Catherine Kamal

Simon Gravelle

Lorenzo Botto
email: l.botto@tudelft.nl

EFFECT OF HYDRODYNAMIC SLIP ON THE ROTATIONAL DYNAMICS OF A THIN BROWNIAN PLATELET IN SHEAR FLOW A PREPRINT

The classical theory by Jeffery predicts that, in the absence of Brownian fluctuations, a thin rigid platelet rotates continuously in a shear flow, performing periodic orbits. However, a stable orientation is possible if the surface of the platelet displays a hydrodynamic slip length λ comparable to or larger than the thickness of the platelet. In this article, by solving the Fokker-Plank equation for the orientation distribution function and corroborating the analysis with boundary integral simulations, we quantify a threshold Péclet number, Pe c , above which such alignment occurs. We found that for Pe smaller than Pe c , but larger than a second threshold, a regime emerges where Brownian fluctuations are strong enough to break the platelet's alignment and induce rotations, but with a period of rotation that depends on the value of λ. For Pe below this second threshold, slip has a negligible effect on the orientational dynamics. We use these thresholds to classify the dynamics of graphene-like nanoplatelets for realistic values of λ and apply our results to the quantification of the orientational contribution to the effective viscosity of a dilute suspension of nanoplatelets with slip. We find a non-monotonic variation of this term, with a minimum occurring when the slip length is comparable to the thickness of the particle.

Introduction

The flow behaviour of thin plate-like particles is of interest in many industrial and environmental applications, ranging from the processing of composite materials [START_REF] Kumar | A review of the mechanical and thermal properties of graphene and its hybrid polymer nanocomposites for structural applications[END_REF] to the transport of clay in natural waters [START_REF] Tawari | Electrical double-layer effects on the Brownian diffusivity and aggregation rate of laponite clay particles[END_REF]. Recently, the emergence of graphene and other 2D nanomaterials and their use in a variety of liquid-based processes [START_REF] White | Effects of aspect ratio and concentration on rheology of epoxy suspensions containing model plate-like nanoparticles[END_REF][START_REF] Del Giudice | Shear rheology of graphene oxide dispersions[END_REF][START_REF] Andrew R Koltonow | Graphene oxide sheets in solvents: to crumple or not to crumple?[END_REF][START_REF] Karagiannidis | Microfluidization of graphite and formulation of graphene-based conductive inks[END_REF] has spurred renewed interest in the dynamics of these extremely thin plate-like colloids when suspended in sheared liquids [START_REF] Xu | Brownian dynamics simulations of nanosheet solutions under shear[END_REF][START_REF] Poulin | Superflexibility of graphene oxide[END_REF][START_REF] Reddy | Rheo-optical analysis of functionalized graphene suspensions[END_REF]. In applications such as graphene inks or polymer nanocomposites, colloidal 2D nanomaterials take the form of platelets of nanometric thickness and lateral size spanning from a few nanometers to a few microns [START_REF] Wick | Classification framework for graphene-based materials[END_REF]. There is a great interest in understanding how the application of an external flow affects the orientation of these highly anisotropic particles in the presence of Brownian fluctuations. The surface of graphene and other 2D nanomaterials can be characterised by a hydrodynamic slip lengths λ of several nanometers [START_REF] Maali | Measurement of the slip length of water flow on graphite surface[END_REF][START_REF] Ortiz-Young | The interplay between apparent viscosity and wettability in nanoconfined water[END_REF][START_REF] Tocci | Friction of water on graphene and hexagonal boron nitride from ab initio methods: very different slippage despite very similar interface structures[END_REF], where λ is the distance within the solid at which the relative solid-fluid velocity extrapolates to zero [START_REF] Bocquet | Flow boundary conditions from nano-to micro-scales[END_REF]; the mathematical definition of λ is given in section 4.3. For comparison, the typical thickness of 2D material is < 1 nm. The aim of the current study is to understand the interplay between hydrodynamic slip and Brownian fluctuations in determining the rotational dynamics of a thin plate in an unbounded, simple shear flow under creeping flow conditions.

According to [START_REF] Jeffery | The motion of ellipsoidal particles immersed in a viscous fluid[END_REF] classical theory, derived for the high-Péclet number limit in which Brownian fluctuations are negligible, an oblate ellipsoid with its normal in the flow plane rotates continuously, performing a characteristic tumbling dynamics (figure 1a). [START_REF] Bretherton | The motion of rigid particles in a shear flow at low Reynolds number[END_REF] generalised the high-Péclet, low-Reynolds numbers theory of Jeffery to axi-symmetric particles of general shape. Bretherton's analysis showed that for axi-symmetric shapes the rotational dynamics is governed by a scalar parameter that depends, for a no-slip particle, only on the shape of the particle. For instance, for an oblate ellipsoidal particle of major semi-axis a and minor semi-axis b, the scalar parameter identified by Bretherton corresponds simply to the geometric aspect ratio b/a. However, when hydrodynamic slip is present, Bretherton's scalar parameter (which we shall name k e in the current article) is not only a function of b/a but must also depend on the slip length λ [START_REF] Zhang | Anisotropic particle in viscous shear flow: Navier slip, reciprocal symmetry, and Jeffery orbit[END_REF], Kamal et al., 2020]. This brings about the question of how slip affects Jeffery's predictions, and how Brownian fluctuations alter the rotational dynamics when the slip length is large in comparison with the particle thickness. This question is particularly relevant considering the nanometric thickness of 2D nanomaterial particles.

Hydrodynamic slip at solid boundaries is known to affect the motion of small rigid particles in fluids. Slip reduces the tangential hydrodynamic stress on the particle surface [START_REF] Sellier | Arbitrary Stokes flow about a fixed or freely-suspended slip particle[END_REF], resulting in a reduction in the translational and rotational drag coefficient of spherical or anisotropic particles [START_REF] Youngren | Rotational friction coefficients for ellipsoids and chemical molecules with the slip boundary condition[END_REF][START_REF] Loyalka | Rotation of non-spherical axi-symmetric particles in the slip regime[END_REF][START_REF] Allison | Low Reynolds number transport properties of axisymmetric particles employing stick and slip boundary conditions[END_REF][START_REF] Keh | Slow motion of axisymmetric slip particles along their axes of revolution[END_REF][START_REF] Keh | Slow motion of a slip spheroid along its axis of revolution[END_REF][START_REF] Sellier | Stokes flow about a slip arbitrary-shaped particle[END_REF][START_REF] Chang | Creeping-flow rotation of a slip spheroid about its axis of revolution[END_REF][START_REF] Sherwood | Resistance coefficients for Stokes flow around a disk with a Navier slip condition[END_REF]. Slip also results in a slowdown of the rotational dynamics of spherical particles freely suspended in a simple shear flow [START_REF] Luo | Effect of surface slip on Stokes flow past a spherical particle in infinite fluid and near a plane wall[END_REF]. A similar slip-dependent slowdown in a shear flow was predicted by [START_REF] Zhang | Anisotropic particle in viscous shear flow: Navier slip, reciprocal symmetry, and Jeffery orbit[END_REF] for a particle with an elliptic cross-section (b/a ∼ 0.5) and infinite extent in the vorticity direction, in the limit of a slip length smaller than the particle thickness (λ ∼ b/5).

Recently, we studied the motion of a thin (b/a 1) rigid graphene-like nanoparticle with hydrodynamic slip [START_REF] Kamal | Hydrodynamic slip can align thin nanoplatelets in shear flow[END_REF], via a combination of molecular dynamics and continuum simulations. In contrast to previous studies [START_REF] Zhang | Anisotropic particle in viscous shear flow: Navier slip, reciprocal symmetry, and Jeffery orbit[END_REF], we considered values of the slip length either larger or smaller than the particle thickness. By using numerical and asymptotic solutions of the low-Reynolds-number boundary integral equations governing the hydrodynamic surface stress distribution, we showed that provided that the half-thickness of the platelet b is smaller than λ, in the limit of infinite Péclet (Pe) numbers the platelet aligns at a small angle φ c with respect to the flow direction, instead of performing full rotational cycles as predicted by the classical Jeffery's theory for Pe → ∞ (figure 1b). Our asymptotic analysis revealed that this behaviour is due to a perturbed balance between two hydrodynamic torque contributions. One contribution is due to tangential hydrodynamic stresses acting on a lever arm proportional to b. The second contribution is due to normal hydrodynamic stresses acting on a lever arm proportional to a. For λ/b 1 the torque due to tangential stresses is larger than that due to normal stresses, resulting in an expected rotation in the same direction of the vorticity vector [START_REF] Singh | Rotational motion of a thin axisymmetric disk in a low Reynolds number linear flow[END_REF]. However, we demonstrated that to leading order in b/a slip induces a reduction in the tangential stresses, without affecting the normal stresses significantly. As a consequence, for λ/b 1 the torque due to normal stresses dominates, resulting in a rotation in the direction opposite to the vorticity vector (for a particle initially aligned in the flow direction, as illustrated in figure 1b) and the attainment of an equilibrium angle φ c , in correspondence to which the two torque contributions balance exactly (an illustration of such arrested dynamics is given in section 4). This surprising result was confirmed with molecular dynamics simulations. While the theory we developed was for infinite values of Pe for which the effect of Brownian fluctuations is negligible, we observed a stable orientation in molecular dynamics simulations of relatively short graphene nanoplatelets for Pe of about 100 [START_REF] Kamal | Hydrodynamic slip can align thin nanoplatelets in shear flow[END_REF]. This raises the question of what is the minimum Pe for which a stable orientation can be observed.

In this paper we first recall the essential elements of the high-Pe number theory we developed previously. Then we analyse the rotational dynamics for finite Pe numbers, by considering solutions of the Fokker-Plank equation governing the single-particle orientational distribution function. To simplify the problem, following [START_REF] Leahy | The effect of shear flow on the rotational diffusion of a single axisymmetric particle[END_REF] we assume that the motion of the particle is in the flow plane, hence the orientational distribution is a function only of the angle with respect to the flow direction and time. Earliest work on the use of the Fokker-Plank equation for predicting the orientational distribution of elongated particles in a shear flow is due to [START_REF] Burgers | On the motion of small particles of elongated form suspended in a viscous liquid[END_REF] and [START_REF] Peterlin | Über die Viskosität von verdÜnnten Lösungen und Suspensionen in Abhängigkeit von der Teilchenform[END_REF], who examined the effects of Brownian fluctuations on the dynamics of ellipsoids and rods for Pe 1. Other authors have examined the range Pe 1 [START_REF] Leal | The effect of weak Brownian rotations on particles in shear flow[END_REF][START_REF] Hinch | Time-dependent shear flows of a suspension of particles with weak Brownian rotations[END_REF][START_REF] Leahy | The effect of shear flow on the rotational diffusion of a single axisymmetric particle[END_REF] and the range of intermediate Pe [START_REF] Sadron | Flow properties of disperse systems[END_REF][START_REF] Scheraga | Non-Newtonian viscosity of solutions of ellipsoidal particles[END_REF][START_REF] Hinch | The effect of Brownian motion on the rheological properties of a suspension of non-spherical particles[END_REF][START_REF] Férec | Numerical solution of the Fokker-Planck equation for fiber suspensions: application to the Folgar-Tucker-Lipscomb model[END_REF], considering ellipsoids (both oblate and prolate) and more general rod-like particles. An important result of these theoretical and numerical analyses is that for elongated particles almost aligned with the flow, Brownian fluctuations increase the average particle's rate of rotation with respect to the high-Pe value, essentially by allowing the particle to escape from the region of small streamwise velocity. Theories for the rotational dynamics of elongated particles based on the seminal work of Jeffery have been confirmed experimentally for a range of Pe numbers by standard rheological measurements [START_REF] Ivanov | Damped oscillations in the viscosity of suspensions of rigid rods. I. Monomodal suspensions[END_REF][START_REF] Mueller | The rheology of suspensions of solid particles[END_REF], rheo-optics [START_REF] Frattini | Rheo-optical studies of the effect of weak Brownian rotations in sheared suspensions[END_REF][START_REF] Fuller | Optical rheometry of complex fluids[END_REF], Vermant et al., 2001, Reddy et al., 2011], and by direct observation of the dynamics of single particles [Anczurowski and Mason, 1967a[START_REF] Stover | Observations of fibre orientation in simple shear flow of semi-dilute suspensions[END_REF][START_REF] Herzhaft | Experimental study of the sedimentation of dilute and semi-dilute suspensions of fibres[END_REF][START_REF] Leahy | Enhancing rotational diffusion using oscillatory shear[END_REF]. We are not aware of experimental work on the rotational dynamics of plate-like particles specifically focusing on the effect of slip.

The structure of the paper is as follows. In section 2 we analyse Bretheton's equation of motion. We show that this equation can be used to describe the rotational dynamics of plate-like particles with slip, provided that when λ/b is larger than a threshold value the effective aspect ratio of the particle is taken to be a complex number. In section 3, we consider numerical solutions of the Fokker-Plank equations for the orientational distribution function. The objective of this section is to illustrate the qualitative features of the particle dynamics and quantify the range of Pe above which the effect of slip-dependent hydrodynamics becomes important. Such analysis enabled us to estimate the threshold Pe giving a stable orientation. In section 4, we apply our theory to the dynamics of nanoplatelets whose geometry mimics that of single and multilayer graphene, for practically relevant values of λ. In this section, we quantify the threshold shear stress needed to prevent the platelet from completing full rotations. Finally, we analyse how the indefinite alignment of the particles affects the orientational contribution to the effective viscosity of a diluted suspension of platelets with slip.

Formulation of the problem

We consider the rotational dynamics of a rigid plate-like particle (henceforth referred to as "platelet") in a simple shear flow. The platelet has length 2a and thickness 2b. We work under conditions of Stokes flow and neglect particle inertia and gravity (i.e., the particle is force and torque free). In our theoretical derivations, we assume that the geometric aspect ratio k = b/a is much smaller than one. A consequence of this is that the rotational dynamics of a tri-dimensional platelet can be well approximated by the rotational dynamics of a bidimensional platelet of infinite extent in the vorticity direction, provided that the motion of the particle occurs in the plane of the flow and the depth of the platelet's extension in the vorticity direction is not small in comparison to its length [START_REF] Kamal | Hydrodynamic slip can align thin nanoplatelets in shear flow[END_REF]. We assume that the particle is symmetric about two planes (lines in 2D) passing through the particle centre. The particle motion is induced by an undisturbed shear flow field u ∞ = γyê x , where êx is the unit normal in the flow direction and γ is the shear rate.

The probability p of finding the platelet at a certain angle with respect to the flow, in response to both hydrodynamic and Brownian stresses, is governed by a Fokker-Planck (Smoluchowski) equation [START_REF] Gardiner | Handbook of stochastic methods: for physics, chemistry & the natural sciences[END_REF]. With just one degree of freedom, the Fokker-Planck equation for the orientational probability distribution function simplifies to a second order P.D.E.:

ṗ = [D r p φ -γΩp] φ , ( 1
)
where φ is the anti-clockwise orientation angle with respect to êx , t is time, ṗ = ∂p/∂t, [ ] φ = ∂[ ]/∂φ, Ω(φ) is the hydrodynamic angular velocity nondimensionalised by γ-1 , and D r is the rotational diffusion coefficient. The ratio of convective and diffusive fluxes in equation ( 1) is the Péclet number Pe = γ/D r .

-

Hydrodynamic angular velocity

The hydrodynamic angular velocity γΩ is required to close equation (1). In the dilute limit, Ω can be evaluated by examining the motion of an isolated freely-suspended platelet in the absence of Brownian fluctuations. For a symmetric platelet, Ω can be calculated exactly by using [START_REF] Bretherton | The motion of rigid particles in a shear flow at low Reynolds number[END_REF] equation of motion [START_REF] Kim | Microhydrodynamics: principles and selected applications[END_REF]. This equation describes the relationship between the time derivatives of the particle orientation vector d, and the rate of strain and vorticity tensors associated to the undisturbed flow field,

Ω ∞ = (∇u ∞ + ∇u T ∞ ) and E ∞ = (∇u ∞ -∇u T ∞ ), respectively: ḋ = γΩ × d = Ω ∞ × d + k 2 e -1 k 2 e + 1 (E ∞ • d -E ∞ : dd d) . (2)
Using spherical polar coordinates, the orientation vector can be expressed as d = sin θ cos φê x +sin θ sin φê y +cos θê z , resulting in the following coupled ODEs [START_REF] Kim | Microhydrodynamics: principles and selected applications[END_REF]:

θ = 1 -k 2 e 1 + k 2 e γ 4 sin 2θ sin 2φ, φ ≡ γΩ = - γ k 2 e + 1
k 2 e cos 2 φ + sin 2 φ .

(3) Note that φ does not depend on θ, hence an analysis of the particular 2D case considered here corresponding to θ = π/2 is representative of the φ-dependence for all values of θ. 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 andPozrikidis, 2008, Kamal et al., 2020]. The parameter k e can be calculated for θ = π/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 (π/2), according to the following expression [START_REF] Cox | The motion of long slender bodies in a viscous fluid. Part 2. Shear flow[END_REF]:

k e = T (0) T (π/2) . ( 4
)
For a platelet satisfying the no-slip boundary condition, T (0)/T (π/2) is positive and therefore k e is a real number [START_REF] Willis | A kinematic model of preferred orientation[END_REF]. In this case, when the geometric aspect ratio k = b/a 1, the parameter k e follows a power-law relationship k e ∝ 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 [START_REF] Jeffery | The motion of ellipsoidal particles immersed in a viscous fluid[END_REF], and a disk with blunt edges has k e ∝ k 3/4 [START_REF] Singh | Rotational motion of a thin axisymmetric disk in a low Reynolds number linear flow[END_REF]. For real k e the solution to equation ( 3) is:

tan θ = Ck e k 2 e cos φ 2 + sin φ 2 , tan φ = -k e tan γt k e + k -1 e , ( 5
)
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

P = 2π γ-1 (k e + k -1 e ). (6)
The value of C depends on θ(0). For θ(0) = π/2, C → ∞ and θ(t) = π/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 θ = π/2 plane are of course possible. The case θ = π/2 for finite Pe is tractable analytically and has been shown, in the no-slip case, to be relevant to the full three-dimensional dynamics [START_REF] Leahy | The effect of shear flow on the rotational diffusion of a single axisymmetric particle[END_REF]). In section 3 we shall consider the general slip case at finite Pe.

An example of an orbit for θ = π/2 and real k e and the corresponding Ω(φ) are given in figure 2 (a) & (b), respectively.

Since k e 1, the time scale for the particle to flip is much smaller than the time spent by the particle near φ = 0 [START_REF] Singh | Rotational motion of a thin axisymmetric disk in a low Reynolds number linear flow[END_REF]. For a no-slip platelet of vanishing thickness, k e → 0 as k → 0 and thus the particle will remain in a fixed position in the ad hoc case k = 0 [START_REF] Bretherton | The motion of rigid particles in a shear flow at low Reynolds number[END_REF]. Otherwise the platelet will rotate (unless the platelet has a specially designed, slightly non-axisymmetric shape, see [START_REF] Borker | Controlling rotation and migration of rings in a simple shear flow through geometric modifications[END_REF]).

For platelets presenting a surface with slip, however, such alignment is possible [START_REF] Kamal | Hydrodynamic slip can align thin nanoplatelets in shear flow[END_REF]. The effect of surface slip is to reduce the value of k 2 e , such that the Ω(φ) curve shifts upwards, reducing the frequency of rotation with respect to the no-slip case [START_REF] Luo | Effect of surface slip on Stokes flow past a spherical particle in infinite fluid and near a plane wall[END_REF][START_REF] Zhang | Anisotropic particle in viscous shear flow: Navier slip, reciprocal symmetry, and Jeffery orbit[END_REF], Kamal et al., 2020]. At or above a critical slip length that scales with the thickness of the platelet, k 2 e becomes negative, and so Ω(φ) will be shifted onto or above the zero line (figure 2 (d)). Such a solution corresponds to a negative value of T (0)/T (π/2) and a purely imaginary complex value of k e [START_REF] Kamal | Hydrodynamic slip can align thin nanoplatelets in shear flow[END_REF]. It is instructive to compare the structure of the φ -θ equations (describing full 3D trajectories) for pure and imaginary values of k e . For a purely imaginary k e , the solution of equation ( 3) is:

tan θ = C|k e | |k e | 2 cos φ 2 + sin φ 2 , tan φ = |k e | tanh γt |k e | -1 -|k e | . (7)
Comparing to equation ( 5), whilst the equation for θ remains unchanged with respect to the case of a real k e , the equation for φ now admits a stable non-periodic solution for t → ∞. The particle relaxes to the stable orientation angle

P R = γ-1 (|k e | -1 -|k e |), (8)
to be compared with the rotational time-period of Eq. ( 6). For |k e | 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. [START_REF] Stuart | Motion of a tank-treading ellipsoidal particle in a shear flow[END_REF]). 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 φ c with the flow. Real k e (k e = 0.11) : 3 Rotational dynamics at finite Péclet numbers

At finite Pe, both rotational diffusion coefficient D r and effective aspect ratio k e change due to slip. However, in the limit k → 0, D r becomes approximately independent of λ [START_REF] Sherwood | Resistance coefficients for Stokes flow around a disk with a Navier slip condition[END_REF]. Therefore, for thin platelets, the rotational dynamics at finite Pe changes primarily due to slip through the dependence of k e on the hydrodynamic angular velocity.

Random walk trajectories

To illustrate the effects of Brownian fluctuations on the platelet's rotational trajectories, we solve numerically the Langevin representation of equation ( 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 [START_REF] Doi | The theory of polymer dynamics[END_REF]]

∆φ = γΩ(φ)∆t + X. (9)
Here, we take X as a Gaussian random variable with zero mean and variance X 2 = 2D r ∆t. Solving numerically this equation for several sequences of random numbers provides a set of trajectories distributed according to equation 1.

Examples of trajectories calculated from equation ( 9) 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 φ ∈ [-π/2, π/2], can always be observed if k e is real. When k e is real, the average time period (in units of the dimensionless time-scale Pe t) for the platelet to perform full rotational cycle decreases as Pe decreases (figure 3 (a)). This behaviour has been predicted in previous theoretical studies on the steady-state rotatory dynamics of plate-like particles [Leal and[START_REF] Leal | The effect of weak Brownian rotations on particles in shear flow[END_REF][START_REF] Hinch | The effect of Brownian motion on the rheological properties of a suspension of non-spherical particles[END_REF], and is in agreement with experiments where the instantaneous rotations of elongated particles have been visualised directly [START_REF] Stover | Observations of fibre orientation in simple shear flow of semi-dilute suspensions[END_REF][START_REF] Herzhaft | Experimental study of the sedimentation of dilute and semi-dilute suspensions of fibres[END_REF][START_REF] Leahy | Enhancing rotational diffusion using oscillatory shear[END_REF], or through rheo-optical or neutron diffraction techniques [START_REF] Frattini | Rheo-optical studies of the effect of weak Brownian rotations in sheared suspensions[END_REF][START_REF] Jogun | Rheology and microstructure of dense suspensions of plate-shaped colloidal particles[END_REF][START_REF] Brown | Orientational order in concentrated dispersions of plate-like kaolinite particles under shear[END_REF]. In contrast, if k e is purely imaginary, for large Pe (Pe = 640), the platelet fluctuates around a stable orientation angle (figure 3 (b)). This angle is approximately the same as the equilibrium angle for Pe → ∞ (φ 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 (b) as compared to figure 3 (a) for Pe = 64. For Pe ∼ 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 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 equation (1).

Time evolution of the orientation distribution function's moments

Taking advantage of p(φ, t), any solution of equation ( 1) can be expressed as a spectral series,

p(φ, t) = a 0 (t) + ∞ n=1 [a n (t) cos (2nφ) + b n (t) sin (2nφ)] . (10)
Normalisation of p requires a 0 = 1/π, and assuming φ to be uniformly distributed at time t = 0 gives an initial uniform distribution p(φ, 0) = 1/π. In spectral coordinates, this is equivalent to having only one non-zero coefficient a 0 = 1/π. Numerically, we solve equation 1 by truncating the series at a large value N , yielding 2N first order differential equations for the spectral coefficients [START_REF] Leahy | The effect of shear flow on the rotational diffusion of a single axisymmetric particle[END_REF]. 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 → 0 for all value of time t (see figure 4 for t → ∞).

The 2N-coupled equations are solved in time by a Runge Kutta method [START_REF] Press | Numerical recipes 3rd edition: The art of scientific computing[END_REF]. 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:

µ = φ = ∞ i=1 π 2n (-1) 1+n b n , σ 2 = (φ -µ) 2 = π 2 12 + ∞ i=1 π 2n 2 (-1) n a n -µ 2 , (11)
respectively. Here, the close brackets represent an average over the orientation distribution function p(φ, t) for fixed t. Many studies discuss solutions to equation 1 for elongated particles with real k e [START_REF] Burgers | On the motion of small particles of elongated form suspended in a viscous liquid[END_REF][START_REF] Peterlin | Über die Viskosität von verdÜnnten Lösungen und Suspensionen in Abhängigkeit von der Teilchenform[END_REF][START_REF] Sadron | Flow properties of disperse systems[END_REF][START_REF] Scheraga | Non-Newtonian viscosity of solutions of ellipsoidal particles[END_REF][START_REF] Leal | The effect of weak Brownian rotations on particles in shear flow[END_REF][START_REF] Hinch | The effect of Brownian motion on the rheological properties of a suspension of non-spherical particles[END_REF][START_REF] Hinch | Time-dependent shear flows of a suspension of particles with weak Brownian rotations[END_REF][START_REF] Férec | Numerical solution of the Fokker-Planck equation for fiber suspensions: application to the Folgar-Tucker-Lipscomb model[END_REF][START_REF] Leahy | The effect of shear flow on the rotational diffusion of a single axisymmetric particle[END_REF], 2017]. 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 (φ -µ) 3 of the orientation distribution for k e = 0.1i and different Pe is shown in figure 5. Similarly to when k e is real [START_REF] Leahy | The effect of shear flow on the rotational diffusion of a single axisymmetric particle[END_REF], 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 = γ-1 (|k e | -1 -|k e |
) of a platelet in absence of Brownian motion.

Steady state probability distribution

We solve equation ( 1) with ṗ = 0 to find the steady state orientation distribution function p(φ) using the method described above. The effect of Pe on p(φ) 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(φ) has a much sharper peak if k e is purely imaginary. This differences can be analysed by considering the limit Pe → ∞. For Pe → ∞, equation (1) reduces at steady state to

pΩ = c, (12)
for some integration constant c. For real k e , normalisation of p gives p = (P γ |Ω|) -1 , where P is the time period (Eq. ( 6)) [Anczurowski and Mason, 1967b]. Such a probability distribution shown by the (red) dotted line in figure 6 has mean orientation angle µ = 0 and finite variance σ 2 by the symmetry of p about φ = 0. If k e is purely imaginary, however, 1/Ω diverges as φ → ±φ c , and thus the integrand of c/Ω will not be finite unless k e = 0 or c = 0, as required by the normalisation constraint for p. Thus, the equation pΩ = c is only satisfied for c = 0 and p(φ) = δ(φ -φ 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 φ c (as observed in figure 3 (b)).

For Pe = 10 the orientation distribution functions for k e = 0.1 and k e = 0.1i are almost identical, and slightly skewed towards positive values of φ (the value of µ, σ 2 and (φ -µ) 3 for k e = 0.1 and k e = 0.1i, 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 equation 1 yields a characteristic angle Φ. From this, we can define a 'local' Pe number, Pe loc = γΩ(φ)Φ/D r [START_REF] Leal | The effect of weak Brownian rotations on particles in shear flow[END_REF]]. The effects of Brownian fluctuations on the orientation of the platelets are considered to be 'weak' compared to the hydrodynamic angular velocity if, for φ near zero, Pe loc 1 [START_REF] Leal | The effect of weak Brownian rotations on particles in shear flow[END_REF]. This condition translates to the Brownian angular velocity being smaller than, using equation 3, |Ω| ∼ k 2 e . For |Ω| ∼ k 2 e then φ can be at most Φ ∼ |k e |, and so |Pe loc | can be at most of the order of |k 3 e Pe|. Thus, for Pe |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. e (λ = 0)|, k -3 e (λ)|) is an important threshold. For Pe Pe m the value of k e does not affect to leading order the orientation distribution p(φ, t). When Pe 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 the statistical moments µ, σ 2 and (φ -µ) 3 . 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 → ∞. In this case the moments depend strongly on the value of k e . For example, if k e is purely imaginary, then the variance σ 2 → 0 and the mean µ → k e for Pe above the Pe m = |k -3 e | line. If instead k e is real then σ 2 is instead finite and µ → 0. Likewise, if Pe is much smaller than |k -3 e | the statistical moments depend only on Pe and become essentially independent of k e (compare, for example, the colour maps for |k e | ≤ 0.1 and Pe ≤ 100 in figure 7).

Finally, for Pe

1, the platelet's rotation is almost entirely due to Brownian diffusion. As Pe → 0, the leading order orientation distribution can be described at all times by a Gaussian distribution with µ = 0 and σ 2 = 1 [START_REF] Burgers | On the motion of small particles of elongated form suspended in a viscous liquid[END_REF][START_REF] Peterlin | Über die Viskosität von verdÜnnten Lösungen und Suspensionen in Abhängigkeit von der Teilchenform[END_REF][START_REF] Scheraga | Non-Newtonian viscosity of solutions of ellipsoidal particles[END_REF]. 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.

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 φ c , from the unstable region in which the platelet performs full rotations. Because the hydrodynamic angular velocity is zero for φ = φ 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 σ 2 is large enough for the fluctuating platelet to "jump out" of the region where Ω(φ) is positive. The width of this "hydrodynamic potential well" is given by the range of values of φ for which Ω(φ) is positive. This width can be approximated as 2 arctan(θ c ) ≈ 2|k e | for |k e | 1. Therefore, we estimate that the probability for a given particle to perform full rotary cycles becomes significant for σ larger than a threshold σ ∼ 2|k e |.

Using our numerical procedure for calculating σ 2 , as given in equation ( 11), we calculate Pe c by first computing σ(Pe, k e ) and then calculating the value of Pe that gives σ = 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 to 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(φ) 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 ∝ |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 equation ( 1). The data are expected to approach this theoretical value for values of |k e | smaller than we could simulate. For |k e | 0.1 the local power-law exponent is definitely not close to -3. In this region, the line Pe c ∝ |k e | -4.8 appears to be a better fit to the data.

Figure 8 does not enable by itself to draw a conclusion about whether a stable orientation occurs for given values of λ 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 λ and a. This will be done in the next section, by considering particles of a specific geometry and slip properties.

Rotational dynamics of model graphene-like nanoplatelets

Our work is motivated by understanding the dynamics of graphene colloids or similar 2D nanomaterials. Applying our theory to realistic graphene-like nanoplatelets requires evaluating k e for geometric and slip characteristics typical of this specific type of anisotropic colloid. The calculation of k e for platelets of atomic thickness (in the following we will use the word "nanoplatelets" for brevity) must be done with care, because the classical framework developed for mesoscale colloidal particles may break down when one of the particle dimensions is comparable with the size of the liquid molecules. In previous studies [START_REF] Gravelle | Liquid exfoliation of multilayer graphene in sheared solvents: A molecular dynamics investigation[END_REF], Kamal et al., 2020], we have performed Molecular Dynamics (MD) of a rigid graphene nanoplatelet suspended in a shear flow of water and other solvents. We showed that k e can still be predicted using a continuum description, provided that the hydrodynamic stress is computed on a suitable reference surface surrounding the platelet and a slip boundary condition is enforced at this surface. Interestingly, we found that using a single slip parameter λ is sufficient to capture the effects of the hydrodynamic slip on the particle rotation and hydrodynamic torque, despite the heterogeneous surface of graphene (which has different slip properties at the edges as compared to the planar region). This feature, which we have proven to be related to the weak sensitivity of the hydrodynamic torque on the tangential components of the traction at the edges, is useful as it simplifies the analysis of the problem by reducing the number of parameters.

Effective nanoplatelet's geometry

Nanoplatelets, made for example of Carbon (C), Boron Nitride (BN), or Molybdenum disulfide (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 = ξ + d gg (n -1)/2, where ξ is the effective radius of a single atom of the nanoplatelet, d gg is the inter-layer spacing, and n is the number of stacks. For multilayer graphene, d gg ≈ 3.35 Å [START_REF] Chung | Review graphite[END_REF]. The effective radius ξ depends on the equilibrium distance between the atoms of the solid and the liquid molecules. For graphene in water, ξ ∼ 1.8 -2.5 Å [START_REF] Gravelle | Large permeabilities of hourglass nanopores: From hydrodynamics to single file transport[END_REF], and comparable values can be expected for BN and MoS 2 due to their similar atomic structure [START_REF] Radisavljevic | Single-layer MoS2 transistors[END_REF][START_REF] Tocci | Friction of water on graphene and hexagonal boron nitride from ab initio methods: very different slippage despite very similar interface structures[END_REF][START_REF] Luan | Wettability and friction of water on a MoS2 nanosheet[END_REF]. The edges form a semi-circle shape for n = 1 (figure 9 (a)) and a flat face with blunt edges as n increases (figure 9 (b,c,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. [START_REF] Singh | Rotational motion of a thin axisymmetric disk in a low Reynolds number linear flow[END_REF] for λ = 0).

Range of slip lengths

For atomically smooth surfaces of 2D nanomaterials such as graphene in contact with common solvents, the hydrodynamics slip length measured at the planar surface of the particle ranges from a few nanometers to tens of nanometers.

As summarised in table 1, experiments performed on graphite [START_REF] Maali | Measurement of the slip length of water flow on graphite surface[END_REF][START_REF] Ortiz-Young | The interplay between apparent viscosity and wettability in nanoconfined water[END_REF], and ab initio calculations [START_REF] Tocci | Friction of water on graphene and hexagonal boron nitride from ab initio methods: very different slippage despite very similar interface structures[END_REF], suggest λ ≈ 10 nm for water and graphene. In the case of NMP or ethanol, the slip length of graphene is also relatively large (> 10 nm) [START_REF] Gravelle | Liquid exfoliation of multilayer graphene in sheared solvents: A molecular dynamics investigation[END_REF]. Ionic Liquid (IL) can also give large slip lengths, but we remark that in this case λ has been found to depend strongly on the shear rate [START_REF] Voeltzel | Orders of magnitude changes in the friction of an Ionic liquid on carbonaceous surfaces[END_REF]. 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 λ can also be significant for atomically smooth and chemically homogeneous materials other than graphene. The interaction of Boron Nitride (BN) and water gives λ ∼ 3 nm [START_REF] Tocci | Friction of water on graphene and hexagonal boron nitride from ab initio methods: very different slippage despite very similar interface structures[END_REF]. 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 λ, consequence of the dependence of MD simulation results on empirical force fields, table 1 suggests that relatively large slip lengths are not uncommon in 2D nanomaterials, which is essentially a consequence of the fact than many 2D 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 λ = 0, 2, 20 and 200 nm.

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

∇ • σ = 0, ∇ • u = 0, (13)
where σ ij = δ ij p + η(∂u i /∂x j + ∂u j /∂x i ) is the hydrodynamic stress with p the pressure, η the viscosity and u is the velocity field. For the reference surface, we use the surface formulated in section 4.1 from MD simulations of a cuboid with rounded edges. At the reference surface of the platelet we prescribed the Navier slip boundary condition

u sl = λ η n × f × n, (14)
where f = σ • n is the hydrodynamic surface traction and n is the unit normal vector [START_REF] Lauga | Microfluidics: the no-slip boundary condition[END_REF]. The definition of the slip length λ remains valid as long as the separation of time and length scales can be assumed. In the present case, the macroscopic time scale P R (equation ( 8)) is given by the rotation of the particles, which is controlled by the particle aspect ratio and by the shear rate, P R = γ-1 (|k e | -1 + |k e |). 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 f corresponding to each given inclination angle φ [START_REF] Pozrikidis | Boundary integral and singularity methods for linearized viscous flow[END_REF]. Denoting the reference surface by S, the boundary integral equation for a point x ∈ S is [START_REF] Luo | Effect of surface slip on Stokes flow past a spherical particle in infinite fluid and near a plane wall[END_REF]] where the integral is over the surface parameterised by the coordinate point x 1 = (x 1 , y 1 ). In equation ( 15), x = x-x 1 , y = y -y 1 ; G and K are Green's functions corresponding to the 2D 'Stokeslet' and 'Stresslet', respectively; u ∞ is the undisturbed simple shear flow field. The traction is used to calculate the hydrodynamic torque T (φ) = êz • [f × x]dS 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 arc length 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 10 (b,c)), we set N = 288 for platelets with a/b ≤ 100 and N = 384 for platelets with a/b > 100. The value of k e is then calculated from equation (4).

S n • K(x , y ) • u sl dS - 1 η S G(x , y ) • f dS = u sl (x) 2 -u ∞ (x), (15)
The equation ( 15) is evaluated numerically by using the method described in [START_REF] Kamal | Hydrodynamic slip can align thin nanoplatelets in shear flow[END_REF]. First, the traction and slip velocity are discretised as N piece-wise constant functions {f 15) becomes [START_REF] Kamal | Hydrodynamic slip can align thin nanoplatelets in shear flow[END_REF]:

N i=1 u sl [i] • si+1 si K(s, s j ) • n(s)dS(s) - f [i] η • si+1 si G(s, s j )dS(s) = u sl (s j ) 2 -u ∞ (s j ). ( 16
)
For i = j, each sub-integral is evaluated using a Gauss-Legendre quadrature method. If i = j, the integrands are singular. The singular integrand containing the G tensor is evaluated by using a specific quadrature method for logarithmic singularities [START_REF] Pozrikidis | A practical guide to boundary element methods with the software library BEMLIB[END_REF], and the singular integrand containing the K tensor is evaluated analytically by Taylor expansion about the singular point [START_REF] Kamal | Hydrodynamic slip can align thin nanoplatelets in shear flow[END_REF]. Equation ( 14) provides a closed relation between u sl and f , and thus equation ( 16) 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 [START_REF] Kamal | Hydrodynamic slip can align thin nanoplatelets in shear flow[END_REF]. 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 u = êz × x in a fluid otherwise at rest. In agreement with the result of [START_REF] Sherwood | Resistance coefficients for Stokes flow around a disk with a Navier slip condition[END_REF] for a disk, we find for our model platelet that F r is independent of slip as k → 0. In this limit we found F r ≈ c 1 ηa 3 , where c 1 = 6.29 ± 0.02 (to be compared to c 1 = 32/3 for a three-dimensional disk of zero thickness [Leal andHinch, 1971, Sherwood, 2012], and to c 1 = 2π for an infinite plate of zero thickness [START_REF] Sherwood | The use of the vane to measure the shear modulus of linear elastic solids[END_REF]). 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 [START_REF] Luo | Effect of surface slip on Stokes flow past a spherical particle in infinite fluid and near a plane wall[END_REF] for a sphere.

Values of k e for our model platelet with ξ = 0.25 nm, slip length λ = 0, 2, 20 and 200 nm, and n = 1 or 10 are shown in figure 11. We limit our calculations to aspect ratios k ≥ 0.001, because using k < 0.001 leads to large numerical error owing to severe resolution requirements. For λ = 0, the effective ratio of the platelet follows the power law relationship k e ∼ 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' [START_REF] Singh | Rigid ring-shaped particles that align in simple shear flow[END_REF]. We find g 1 ≈ 0.91 for n = 1 and g 10 ≈ 1.0 for n = 10. As λ increases, however, the relation between k e and k is not necessary a power law, as shown for λ = 2, 20 and 200 nm for n = 1 and λ = 20 and 200 nm for n = 10. In the limit λ/a 1, the value of k e can be analytically approximated as k e ∝ i √ k [START_REF] Kamal | Hydrodynamic slip can align thin nanoplatelets in shear flow[END_REF], as confirmed in figure 11 for platelet length 2a 20 nm, n = 1 and λ = 20 nm and 200 nm (we remark, however, that most applications of nanoplatelets typically involve λ < a).

k e ∈ R 2a [nm] -1/2 -1/2 2a [nm] -3/4 -3/4 k e ∈ iR k e ∈ R
As the slip length increases, the value of k 2 e decreases and changes sign at some critical value λ c (figure 12). An explanation to this is the following. Calculating k e requires the evaluation of the total hydrodynamic torque on a fixed platelet for two angles: φ = 0 and φ = π/2. The total torque for φ = 0 is more sensitive to λ than the torque for φ = π/2, because slip affects primarily the flow moving in the direction tangential to the surface of the particle. Thus, predicting λ c essentially requires predicting the angle for which T (0) changes sign. When λ = 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 ∝ b 2 ; the second-order tangential traction determining this torque is f

(2)
x ∝ b [START_REF] Singh | Rotational motion of a thin axisymmetric disk in a low Reynolds number linear flow[END_REF]. 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 λ/a 1 shows that the effect of slip is to reduce f x by an amount of O(λ), resulting in a second-order traction f

(2,sl) x ∝ λ [START_REF] Kamal | Hydrodynamic slip can align thin nanoplatelets in shear flow[END_REF], where "sl" indicates that this traction is due to slip. The total torque T (0) changes sign when f

(2,sl) x > f (2)
x . This condition occurs at a threshold value λ c ≈ h n b, where h n is a numerical prefactor that depends on the number of layers n [START_REF] Kamal | Hydrodynamic slip can align thin nanoplatelets in shear flow[END_REF]. For n = 1, our numerical procedure gives h 1 ≈ 0.8, and b ≈ 0.25 nm so λ c ≈ 0.2 nm. For n = 10, h 10 ≈ 2.2 and b ≈ 1.75 nm so λ c ≈ 3.85 nm (figure 12). For moderate-to-large values of λ/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 λ so that k e is in general a non-trivial function of λ. 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 λ/b is large [START_REF] Kamal | Hydrodynamic slip can align thin nanoplatelets in shear flow[END_REF]. In the following, since k e is a non-trivial function of λ, b, and a (figure 12), we will use the numerically computed values of k e .

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 λ = 0 (no-slip) and λ = 2 nm (slip) for fixed a/b = 100 and n = 1. The corresponding effective aspect ratios are k e = 0.028 and k e = 0.063i for λ = 0 and λ = 2 nm, respectively. In figure 14, we plot the mean µ and variance σ 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 µ and σ 2 at large Pe. To describe the qualitative differences in the trends of µ and σ 2 between the slip and no-slip cases, we subdivide the data into three different regions (I, II, and III, figure 14). Region III corresponds to Pe > Pe c , where Pe c is the critical value introduced in section 3.3 above which no full rotational cycles occur for λ = 2 nm. Region III is a 'large slip effects' regime. In this regime the platelet with slip performs small fluctuations around φ c . In this region the curves approach smoothly the values expected for Pe → ∞ (section 3.2): for λ = 2 nm, µ is approximately constant (µ ≈ |k e |), and σ 2 → 1/(2Pe|k e |) as Pe → ∞. This latter asymptotic solution for σ 2 can be obtained by inserting into equation (1) the linearisation of the angular velocity for angles close to φ c , Ω = 2k e (φ -φ c ), and assuming that p is Gaussian. For λ = 0, µ vanishes as Pe → ∞ and σ 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 µ or σ 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 to region I. This reduction in frequency indicates that in this region hydrodynamics mitigates the randomising effect of Brownian motion.

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 = 10000, a small slip of λ = 2 nm leads to a reduction by about one order of magnitude of the variance.

As explained in section 3.2, the boundary Pe I between the "negligible slip effects" regime and the "moderate slip effects" regime must satisfies the condition Pe I Pe m = min(|k -3 e (λ = 0)|, k -3 e (λ)|). Our numerical simulations (shown in both figure 7 and figure 14) indicate that the correlation Pe I 0.1Pe 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 section 3 is that now Pe c and Pe m can be evaluated as a function of the known variables λ and 2a, rather than k e .

We add to figure 15 red dash-dot lines corresponding to the Pe number for three shear stress values: γη = 10 7 Pa, γη = 2 × 10 4 Pa, γη = 10 Pa (the Péclet number scales proportionally to the stress according to Pe ≈ c 1 γηa 3 /(k B T ), where 'c 1 = 6.29 ± 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 subsection 4.3. These characteristic shear stresses are, respectively, typical of: exfoliation processes in a low viscosity fluid [START_REF] Keith R Paton | Scalable production of large quantities of defect-free few-layer graphene by shear exfoliation in liquids[END_REF]; mixing in very viscous fluids [START_REF] Huang | Dispersion of carbon nanotubes: mixing, sonication, stabilization, and composite properties[END_REF] and dispersion by microfluidisation [START_REF] Paton | Production of few-layer graphene by microfluidization[END_REF][START_REF] Karagiannidis | Microfluidization of graphite and formulation of graphene-based conductive inks[END_REF]; fast lubrication processes [START_REF] Jonsson | Measurement of rheological properties of ultrathin lubricant films at very high shear rates and near-ambient pressure[END_REF], high-speed blade coating [START_REF] Willenbacher | High shear rheology of paper coating colors-more than just viscosity[END_REF], and MD simulations [START_REF] Voeltzel | Orders of magnitude changes in the friction of an Ionic liquid on carbonaceous surfaces[END_REF], Gravelle et al., 2020]. 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 λ, a and n we considered. For the intermediate shear stress (dotted-dashed line in figure 15) the 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 λ at 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 φ c if λ > λ 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 100 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, λ = 200 nm, and η γ = 2 × 10 4 Pa as an example, the rotation of the platelet is not affected by slip for 2a < 40 nm, whereas platelets with 2a > 40 nm enter the "moderate slip effects region" where slip effects start becoming significant.

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 η eff of a dilute suspension of nanoplatelets for finite values of λ. The effective shear viscosity of a dilute suspension of particles can be calculated as

η eff = η + σ xy γ , (17)
where σ xy is the particle stress [START_REF] Jeffery | The motion of ellipsoidal particles immersed in a viscous fluid[END_REF][START_REF] Giesekus | Elasto-viskose Flüssigkeiten, für die in stationären Schichtströmungen sämtliche Normalspannungskomponenten verschieden groß sind[END_REF][START_REF] Brenner | Rheology of a dilute suspension of axisymmetric Brownian particles[END_REF][START_REF] Leal | The effect of weak Brownian rotations on particles in shear flow[END_REF]]. 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

σ xy = cη γ A 2 1 -cos 4φ + B + C 2Pe sin 2φ , (18)
where A, B, and C are dimensionless coefficients [START_REF] Rallison | The effects of Brownian rotations in a dilute suspension of rigid particles of arbitrary shape[END_REF] and c is the solid fraction. The first two terms on the right side of equation ( 18) are due to the hydrodynamic stress acting on each platelet. The third term is the contribution to the particle stress due to Brownian fluctuations.

As seen in Section 3.2, slip has a strong influence of the time-averaged orientation of the particle, with consequences on the the statistical quantities 1 -cos 4φ and sin 2φ in equation ( 18). A map quantifying how 1 -cos 4φ and sin 2φ vary in the (Pe, k e ) space is shown in figure 16. For Pe |k e | -3 , the quantities 1 -cos 4φ and sin 2φ are practically independent of whether k e is real or purely imaginary. The term 1 -cos 4φ becomes significantly larger than sin 2φ for small Pe. Thus, the effective viscosity depends increasingly on 1 -cos 4φ as Pe decreases, in agreement with the theoretical results of [START_REF] Hinch | The effect of Brownian motion on the rheological properties of a suspension of non-spherical particles[END_REF] and [START_REF] Leahy | The effect of shear flow on the rotational diffusion of a single axisymmetric particle[END_REF] (for elongated particles with λ = 0) and the experimental results of Del Giudice and Shen [2017] for graphene oxide.

For Pe |k e | -3 , hydrodynamic stresses become dominant. Figure 16 indeed reveals a strong dependency of 1 -cos 4φ and sin 2φ on k e in this region. Since the prefactor of sin 2φ tends to zero for large Pe, here we focus on analysing the term 1 -cos 4φ . The leading-order values of 1 -cos 4φ in this region can be estimated directly by using the probability distributions calculated in section 3.2 for Pe → ∞:

1 -cos 4φ = k e 2(k 2 e + 2k e + 1) -1 , if k e ∈ R, 1 -cos 4φ c , if k e ∈ iR. (19)
For |k e | → 0, 1 -cos 4φ tends to k e /2 and 8|k e | 2 for real or purely imaginary k e , respectively. For λ/a = 0 and λ/a → ∞, 1 -cos 4φ can be estimated in terms of the geometric aspect ratio k. For λ = 0, k e ∝ k, and so 1 -cos 4φ ∝ k. For λ/a → ∞ we have k e ∝ i √ k, giving again 1 -cos 4φ ∝ k. Thus, quite interestingly, the limits λ/a → 0 and λ/a → ∞ yield a similar scaling relationship of 1 -cos 4φ with k.

For λ = λ c , we have k e = 0 and therefore 1 -cos 4φ = 0. When 1 -cos 4φ vanishes, the particle stress reduces to σ xy = cη γB. If the variation of B on λ is sufficiently small in the neighbourhood of λ c (compared to A 1 -cos 4φ ), then the value λ = λ c will corresponds to a local minimum in the σ xy vs λ curve, because A 1 -cos 4φ = 0 ≥ 0 for all values of λ. Thus, the decrease in the value of σ xy with λ may not be always monotonic for large Pe.

Discussion

We have shown that the classical Bretherton's equation of motion can describe the rotational dynamics of thin rigid particles with surface slip, as long as an effective aspect ratio k e is introduced to account for the hydrodynamic slip length λ in addition to the particle's shape. For a fixed geometric aspect ratio b/a and a small slip length λ < b (with b the platelet's half-thickness), k e is real, and an increase of λ simply leads to a decrease of k e . However, when λ is sufficiently large in comparison to b, k e becomes a purely imaginary number. In the limit of infinite Peclet numbers, a platelet with an imaginary k e does not perform the periodic orbits predicted by the classical theory of Jeffery. Instead, the platelet's orientation fluctuates in time around a small angle φ c with respect to the flow direction. We have identified a critical Péclet number Pe c above which such stabilisation occurs, and below which rotations due to Brownian fluctuations appear. The numerical results suggest Pe c ∼ 0.45|k e | -3.4 for |k e | 0.1, and Pe c ∼ 0.02|k e | -4.8 for |k e | > 0.1, with a smooth transition between the two scalings. An alternative critical Péclet number Pe m , based on balancing the orders of magnitude of the convective and diffusive terms in equation ( 1), predicts the hydrodynamic stresses acting on the platelet to be dominant with respect to the Brownian stress when Pe m = min(|k -3 e (λ = 0)|, k -3 e (λ)|). If Pe is smaller than Pe I ≈ 0.1Pe m , we find a negligible difference between the rotational behaviour of slip and no-slip platelets. In this regime, Brownian stress dominates over hydrodynamic stress in the regime where slip effects are important. For Pe smaller than Pe c , Brownian fluctuations force the platelet with slip to complete full rotations, but with a period larger than the one expected in the no-slip case. Finally, we examined the effect of stable alignment on the orientational contribution to the effective viscosity by analysing how the term 1 -cos 4φ changes due to slip. At large Pe, we found that this term does not decrease monotonically with λ but instead attains a minimum value at λ = λ c . Some assumptions in our mathematical model require a discussion in view of the results. One aspect regards the rigidity of the particle, as thin particles are intuitively expected to be flexible. Assuming that the bending rigidity follows B ∼ Eb 3 with E the Young's modulus [START_REF] Poot | Nanomechanical properties of few-layer graphene membranes[END_REF], where E ∼ 10 11 Pa for pure graphene [START_REF] Lindahl | Determination of the bending rigidity of graphene via electrostatic actuation of buckled membranes[END_REF], for platelets aligned almost parallel to the direction of the shear flow, buckling should occur when η γ(a/b) 2 /E ≥ 1 [START_REF] Lingard | The deformation of disc-shaped particles by a shearing fluid with application to the red blood cell[END_REF]Whitmore, 1974, Kamal et al., 2020]. Using the typical shear stress γη ∼ 10 4 Pa for which slip effects become important (section 4.4), one finds that the platelet behaves as rigid as long as the geometric aspect ratio k ≥ 10 -7/2 0.0003. Therefore, the assumption of rigidity in our work may not be very restrictive for particles whose aspect ratio is not too extreme. Our results have been obtained for a 2D geometry, which strictly speaking corresponds to nanoplatelets with an infinite extension in the direction of the vorticity vector. Real nanoplatelets have, of course, a finite lateral size, with a shape that can vary considerably, from irregular 'pentagon' plate-shape [START_REF] Del Giudice | Filling the gap between transient and steady shear rheology of aqueous graphene oxide dispersions[END_REF] to 2D 'ribbon' like [START_REF] Hao | Aqueous dispersions of tcnq-anion-stabilized graphene sheets[END_REF]). However, it can be shown by comparing the hydrodynamic torque exerted on a 2D plate-like particle with that on a three-dimensional disk that these different geometries lead to values of the effective aspect ratio k e that are very similar to each other [START_REF] Kamal | Hydrodynamic slip can align thin nanoplatelets in shear flow[END_REF]. Therefore, the rotational dynamics is expected to be qualitatively similar for different plate-like objects, as long as the platelet's extension in the vorticity direction is not too small. Finally, a 3D object also possesses extra degrees of freedoms, i.e. its rotation is not confined to a plane. However, as discussed by [START_REF] Leahy | The effect of shear flow on the rotational diffusion of a single axisymmetric particle[END_REF] and [START_REF] Hinch | The effect of Brownian motion on the rheological properties of a suspension of non-spherical particles[END_REF], the qualitative features of the rotational dynamics of a particle whose motion is confined to a plane are expected to be similar to those of a particle whose motion is in the full three-dimensional space. Therefore, we expect our model to have relevance to realistic systems. Additional insights would require the solution of full three-dimensional trajectories. Preliminary Molecular Dynamics results we have carried out with plate-like slip molecules rotating in the full three-dimensional space reveal that the 2D approximation used here give rotational statistics that are comparable in trend and magnitude -with reasonably good approximation -to the ones computed from Molecular Dynamics, giving us confidence in the value of the 2D approximation employed here.

To illustrate how our theoretical results could be applied in practice, in section 4 we have considered geometric and slip parameters relevant to graphene and other 2D nanomaterials. By calculating the values of Pe m and Pe c for typical values of k, b and λ, we have found that stresses γη of the order of 10 4 Pa are required for hydrodynamic slip to affect quantitatively the rotational dynamics.

The use of fluid more viscous than water would allow for such large shear stress values. For instance, some ionic liquids have a viscosity about 3 orders of magnitude larger than that of water, while also showing slip length as large as 100 nm on graphene [START_REF] Voeltzel | Orders of magnitude changes in the friction of an Ionic liquid on carbonaceous surfaces[END_REF]. The use of very viscous fluids would have the further benefit of reducing the possibility of hydrodynamic instabilities and turbulences. Large viscous stresses may be accompanied by heating, but one could account for this effect by evaluating the viscosity and the slip length at the corresponding temperature.

An experimental verification of our results is not without challenges. For example, surface chemical modification of graphene nanoparticles is generally needed to obtain good dispersion of graphene in water, and when functionalization is applied to the planar region of the platelet, the slip length is expected to be reduced [START_REF] Wei | Breakdown of fast water transport in graphene oxides[END_REF]. Particles for which the functionalization is limited to the particle edges may not have this problem [START_REF] Park | Highly dispersible edge-selectively oxidized graphene with improved electrical performance[END_REF][START_REF] Aliyeva | Recent developments in edge-selective functionalization of surface of graphite and derivatives-a review[END_REF], and could be more suitable to confirm the theory. Detecting rotation may not be trivial. For example, techniques that measure the average orientation of the particles may not enable to clearly distinguish between a suspension in which very slender particles rotate but are oriented in a time-average sense, from a suspension in which the particles orientation angle is constant, since in both cases φ is expected to be small. Because the occurrence of a stable orientation depends only on the ratio λ/b, and not on the particle length, a solution could be to use plate-like particles that are thin (b ∼ 1 nm) but not too long. For a given shear stress the ideal particle to evidence slip effects would be the shortest particle whose value of a is in the "moderate slip effects" region, since this choice would give the largest variation in k e due to slip, and thus the largest variation in φ. With this choice of particle length, the different rotational behaviours of slip and no-slip particles should be particularly evident.

In experiments, different statistical measures of particle orientation could be adopted, and some of them may be more sensitive to slip than others. To illustrate this dependence and the impact slip might have on orientational statistics that are evaluated in practice, we discuss two quantities that are usually measured in rheo-optical studies [START_REF] Frattini | Rheo-optical studies of the effect of weak Brownian rotations in sheared suspensions[END_REF][START_REF] Fuller | Optical rheometry of complex fluids[END_REF][START_REF] Vermant | Rheooptical determination of aspect ratio and polydispersity of nonspherical particles[END_REF][START_REF] Reddy | Flow dichroism as a reliable method to measure the hydrodynamic aspect ratio of gold nanoparticles[END_REF]: the average orientation angle χ and the degree of aligment ∆n /(∆n max ), defined as:

χ = 1 2 arctan sin θ 2 sin 2φ sin θ 2 cos 2φ , ∆n ∆n max = sin θ 2 sin 2φ 2 + sin θ 2 cos 2φ 2 . ( 20
)
Figure 17 compares our theoretical prediction for χ and ∆n /(∆n max ) (in the case θ = π/2) to the experimental data of [START_REF] Reddy | Rheo-optical analysis of functionalized graphene suspensions[END_REF]. The experiments are for dilute polydisperse suspensions of functionalised graphene platelets, with average half-length a = 250 nm and half thickness b(n = 5) = 0.86 nm, and slightly oblate gold nano-spheroids, with b/a = 0.55 and lengths of either 170 nm or 290 nm. The functionalised graphene sheets were dispersed in mineral oil (viscosity η = 100 mPas) and the oblate gold spheroids were dispersed in a glycerol/water mixture at 99.5 % glycerol (η = 1.2 Pa.s). The theoretical values (lines) are for particles having the same average aspect ratio as in the experiments, and considering the limiting cases of λ = 0 and λ = 20 nm (figure 17). For large Pe, the functionalised graphene platelets where found experimentally to converge to a finite angle χ = 5 • , instead of approaching 0 (as one would expect for a platelet with a no-slip boundary in the limit Pe → ∞). The plot shows that our theory raises the χ -Pe curve closer to χ = 5 • at large Pe, although probably a value λ = 20 nm is larger than the one occurring in the experiment. Importantly, the plots show that for the slightly oblate spheroids, slight changes in k e due to slip makes only a slight change to χ in figure 17, but a more marked change in ∆n /(∆n max ), suggesting that the second quantity is a more sensitive measure of slip effects.

We suggest that experiments similar to those of [START_REF] Reddy | Rheo-optical analysis of functionalized graphene suspensions[END_REF] could be done in which both χ and ∆n /(∆n max ) are compared for a range of slip lengths and values of particle thickness. Rather than plotting χ or ∆n /(∆n max ), one could also plot sin θ 2 sin 2φ /2 and 2 -2 sin θ 2 cos 2φ -sin θ 2 sin 2φ 2 /4. For small angles these quantities approximate the mean φ and variance (φ -µ) 2 , and should therefore have a marked variation in the neighbourhood of λ/b = 1 when plotted against λ/b. Another approach to evidence the effect of slip would be to conduct steady-state viscosity measurements [White et al., 2015, Del Giudice and[START_REF] Del Giudice | Shear rheology of graphene oxide dispersions[END_REF], by focusing on the high-Pe regime where the effect of slip is maximum. The change in viscosity we predict for a range of values of λ appears to be large enough to be measurable.

Our theory is not limited to nanometrically thin particles. Slip lengths of order 1 µm could be achieved for example, through the creation of a wetted/gas layers over the particle's surface [START_REF] Lee | Interfacial slip on rough, patterned and soft surfaces: A review of experiments and simulations[END_REF], depletion layers [Tuinier andTaniguchi, 2004, Fan et al., 2007] or surface nanobubbles [START_REF] Neto | Boundary slip in Newtonian liquids: a review of experimental studies[END_REF][START_REF] Yang | Correlation between geometry and nanobubble distribution on HOPG surface[END_REF]. This raises the possibility of conducting experiments with relatively large plate-like particles whose dynamics can be accessed optically.

Applications of the theory developed in the current paper may include: i) the development of rheological models for 2D nanomaterials in suspension (e.g. graphene liquid crystals); ii) the design of experimental methods to measure the slip properties of plate-like particles or macromolecules, which could yield insights into molecular interactions at the solid-liquid interface; iii) the use of hydrodynamic slip to favour the flow of 2D nanomaterials in narrow channel (e.g., to avoid clogging). Regarding i), because we have shown that slip plate-like particles have a smaller propensity to rotate in shear flow than no-slip particles, and thus interactions with other particles are more limited than in the no-slip case, theories developed for the dilute case could also apply to more concentrated systems, as pointed out previously for ring-shaped particles [START_REF] Singh | Rigid ring-shaped particles that align in simple shear flow[END_REF][START_REF] Borker | Controlling rotation and migration of rings in a simple shear flow through geometric modifications[END_REF]. Finally, alignment is an essential ingredient to impart superior properties to nanocomposite materials. Surface modifications producing substantial slip could be used to align particles produced from the exfoliation of 2D layered materials, which can be easily produced on mass scales by liquid-phase exfoliation [START_REF] Botto | Towards nanomechanical models of liquid-phase exfoliation of layered 2d nanomaterials: analysis of a π-peel model[END_REF][START_REF] Salussolia | Micromechanics of liquid-phase exfoliation of a layered 2d material: a hydrodynamic peeling model[END_REF], Gravelle et al., 2020] and have therefore potential for applications. Because the ideal infinite Peclet number regime may not be achievable in practice, our results provide theoretical guidelines for deciding in which cases slip will have a dominant effect on the alignment of plate-like nanoparticles in the presence of Brownian motion.

Figure 1 :

 1 Figure 1: Sketch of the rotational dynamics of a platelet having (a) zero slip length, as described by Jeffery's theory, and (b) slip length λ significantly greater than the platelet's half-thickness b. The behaviours described in (a) and (b) are characteristic of the infinite Peclet number limit. In this paper we enquire about the effect of Brownian fluctuations on the rotational dynamics.

Figure 2 :

 2 Figure 2: (a) Rotation angle φ for a particle with real effective aspect ratio k e = 0.16 and Pe = ∞. (b) Associated angular velocity γΩ. Notice that Ω < 0 ∀φ. (c) Rotation angle φ for a particle with imaginary effective aspect ratio k e = 0.28i and Pe = ∞, showing stabilisation at φ c ≈ 0.27. (d) Corresponding non-dimensional angular velocity, with the two equilibrium points at φ = ±φ c . The ' ' symbol marks the stable angle φ c . The angle denoted by ' ' is unstable.

  φ c = arctan(|k e |), as shown in figure 2 (c). For this angle, Ω has a positive zero root, indicated by the open circle in figure 2 (d). The relaxation to equilibrium occurs on a timescale

  imaginary k e (k e = 0.21i) :

Figure 3 :

 3 Figure 3: Time evolution of the rotation angle for different Pe, as calculated from equation (9), comparing (a) real k e to (b) purely imaginary k e . The red dashed line is the value of φ c obtained for Pe → ∞.

Figure 4 :

 4 Figure 4: Spectral coefficients a n and b n for k e = 0.05i and Pe= 10 6 and t → ∞. The spectral coefficients are truncated at n = 1000, since a n , b n → 0 above this threshold.

Figure 5 :

 5 Figure 5: Mean (a), variance (b), and skewness (c) of the angular probability distribution versus time, for a nanoplatelet with k e = 0.1i and different values of Pe.

Figure 6 :

 6 Figure 6: Steady state orientation distribution function p(φ) for a real or a purely imaginary value of k e for different Pe numbers. The red dotted line marks the orientation distribution function for Pe → ∞ and k e = 0.1 (left), and the location of φ c for k e = 0.1i (right). The values [µ, σ 2 , (φ -µ) 3 ] 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 = 0.1i.

Figure 7 :Figure 8 :

 78 Figure 7: Colour maps of the mean µ, variance σ 2 , and skewness (φ -µ) 3 of the steady-state orientation distribution in the (Pe, k e ) space. 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.

Figure 9 :

 9 Figure 9: (a-c) Water density profiles as extracted from molecular dynamics 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. [2020]. 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 λ, the effective radius ξ of the carbon atoms, and the inter-layer spacing d gg .

Figure 10 :

 10 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 λ = 20 nm, a/b = 10, or a/b = 100 and either n = 1 (b), or n = 10 (c).

  [1] . . . f [N ]} and {u sl [1] . . . u sl [N ]}, respectively, associated to a surface element S i = si+1 si dS. With this discretisation, for each point s j for j = {1 • • • N }, the discretised form of equation (

Figure 11 :

 11 Figure 11: Absolute value of the effective aspect ratio k e versus the platelet length 2a for ξ = 0.25 nm and for n = 1 (left) and n = 10 (right). The black lines correspond to the purely imaginary values of k e , and the red lines to real values of k e .

Figure 12 :Figure 13 :Figure 14 :

 121314 Figure 12: Effective aspect ratio k 2 e versus the slip length λ for different platelet lengths 2a. The half-thickness of the platelet is b ≈ 0.25 nm and b ≈ 1.75 nm for n = 1 and n = 10, respectively. Red dashed lines mark the positions where k 2 e changes sign.

Figure 15 :

 15 Figure 15: Effect of slip on the rotational dynamics of a nanoplatelet with length = depth = 2a and width = 2b in the (Pe, 2a) space. Upper black curve marks the threshold Péclet number Pe c for which σ 2 = 2k 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 γη = 10 7 Pa, γη = 2 × 10 4 Pa, γη = 10 Pa, respectively.

Figure 16 :Figure 17 :

 1617 Figure 16: Colour maps of 1 -cos (4φ) and sin (2φ) in the (Pe, k e ) space for (left) an imaginary value of k e and (right) a real value of k e . The black line marks the equation Pe = |k e | -3 .

Table 1 :

 1 Literature values of the slip length for different solid material/solvent combinations. DlC is for Diamond-like Carbon, GO is for Graphene Oxide, BN is for Boron Nitride, MoS

	Material Liquid (η [mPa s])	λ [nm]	Method	Reference
	graphene	water (1.0)	10.4 ± 2.2	ab initio MD	Tocci et al. [2014]
	graphene	water (1.0)	60 ± 5	MD	Kamal et al. [2020]
	graphene	water (1.0)	[1 -80]	MD	Kannam et al. [2013]
	graphene	water (1.0)	83	MD	Falk et al. [2012]
	graphene	decane (0.85)	103	MD	Falk et al. [2012]
	graphene	OMCTS (1.0)	18	MD	Falk et al. [2012]
	graphene	ethanol (1.1)	250	MD	Falk et al. [2012]
	graphene	ethanol (1.1)	30 ± 3	MD	Falk et al. [2012]
	graphene	NMP (1.6)	14 ± 2	MD	Kamal et al. [2020]
	graphene	CPO (1.3)	45 ± 2	MD	Kamal et al. [2020]
	graphite	IL ([1 -200])	[300 -10 000]	MD	Voeltzel et al. [2018]
	graphite	water (1.0)	8	experiment	Maali et al. [2008]
	graphite	water (1.0)	12 ± 3.3	experiment	Ortiz-Young et al. [2013]
	DlC	water (1.0)	0.55 ± 1.37	experiment	Ortiz-Young et al. [2013]
	DlC	IL ([1 -200])	[1 -100]	MD	Voeltzel et al. [2018]
	MoS 2	water (1.0)	5.6	MD	Luan and Zhou [2016]
	mica	water (1.0)	< 2	experiment	Maali et al. [2008]
	mica	water (1.0)	∼ 0	experiment	Ortiz-Young et al. [2013]
	BN	water (1.0)	3.3 ± 0.6	ab initio MD	Tocci et al. [2014]
	GO	water (1.0)	0.34 ± 0.38	experiment	Ortiz-Young et al. [2013]
	silicon	water (1.0)	1 ± 1.7	experiment	Ortiz-Young et al. [2013]

2 is for Molybdenum disulfide, IL is for Ionic Liquid, NMP is for N-Methyl-2-Pyrrolidone, OMCTS is for Octamethylcyclotetrasiloxane, CPO is for Cyclopentanone.

Funding

This work was supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program through project FlexNanoFlow (Grant agreement no. 715475).