[\boldsymbol{u}_{{sl}}[i]\boldsymbol{\cdot}\int_{s_i}^{s_{i+1}} \boldsymbol{K}(s,s_j)\boldsymbol{\cdot} \boldsymbol{n}(s)\,\mathrm{d} S(s) - \frac{\boldsymbol{f}[i]}{\eta}\boldsymbol{\cdot} \int_{s_i}^{s_{i+1}} \boldsymbol{G}(s,s_j)\,\mathrm{d} S(s) \right] =\frac{\boldsymbol{u}_{sl}(s_j)}{2} -\boldsymbol{u}_{\infty}(s_j). \end{align}

For $i\neq j$, each subintegral is evaluated using a Gauss–Legendre quadrature method. If $i=j$, the integrands are singular. The singular integrand containing the $\boldsymbol {G}$ tensor is evaluated by using a specific quadrature method for logarithmic singularities (Pozrikidis Reference Pozrikidis2002), and the singular integrand containing the $\boldsymbol {K}$ tensor is evaluated analytically by Taylor expansion about the singular point (Kamal et al. Reference Kamal, Gravelle and Botto2020). Equation (4.2) provides a closed relation between $\boldsymbol {u}_{{sl}}$ and $\boldsymbol {f}$, and thus (4.4) can be arranged into a closed system of $N$ linear equations for each component of the traction $f_{i}[j]$. This system is solved for $f_{i}[j]$ by using Gaussian elimination. To test the accuracy of our implementation, we compared the predicted value of $k_{e}$ with the corresponding exact solution for circular cylinder in shear flow (Kamal et al. Reference Kamal, Gravelle and Botto2020). This test confirms the expected second-order spatial convergence.

The rotational diffusion coefficient of a particle is given by $D_{r}=k_{B} T/ F_r$, where $k_{B}$ is the Boltzmann's constant, $T$ is the absolute temperature and $F_r$ is the rotational drag coefficient. The rotational drag coefficient is also calculated via the boundary integral method, by calculating the hydrodynamic torque exerted on the fluid by a platelet rotating with velocity $\boldsymbol {u}= \boldsymbol {\hat {e}}_{z} \times \boldsymbol {x}$ in a fluid otherwise at rest. In agreement with the result of Sherwood (Reference Sherwood2012) for a disk, we find for our model platelet that $F_r$ is independent of slip as $k \to 0$. In this limit we found $F_r \approx c_1 \eta a^3$, where $c_1 = 6.29 \pm 0.02$ (to be compared with $c_1 = 32/3$ for a 3-D disk of zero thickness (Leal & Hinch Reference Leal and Hinch1971; Sherwood Reference Sherwood2012), and to $c_1 = 2 {\rm \pi}$ for an infinite plate of zero thickness (Sherwood & Meeten Reference Sherwood and Meeten1991)). The second-order accuracy in the computation of $D_{r}$ was assessed by comparing against the exact solution for $D_{r}$ for a cylinder with a Navier slip boundary condition, as done by Luo & Pozrikidis (Reference Luo and Pozrikidis2008) for a sphere.

Values of $k_{e}$ for our model platelet with $\xi =0.25\ \textrm {nm}$, slip length $\lambda = 0$, $2$, $20$ and $200\ \textrm {nm}$, and $n = 1$ or $10$ are shown in figure 11. We limit our calculations to aspect ratios $k \ge 0.001$, because using $k<0.001$ leads to large numerical error owing to severe resolution requirements. For $\lambda =0$, the effective ratio of the platelet follows the power law relationship $k_{e} \sim g_{n} k^{3/4}$, with $g_{n}$ a prefactor that depends on $n$. This power law relationship is the same as for disks with ‘blunt edges’ (Singh, Koch & Stroock Reference Singh, Koch and Stroock2013). We find $g_{1} \approx 0.91$ for $n=1$ and $g_{10} \approx 1.0$ for $n=10$. As $\lambda$ increases, however, the relation between $k_{e}$ and $k$ is not necessary a power law, as shown for $\lambda =2,\;20$ and 200 nm for $n=1$, and $\lambda =20$ and 200 nm for $n=10$. In the limit $\lambda /a \gg 1$, the value of $k_{e}$ can be analytically approximated as $k_{e} \propto \text {i} \sqrt {k}$ (Kamal et al. Reference Kamal, Gravelle and Botto2020), as confirmed in figure 11 for platelet length $2 a \lessapprox 20\ \textrm {nm}$, $n=1$ and $\lambda =20\ \textrm {nm}$ and $200\ \textrm {nm}$ (we remark, however, that most applications of nanoplatelets typically involve $\lambda < a$).

Figure 11. Absolute value of the effective aspect ratio $k_{e}$ versus the platelet length $2 a$ for $\xi =0.25\ \textrm {nm}$ and for $n=1$ (a) and $n=10$ (b). The black lines correspond to the purely imaginary values of $k_{e}$ and the red lines to real values of $k_{e}$.

As the slip length increases, the value of $k_{e}^2$ decreases and changes sign at some critical value $\lambda _c$ (figure 12). An explanation for this is the following. Calculating $k_{e}$ requires the evaluation of the total hydrodynamic torque on a fixed platelet for two angles: $\phi =0$ and $\phi ={\rm \pi} /2$. The total torque for $\phi =0$ is more sensitive to $\lambda$ than the torque for $\phi ={\rm \pi} /2$, because slip affects primarily the flow moving in the direction tangential to the surface of the particle. Thus, predicting $\lambda _c$ essentially requires predicting the angle for which $T(0)$ changes sign. When $\lambda =0$, in an asymptotic expansion in powers of $b$, the $O(b)$ contribution to $T(0)$ turns out to be zero, because of an exact cancellation between torque contributions due to normal and tangential stresses. This cancellation results in a ‘clockwise’ torque that is second-order in the thickness, $T\propto b^2$; the second-order tangential traction determining this torque is $f_x^{(2)} \propto b$ (Singh et al. Reference Singh, Koch, Subramanian and Stroock2014). The effect of slip over the flat surface of the platelet is to reduce the tangential traction $f_x$, without changing significantly the normal traction $f_y$ (figure 13). An analysis for $\lambda /a \ll 1$ shows that the effect of slip is to reduce $f_x$ by an amount of $O(\lambda )$, resulting in a second-order traction $f_x^{(2,{sl})} \propto \lambda$ (Kamal et al. Reference Kamal, Gravelle and Botto2020), where ‘$sl$’ indicates that this traction is due to slip. The total torque $T(0)$ changes sign when $f_x^{(2,{sl})} > f_x^{(2)}$. This condition occurs at a threshold value $\lambda _c \approx h_n b$, where $h_n$ is a numerical prefactor that depends on the number of layers $n$ (Kamal et al. Reference Kamal, Gravelle and Botto2020). For $n=1$, our numerical procedure gives $h_1 \approx 0.8$, and $b \approx 0.25\ \textrm {nm}$ so $\lambda _c \approx 0.2\ \textrm {nm}$. For $n=10$, $h_{10}\approx 2.2$ and $b \approx 1.75\ \textrm {nm}$ so $\lambda _c \approx 3.85\ \textrm {nm}$ (figure 12). For moderate-to-large values of $\lambda /b$, slip can have a marked effect on the traction at and near the edges of the platelet (the position of the edge is marked by the dotted vertical line in figure 13). This traction can significantly affect $k_e$ for larger values of $\lambda$ so that $k_e$ is in general a non-trivial function of $\lambda$. Comparison of the traction and $T(0)$ with MD simulations reveals that the approximation of a uniform surface slip over the surface of the platelet still gives accurate predictions of $k_e$ even when $\lambda /b$ is large (Kamal et al. Reference Kamal, Gravelle and Botto2020). In the following, since $k_e$ is a non-trivial function of $\lambda$, $b$ and $a$ (figure 12), we will use the numerically computed values of $k_{e}$.

Figure 12. Effective aspect ratio $k_{e}^2$ versus the slip length $\lambda$ for different platelet lengths $2a$. The half-thickness of the platelet is $b\approx 0.25\ \textrm {nm}$ and $b\approx 1.75\ \textrm {nm}$ for $n=1$ and $n=10$, respectively. Red dashed lines mark the positions where $k_{e}^2$ changes sign.

Figure 13. The streamwise (a) and normal (b) hydrodynamic traction distribution along the surface of a platelet fixed at an orientation $\phi =0$ for $n=1$ and $a/b=10$. The vertical dotted line marks the position of the edge of the platelet.

4.4. Regimes of rotation

Here we analyse the effect of the Péclet number, comparing the rotational dynamics of nanoplatelets with and without surface slip. We first compare the cases $\lambda =0$ (no-slip) and $\lambda =2\ \textrm {nm}$ (slip) for fixed $a/b=100$ and $n=1$. The corresponding effective aspect ratios are $k_{e}=0.028$ and $k_{e}=0.063$i for $\lambda =0$ and $\lambda =2\ \textrm {nm}$, respectively. In figure 14, we plot the mean $\mu$ and variance $\sigma ^2$ of the rotation angle for these two cases as a function of $Pe$. In agreement with figure 7, the results show that slip has no effect on particle orientation at low $Pe$, but has a strong influence on both $\mu$ and $\sigma ^2$ at large $Pe$. To describe the qualitative differences in the trends of $\mu$ and $\sigma ^2$ between the slip and no-slip cases, we subdivide the data into three different regions (I, II and III, see figure 14). Region III corresponds to ${Pe} > {Pe}_{c}$, where $Pe_{c}$ is the critical value introduced in § 3.3 above which no full rotational cycles occur for $\lambda =2\ \textrm {nm}$. Region III is a ‘large slip effects’ regime. In this regime the platelet with slip performs small fluctuations around $\phi _{c}$. In this region the curves approach smoothly the values expected for ${Pe} \to \infty$ (§ 3.2): for $\lambda =2\ \textrm {nm}$, $\mu$ is approximately constant ($\mu \approx |k_{e}|$), and $\sigma ^2 \to 1/(2 {Pe} |k_{e}|)$ as ${Pe} \to \infty$. This latter asymptotic solution for $\sigma ^2$ can be obtained by inserting into (2.1) the linearisation of the angular velocity for angles close to $\phi _{c}$, $\varOmega =2 k_{e} (\phi -\phi _{c})$ and assuming that $p$ is Gaussian. For $\lambda =0$, $\mu$ vanishes as ${Pe} \to \infty$ and $\sigma ^2$ is finite, as predicted by Jeffery's theory. Regions I and II correspond to ${Pe}<{Pe}_{c}$. In region I, slip and no-slip curves for $\mu$ or $\sigma ^2$ are practically indistinguishable, as apparently in this region the effect of the hydrodynamic stresses is subdominant with respect to Brownian stresses in setting the orientational dynamics. In region II, differences in the curves due to slip exists, but are not as marked as in region III. Region II corresponds to a ‘moderate slip effects’ regime, in which full rotational cycles occur even for platelets with an imaginary $k_{e}$, but with a significantly reduced frequency as compared with region I. This reduction in frequency indicates that in this region hydrodynamics mitigates the randomising effect of Brownian motion.

Figure 14. Comparison of mean and variance of the fluctuating rotation angle for a nanoplatelet with slip ($k_{e}=0.063\text {i}$) and without slip ($k_{e}=0.028$). The red dashed line marks the asymptotic solution for $\sigma ^2$ for purely imaginary values of $k_{e}$. The nanoplatelet has half-width $b=0.25\ \textrm {nm}$ (i.e. $n=1$) and half-length $a=25\ \textrm {nm}$. Region I: ${Pe} < {Pe}_{I}$, negligible slip effects for both $\lambda = 0$ and $\lambda = 2$  nm. Region II: ${Pe}_{I} \leq {Pe}<{Pe}_{c}$, moderate slip effects region characterised by changes in the frequency of rotation due to slip. Region III: ${Pe}\geq {Pe}_{c}$, large slip effects region where slip completely suppresses the rotation of the nanoplatelet.

In regions II and III the effect of slip is to increase the mean and reduce the variance of the orientation distribution. The effect of slip on the variance is particularly evident: for ${Pe} = 10\,000$, a small slip of $\lambda = 2\ \textrm {nm}$ leads to a reduction by approximately one order of magnitude of the variance.

As explained in § 3.2, the value ${Pe}_{I}$ separating the ‘negligible slip effects’ regime and the ‘moderate slip effects’ regime must satisfy the condition ${Pe}_{I}\ll {Pe}_{m}=\min (|k_{e}^{-3}(\lambda =0)|,k_{e}^{-3}(\lambda )|)$. Our numerical simulations (shown in both figures 7 and 14) indicate that the correlation ${Pe}_{I}\simeq 0.1 {Pe}_{m}$ provides a good approximation to the data for practical purposes.

In figure 15, we extend the analyses of figure 14 to assess the extent of the regions of large slip effects, moderate slip effects and negligible slip effects by evaluating ${Pe}_{c}$ and ${Pe}_{I}$ in the ($Pe$, $2a$) space. For our comparison, we use the same platelets and corresponding values of $k_{e}$ used in figure 11. The difference to the analysis presented in § 3 is that now ${Pe}_{c}$ and ${Pe}_{m}$ can be evaluated as a function of the known variables $\lambda$ and $2a$, rather than $k_{e}$.

Figure 15. Effect of slip on the rotational dynamics of a nanoplatelet with $\text {length} = \text {depth} = 2a$ and $\text {width} = 2b$ in the ($Pe$, $2a$) space. Upper black curve marks the threshold Péclet number ${Pe}_{c}$ for which $\sigma ^2=2 k_{e}$; above this threshold the hydrodynamic slip has a dramatic effect on the particle dynamics. Lower black curve marks the threshold Péclet number ${Pe}_{I}$; above this threshold, slip effects become important. The dashed, dashed–dot and dash–double-dot lines correspond to $\dot {\gamma }\eta = 10^7\ \text {Pa}$, $\dot {\gamma }\eta = 2\times 10^4\ \text {Pa}$, $\dot {\gamma }\eta = 10\ \text {Pa}$, respectively.

We add to figure 15 red dash–dot lines corresponding to the $Pe$ number for three shear stress values: $\dot {\gamma }\eta = 10^7\ \textrm {Pa}$, $\dot {\gamma }\eta = 2\times 10^4\ \textrm {Pa}$, $\dot {\gamma }\eta = 10\ \textrm {Pa}$ (the Péclet number scales proportionally to the stress according to ${Pe}\approx c_1 \dot {\gamma }\eta a^3/(k_B T)$, where ‘$c_1 = 6.29 \pm 0.02$’ was calculated from a boundary integral simulation of the hydrodynamic torque on a particle rotating with assigned velocity in a still fluid, see § 4.3). These characteristic shear stresses are, respectively, typical of: exfoliation processes in a low viscosity fluid (Paton et al. Reference Paton2014); mixing in very viscous fluids (Huang & Terentjev Reference Huang and Terentjev2012) and dispersion by microfluidisation (Karagiannidis et al. Reference Karagiannidis2017; Paton et al. Reference Paton, Anderson, Pollard and Sainsbury2017); fast lubrication processes (Jonsson & Bhushan Reference Jonsson and Bhushan1995), high-speed blade coating (Willenbacher, Hanciogullari & Wagner Reference Willenbacher, Hanciogullari and Wagner1997) and MD simulations (Voeltzel et al. Reference Voeltzel, Fillot, Vergne and Joly2018; Gravelle et al. Reference Gravelle, Kamal and Botto2020; Gravelle, Kamal & Botto Reference Gravelle, Kamal and Botto2021). The effects of slip for each of these characteristic shear stresses can be assessed by identifying, for a given value of $a$, the region where the corresponding $Pe$ lies. For example, the lowest shear stress (red dashed line in figure 15) is within the region where slip effects are negligible for all the values of $\lambda$, $a$ and $n$ we considered. For the intermediate shear stress (dotted–dashed line in figure 15) slip has a noticeable impact on the rotational dynamics: thicker platelets ($n=10$) are likely to align if the slip length is sufficiently large so that $k_{e}$ is purely imaginary. Otherwise for the thinner platelets, such as for $n=1$, then there is a range of $a$ and $\lambda$ for which the average frequency for the platelets to complete a full rotational cycle is smaller than for a no-slip platelet. Finally, the highest shear stress (double-dotted–dashed line in figure 15) produces a value of $Pe$ well within the ‘large slip effects’ region where the platelet fluctuates around $\phi _{c}$ if $\lambda > \lambda _c$.

The length of the platelet also affects the rotational dynamics. The local power law exponent characterising the dependence of ${Pe}_{c}$ and ${Pe}_{I}$ on $a$ is smaller than $3$ for comparatively small values of $a$ ($a \ll 100\ \textrm {nm}$ approximately in figure 15), and approaches $3$ as $a$ increases. A consequence of the fact that the ‘moderate slip effects region’ is not bounded by parallel lines in a log–log plot is that for given particle thickness and shear stress value longer platelets are more likely to be affected by slip. Taking the case $n=1$, $\lambda =200\ \textrm {nm}$, and $\eta \dot {\gamma } = 2\times 10^4\ \textrm {Pa}$ as an example, the rotation of the platelet is not affected by slip for $2a<40\ \textrm {nm}$, whereas platelets with $2a>40\ \textrm {nm}$ enter the ‘moderate slip effects region’ where slip effects start becoming significant.

4.5. Effective viscosity of a dilute suspension of nanoplatelets with slip

As an illustration of a macroscopic effect due to slip, in the current section we analyse the effects of stable alignment on the orientational contribution to the effective viscosity $\eta _{{eff}}$ of a dilute suspension of nanoplatelets for finite values of $\lambda$. The effective shear viscosity of a dilute suspension of particles can be calculated as

(4.5)\begin{equation} \eta_{{eff}}=\eta+\frac{\sigma'_{xy}}{\dot{\gamma}}, \end{equation}

where $\sigma '_{xy}$ is the particle stress (Jeffery Reference Jeffery1922; Giesekus Reference Giesekus1962; Brenner Reference Brenner1974; Leal & Hinch Reference Leal and Hinch1971). For a suspension in which the motion of the particles is restricted to the flow plane, to first order in the particle concentration, the particle stress can be written as

(4.6)\begin{equation} \sigma'_{xy}=c\eta \dot{\gamma}\left(\frac{A}{2}\left\langle1-\cos{4\phi}\right\rangle+B +\frac{C}{2 {Pe}} \left\langle \sin{2\phi} \right\rangle \right), \end{equation}

where $A,\;B$ and $C$ are dimensionless coefficients (Rallison Reference Rallison1978) and $c$ is the solid fraction. The first two terms on the right-hand side of (4.6) are due to the hydrodynamic stress acting on each platelet. The third term is the contribution of Brownian fluctuations to the particle stress.

As seen in § 3.2, slip has a strong influence on the time-averaged orientation of the particle, with consequences on the statistical quantities $\left \langle 1- \cos {4\phi } \right \rangle$ and $\left \langle \sin {2\phi } \right \rangle$ in (4.6). A map quantifying how $\left \langle 1- \cos {4\phi } \right \rangle$ and $\left \langle \sin {2\phi } \right \rangle$ vary in the ($Pe$, $k_{e}$) space is shown in figure 16. For $Pe$ $\ll |k_{e}|^{-3}$, the quantities $\left \langle 1- \cos {4\phi } \right \rangle$ and $\left \langle \sin {2\phi } \right \rangle$ are practically independent of whether $k_{e}$ is real or purely imaginary. The term $\left \langle 1- \cos {4\phi } \right \rangle$ becomes significantly larger than $\left \langle \sin {2\phi } \right \rangle$ for small $Pe$. Thus, the effective viscosity depends increasingly on $\left \langle 1- \cos {4\phi } \right \rangle$ as $Pe$ decreases, in agreement with the theoretical results of Hinch & Leal (Reference Hinch and Leal1972) and Leahy et al. (Reference Leahy, Koch and Cohen2015) (for elongated particles with $\lambda =0$) and the experimental results of Del Giudice & Shen (Reference Del Giudice and Shen2017) for graphene oxide.

Figure 16. Colour maps of $\left \langle 1-\cos {(4\phi )}\right \rangle$ and $\left \langle \sin {(2\phi )}\right \rangle$ in the ($Pe$, $k_{e}$) space for (a,c) an imaginary value of $k_{e}$ and (b,d) a real value of $k_{e}$. The black line marks the equation ${Pe}=|k_{e}|^{-3}$.

For $Pe \gg |k_{e}|^{-3}$, hydrodynamic stresses become dominant. Figure 16 indeed reveals a strong dependency of $\left \langle 1- \cos {4\phi } \right \rangle$ and $\left \langle \sin {2\phi } \right \rangle$ on $k_{e}$ in this region. Since the prefactor of $\left \langle \sin {2\phi } \right \rangle$ tends to zero for large $Pe$, here we focus on analysing the term $\left \langle 1- \cos {4\phi } \right \rangle$. The leading-order values of $\left \langle 1- \cos {4\phi } \right \rangle$ in this region can be estimated directly by using the probability distributions calculated for $Pe$ $\to \infty$ in § 3.2:

(4.7) \begin{equation} \left\langle1-\cos{4\phi}\right\rangle= \left\{ \begin{array}{@{}ll} 4 k_e/(k_e +1)^2 & \text{if }k_{e} \in \mathbb{R}, \\ 1-\cos{4\phi_{c}} & \text{if }k_{e} \in i\mathbb{R}. \end{array} \right. \end{equation}

For $|k_{e}| \to 0$, $\left \langle 1-\cos {4\phi }\right \rangle$ tends to $4 k_e$ and $8 |k_{e}|^2$ for real or purely imaginary $k_{e}$, respectively. For $\lambda /a = 0$ and $\lambda /a \to \infty$, $\left \langle 1-\cos {4\phi }\right \rangle$ can be estimated in terms of the geometric aspect ratio $k$. For $\lambda =0$, $k_{e} \propto k$, and so $\left \langle 1-\cos {4\phi }\right \rangle \propto k$. For $\lambda /a \to \infty$ we have $k_{e} \propto \text {i} \sqrt {k}$, giving again $\left \langle 1-\cos {4\phi }\right \rangle \propto k$. Thus, quite interestingly, the limits $\lambda /a \to 0$ and $\lambda /a \to \infty$ yield a similar scaling relationship of $\left \langle 1-\cos {4\phi }\right \rangle$ with $k$.

For $\lambda =\lambda _c$, we have $k_{e}=0$ and therefore $\left \langle 1-\cos {4\phi }\right \rangle =0$. When $\left \langle 1-\cos {4\phi }\right \rangle$ vanishes, the particle stress reduces to $\sigma '_{xy}=c\eta \dot {\gamma }B$. If the variation of $B$ on $\lambda$ is sufficiently small in the neighbourhood of $\lambda _c$ (compared with $A\left \langle 1-\cos {4\phi }\right \rangle$), then the value $\lambda =\lambda _c$ will correspond to a local minimum in the $\sigma '_{xy}$ versus $\lambda$ curve, because $A\left \langle 1-\cos {4\phi }\right \rangle \ge 0$ for all values of $\lambda$. Thus, the decrease in the value of $\sigma '_{xy}$ with $\lambda$ may not be always monotonic for large $Pe$.