1. Introduction
Artificial micro- and nanomotors have attracted significant attention over the past two decades owing to their potential applications in targeted drug delivery (Sun et al. Reference Sun, Fan, Meng, Song, Chen, Sun and Xie2019), medical diagnostics (Chałupniak et al. Reference Chałupniak, Morales-Narváez and Merkoçi2015), assisted fertilisation (Benhal Reference Benhal2024), energy harvesting (Singh et al. Reference Singh, Soto, Kaufmann and Wang2015), environmental remediation (Jurado-Sánchez & Wang Reference Jurado-Sánchez and Wang2018) and microrobotics (Dey et al. Reference Dey, Wong, Altemose and Sen2016). These micromotors convert chemical, optical, electrical or thermal energy into mechanical motion, enabling autonomous locomotion in the absence of moving appendages (Paxton et al. Reference Paxton, Sundararajan, Mallouk and Sen2006; Gangwal et al. Reference Gangwal, Cayre, Bazant and Velev2008; Jiang et al. Reference Jiang, Yoshinaga and Sano2010; Palagi et al. Reference Palagi2016; Poddar et al. Reference Poddar, Bandopadhyay and Chakraborty2019). Additional thermophoretic, electrical or chemical contributions have been shown to provide superior control over particle motion (Simoncelli et al. Reference Simoncelli, Summer, Nedev, Kühler and Feldmann2016; Poddar Reference Poddar2023; Damor et al. Reference Damor, Ghosh and Poddar2023). Among the different propulsion strategies, self-phoretic Janus particles, which harness asymmetric interactions with their chemical environment, have emerged as a robust model system for autonomous locomotion at low Reynolds numbers.
In self-diffusiophoretic propulsion, a chemical reaction catalysed on one hemisphere of a Janus particle generates spatial solute concentration gradients. These gradients induce tangential slip flows along the particle surface, giving rise to self-sustained motion without any external forcing (Sabass & Seifert Reference Sabass and Seifert2012; Michelin et al. Reference Michelin, Lauga and Bartolo2013; Mondal et al. Reference Mondal, Ghosh, Poddar and Chakraborty2026). Experiments have demonstrated that even electrically neutral platinum-coated polystyrene spheres can self-propel through the decomposition of hydrogen peroxide (Baraban et al. Reference Baraban, Tasinkevych, Popescu, Sanchez, Dietrich and Schmidt2012). Subsequent studies revealed the influence of solute advection, surface reaction heterogeneity and electrokinetic effects on particle trajectories and rotation (Reigh et al. Reference Reigh, Huang, Löwen, Lauga and Kapral2020; Asmolov et al. Reference Asmolov, Nizkaya and Vinogradova2022; Zhu et al. Reference Zhu, Yang and Guo2024). More recently, direct experimental quantification of diffusiophoretic forces has become possible with optical tweezer-based techniques (Raj et al. Reference Raj, Roy, Kumar, Roy, Mani and Sudhakar2025).
While the free-space dynamics of self-phoretic swimmers is now well established (Michelin et al. Reference Michelin, Lauga and Bartolo2013; Frankel & Khair Reference Frankel and Khair2014; Uspal Reference Uspal2019; Auschra et al. Reference Auschra, Bregulla, Kroy and Cichos2021), their behaviour near confining boundaries remains an area of active exploration. Confinement fundamentally alters both the chemical and hydrodynamic fields, giving rise to complex near-wall states such as collision, reorientation, steady sliding or hovering (Crowdy Reference Crowdy2013; Kreuter et al. Reference Kreuter, Siems, Nielaba, Leiderer and Erbe2013; Uspal et al. Reference Uspal, Popescu, Dietrich and Tasinkevych2015b ). Theoretical works have incorporated chemical and hydrodynamic boundary effects for both two-dimensional (Crowdy Reference Crowdy2013) and three-dimensional (Ibrahim & Liverpool Reference Ibrahim and Liverpool2015; Mozaffari et al. Reference Mozaffari, Sharifi-Mood, Koplik and Maldarelli2016) systems. The near-wall propulsion due to confinement-induced localisation of chemical gradients and stability against the non-axisymmetric perturbations of active droplets were extensively studied in Desai & Michelin (Reference Desai and Michelin2021). Similarly, in an axisymmetric setting, Lippera et al. (Reference Lippera, Morozov, Benzaquen and Michelin2020b ) showed that both droplet–wall and droplet–droplet collisions and rebound dynamics depend strongly on the Péclet number through the interplay of solute transport and hydrodynamic flows. Furthermore, Lippera et al. (Reference Lippera, Benzaquen and Michelin2020a ) demonstrated that even slight size asymmetries can significantly alter droplet–droplet collision dynamics and subsequent trajectories.
Similarly, experiments have confirmed that wall proximity can modify swimming speed, stability and orientation (Yang et al. Reference Ybert, Barentin, Cottin-Bizonne, Joseph and Bocquet2016; Shen et al. Reference Shen, Würger and Lintuvuori2018; Leeth Holterhoff et al. Reference Leeth Holterhoff, Li and Gibbs2018). Similar wall-induced behaviours are observed in biological microswimmers like E. coli, which display circular or oscillatory trajectories depending on boundary interactions (Lauga et al. Reference Lauga, DiLuzio, Whitesides and Stone2006). A short-ranged repulsive wall potential has been shown to induce steady oscillations in chemotactic microbots (Bayati & Najafi Reference Bayati and Najafi2019). Shen et al. (Reference Shen, Würger and Lintuvuori2018) demonstrated that the near-wall speed of Janus particles exceeds their bulk velocity, while Yang et al. (Reference Ybert, Barentin, Cottin-Bizonne, Joseph and Bocquet2016) reported that particle self-alignment within micropores arises from differential rotational motion governed by concentration gradients of surrounding chemical species. Furthermore, Leeth Holterhoff et al. (Reference Leeth Holterhoff, Li and Gibbs2018) found that reducing the surface zeta potential or increasing surface roughness can further enhance the propulsion of photoactive micromotors. Recent studies have also revealed that self-thermophoretic micromotors can switch their motion from escaping to approaching the wall (Poddar et al. Reference Poddar, Bandopadhyay and Chakraborty2021) and even exhibit backward motion contrary to their director orientation due to wall interactions (Poddar Reference Poddar2023).
At microscale separations, the nature of the confining surface strongly influences near-wall hydrodynamics. In lab-on-a-chip devices, hydrophobic or polymer-coated surfaces can sustain measurable interfacial slip lengths, ranging from tens of nanometres to several micrometres, due to the formation of gas nanolayers or self-assembled monolayers (Tretheway & Meinhart 2002, 2004; Chakraborty Reference Chakraborty2008; Maduar et al. Reference Maduar, Belyaev, Lobaskin and Vinogradova2015; Dey et al. Reference Dey, Saha and Chakraborty2020). In biologically relevant environments, surfaces such as mucosal linings are naturally hydrophobic and coated with complex macromolecular layers, leading to significant deviations from the classical no-slip assumption (Kasdorf et al. Reference Kasdorf, Weber, Petrou, Srivastava, Crouzier and Lieleg2017; Leal et al. Reference Leal, Smyth and Ghosh2017). For smooth hydrophobic surfaces, the slip length is typically small, of the order of a few tens of nanometres (Huang et al. Reference Huang, Sendner, Horinek, Netz and Bocquet2008; Bocquet & Charlaix Reference Bocquet and Charlaix2010). However, micrometre-scale effective slip lengths (
$l_s \sim 1$
–
$10 \, {\unicode[Arial]{x03BC}}$
m) are well documented in several practical situations. Surface chemical modifications, such as self-assembled monolayers of hydrophobic molecules, have been shown to increase slip lengths into the micrometre range (Tretheway & Meinhart Reference Tretheway and Meinhart2004; Lauga et al. Reference Lauga, Brenner and Stone2007; Trefethen & Bau Reference Trefethen and Bau2022). Moreover, in the case of specially treated nano- or microstructured surfaces, air bubbles can become trapped within the surface asperities, establishing a Cassie–Baxter state. Under this condition, the fluid experiences alternating patches of the solid wall (modelled as no-slip) and air–liquid interfaces (modelled as free-slip or low-friction partial-slip). The macroscopically averaged, effective hydrodynamic boundary condition can then be treated as a uniform partial slippage with an apparent slip length spanning up to several micrometres (Joseph et al. Reference Joseph, Cottin-Bizonne, Benoit, Ybert, Journet, Tabeling and Bocquet2006; Choi & Kim Reference Choi and Kim2006; Lee et al. Reference Lee, Choi and Kim2008; Asmolov et al. Reference Asmolov, Zhou, Schmid and Vinogradova2013; Nizkaya et al. Reference Nizkaya, Asmolov, Zhou, Schmid and Vinogradova2015). The growing interest in nanostructured and superhydrophobic surfaces for drag-reduction applications (Zhu et al. Reference Zhu, Yang and Guo2021; Feng et al. Reference Feng, Sun and Tian2022) increases the applicability of the present work. Importantly, the magnitude of this effective slip can be tuned through surface design parameters such as roughness geometry and wettability (Ybert et al. Reference Zhu and Granick2007; Asmolov et al. Reference Asmolov, Zhou, Schmid and Vinogradova2013).
The interaction between microswimmers, e.g. E. coli or synthetic Janus particles, and high-slip surfaces can be encountered in several practical applications. For example, superhydrophobic coatings can be used to prevent bacteria from settling on medical devices. Similarly, in lab-on-a-chip devices, ‘slippery’ and ‘sticky’ patches can be fabricated to steer or trap swimmers based on their hydrodynamic signatures. Also, for synthetic swimmers navigating the mucus-lined walls of the gastrointestinal tract or blood vessels, the biological coating acts as a slip-inducing lubricant (Wang et al. Reference Wei, Zhou, Tang and Wang2020). The interfacial slip can fundamentally alter near-wall flow fields and microswimming characteristics. For instance, it has been shown to reverse the circular trajectories of bacteria (Lemelle et al. Reference Lemelle, Palierne, Chatre, Vaillant and Place2013; Hu et al. Reference Hu, Wysocki, Winkler and Gompper2015). Experimental investigations have further demonstrated that micromotors exhibit substantially lower speed when the confining surface is coated with hydrophilic polyelectrolyte (Wei et al. Reference Willmott2018). The dependence of catalytic swimmer speed on the solution contact angle further indicates the role of a slippery surface in governing self-propulsion (Ketzetzi et al. Reference Ketzetzi, De Graaf, Doherty and Kraft2020).
Recent theoretical and numerical studies on microswimmer dynamics based on the spherical squirmer model have revealed slip-induced steady sliding, oscillations or rheotactic states (Poddar et al. Reference Poddar, Bandopadhyay and Chakraborty2020; Ghosh & Poddar Reference Ghosh and Poddar2023; Ghosh et al. Reference Ghosh, Ghoshal and Poddar2024). Numerical study on elongated swimmers reported that wall-bound sliding or oscillating states appear beyond a critical slip length (Guan et al. Reference Guan, Ying and Lin2024). However, a key limitation of the squirmer model used in theoretical studies is that it represents self-propulsion through prescribed surface squirming modes, without accounting for the near-wall redistribution of chemical species that is intrinsic to catalytic micromotors. Popescu et al. (Reference Popescu, Uspal, Eskandari, Tasinkevych and Dietrich2018) pointed out that, in the presence of other particles or boundaries, the behaviour of chemically active colloids can differ qualitatively. Even in the far field, their behaviour differs from that of the corresponding ‘effective squirmer’ obtained through mapping in an unbounded fluid. It therefore remains an open question as to how wall-induced distortions of the chemical field, together with hydrodynamic slip at the boundary, influence the near-wall behaviour of self-diffusiophoretic swimmers.
In this work, we address the aforementioned gaps in understanding by developing an exact theoretical model based on bispherical eigenfunction expansions for a self-diffusiophoretic Janus particle near an impenetrable slippery wall. The solute concentration field has been resolved in detail by imposing boundary conditions at both the particle surface and the wall. Instead of resolving the complete velocity profile, the Lorentz reciprocal theorem (Happel et al. Reference Happel and Brenner1983) is utilised to obtain the phoretic thrust force and torque responsible for particle motion. Subsequently, the translational and rotation velocities are obtained by invoking the force-free and torque-free conditions. Finally, the dynamics of the micromotors has been analysed using the dynamic systems theory and the quasi-steady trajectories are plotted for different initial conditions, catalytic coverage angles
$ (\theta _{\textit{cap}})$
and dimensionless hydrodynamic slip lengths
$ (l_s)$
.
Compared with the classical no-slip case for active particles (Uspal et al. Reference Uspal, Popescu, Dietrich and Tasinkevych2015b ; Mozaffari et al. Reference Mozaffari, Sharifi-Mood, Koplik and Maldarelli2016) or droplets (Lippera et al. Reference Lippera, Benzaquen and Michelin2020a ,Reference Lippera, Morozov, Benzaquen and Michelin b ), the inclusion of a Navier slip boundary condition reveals an array of rich dynamic characteristics, including an early onset of steady sliding, slip-induced particle–wall collisions and stable trapping states arising solely from wall slip. Moreover, several essential features of the dynamics cannot be addressed by lubrication or method-of-reflections arguments alone. Firstly, the critical slip length at which a micromotor transitions from a wall-reflected trajectory to a wall-collision trajectory occurs at intermediate separations (figure 3), beyond the domain of validity of either asymptotic method. Secondly, the topology of the regime map (figure 2), including the separatrix between distinct dynamical regimes, requires knowledge of the swimmer dynamics across all separations simultaneously, thereby remaining beyond the scope of lubrication-based or reflection-based analysis. Thirdly, the stable fixed points corresponding to hovering of the micromotor at a finite height above the wall (figure 9) emerge from a balance of hydrodynamic and phoretic effects at intermediate separations and cannot be predicted by lubrication theory alone.
In addition, we report that contrary to the earlier experimental observation of increased particle sliding speeds in the presence of wall slip (Ketzetzi et al. Reference Ketzetzi, De Graaf, Doherty and Kraft2020), the swimming speed in the present system may either increase or decrease depending on the relative contributions of slip-induced hydrodynamic effects and phoretically driven flows. We point out that the presently reported novel slip-mediated trapping states are observed without the need for an externally controllable scalar field (Poddar Reference Poddar2023) or shear flow (Ghosh & Poddar Reference Ghosh and Poddar2023; Ghosh et al. Reference Ghosh, Ghoshal and Poddar2024) and are an exclusive consequence of altered hydrodynamic resistance to flow due to surface hydrophobicity.
2. Mathematical formulation
We consider a spherical Janus micromotor of radius
$ a$
, suspended in a quiescent fluid of constant viscosity
$\mu$
, in proximity to a flat solid substrate, which deviates from a no-slip surface due to complex wettability characteristics. The micromotor catalytically generates a neutral solute over the active portion of its surface, defined by a polar angle
$ \theta _{\textit{cap}}$
. The remaining surface is inert and does not contribute to solute production. This asymmetric surface activity results in a non-uniform solute concentration field around the particle, as illustrated schematically in figure 1. The interaction between the solute and the particle surface drives self-diffusiophoretic propulsion. In the regime where the interaction layer thickness
$ \lambda$
is much smaller than the particle radius
$ a$
, the tangential gradient of solute concentration along the surface induces an effective surface flow velocity (Anderson Reference Anderson1989). For repulsive solute–surface interactions, this surface flow velocity is directed towards the catalytic cap, causing the micromotor to propel itself with the inert side leading. It is important to distinguish this surface flow velocity from the hydrodynamic slip velocity at a hydrophobic wall, so the term ‘slip velocity’ is avoided in this context to prevent confusion.
As shown in figure 1, the particle centre
$O'$
is positioned at a height
$\widetilde {h}$
. The diffusiophoretic thrust that acts on the micromotor is directed along
$ \boldsymbol{p}$
, which is inclined at a polar orientation angle
$ \theta _{p}$
with respect to the
$ z$
axis. The azimuthal orientation angle
$ \phi _p$
refers to the angle between the horizontal projection of
$ \boldsymbol{p}$
and the
$ x$
axis. Accordingly, the orientation vector in the fixed coordinate system can be expressed as
$ \boldsymbol{p} = \sin \theta _{p} \cos \phi _p \boldsymbol{e}_x + \sin \theta _{p} \sin \phi _p \boldsymbol{e}_y + \cos \theta _{p} \boldsymbol{e}_z$
. Here, we consider that the micromotor exhibits a purely axisymmetric self-diffusiophoresis in the unbounded domain. Thus, the azimuthal angle
$ \phi _p$
remains zero throughout the analysis. By convention, clockwise rotation about the
$ y$
axis is considered positive.
Schematic of a self-diffusiophoretic micromotor near a slippery surface. The angle
$ \angle AO'B = \theta _{\textit{cap}}$
denotes the extent of catalyst coverage on the inert particle. The parameter
$ \widetilde {l}_s$
represents the slip length at the boundary. The orientation of the micromotor is described by the unit vector
$ \boldsymbol{p}$
, which is inclined at an angle
$ \theta _p$
from the vertical axis.
Figure 1. Long description
A schematic of a self-diffusiophoretic micromotor near a slippery surface. The diagram includes a spherical particle with a catalytic surface and an inert surface. The angle theta_cap denotes the extent of catalyst coverage on the inert particle. The parameter tilde(l)_s represents the slip length at the boundary. The orientation of the micromotor is described by the unit vector p, which is inclined at an angle theta_p from the vertical axis. The micromotor is positioned near a slippery wall, and the diagram includes axes labeled x, y, and z. The catalytic surface is shown interacting with the environment, and the inert surface is depicted as non-reactive. The diagram illustrates the relationship between the micromotor’s orientation, the slippery surface, and the extent of catalyst coverage.
A key challenge in this problem arises from the simultaneous presence of planar and spherical geometries, which makes it difficult to enforce boundary conditions exactly within a single conventional coordinate system. To address this, we employ the bispherical coordinate system
$(\xi , \eta , \phi )$
, which is particularly well suited to problems involving a spherical body near a plane wall (Lee & Leal Reference Lee and Leal1980). In this framework, both the particle surface and the wall are represented by surfaces of constant coordinates, namely
$ \xi = \xi _0$
for the particle and
$\xi = 0$
for the wall. This feature allows the exact imposition of boundary conditions on both surfaces without requiring any geometric approximations. The transformation between bispherical and cylindrical coordinates
$ (\rho , \phi , z)$
is given by (Happel et al. Reference Happel and Brenner1983; Behera et al. Reference Behera, Poddar and Chakraborty2023)
\begin{align} {z} + {\rm i} {\rho } = {\rm i} \,c \cot \!\left ( \frac {1}{2} (\eta + {\rm i} \,\xi ) \right ) \!, \end{align}
where
$ c$
is a positive scale factor and
$ {\rm i} = \sqrt {-1}$
. The sphere has its centre located at
$ z=c \, \coth (\xi _0)$
and radius
$ c/\sinh (\xi _0)$
. Moreover, the radius of the spherical micromotor is given by
$a=c/\sinh (\xi _0)$
, while its centre
$\mathcal{O'}$
is located at a height
$\widetilde {h}=c\coth (\xi _0)$
from the plane surface. In addition,
$ \widetilde {\delta } = \widetilde {h}-a$
denotes the smallest distance (dimensionless) between the micromotor surface and the plane wall.
This approach eliminates the need to combine disparate asymptotic methods across different separation regimes, such as far-field image-singularity techniques (Spagnolie & Lauga Reference Spagnolie and Lauga2012; Lopez & Lauga Reference Lopez and Lauga2014) and near-field lubrication theory (Ishikawa, Simmonds & Pedley Reference Ishikawa, Simmonds and Pedley2006; Ruangkriengsin et al. Reference Ruangkriengsin, Turk and Stone2026). Lubrication theory is most effective when the gap
$\widetilde {\delta }$
between the particle and the wall is much smaller than the particle radius, i.e.
$\widetilde {\delta } \ll a$
. However, the reorientation mechanisms and the transition between reflection and collision often occur at intermediate separations (
$\widetilde {\delta }/a \sim 1$
) (Uspal et al.
Reference Uspal, Popescu, Dietrich and Tasinkevych2015b
; Mozaffari et al. Reference Mozaffari, Sharifi-Mood, Koplik and Maldarelli2016). On the other hand, the method of reflections typically relies on far-field approximations in terms of certain leading-order singularities (Ibrahim & Liverpool Reference Ibrahim and Liverpool2015) and is accurate only in the limit
$\widetilde {\delta } /a \gg 1$
. In contrast, the bispherical coordinate formulation provides a uniformly valid description across the full range of particle–wall separations.
Previous studies (Lippera et al. Reference Lippera, Benzaquen and Michelin2020a ,Reference Lippera, Morozov, Benzaquen and Michelin b ; Desai & Michelin Reference Desai and Michelin2021) have successfully captured the hydrodynamic interactions between active droplets and between an active droplet/particle and a non-slippery confining boundary using bispherical coordinate-based analyses. The corresponding dynamics in the presence of wall slip therefore remains unexplored, albeit of considerable practical relevance to superhydrophobic surfaces.
In addition, the regime of moderate-to-large slip lengths considered here cannot be captured by asymptotic perturbation approaches that assume vanishingly small slip lengths, as adopted in several earlier studies (Willmott Reference Yang, Qian, Zhao and Qiao2008; Swan & Brady Reference Swan and Brady2010; Lopez & Lauga Reference Lopez and Lauga2014). Furthermore, it has been noted in the literature that far-field approximations can overestimate near-field hydrodynamic interactions in certain cases, as highlighted by the numerical study of Hu et al. (Reference Hu, Wysocki, Winkler and Gompper2015). The present formulation therefore extends the applicability of bispherical methods to a broader and practically relevant parameter regime, where slip-induced modifications to swimmer dynamics can be substantial.
2.1. Concentration field
Two characteristic time scales govern the dynamics of transport of solutes in this system, namely (i) the diffusive time scale:
$ \tau _{\textit{diff}} = a^2 / D$
; and (ii) the advective time scale:
$ \tau _{\textit{ad}v} = a / \widetilde {u}_\infty$
, where
$ D$
is the diffusivity of the solute species and
$ \widetilde {u}_\infty$
is the self-propulsion velocity in the unbounded domain. Based on earlier experimental observations (Simmchen et al. Reference Simmchen, Katuri, Uspal, Popescu, Tasinkevych and Sánchez2016; Katuri et al. Reference Katuri, Uspal, Simmchen, Miguel-López and Sánchez2018), the micromotor radius typically is
$ a \sim 10^{-6} \, \text{m}$
, while its self-propulsion velocity in an unbounded fluid domain is of the order of
$ \tilde {u}_\infty \sim 10^{-6} \ \text{ to } 10^{-5}\, \text{m s}^{-1}$
. The diffusivity of the solute species involved is approximately
$ {D} \sim 10^{-9} \, \text{m}^2\,\text{s}^{-1}$
. To quantify the relative importance of the advection and diffusion in the transport process, we define the Péclet number as
$ \textit{Pe} =\tau _{\textit{diff}}/\tau _{\textit{ad}v} = {\tilde {u}_\infty a}/{D}$
. For a typical experimental scenario
$\textit{Pe} \sim 10^{-3} \text{ to } 10^{-2}$
. This low Péclet number confirms that the transport of solute around the micromotor is predominantly governed by diffusion. Further considering a quasi-steady-state, the solute concentration field
$ \widetilde {C}$
can be described by the Laplace equation:
\begin{align} \widetilde{\boldsymbol{\nabla}}^{2} \widetilde {C}= 0. \end{align}
The solute flux is non-zero only on the chemically active region of the micromotor, where it maintains a uniform magnitude
$ \widetilde {\mathcal{J}}$
. However, the flux vanishes on the inert region. Accordingly, the solute boundary condition at the micromotor surface can be expressed as
\begin{align} \text{at } \xi = \xi _0, \quad - \boldsymbol{n} \boldsymbol{\cdot } {\widetilde{\boldsymbol{\nabla}}} \widetilde {C} =\begin{cases} 0, &\text{if} \, \cos (\pi - \theta _{\textit{cap}}) \le \boldsymbol{{p} \boldsymbol{\cdot }{n}} \le 1 \\[4pt]\widetilde {\mathcal{J}}, &\text{otherwise}. \end{cases} \end{align}
Similarly, the inert nature of the nearby plane wall enforces a zero solute flux condition:
\begin{align} \text{at} \,\, \xi = 0, \quad \boldsymbol{e}_\xi \boldsymbol{\cdot } \widetilde{\boldsymbol{\nabla }} \widetilde {C} = 0. \end{align}
The above boundary conditions can be expressed in bispherical coordinates as
\begin{align} \text{at } \xi = \xi _0, \quad - \boldsymbol{n} \boldsymbol{\cdot } \widetilde{\boldsymbol{\nabla }} \widetilde {C} =\begin{cases} 0, &\text{if} \,\, \cos (\pi - \theta _{\textit{cap}}) \le \boldsymbol{{p} \boldsymbol{\cdot }{n}} \le 1 \\[4pt]\widetilde {\mathcal{J}}, &\text{otherwise} \end{cases} \end{align}
and
\begin{align} \text{at} \,\, \xi = 0, \quad \frac {\partial \widetilde {C}}{\partial \xi } = 0, \end{align}
respectively.
In order to non-dimensionalise the problem, we adopt the following reference scales:
$ \text{length} \sim a \, (\text{radius of the swimmer}) \, \text{and concentration} \sim { \widetilde {\mathcal{J}} a}/{D}$
, and subsequently drop the tilde symbol to denote the corresponding dimensionless variables. In bispherical coordinates, the general solution for the solute concentration field satisfying the Laplace equation (2.2) can be expressed as
\begin{align} C (\xi ,\eta ,\phi ) = \sinh (\xi _0) \sqrt {\cosh \xi - \cos \eta } & \sum _{m=0}^{\infty } \sum _{n=m}^{\infty } \Bigg \{ a_{n}^m \sinh \left ( \left ( n + \tfrac {1}{2} \right ) \xi \right ) \nonumber \\ &\quad + b_{n}^m \cosh \left ( \left ( n + \tfrac {1}{2} \right ) \xi \right ) \Bigg \} P_{n}^m(\cos \eta ) \cos (m\phi ), \end{align}
where
$P_n^m(\cos \eta )$
is the associated Legendre polynomial of order
$m$
and degree
$n$
.
The solute impermeability boundary condition at the plane wall (2.6) is satisfied for
\begin{align} a_n^m=0 \quad ( m,n \ge 0). \end{align}
We subsequently invoke the asymmetric reaction flux boundary condition at the particle–fluid interface (2.5) along with the recursion relations of the associated Legendre polynomials and appropriate orthogonality condition (Abramowitz & Stegun Reference Abramowitz and Stegun1948). This results in the following system of linear algebraic equations for the remaining unknown coefficients
$b_{n}^m$
:
\begin{align} &\Big [ {-} n \sinh \left (n - \tfrac {1}{2}\right )\xi _0 \, b_{n-1}^0 + \left (n \sinh \left (n - \tfrac {1}{2}\right )\xi _0 + (n+1)\sinh \left (n + \tfrac {3}{2}\right )\xi _0 \right )b_{n}^0 \nonumber \\ &\quad - (n+1)\sinh \left (n + \tfrac {3}{2}\right )\xi _0 \, b_{n+1}^0 \Big ] \nonumber \\ &= \frac {2n+1}{2\pi } \int _0^{2\pi } \int _0^\pi \sqrt {\cosh \xi _0 - \cos \eta } \sin \eta \, P_n^0(\cos \eta )\, \text{d}\eta \, \text{d}\phi \quad (m=0), \\[-12pt] \nonumber \end{align}
\begin{align} &\Big [ {-}(n-m)\sinh \left (n - \tfrac {1}{2}\right )\xi _0 \, b_{n-1}^m + \left (n \sinh \left (n - \tfrac {1}{2}\right )\xi _0 + (n+1)\sinh \left (n + \tfrac {3}{2}\right )\xi _0 \right )b_{n}^m \nonumber \\&\quad - (n+m+1)\sinh \left (n + \tfrac {3}{2}\right )\xi _0 \, b_{n+1}^m \Big ] \nonumber \\&= \frac {(2n+1)(n-m)!}{\pi (n+m)!} \!\! \int _0^{2\pi } \!\! \int _0^\pi \!\! \sqrt {\cosh \xi _0 - \cos \eta } \cos (m\phi ) \sin \eta \, P_n^m(\cos \eta )\, \text{d}\eta \, \text{d}\phi \quad (m \geqslant 1). \\[10pt] \nonumber \end{align}
In the axisymmetric orientations (
$\theta _p= 0^{\circ }$
and
$180^{\circ }$
), the solute concentration field does not depend on the azimuthal angle (
$\phi$
), and the expansion reduces to the
$m=0$
mode. The solved coefficients (
$b_n^m$
) from (2.9) and (2.10) become negligible at higher modes, which allows us to truncate the infinite series solution at sufficiently large values of
$m$
and
$n$
. For numerical implementation, the infinite series in (2.7) is truncated at a suitably large number
$ n=N_c$
, such that the relative difference between adjacent coefficients is smaller than
$ 10^{-6}$
, thereby ensuring numerical accuracy. This procedure gives rise to a sparse, banded system of linear algebraic equations of dimension
$3N_c \times 3N_c$
, which is solved numerically to obtain the coefficients
$b_{n}^{m}$
and thereby reconstruct the solute concentration field using (2.7).
2.2. Hydrodynamics of near-wall self-diffusiophoresis
The flow around micromotors occurs in the low-Reynolds-number regime, where inertial effects are negligible and viscous forces dominate. Under these conditions, the flow of the surrounding fluid is accurately described by the Stokes flow approximation (Lauga & Powers Reference Lauga and Powers2009), combined with the incompressibility condition:
\begin{align} - \widetilde {\boldsymbol{\nabla }} p + \mu \boldsymbol{\nabla} ^2 \widetilde{\boldsymbol{u}} = \boldsymbol{0} \quad \text{and} \quad \widetilde{\boldsymbol{\nabla }} \boldsymbol{\cdot } \widetilde {\boldsymbol{u}} = 0. \end{align}
The no-penetration condition holds at the plane surface due to the impermeability condition. Therefore, the fluid velocity at the plane wall satisfies
\begin{align} \text{at} \,\, \xi = 0, \quad \widetilde {\boldsymbol{u}}\boldsymbol{\cdot }\boldsymbol{n} = 0. \end{align}
In the cylindrical coordinates, the above condition can be stated as
\begin{align} \text{at} \,\, \xi = 0, \quad \widetilde {u}_z = 0. \end{align}
We note that a key advantage of the bispherical eigenfunction approach is the availability of an exact analytical solution to the Stokes equations in the literature (Lee & Leal Reference Lee and Leal1980). This solution naturally provides the velocity components in cylindrical coordinates, with the spatial dependence expressed through bispherical coordinates, as presented in (2.26)–(2.28). Therefore, it is straightforward to use the velocity components in cylindrical coordinates while enforcing the boundary conditions in the bispherical eigenfunction approach. Notably, recasting the velocity field in bispherical coordinate components would offer no additional computational advantage. This approach is widely adopted in related studies (Mozaffari et al. Reference Mozaffari, Sharifi-Mood, Koplik and Maldarelli2016; Shaik & Ardekani Reference Shaik and Ardekani2017; Poddar et al. Reference Poddar, Bandopadhyay and Chakraborty2020).
The velocity boundary condition at the micromotor surface, defined with respect to the stationary (laboratory) reference frame, is given by
\begin{align} \text{at } \xi = \xi _0, \qquad \widetilde {\boldsymbol{u}} = \widetilde {\boldsymbol{V}} + \widetilde {\boldsymbol{\varOmega }} \times \widetilde {\boldsymbol{r}}_{O'} + \widetilde {\boldsymbol{u}}_s. \end{align}
In the above equation,
$ \widetilde {\boldsymbol{r}}_{O'}$
denotes the position vector with respect to the centre of the micromotor, whereas
$ \widetilde {\boldsymbol{V}}$
and
$ \widetilde {\boldsymbol{\varOmega }}$
denote the translational and rotational velocities of the micromotor, respectively. A tangential gradient in solute concentration along the micromotor surface induces an osmotic pressure imbalance in the surrounding fluid, giving rise to a localised diffusioosmotic slip velocity (Moran & Posner Reference Moran and Posner2017). This surface slip is linearly related to the tangential concentration gradient and can be expressed as
\begin{align} \widetilde {\boldsymbol{u}}_s = \mathcal{M} (\boldsymbol{\mathbb{I}-{n}{n}}) \boldsymbol{\cdot }\widetilde{\boldsymbol{\nabla }} \widetilde {C}, \end{align}
where
$ \mathbb{I}$
is the identity matrix and
$ \mathcal{M}$
is the diffusiophoretic mobility. The sign of
$ \mathcal{M}$
depends on the nature of solute–surface interactions. In this study, we adopt
$ \mathcal{M} \lt 0$
(Uspal et al.
Reference Uspal, Popescu, Dietrich and Tasinkevych2015b
; Mozaffari et al. Reference Mozaffari, Sharifi-Mood, Koplik and Maldarelli2016), indicating a repulsive interaction. Under this condition, the resulting slip velocity is oriented from regions of lower solute concentration towards regions of higher concentration along the micromotor surface. In the bispherical coordinates, (2.15) can be written as
\begin{align} \widetilde {\boldsymbol{u}}_s = \frac {\mathcal{M}(\cosh \xi -\cos \eta )}{c} \left (\frac {\partial \widetilde {C} }{\partial \eta }\,\boldsymbol{e}_\eta + \frac {1}{\sin \eta } \frac {\partial \widetilde {C} }{\partial \phi }\,\boldsymbol{e}_\phi \right ) \!. \end{align}
Thus, the above equation immediately provides the surface velocity components in bispherical coordinates. The velocity components in the bispherical coordinate system can be converted to cylindrical coordinates
$ (\rho , \phi , z)$
by using the following relations:
\begin{align} \begin{aligned} \widetilde {u}_{\xi } &= - \frac {\widetilde {u}_{\rho } \sinh (\xi ) \sin (\eta )}{\cosh (\xi ) - \cos (\eta )} + \frac {\widetilde {u}_{z} \left (1 - \cosh (\xi ) \cos (\eta )\right )}{\cosh (\xi ) - \cos (\eta )}, \\[1em] \widetilde {u}_{\eta } & = - \frac {\widetilde {u}_{z} \sinh (\xi ) \sin (\eta )}{\cosh (\xi ) - \cos (\eta )} - \frac {\widetilde {u}_{\rho } \left (1 - \cosh (\xi ) \cos (\eta )\right )}{\cosh (\xi ) - \cos (\eta )}. \end{aligned} \end{align}
The hydrodynamic slippage at the solid boundary is described through the Navier slip boundary condition (Navier Reference Navier1823), given by
\begin{align} \text{at } \xi = 0, \qquad \widetilde {\boldsymbol{u}}_t = \widetilde {l}_s \boldsymbol{n}_w \boldsymbol{\cdot } \left ( \widetilde {\boldsymbol{\nabla }} \widetilde {\boldsymbol{u}} + \left (\widetilde {\boldsymbol{\nabla }} \widetilde {\boldsymbol{u}}\right )^{T} \right ) \boldsymbol{\cdot } \left ( \mathbb{I} - \boldsymbol{n}_w \boldsymbol{n}_w \right ), \end{align}
where
$\boldsymbol{n}_w$
is the outward unit normal at the wall and
$\widetilde {\boldsymbol{u}}_t$
denotes the tangential velocity component at the boundary. The parameter
$\widetilde {l}_s$
is known as the slip length. It quantifies the degree of hydrodynamic slippage by indicating the hypothetical distance beneath the wall at which the extrapolated tangential velocity would reduce to zero (see figure 1). Together with the no-penetration condition at the wall, this yields
\begin{align} \text{at } \xi = 0, \qquad \widetilde {u}_{\rho }= \widetilde {l}_{\!_S}\frac {\partial {\widetilde {u}_{\rho }}}{\partial \widetilde {z}} \qquad \text{and} \qquad \widetilde {u}_{\phi }= \widetilde {l}_{\!_S}\frac {\partial {\widetilde {u}_{\phi }}}{\partial \widetilde {z}}. \end{align}
These expressions indicate that the tangential velocity components are not constants but satisfy mixed (Robin-type) boundary conditions, fundamentally distinguishing the Navier slip condition from a simple sliding boundary. Equation (2.19) is subsequently written as
\begin{align} \begin{aligned} \text{at } \xi = 0, \qquad \widetilde {u}_\rho &= \frac {\widetilde {l}_{\!_S}}{a} \left [ \left ( 1 - \cos (\eta ) \cosh (\xi )\right ) \frac {\partial \widetilde {u}_\rho }{\partial \xi } - \sin (\eta ) \sinh (\xi ) \frac {\partial \widetilde {u}_\rho }{\partial \eta } \right ] \!, \\[1em]\widetilde {u}_\phi &= \frac {\widetilde {l}_{\!_S}}{a} \left [ \left ( 1 - \cos (\eta ) \cosh (\xi )\right ) \frac {\partial \widetilde {u}_\phi }{\partial \xi } - \sin (\eta ) \sinh (\xi ) \frac {\partial \widetilde {u}_\phi }{\partial \eta } \right ]\!. \end{aligned} \end{align}
In is noteworthy that the Faxén-law-based reflection approach for a no-slip plane wall (Ibrahim & Liverpool Reference Ibrahim and Liverpool2015) cannot be extended to the present problem. In such an approach, the Faxén laws are used to calculate the velocity components using a reflection-theorem-based analysis, where the background flow originates from the distorted flow field due to the wall at each order of reflection (Kim & Karrila Reference Kim and Karrila2013). Subsequently, corrections to the bulk velocities are calculated at each order of reflection (Ibrahim & Liverpool Reference Ibrahim and Liverpool2015). The practical hurdle in adopting a similar formulation in the present case is the satisfying of the mixed type of boundary condition at the plane wall with an arbitrary slip length
$ (l_s)$
and deriving the required Green’s function for the Stokes equations in a half-space with Navier slip. Consequently, a simple image singularity system cannot be obtained. In stark contrast, the present work employs an exact bispherical coordinate method, in which the boundary conditions are imposed simultaneously and exactly at both the spherical surface
$(\xi = \xi _0)$
and the planar wall
$(\xi = 0)$
, going beyond the asymptotic approximation used in a reflection method.
In this study, the governing equations are non-dimensionalised using the characteristic velocity of the freely moving micromotor in an unbounded medium (Mozaffari et al. Reference Mozaffari, Sharifi-Mood, Koplik and Maldarelli2016)
$ \widetilde {u}_\infty = ( {|\mathcal{M}| \widetilde {C}_0 (1 - \cos ^2 \theta _{\textit{cap}} )})/({4D}),$
while the pressure is scaled by
$\mu \widetilde {u}_\infty / a$
. Subsequently, the tilde notation is omitted for convenience, and all variables are expressed in their dimensionless form.
It is worth mentioning that mapping the present system onto a standard squirmer model (Ishikawa et al. Reference Ishikawa, Simmonds and Pedley2006; Poddar et al. Reference Poddar, Bandopadhyay and Chakraborty2020) is insufficient for capturing the essential features of the dynamics near a wall for the following reasons. In the squirmer model, the velocity boundary condition on the particle surface is fixed and prescribed using a squirmer parameter
$ (\beta )$
. However, for a diffusiophoretic swimmer near a boundary, the wall introduces a dual effect. (i) Hydrodynamic interaction: the wall reflects and modifies the flow field generated by the swimmer. This aspect is indeed captured within the squirmer framework. (ii) Chemical field distortion: the wall imposes a boundary condition on the diffusion equation (
$\boldsymbol{\nabla} ^2 C = 0$
), which alters the concentration distribution relative to the unbounded case. Since the phoretic slip satisfies
$u_s \propto \boldsymbol{\nabla }C$
, the propulsion mechanism is directly affected by this wall-induced modification. As a result, the effective squirmer parameter
$\beta$
is no longer an intrinsic property of the swimmer but becomes a function of the swimmer–wall separation
$ (\delta )$
. This dependence is absent in the classical squirmer model, where the surface slip is independent of the environment. The chemical gradient induced by the wall can create long-range torques and forces that are absent in the prescribed-slip squirmer model (Ibrahim & Liverpool Reference Ibrahim and Liverpool2015). Therefore, although a mapping
$\theta _{\textit{cap}} \rightarrow \beta$
is valid in an unbounded domain (Bickel et al. Reference Bickel, Majee and Würger2013; Michelin & Lauga Reference Michelin and Lauga2014), prior studies (Ibrahim & Liverpool Reference Ibrahim and Liverpool2015; Popescu et al. Reference Popescu, Uspal, Eskandari, Tasinkevych and Dietrich2018) have shown that near boundaries, chemically active particles exhibit qualitative deviations from their squirmer counterparts. Thus, the present work cannot be reduced to our earlier squirmer-based works (Poddar et al. Reference Poddar, Bandopadhyay and Chakraborty2020; Ghosh et al. Reference Ghosh, Ghoshal and Poddar2024) just by switching off some terms.
2.3. Determination of velocity components of the micromotor
The unknown velocity components of the micromotor
$ \boldsymbol{V}$
and
$ \boldsymbol{\varOmega }$
can be calculated by considering the equation of motion of the micromotor. In the case of negligible inertia and neutral buoyancy in the surrounding fluid, the micromotor satisfies the following force- and torque-free conditions:
\begin{align} \boldsymbol{F}= \iint \limits _{S_p}^{} \boldsymbol{\sigma } \boldsymbol{\cdot }\boldsymbol{n}\, \text{d}S=\boldsymbol{0} \quad \text{and} \quad \boldsymbol{G}= \iint \limits _{S_p}^{} \boldsymbol{r}_{O'} \times (\boldsymbol{\sigma }\boldsymbol{\cdot }\boldsymbol{n})\, \text{d}S=\boldsymbol{0}, \end{align}
where
$ S_p$
represents the surface of the swimmer,
$ \boldsymbol{n}$
denotes the outward unit normal vector on
$ S_p$
and
$ \boldsymbol{\sigma }$
signifies the stress tensor of the surrounding fluid.
The hydrodynamics is split into two subproblems:
-
(i) Thrust problem. This considers the flow produced by the diffusioosmotic slip velocity on the swimmer surface while the swimmer is held fixed.
-
(ii) Drag problem. This treats the swimmer as a rigid body undergoing translation and rotation and accounts only for the viscous resistance (drag) exerted by the surrounding fluid.
The total force and torque are therefore the sum of the thrust and drag contributions. Therefore, the force- and torque-free conditions (2.21) can be written as
\begin{align} \boldsymbol{F}^{(\textit{Drag})}+\boldsymbol{F}^{(\textit{Thrust})}= \boldsymbol{0} \quad \mathrm{and} \quad \boldsymbol{G}^{(\textit{Drag})}+\boldsymbol{G}^{(\textit{Thrust})}= \boldsymbol{0}. \quad \end{align}
The drag force and torque on a particle translating and rotating near a wall may be obtained following earlier treatments of passive particle motion (Dean & O’Neill Reference Dean and O’Neill1963; O’Neill Reference O’Neill1964; Pasol et al. Reference Pasol, Chaoui, Yahiaoui and Feuillebois2005), and are therefore not reproduced here. These results for unit translation and unit rotation velocities
$ (V_x=V_z=\varOmega _y=1)$
are used as the resistance factors
$\boldsymbol{\mathcal{R}}_T$
,
$\boldsymbol{\mathcal{R}}_C$
and
$\boldsymbol{\mathcal{R}}_R$
in the present problem. Thus, the drag force and torque acting on a swimmer with unknown velocity components can be expressed as
\begin{align} \boldsymbol{F}^{(\textit{Drag})} &= \boldsymbol{\mathcal{R}}_T \boldsymbol{\cdot }\boldsymbol{V} + \boldsymbol{\mathcal{R}}_C^{T} \boldsymbol{\cdot }\boldsymbol{\varOmega }, \\[-12pt] \nonumber \end{align}
\begin{align} \boldsymbol{G}^{(\textit{Drag})} &= \boldsymbol{\mathcal{R}}_C \boldsymbol{\cdot }\boldsymbol{V} + \boldsymbol{\mathcal{R}}_R \boldsymbol{\cdot }\boldsymbol{\varOmega }. \\[10pt] \nonumber \end{align}
We employ the Lorentz reciprocal theorem (Happel et al. Reference Happel and Brenner1983) to directly compute the thrust components without solving the complete Stokes flow field associated with the ‘thrust’ problem. According to the Lorentz reciprocal theorem (Happel et al. Reference Happel and Brenner1983), the relation between the two flow fields of the ‘thrust’ problem and an ‘auxiliary’ Stokes problem having the same geometry but different boundary condition can be expressed as
\begin{align} \iint _{\partial S} \boldsymbol{n} \boldsymbol{\cdot }\boldsymbol{\sigma }^{(\textit{Thrust})} \boldsymbol{\cdot }\boldsymbol{u}^{(\textit{Aux})}\, \text{d}S = \iint _{\partial S} \boldsymbol{n} \boldsymbol{\cdot }\boldsymbol{\sigma }^{(\textit{Aux})} \boldsymbol{\cdot }\boldsymbol{u}^{(\textit{Thrust})}\, \text{d}S, \end{align}
where
$\partial S$
denotes the boundary enclosing the fluid domain. Considering the no-slip condition for the fluid velocity at the planar wall and the decay of the flow field at large distances from the micromotor, the surface integral can be evaluated solely over the particle surface
$S_p$
, replacing
$\partial S$
in (2.24).
Further, owing to the symmetric distribution of the metallic coating, the translation of the micromotor is confined to the plane defined by the wall normal and the particle symmetry axis. At the same time, the particle is free to rotate about an axis that is perpendicular to both the wall normal (
$\boldsymbol{e}_z$
) and the micromotor orientation (
$\boldsymbol{p}$
). Hence, the micromotor motion can be completely described by the velocity components
$V_x$
,
$V_z$
and
$\varOmega _y$
. Accordingly, to evaluate the thrust components, we consider the following three auxiliary Stokes problems corresponding to a passively driven spherical particle near a wall with a hydrodynamic slip boundary condition:
-
(i) translation of the particle in the
$+x$
direction with a unit translational velocity, -
(ii) translation of the particle in the
$+z$
direction with a unit translational velocity, -
(iii) rotation of the particle in the
$+y$
direction with a unit rotational velocity.
A detailed analysis of these problems can be found in earlier works (Goren Reference Goren1973; Loussaief et al. Reference Loussaief, Pasol and Feuillebois2015).
Exploiting the reciprocal theorem (2.24), we can calculate the thrust components as follows:
\begin{align} \begin{aligned} F^{(\textit{Thrust})}_{\textit{Aux},\,A} &= \iint _{S_p} \boldsymbol{n} \boldsymbol{\cdot }\boldsymbol{\sigma }_{\textit{Aux},\,A} \boldsymbol{\cdot }\boldsymbol{u}_s \, \text{d}S, \\F^{(\textit{Thrust})}_{\textit{Aux},\,B} &= \iint _{S_p} \boldsymbol{n} \boldsymbol{\cdot }\boldsymbol{\sigma }_{\textit{Aux},\,B} \boldsymbol{\cdot }\boldsymbol{u}_s \, \text{d}S, \\G^{(\textit{Thrust})}_{\textit{Aux},\,C} &= \iint _{S_p} \boldsymbol{n} \boldsymbol{\cdot }\boldsymbol{\sigma }_{\textit{Aux},\,C} \boldsymbol{\cdot }\boldsymbol{u}_s \, \text{d}S, \end{aligned} \end{align}
where
$\boldsymbol{u}_s$
the represents the slip velocity on the surface of a fixed swimmer (2.15). We can calculate the velocity components of the micromotor using (2.25) and (2.23) in (2.22).
We solve the auxiliary problems by using the solution of Stokes flow in bispherical coordinates. The expressions of the pressure and velocity field of the fluid in the cylindrical coordinates
$ (u_{\rho },u_{\phi },u_{z})$
in terms of eigenfunctions in the bispherical coordinates are given by (Lee & Leal Reference Lee and Leal1980)
\begin{align} \begin{aligned} u_\rho (\xi ,\eta ,\phi ) &= \frac {\rho }{2}\sum _{m=0}^{\infty }\Bigg \{\frac {1}{c}\sqrt {\cosh \xi -\zeta } \sum _{n=m}^{\infty }\big [A_n^{m}\sinh (\beta _n\xi )+B_n^{m}\cosh (\beta _n\xi )\big ]P_n^{m}(\zeta )\Bigg \}\cos \big (m\phi \big )\\ &\quad + \,\sqrt {\cosh \xi -\zeta }\sum _{n=1}^{\infty }\big [E_n^{0}\sin (\beta _n\xi )+F_n^{0}\cosh (\beta _n\xi )\big ]P_n^{1}(\zeta )\\ &\quad + \frac {\sqrt {\cosh \xi -\zeta }}{2}\sum _{m=1}^{\infty }\Bigg \{\sum _{n=m+1}^{\infty }\big [E_n^{m}\sin (\beta _n\xi )+F_n^{m}\cosh (\beta _n\xi )\big ]P_n^{m+1}(\zeta ) \\ &\qquad \qquad \qquad \qquad \qquad \qquad +\ \sum _{n=m-1}^{\infty }\big [G_n^{m}\sin (\beta _n\xi )+H_n^{m}\cosh (\beta _n\xi )\big ]P_n^{\,m-1}(\zeta )\Bigg \}\\ &\qquad \qquad \qquad \qquad \qquad \qquad \times \cos \big (m\phi \big ), \end{aligned} \\[-28pt] \nonumber \end{align}
\begin{align} \begin{aligned} u_\phi (\xi ,\eta ,\phi ) = &\frac {\sqrt {\cosh \xi -\zeta }}{2} \sum _{m=1}^{\infty }\Bigg \{\sum _{n=m+1}^{\infty }\big [E_n^{m}\sin (\beta _n\xi )+F_n^{m}\cosh (\beta _n\xi )\big ]P_n^{m+1}(\zeta ) \\ &\qquad \qquad \qquad \qquad \qquad \qquad -\ \sum _{n=m-1}^{\infty }\big [G_n^{m}\sin (\beta _n\xi )+H_n^{m}\cosh (\beta _n\xi )\big ]P_n^{\,m-1}(\zeta )\Bigg \}\\ &\qquad \qquad \qquad \qquad \qquad \qquad \times \sin \big (m\phi \big ), \end{aligned} \\[-28pt] \nonumber \end{align}
\begin{align} \begin{aligned} u_z(\xi ,\eta ,\phi ) &=\sqrt {\cosh \xi -\zeta }\sum _{m=0}^{\infty }\Bigg \{\frac {z}{2c} \sum _{n=m}^{\infty }\big [A_n^{m}\sinh (\beta _n\xi )+B_n^{m}\cosh (\beta _n\xi )\big ]P_n^{m}(\zeta )\\ &\qquad \qquad \qquad \qquad \qquad \qquad +\ \sum _{n=m}^{\infty }C_n^{m}\sin (\beta _n\xi )P_n^{m}(\zeta )\Bigg \}\cos \big (m\phi \big ), \end{aligned} \\[-12pt] \nonumber \end{align}
where
$ \zeta =\cos (\eta )$
.
The general solution involves seven sets of unknown coefficients, namely
$A_n^m, B_n^m, C_n^m, E_n^m, F_n^m, G_n^m$
and
$H_n^m$
, which are determined by enforcing the boundary conditions at the swimmer surface (2.14), the Navier slip condition at the planar wall (2.20), together with the incompressibility constraint. To impose the swimmer surface boundary condition for the auxiliary Stokes problems, the surface velocity components are expanded in a bispherical eigenfunction basis:
\begin{align} u_{s,\rho }= \sum _{m} u_{s,\rho }^m (\xi ,\eta ) \cos (m\phi +\alpha _m), \\[-28pt] \nonumber \end{align}
\begin{align} u_{s,\phi }= \sum _{m} u_{s,\phi }^m (\xi ,\eta ) \sin (m\phi +\alpha _m), \\[-28pt] \nonumber \end{align}
\begin{align} u_{s,z}= \sum _{m} u_{s,z}^m (\xi ,\eta ) \cos (m\phi +\alpha _m). \\[-12pt] \nonumber \end{align}
For the axisymmetric mode (
$m = 0$
), the velocity components take the form
\begin{align} u_{s,\rho }^0 &= \sqrt {\cosh (\xi _0) - \zeta }\,\, \sum _{n} X_n^0(\xi )\, P_n^{1}(\zeta ), \\[-12pt] \nonumber \end{align}
\begin{align} u_{s,\phi }^0 &= \sqrt {\cosh (\xi _0) - \zeta }\,\, \sum _{n} Y_n^0(\xi )\, P_n^{1}(\zeta ). \\[8pt] \nonumber \end{align}
For non-axisymmetric modes (
$m \ge 1$
), it is convenient to consider the following expansions:
\begin{align} u_{s,\rho }^m + u_{s,\phi }^m &= \sqrt {\cosh (\xi _0) - \zeta }\,\, \sum _{n} X_n^m(\xi )\, P_n^{m+1}(\zeta ), \\[-12pt] \nonumber \end{align}
\begin{align} u_{s,\rho }^m - u_{s,\phi }^m &= \sqrt {\cosh (\xi _0) - \zeta }\,\, \sum _{n} Y_n^m(\xi )\, P_n^{m-1}(\zeta ), \\[-12pt] \nonumber \end{align}
\begin{align} u_{s,z}^m & = \sqrt {\cosh (\xi _0) - \zeta }\,\, \sum _{n} Z_n^m(\xi )\, P_n^{m}(\zeta ). \\[10pt] \nonumber \end{align}
The coefficients
$X_n^m$
,
$Y_n^m$
and
$Z_n^m$
are determined by enforcing the boundary conditions at the swimmer surface, and thus do not depend on the hydrodynamic slip length at the plane surface. For brevity, the explicit forms of these constants for all the auxiliary problems are omitted, and the reader is referred to Lee & Leal (Reference Lee and Leal1980). It is important to note that the bispherical basis functions do not automatically satisfy the boundary conditions at
$\xi = 0$
and
$\xi = \xi _0$
and the modal coefficients are coupled with each other. Thus, we use a spectral convergence approach, as described below. By exploiting the orthogonality of the associated Legendre polynomials, the boundary conditions are projected onto the basis of associated Legendre polynomials. This procedure results in an infinite system of coupled linear algebraic equations for the unknown modal coefficients. In practice, this system is truncated at a finite order
$N_h$
, such that a prescribed convergence tolerance (typically of
$\mathcal{O}(10^{-4})$
) is achieved based on successive estimates of the coefficients. The system of equations has a banded matrix structure
$ (7 N_h\times 7 N_h)$
, which can then be solved numerically. The well-posedness of the problem is ensured by the physical boundary conditions imposed on each surface, which together uniquely determine the solution. In practice, we verified that the truncated system yields a well-conditioned coefficient matrix for the range of parameters considered, and that the results are numerically converged. No additional constraints beyond the standard boundary conditions were required to close the system.
Notably, in the case of a no-slip planar wall, the
$7N_h \times 7N_h$
system can be significantly simplified, as all unknown constants can be expressed in terms of a single coefficient. This reduces the problem to solving a system of size
$N_h \times N_h$
(O’Neill Reference O’Neill1964). In contrast, the presence of a Navier slip boundary condition at the plane wall prevents such a simplification. Nevertheless, the system can still be reduced to a smaller linear system of size
$4N_h \times 4N_h$
by following the procedure outlined in Loussaief et al. (Reference Loussaief, Pasol and Feuillebois2015). As a result, the four coupled coefficients
$A_n^m$
,
$C_n^m$
,
$E_n^m$
and
$G_n^m$
are determined directly from the matrix inversion, while the remaining coefficients are obtained through explicit relations.
Moreover, inclusion of wall slip
$ (l_s)$
weakens the damping of higher-order modes, which leads to a slower decay of the series coefficients for the narrow gaps (
$ \delta$
). As a result, the convergence rate deteriorates, and a larger number of terms (
$N_h$
) must be retained to achieve the desired accuracy, resulting in larger computational time. In addition, the convergence of the micromotor velocity components is sensitive to the truncation orders used for the concentration and flow-field expansions, denoted by
$N_c$
and
$N_h$
, respectively. Too high a value of the highest mode
$ N_k (k=c,h)$
may not ensure convergence due to the onset of round-off error. For small values of
$ N_k$
, the total numerical error is dominated by truncation and increasing
$ N_k$
reduces total error. After a certain point, a further increase of
$ N_k$
yields diminishing improvements in truncation error while round-off starts to grow faster, causing the total error to increase eventually. Thus the total numerical error, being the sum of truncation and round-off contributions (Saad Reference Saad2003; Trefethen & Bau Reference Trefethen and Bau2022), exhibits an intermediate optimal range of
$N_k$
, where the error is minimised. This phenomenon is shown in figure 13 through some example cases. Accordingly, we performed a systematic mode-independence study by varying both
$N_c$
and
$N_h$
. We found that the required
$N_c$
increases primarily as the particle–wall separation
$\delta$
decreases, whereas the required
$N_h$
depends additionally on the hydrodynamic slip length
$l_s$
: larger
$l_s$
necessitates larger
$N_h$
. In the computations reported here,
$N_c=60$
was sufficient to attain the target accuracy, while
$N_h$
needed to be increased to 3000 to ensure convergence in cases with large slip.
To perform the calculations in (2.25), we make use of the relations in (2.20) to convert the velocity components obtained in cylindrical coordinates
$ (\rho , \phi , z)$
to the bispherical coordinate system
$ (\xi , \eta , \phi )$
. Subsequently, the deviatoric stress tensor components in the bispherical system are derived as
\begin{align} \begin{aligned} \sigma _{\xi \eta } &= \mu \left [ \frac {\partial }{\partial \eta } \left ( \frac {\cosh \xi - \cos \eta }{c} u_{\xi } \right ) + \frac {\partial }{\partial \xi } \left ( \frac {\cosh \xi - \cos \eta }{c} u_{\eta } \right ) \right ]\!, \\[1em] \sigma _{\xi \phi } &= \mu \left [ \frac {\cosh \xi - \cos \eta }{c \sin \eta } \frac {\partial u_{\xi }}{\partial \phi } + \frac {\partial }{\partial \xi } \left ( \frac {\cosh \xi - \cos \eta }{c} u_{\phi } \right ) \right ]\!. \end{aligned} \end{align}
We substitute (2.26)–(2.28) in (2.17), and then calculate the expansions of the stress tensor components shown in (2.35). These surface velocity and stress tensor components are subsequently used in (2.25) to calculate the phoretic thrust components on the micromotor.
The vertical component of the translational velocity (
$V_z$
) is obtained from the force-free condition (2.22) in the form
\begin{align} V_z = -\frac {F_z^{(\textit{Thrust})}}{f_z^{T}}, \end{align}
where
$f_z^{T}$
represents the translational resistance coefficient for motion perpendicular to the wall. Since
$f_z^{T}$
is strictly positive, any reversal in the direction of
$V_z$
is determined entirely by the sign of the vertical phoretic thrust
$F_z^{(\textit{Thrust})}$
only. In contrast, the translational and rotational velocities parallel to the wall,
$V_x$
and
$\varOmega _y$
, are calculated by solving a coupled system of force- and torque-free conditions (2.22), which yield
\begin{gather} V_x = \frac {F_x^{(\textit{Thrust})}\, f_{y}^{R} - C_y^{(\textit{Thrust})}\, f_{x}^{R}}{f_{x}^{R} f_{y}^{T} - f_{x}^{T} f_{y}^{R}}, \\[-24pt] \nonumber \end{gather}
\begin{gather} \varOmega _y = \frac {C_y^{(\textit{Thrust})}\, f_{x}^{T} - F_x^{(\textit{Thrust})}\, f_{y}^{T}}{f_{x}^{R} f_{y}^{T} - f_{x}^{T} f_{y}^{R}}. \\[0pt] \nonumber \end{gather}
Here, the hydrodynamic resistance coefficients entering the above expressions relate the drag forces and torques to the micromotor velocity components through the relations
\begin{align} F_{x}^{(\textit{Drag},T)} = f_{x}^{T} V_x, \quad F_{x}^{(\textit{Drag},R)} = f_{x}^{R} \varOmega _y, \quad C_{y}^{(\textit{Drag},T)} = f_{y}^{T} V_x, \quad C_{y}^{(\textit{Drag},R)} = f_{y}^{R} \varOmega _y. \end{align}
2.4. Phase-plane dynamics and trajectories
The
$x$
–
$z$
plane symmetry eliminates all out-of-plane degrees of freedom, and the configuration of the micromotor is fully described by
\begin{align} \boldsymbol{q}(t) = \bigl (x(t),\,\, h(t),\,\, \theta _p(t)\bigr ). \end{align}
Now, the dynamics of the micromotor is derived from the equations of motion applied to an inertialess micromotor at a position
$ (x(t), h(t))$
and orientation
$ (\theta _p(t))$
(Shum et al. Reference Shum, Gaffney and Smith2010; Ishimoto & Gaffney Reference Ishimoto and Gaffney2013):
\begin{align} \dot {x} &= V_x(h, \theta _p;\, l_s, \theta _{\textit{cap}}), \\[-10pt] \nonumber \end{align}
\begin{align} \dot {h} &= V_z(h, \theta _p;\, l_s, \theta _{\textit{cap}}), \\[-10pt] \nonumber \end{align}
\begin{align} \dot {\theta }_p &= \varOmega _y(h, \theta _p;\, l_s, \theta _{\textit{cap}}), \\[10pt] \nonumber \end{align}
where
$l_s$
and
$\theta _{\textit{cap}}$
appear as parameters. For calculating the time evolution of the micromotor configuration, the above equations have to be integrated for a given initial state
$ (x_0,z_0,\theta _{p,0})$
. At each time
$ t$
, expressions on the right-hand sides of the above equations are calculated using the exact solution approach described in §§ 2.2 and 2.3. In this analysis, we disregard stochastic perturbations such as thermal noise and focus solely on deterministic dynamics, consistent with previous studies on near-wall swimmer kinematics (Shum et al. Reference Shum, Gaffney and Smith2010; Spagnolie & Lauga Reference Spagnolie and Lauga2012; Mozaffari et al. Reference Mozaffari, Sharifi-Mood, Koplik and Maldarelli2016; Poddar et al. Reference Poddar, Bandopadhyay and Chakraborty2021).
A comprehensive analysis of micromotor dynamics depends on the key governing parameters, namely the hydrodynamic slip length (
$l_s$
) and the metallic cap angle (
$\theta _{\textit{cap}}$
) as well as on various initial configurations
$(h_0, \theta _{p,0})$
. This would, in principle, require an extensive number of time-resolved simulations, as reported earlier (Mozaffari et al. Reference Mozaffari, Sharifi-Mood, Koplik and Maldarelli2016; Poddar et al. Reference Poddar, Bandopadhyay and Chakraborty2020). Moreover, as there is no a priori information regarding when each trajectory has reached its long-time steady behaviour, it leads to prohibitively large computational time.
To overcome the above limitation, we invoke concepts from dynamical systems theory and leverage the geometric and symmetry properties inherent in the problem. We note that since the wall is infinite and uniform, the right-hand sides in (2.40a–c), i.e.
$V_x, V_z$
and
$\varOmega _y$
, are all independent of
$x$
. As a result, the equation for
$\dot {x}$
decouples entirely from the system and is determined entirely by the instantaneous values of
$(h, \theta _p)$
. This approach enables a substantial reduction in complexity by collapsing the high-dimensional swimmer dynamics into a two-dimensional autonomous system in the
$(h, \theta _p)$
phase space (Shum et al. Reference Shum, Gaffney and Smith2010; Uspal et al. Reference Uspal, Popescu, Dietrich and Tasinkevych2015a
):
\begin{align} \dot {h} &= V_z(h, \theta _p; l_s, \theta _{\textit{cap}}), \\[-10pt] \nonumber \end{align}
\begin{align} \dot {\theta }_p &= \varOmega _y(h, \theta _p; l_s, \theta _{\textit{cap}}), \\[10pt] \nonumber \end{align}
with initial conditions
$h(0) = h_0$
and
$\theta _p(0) = \theta _{p,0}$
.
In the subsequent analysis, we employ phase-plane representations to visualise the reduced dynamical system. Additionally, to provide a more intuitive understanding of the swimmer kinematics, representative trajectories are projected onto the physical
$x$
–
$z$
plane.
3. Results and discussion
In this section, we examine in detail the diverse locomotion behaviours exhibited by a self-diffusiophoretic micromotor in a quiescent fluid medium near a slippery wall. The proximity of a solid boundary introduces complex hydrodynamic and physicochemical interactions that strongly influence the propulsion characteristics of the micromotor. Different physical approximations to simulate hydrodynamic forces and phoretic surface velocity are expected to lose validity very close to the boundary due to the influence of various non-hydrodynamic nanoscale interaction forces. For a representative micromotor of radius
$ 10 \, {\unicode[Arial]{x03BC}} \mathrm{m}$
, a dimensionless cutoff separation of
$\delta =0.01$
(corresponding to a physical gap of approximately
$ 100 \, \mathrm{nm}$
) is imposed in the present simulations (Klein et al. Reference Klein, Clapp and Dickinson2003; Uspal et al. Reference Uspal, Popescu, Dietrich and Tasinkevych2015b
; Shum et al. Reference Shum, Gaffney and Smith2010; Poddar et al. Reference Poddar, Bandopadhyay and Chakraborty2021). Thus, we consider a collision state beyond this limit (Uspal et al. Reference Uspal, Popescu, Dietrich and Tasinkevych2015b
; Ghosh & Poddar Reference Ghosh and Poddar2023). Conversely, when the micromotor moves sufficiently far from the boundary, hydrodynamic interactions weaken, and the particle experiences an effective decoupling from the wall. In such cases, the trajectory typically transitions to an escaping state, where the micromotor continues to swim away from the surface without returning. Based on previous analyses (Uspal et al. Reference Uspal, Popescu, Dietrich and Tasinkevych2015b
; Poddar et al. Reference Poddar, Bandopadhyay and Chakraborty2021), this transition is observed when the dimensionless particle–wall distance exceeds approximately
$ h = 8$
. Moreover, in view of the mirror-symmetric nature of in-plane dynamics about
$\theta _p=180^{\circ }$
in the present study, it is sufficient to demonstrate the phase portraits for the range
$0^{\circ } \le \theta _{p} \le 180^{\circ }$
.
3.1. Summary of various swimming states
To capture the combined effect of the slippery wall and catalytic coverage on the motility characteristics of the micromotor, we study the in-plane dynamics from the phase portraits obtained by solving the autonomous system presented in (2.41). Consistent with the observations of Crowdy (Reference Crowdy2013), the phase portraits corresponding to non-slippery walls reveal that when the inert face of the particle is directed towards the wall, the swimmer undergoes a direct collision, whereas an initially upward-oriented configuration (
$\theta _{p,0} \le 60^{\circ }$
) results in an immediate escape from the boundary. A comprehensive regime map (figure 2) for catalytic coverage angles in the range
$5^{\circ } \le \theta _{\textit{cap}} \le 175^{\circ }$
illustrates the diverse locomotion modes arising from the diffusiophoretic nature of the particle. Experimental evidence suggests that the hydrophobicity of slippery walls can range from nanometres to
$2\,{\unicode[Arial]{x03BC}}$
m (Tretheway & Meinhart Reference Tretheway and Meinhart2002; Hu et al. Reference Hu, Wysocki, Winkler and Gompper2015) when a surface is coated with materials like octadecyltrichlorosilane. The slip length of superhydrophobic walls can reach beyond
$100 \,{\unicode[Arial]{x03BC}}$
m (Choi & Kim Reference Choi and Kim2006; Lee et al. Reference Lee, Choi and Kim2008). Although the mucus layers are only weakly hydrophobic, experimental analysis or lab-on-a-chip applications highlight that superhydrophobic confinements are also prevalent in various real-life applications (Pimponi et al. Reference Pimponi, Chinappi, Gualtieri and Casciola2014; Hu et al. Reference Hu, Wysocki, Winkler and Gompper2015; Nizkaya et al. Reference Nizkaya, Asmolov, Zhou, Schmid and Vinogradova2015, Reference Nizkaya, Asmolov, Harting and Vinogradova2020). Following previous experimental investigations (Zhu & Granick Reference Zhu, van Gogh, Zhu, Pak and Man2001; Huang et al. Reference Huang, Sendner, Horinek, Netz and Bocquet2008), the dimensionless slip length is varied within the range
$0 \le l_s \le 10$
to examine the influence of hydrodynamic slippage on the swimmer dynamics. Given that microswimmers usually have sizes of the order of
$10$
–
$100\,{\unicode[Arial]{x03BC}} \text{m}$
and prior measurements of wall slip lengths (Zhu & Granick Reference Zhu, van Gogh, Zhu, Pak and Man2001; Tretheway & Meinhart Reference Tretheway and Meinhart2002; Huang et al. Reference Huang, Sendner, Horinek, Netz and Bocquet2008), we consider
$l_s$
values between
$0$
and
$10$
to represent physically relevant hydrodynamic slippage on the swimmer dynamics.
Regime map illustrating the combined influence of catalytic coverage (
$\theta _{\textit{cap}}$
) and dimensionless slip length (
$l_s$
) on the swimmer dynamics. Four distinct behaviours are identified: escape (blue), collision (red), sliding (green) and hovering (black). Based on the characteristic dynamics, three zones (low, intermediate and high coverage) are demarcated. The results are not limited to a specific initial condition
$(h_0, \theta _{p,0})$
; instead, they are obtained by analysing all phase portraits computed for
$5^{\circ } \le \theta _{\textit{cap}} \le 175^{\circ }$
in increments of
$5^{\circ }$
, and for 20 logarithmically spaced values of
$l_s$
within
$0.01 \le l_s \le 10$
. The
$l_s = 0.01$
cases effectively reproduce the no-slip limit. The magenta dashed line showing
$ \theta _{cap, cr}$
marks the onset of stable swimming states. The slip-induced sliding states are highlighted with yellow.
In the regime map (figure 2), four distinct characteristics, escaping, sliding, collision and hovering states, are highlighted using blue, green, red and black backgrounds, respectively. In contrast to previous self-diffusiophoretic studies (Ibrahim & Liverpool Reference Ibrahim and Liverpool2015; Uspal et al. Reference Uspal, Popescu, Dietrich and Tasinkevych2015b
; Mozaffari et al. Reference Mozaffari, Sharifi-Mood, Koplik and Maldarelli2016) that primarily emphasised the role of catalytic coverage, wall proximity and chemical gradients, the present work highlights the crucial role of surface wettability in regulating micromotor motion. The findings reveal that even moderate slip length
$ (l_s)$
can qualitatively alter swimmer trajectories and stability. In figure 2, we highlight three distinct coverage zones based on the locomotion characteristics of the swimmer. A micromotor always escapes (blue square markers) from the wall when the coverage angle is in the range
$5^{\circ } \le \theta _{\textit{cap}} \le 60^{\circ }$
(‘low coverage zone’) irrespective of the slip length value. Contrastingly, when the coverage angle is in the range
$60^{\circ } \lt \theta _{\textit{cap}} \le 140^{\circ }$
(‘intermediate coverage zone’), the slip length in association with the catalytic coverage brings out diverse locomotion characteristics, ranging from the ‘escaping’ trajectories, slip-induced colliding states (red zone) to stable near-wall sliding states (green zone). To highlight the emergence of a stable sliding state for
$\theta _{\textit{cap}} \gt 90^{\circ }$
, a dashed purple line is included to mark the critical catalytic coverage
$(\theta _{cap,cr})$
. Additionally, beyond
$\theta _{\textit{cap}} \gt 140^{\circ }$
(‘high coverage zone’), only the ‘hovering’ states (black square markers) are present, thereby indicating the insignificant role of slip in this domain. A detailed discussion on the contrasting characteristics of the micromotor controlled by these parameters is presented in the upcoming subsections.
It is to be noted that the hydrodynamic slip does not lead to any significant qualitative modification in the dynamics of the micromotor in the ‘low coverage zone’, i.e.
$ \theta _{\textit{cap}} \le 60^{\circ }$
. The observed enhancement in wall-detention behaviour in this regime is qualitatively similar to the findings of our earlier study (Poddar et al. Reference Poddar, Bandopadhyay and Chakraborty2020). Hence, the corresponding results are not presented in the current work.
3.2. Enhanced trapping due to wall slip
The alterations in the swimming behaviour of a self-diffusiophoretic micromotor, arising from the combined effects of catalytic coverage and wall slip, become most pronounced in the intermediate coverage regime
$(60^{\circ } \lt \theta _{\textit{cap}} \lt 145^{\circ })$
. Within this range, the particle exhibits a distinct transition in its locomotion from escaping after near-wall repulsion to undergoing direct collision or attaining a steady sliding state above the boundary. Based on the diverse slip-induced transitions presented in figure 2, this intermediate regime can be further divided into three subdomains: (i) a low-intermediate zone (
$60^{\circ } \lt \theta _{\textit{cap}} \lt 95^{\circ }$
), (ii) a mid-intermediate zone (
$95^{\circ } \le \theta _{\textit{cap}} \lt 115^{\circ }$
) and (iii) a high-intermediate zone (
$115^{\circ } \le \theta _{\textit{cap}} \lt 145^{\circ }$
).
Diverse in-plane dynamics of a self-diffusiophoretic micromotor are highlighted using phase portraits in (a)
$l_s=0$
and (b)
$l_s=3.6$
for
$\theta _{\textit{cap}}=75^{\circ }$
; (d)
$l_s=0$
, (e)
$l_s=0.2$
and (f)
$l_s=1.35$
for
$\theta _{\textit{cap}}=105^{\circ }$
; and (h)
$l_s=0$
and (i)
$l_s=1$
for
$\theta _{\textit{cap}}=130^{\circ }$
. Corresponding trajectories are shown for (c)
$\theta _{\textit{cap}}=75^{\circ }$
, (g)
$\theta _{\textit{cap}}=105^{\circ }$
and (j)
$\theta _{\textit{cap}}=130^{\circ }$
. Orange cross and green square markers denote the saddle point and stable sliding states. Escaping, sliding and collision states are highlighted in blue, green and red trajectories. The yellow circles show the initial location of the micromotor in the phase space.
Figure 3. Long description
Panel A: A phase portrait graph showing the dynamics of a self-diffusiophoretic micromotor. The horizontal axis is labeled θp (degrees) ranging from 0 to 180, and the vertical axis is labeled h-1 ranging from 0 to 6. The graph includes blue arrows indicating the direction of motion, a yellow circle marking the initial location of the micromotor, and an orange cross denoting the saddle point. Panel B: Another phase portrait graph similar to Panel A but with different dynamics. The axes and markers are the same as in Panel A. Panel C: A line graph showing the corresponding trajectories for the micromotor. The horizontal axis is labeled x ranging from 0 to 80, and the vertical axis is labeled h-1 ranging from 0 to 8. The graph includes blue, green, and red trajectories indicating escaping, sliding, and collision states, respectively. Panel D: A phase portrait graph similar to Panel A but for different parameters. The axes and markers are the same as in Panel A. Panel E: Another phase portrait graph similar to Panel D but with different dynamics. The axes and markers are the same as in Panel D. Panel F: A phase portrait graph similar to Panel E but with different dynamics. The axes and markers are the same as in Panel E. Panel G: A line graph showing the corresponding trajectories for the micromotor. The horizontal axis is labeled x ranging from 0 to 500, and the vertical axis is labeled h-1 ranging from 0 to 5. The graph includes blue, green, and red trajectories indicating escaping, sliding, and collision states, respectively. Panel H: A phase portrait graph similar to Panel A but for different parameters. The axes and markers are the same as in Panel A. Panel I: Another phase portrait graph similar to Panel H but with different dynamics. The axes and markers are the same as in Panel H. Panel J: A line graph showing the corresponding trajectories for the micromotor. The horizontal axis is labeled x ranging from 0 to 80, and the vertical axis is labeled h-1 ranging from 0 to 3. The graph includes green and red trajectories indicating sliding and collision states, respectively.
Schematics showing mechanism of slip-induced trapping. (a) The directions of wall-normal velocity
$ (V_z)$
and the wall-parallel rotational velocity
$ (\varOmega _y)$
near a no-slip wall. (b) The additional components introduced by wall slip
$\Delta V_z |_{{slip}}$
and
$\Delta \varOmega _y |_{{slip}}$
.
Distribution of solute concentration surrounding the micromotor for inclination angles (a)
$\theta _{p}=120^{\circ }$
and (b)
$\theta_p = 105^{\circ }$
at
$h=2$
. (c–e) Variation of the velocity components (
$V_x$
,
$V_z$
and
$\varOmega _y$
) with
$h-1$
for different
$l_s$
at
$\theta _{p}=120^{\circ }$
(solid lines) and
$105^{\circ }$
(dashed lines). In all cases,
$ \theta _{\textit{cap}}=75^{\circ }$
.
The phase portrait in figure 3(a) shows that a self-diffusiophoretic micromotor with a catalytic coverage of
$\theta _{\textit{cap}} = 75^{\circ }$
initially moves towards a no-slip wall (deep-blue trajectory). During its approach, the orientation vector
$\boldsymbol{p}$
undergoes a counterclockwise (CCW) rotation, which eventually leads to reflection from the no-slip wall. In contrast, the presence of a slippery wall with
$l_s = 3.6$
suppresses this reorientation mechanism, and the micromotor instead collides with the wall, as illustrated by the red trajectories in figures 3(b) and 3(c). Figures 5(a) and 5(b) reveal a strong accumulation of solutes near the wall. The intensified solute gradient enhances the phoretic surface flow around the particle, and generates an enhanced CCW torque that initially favours reflection when the active cap faces the wall. However, wall slip alters this mechanism significantly. As seen in figure 5(e), the red dashed curve indicates that the CCW torque diminishes (
$\varOmega _y \approx 0$
) when the particle begins to orient away from the slippery surface. This reduction in torque arises because the slip condition weakens the hydrodynamic interaction between the micromotor and the wall.
For a chemically active micromotor near a wall, the solute concentration field
$(\widetilde {C}(\boldsymbol{r}))$
is strongly distorted by the boundary. This distortion creates local, high-order gradients across the particle surface that are not captured by far-field image systems. These gradients are crucial for determining the phoretic torque that leads to the observed CCW or clockwise (CW) rotations. Thus, they dictate the precise tipping point of the trajectory. As a consequence, the study by Ibrahim & Liverpool (Reference Ibrahim and Liverpool2015) could not capture the ‘cushioning effect’ due to solute accumulation near a no-slip wall, which prevents a direct collision of a micromotor against a no-slip wall (Mozaffari et al. Reference Mozaffari, Sharifi-Mood, Koplik and Maldarelli2016). This discrepancy underscores the unreliability of far-field techniques in near-wall regimes, even under standard no-slip conditions. Notably, this mechanism is also absent in the ‘squirmer’ model of microswimmers, where the collision against the plane wall is inevitable in the absence of a non-hydrodynamic repulsive force (Poddar et al. Reference Poddar, Bandopadhyay and Chakraborty2020).
In the present problem, the slip length increases the magnitude of the wall-normal velocity component
$V_z$
, which remains negative even at
$\theta _p = 105^{\circ }$
(figure 5
d). This persistent downward motion, combined with the reduced reorienting torque, disrupts the near-wall ‘cushioning effect’ (Mozaffari et al. Reference Mozaffari, Sharifi-Mood, Koplik and Maldarelli2016; Poddar et al. Reference Poddar, Bandopadhyay and Chakraborty2021). Consequently, the micromotor collides with the boundary.
It is noteworthy that the incorporation of a cutoff separation distance of
$ \delta =0.01$
makes the existence of any equilibrium configuration beyond this separation untraceable within the current numerical framework. However, introducing an explicit repulsive potential at the wall, such as those arising from steric or electrostatic interaction, could sustain and stabilise these near-wall states by balancing the hydrodynamic attraction (Li & Ardekani Reference Li and Ardekani2014; Lintuvuori et al. Reference Lintuvuori, Brown, Stratford and Marenduzzo2016; Poddar et al. Reference Poddar, Bandopadhyay and Chakraborty2020). Thus, the transitions from ‘sliding’ to ‘collision’ states can be attributed to the mathematical limitation of pure hydrodynamic models (Uspal et al. Reference Uspal, Popescu, Dietrich and Tasinkevych2015a
; Poddar Reference Poddar2023). As a result, both the reported ‘stable’ sliding and ‘collision’ states can be regarded as the conditions where the micromotor is trapped near the wall.
As shown in figure 2, within the low-intermediate coverage zone (
$60^{\circ } \lt \theta _{\textit{cap}} \lt 95^{\circ }$
), an increase in the catalytic coverage
$\theta _{\textit{cap}}$
reduces the critical slip length required for the micromotor to undergo collision with the wall. For instance, while a relatively large slip length of
$l_s = 10$
is needed to induce collision at
$\theta _{\textit{cap}} = 65^{\circ }$
, the transition from escaping to colliding behaviour occurs at a much lower value of
$l_s = 0.6$
when
$\theta _{\textit{cap}}=90^{\circ }$
.
Further enhancement of the catalytic coverage beyond a critical limit (dashed magenta line of
$\theta _{cap, cr}$
in figure 2) unveils the early emergence of stable sliding states due to the presence of a slippery wall. Figure 2 shows that the escaping trajectories at smaller
$l_s$
for
$\theta _{\textit{cap}}=95^{\circ }$
start getting trapped near the wall at
$l_s=0.5$
and exhibit stable sliding motion until a sufficiently high
$l_s$
directs the particle towards the wall for collision. Notably, similar sliding states near a no-slip wall were reported earlier for much higher values of
$\theta _{\textit{cap}}$
, i.e.
$\approx 113^{\circ }$
in Uspal et al. (Reference Uspal, Popescu, Dietrich and Tasinkevych2015b
) and
$\theta _{\textit{cap}} = 115^{\circ }$
in Mozaffari et al. (Reference Mozaffari, Sharifi-Mood, Koplik and Maldarelli2016). Therefore, the slippery nature of the wall broadens the scope of trapping the micromotors without requiring any change in the design strategy of micromotors. This comparison highlights that introducing wall slip effectively lowers the threshold of activity required to achieve stable near-wall motion, thereby widening the parameter regime where micromotors can be hydrodynamically trapped without modifying their intrinsic design.
Figure 3(d–g) elaborates the slip-driven transition from escaping (blue trajectory) to near-wall sliding states (green trajectory) of a micromotor, followed by collision (red trajectory) for
$\theta _{\textit{cap}}=105^{\circ }$
. The mechanism of slip-induced trapping due to the additional components of velocity imparted by finite slip length is schematically shown in figure 4. Near a no-slip wall, the micromotor experiences an upward wall-normal force and a CCW torque, as shown in figure 4(a). However, with the introduction of wall slip, a counteracting downward wall-normal force and a CW torque emerge (figure 4
b), as evidenced through the signs of
$ \varOmega _y$
in figure 7(g–i). The CW torque near a slippery wall counters the CCW torque and arrests the rotational motion at the equilibrium point. At the same time, the counteracting downward wall-normal force intensifies with
$l_s$
, and the wall-normal velocity is suppressed. As a result, the micromotor attains a stable height from the confinement at an intermediate
$l_s$
. Further increment of
$l_s$
strengthens the downward wall-normal force, which increases the negative magnitude of
$V_z$
, resulting in collision states.
(a–c) Variation of velocity components with slip length for
$\theta _{\textit{cap}}=105^{\circ }$
coverage at different inclinations.
Figure 6. Long description
Three line graphs show the variation of velocity components with slip length for different inclinations. Panel A: The line graph displays the variation of Vx with time (t) for different slip lengths (ls). The x-axis represents time (t) in arbitrary units, and the y-axis represents the velocity component Vx. The graph includes three lines: a blue dashed line for ls = 0.01, a green dashed line for ls = 0.2, and a solid red line for ls = 1.35. Panel B: The line graph shows the variation of Vz with time (t) for different slip lengths (ls). The x-axis represents time (t) in arbitrary units, and the y-axis represents the velocity component Vz. The graph includes three lines: a blue dashed line for ls = 0.01, a green dashed line for ls = 0.2, and a solid red line for ls = 1.35. Panel C: The line graph illustrates the variation of Ωy with time (t) for different slip lengths (ls). The x-axis represents time (t) in arbitrary units, and the y-axis represents the angular velocity component Ωy. The graph includes three lines: a blue dashed line for ls = 0.01, a green dashed line for ls = 0.2, and a solid red line for ls = 1.35.
(a–c) Catalytic concentration gradient for different inclination angles at
$\theta _{\textit{cap}}=105^{\circ }$
and
$h=1.5$
. (d–l) Modulations in velocity components (
$V_x$
,
$V_z$
and
$\varOmega _y$
) with
$h-1$
for different
$l_s$
.
In order the investigate the time evolution of the dynamics of a micromotor leading to a stable state (
$h^*=1.3$
,
$\theta _{p}^*=117^{\circ }$
), we have presented the variation of velocity components along its trajectory in figure 6. Figure 6(c) shows that
$\varOmega _y$
remains nearly zero when the particle moves near a slippery wall with a slip length of
$l_s = 0.2$
, confirming the attainment of a steady sliding configuration. In contrast, figure 3(f,g) demonstrates a collision state (red trajectory) at
$l_s = 1.35$
after
$t = 18.7$
. The corresponding curves for
$l_s = 1.35$
and
$5$
in figures 7(g) and 7(h) further highlight a strong reversal of rotation from CCW to CW with increasing slip length, which promotes collision. As shown by the red curve in figure 6(b), the negative wall-normal velocity (
$V_z \lt 0$
) gradually decreases in magnitude over time, but the particle collides with the wall before
$V_z$
turns positive.
These slip-induced effects give rise to new stable sliding states beyond the critical coverage angle
$\theta _{cap,cr}$
for
$ 95^{\circ } \le \theta _{\textit{cap}} \lt 115^{\circ }$
, as indicated in the yellow-shaded region of figure 2. A greater amount of solute accumulates as we increase
$\theta _{\textit{cap}}$
, which facilitates the slip-modulated redistribution of flow field in the particle–wall gap. The negative wall-normal velocity (
$V_z \lt 0$
) persists across all slip lengths and wall separations at
$\theta _p = 135^{\circ }$
(figure 7
l). In contrast, at smaller inclination angles (
$\theta _p = 120^{\circ }$
and
$105^{\circ }$
; figures 7
k and 7
j),
$V_z$
transitions from positive to negative as the micromotor moves away from the wall. This indicates the enhanced competition between slip-modulated hydrodynamic forces and phoretic surface flow at high
$ \theta _{\textit{cap}}$
.
(a–c) Catalytic concentration gradient for different inclination angles at
$\theta _{\textit{cap}}=130^{\circ }$
and
$h=1.5$
. (d–f) Modulations in velocity components (
$V_x$
,
$V_z$
and
$\varOmega _y$
) with
$h-1$
for different
$l_s$
.
Figure 8. Long description
Panel A: A contour plot shows the catalytic concentration gradient for an inclination angle of 120 degrees. The x and y axes are labeled with numerical values, and the color gradient indicates concentration levels. Panel B: Another contour plot displays the catalytic concentration gradient for an inclination angle of 140 degrees, with similar axes and color gradient. Panel C: A third contour plot illustrates the catalytic concentration gradient for an inclination angle of 160 degrees, maintaining the same axes and color gradient. Panel D: A line graph plots the velocity component Vx against the variable h-1 for different values of ls and theta_p. The x-axis is labeled h-1 and the y-axis is labeled Vx. Panel E: Another line graph shows the velocity component Vz against h-1 for different values of ls and theta_p, with corresponding axes labels. Panel F: A final line graph depicts the rotational velocity component Omega_y against h-1 for different values of ls and theta_p, with the respective axes labels.
The micromotor does not escape when
$\theta _{\textit{cap}}\ge 115^{\circ }$
, and steady sliding states can be obtained even near a no-slip wall, as shown in figure 2. The red trajectories in figures 3(i) and 3(j) indicate that sliding trajectories at
$\theta _{\textit{cap}}=130^{\circ }$
collide with the confinement when the slip length is enhanced to
$l_s=1$
. The inclusion of slippery confinement provides a CW torque, which makes
$\varOmega _y$
positive near the confinement as shown by the red trajectories with diamond markers in figure 8(f). The red dashed curve in figure 8(e) for
$\theta _p=160^{\circ }$
also highlights that
$V_z$
remains negative with an increase in
$\theta _p$
. Thus, in this regime, the already dominating CW torque and downward movement due to enhanced coverage are further reinforced by the slip length, leading to collision.
3.3. Slip-induced increase in hovering height
The phase portrait in figure 9(a) depicts that a catalytic Janus micromotor with a coverage angle
$\theta _{\textit{cap}}=150^{\circ }$
attains a ‘hovering’ state (brown circular marker in figures 9
a and 9
b) near a slippery confinement with slip length
$l_s=1$
at a stable height
$h=1.49$
. In this stable configuration, the orientation is fixed at
$\theta _{p}=180^{\circ }$
. It can be observed from the trajectory in figure 9(b) that the micromotor initially advances towards the impenetrable wall, but the surrounding concentration field eventually restricts its motility. In this condition, the large active area leads to more accumulation of the solute between the swimmer and the wall, which gets distributed symmetrically surrounding the micromotor when it is close to the wall, as shown in figure 9(c). Here, the phoretically induced CW torque dominates up to the condition where the micromotor attains a stable axisymmetric orientation of
$\theta _{p}=180^{\circ }$
for sufficiently large
$\theta _{\textit{cap}}$
. The inset to figure 9(b) further depicts that, at this configuration, the resultant
$V_x$
,
$V_z$
and
$\varOmega _y$
together become
$0$
. Once the micromotor settles into the orientation
$\theta _{p}=180^{\circ }$
the wall slippage becomes insufficient to alter the hydrodynamics and break the symmetry in the flow field induced by a strong symmetric concentration field (figure 9
c), thereby sustaining the hovering state.
Although the wall slip does not lead to any qualitative alteration in stability for high coverage angles
$ (\theta _{\textit{cap}} \ge 145^\circ)$
, raising
$l_s$
shifts the hovering height (
$h^*$
) upward. This slip-induced effect is highlighted in the green-shaded region of figure 9(d) (
$\theta _{\textit{cap}}=150^{\circ }$
) and in the corresponding inset (
$\theta _{\textit{cap}}=145^{\circ }$
). However, the effect of slip length on hovering height
$(h^*)$
saturates at very high slip lengths (e.g.
$l_s\gt 1$
for
$\theta _{\textit{cap}}=150^{\circ }$
).
For a physical explanation of the above observations, we appeal to figure 10, where the variations of different velocity components of the micromotor are shown with increasing distance from the wall, for different slip lengths and a fixed orientation of
$ \theta _p=120^\circ$
and
$ \theta _{\textit{cap}}=150^\circ$
. The close agreement between our results for a no-slip wall and those reported by Mozaffari et al. (Reference Mozaffari, Sharifi-Mood, Koplik and Maldarelli2016) has been highlighted. As shown in figure 10, close to the wall,
$V_x$
decreases with
$l_s$
while
$V_z$
and
$\varOmega _y$
increase. This enhanced
$V_z$
at higher
$ l_s$
drives the upward shift of
$h^*$
. Additionally, the increment in
$V_z$
is very small beyond
$l_s = 1$
, which explains the nearly constant hovering height at high slip lengths in figure 9(d). Figures 10(a) and 10(c) also indicate that after attaining an orientation of
$\theta _{p}=180^{\circ }$
,
$V_x$
and
$\varOmega _y$
remains zero for all wall–particle separations even in the case of
$ l_s =1$
, thereby confirming the stability of the hovering states near a slippery wall.
Characteristics of a self-diffusiophoretic micromotor for
$\theta _{\textit{cap}}=150^{\circ }$
and
$l_s=1$
. In-plane dynamics of a self-diffusiophoretic micromotor is shown in the (a) phase portrait and its (b) trajectory when released from
$h_0=4$
and
$\theta _{p,0}=115^{\circ }$
. Inset in (b) represents the evolution of various velocity components along the trajectory presented in (b). (c) The concentration field surrounding the micromotor at
$h=1.5$
at its hovering state. (d) Variation of hovering height (
$h^*$
) with
$l_s$
for
$\theta _{\textit{cap}}=150^{\circ }$
and (in the inset) for
$\theta _{\textit{cap}}=145^{\circ }$
.
(a–c) Variations of different velocity components with
$l_s=0$
,
$1$
and
$5$
for a micromotor with coverage
$\theta _{\textit{cap}}=150^{\circ }$
for an orientation
$\theta _{p}=120^{\circ }$
. The velocity variations with
$\delta$
for the hovering state at
$\theta _p=180^{\circ }$
are highlighted by the orange dashed line.
Figure 10. Long description
Panel A: A line graph shows the variation of the velocity component Vx with respect to h-1. The x-axis is labeled h-1 and ranges from 10^-2 to 10^1. The y-axis is labeled Vx and ranges from 0 to 2.5. Different lines represent various conditions with different colors and styles. Panel B: A line graph shows the variation of the velocity component Vz with respect to h-1. The x-axis is labeled h-1 and ranges from 10^-2 to 10^1. The y-axis is labeled Vz and ranges from 0 to 15. Different lines represent various conditions with different colors and styles. An orange dashed line highlights the hovering state. Panel C: A line graph shows the variation of the angular velocity component Ωy with respect to h-1. The x-axis is labeled h-1 and ranges from 10^-2 to 10^1. The y-axis is labeled Ωy and ranges from -1.5 to 0.5. Different lines represent various conditions with different colors and styles.
3.4. Effect of slip on sliding speed
Although the stability characteristics of the motion do not depend on the wall-parallel velocity component
$V_x$
due to translational invariance of the dynamic system in (2.41), it is crucial to estimate the speed of transport in the wall-parallel direction, as highlighted in the experiments of Ketzetzi et al. (Reference Ketzetzi, De Graaf, Doherty and Kraft2020).
The velocity component
$V_x$
for different
$h$
plotted at (a)
$\theta _{p}=120^{\circ }$
and (b)
$\theta_p = 90^{\circ }$
for a low catalytic coverage,
$\theta _{\textit{cap}}=40^{\circ }$
, and varying slip lengths,
$l_s$
.
Figure 11 illustrates
$V_x$
for different slip lengths for a small coverage angle
$ \theta _{\textit{cap}} = 40^{\circ }$
. The figure describes an enhancement in
$V_x$
near the wall due to the enhancement in slip length. However,
$V_x$
gets enhanced as the micromotor rotates CCW and orients away from the wall (figure 11
b). To consolidate the slip-induced trends in
$V_x$
reported in figures 5(c–e), 7(j–l), 10(a) and 11(a,b), we present a regime map in figure 12. This map presents the combinations of catalytic coverage (
$\theta _{\textit{cap}}$
) and particle inclination (
$\theta _p$
) associated with either speed-up or retardation. The green (orange) regions indicate parametric zones where
$V_x$
consistently increases (decreases) with slip length
$l_s$
, while the yellow patches denote transitional regimes in which both behaviours may arise depending on the particle–wall separation. It also indicates that a micromotor with fixed catalytic coverage
$ (\theta _{\textit{cap}})$
may either increase or decrease its speed depending on its director orientation
$ (\theta _p)$
with respect to the wall.
The observed decrease in the longitudinal swimming velocity
$V_x$
with increasing slip length
$l_s$
in figure 7(d–f) appears counterintuitive when compared with the behaviour of a passively driven sphere translating parallel to a slippery wall. In the latter case, the reduced hydrodynamic drag associated with wall slip leads to an increase in migration velocity relative to a no-slip surface (Loussaief et al. Reference Loussaief, Pasol and Feuillebois2015). For a self-diffusiophoretic micromotor, however, the swimming velocity is determined not only by the hydrodynamic resistance associated with translation and rotation parallel to the wall, but also by the phoretic thrust generated through surface activity, as evident from (2.22) and (2.23). In addition to the resistance for translation and rotation, both the thrust force,
$F_x^{(\textit{Thrust})}$
, and the associated torque,
$G_y^{(\textit{Thrust})}$
, are strongly influenced by the wall slip. Consequently, the resultant value of
$V_x$
is determined by the competition between phoretic thrust and the slip-induced modifications to hydrodynamic resistance coefficients, as shown in (2.37
a). This mechanism is absent in squirmer-based models (Poddar et al. Reference Poddar, Bandopadhyay and Chakraborty2020).
The intricate dependence of
$V_x$
on wall slip revealed here extends the understanding provided by Ketzetzi et al. (Reference Ketzetzi, De Graaf, Doherty and Kraft2020), who observed speed-up for half-coated particles near hydrophobic walls. We conjecture that their experiments correspond to a specific subset of the geometric and chemical parameter space. This is further supported by the following key contrast. In their experiments, micromotors rapidly approached the lower surface and subsequently propelled stably along it, whereas our analysis also predicts wall-induced repulsion, escape and a variety of transitional behaviours arising under different initial orientations and combinations of slip length
$ l_s$
and cap coverage
$ \theta _{\textrm{cap}}$
. Moreover, the experimental conditions of Ketzetzi et al. (Reference Ketzetzi, De Graaf, Doherty and Kraft2020) were restricted to hydrophobic coatings on glass substrates, which typically produce slip lengths of only a few tens of nanometres (Huang et al. Reference Huang, Sendner, Horinek, Netz and Bocquet2008; Bocquet & Charlaix Reference Bocquet and Charlaix2010). However, for nano-engineered surfaces, the effective hydrodynamic boundary condition at the fluid–solid interface can be modelled as a uniform partial slippage with high slip length, even reaching up to several micrometres (Choi & Kim Reference Choi and Kim2006; Joseph et al. Reference Joseph, Cottin-Bizonne, Benoit, Ybert, Journet, Tabeling and Bocquet2006; Lee et al. Reference Lee, Choi and Kim2008; Nizkaya et al. Reference Nizkaya, Asmolov, Zhou, Schmid and Vinogradova2015). Thus, the present analysis reveals that, with suitable control over micromotor orientation and surface patterning, one can design systems where micromotors are accelerated, decelerated or tuned for targeted dynamical responses.
Regime map highlighting the behaviour of
$V_x$
with
$l_s$
depending upon different combinations catalytic coverage (
$\theta _{\textit{cap}}$
) and inclination angles (
$\theta _p$
).
4. Conclusions and remarks
To summarise, we have presented a theoretical investigation into the hydrodynamics of a self-diffusiophoretic Janus micromotor near a slippery confinement. By simultaneously incorporating the Navier slip boundary condition alongside the self-diffusiophoretic contribution to the swimmer motion, the model captures a broad spectrum of near-wall locomotion behaviours that were previously inaccessible to no-slip analyses (Crowdy Reference Crowdy2013; Uspal et al. Reference Uspal, Popescu, Dietrich and Tasinkevych2015a
; Ibrahim & Liverpool Reference Ibrahim and Liverpool2015; Mozaffari et al. Reference Mozaffari, Sharifi-Mood, Koplik and Maldarelli2016). The coupled physics of concentration distribution and the hydrodynamics of wall–swimmer interactions have been resolved using bispherical eigenfunction expansions in the creeping flow regime. The current formulation accommodates the complex wettability characteristics of a confining substrate through arbitrary slip lengths
$ (l_s)$
, thereby uncovering new hydrodynamic regimes arising solely from wall slippage.
In the intermediate catalytic coverage regime (
$60^{\circ } \lt \theta _{\textit{cap}} \lt 145^{\circ }$
), the presence of wall slip gives rise to a qualitatively new class of near-wall behaviours that are absent for no-slip boundaries. Within this range, the micromotor undergoes a sharp transition from escaping trajectories to either wall-collision or stable sliding states. Increasing
$\theta _{\textit{cap}}$
beyond a critical limit (
$\theta _{\textit{cap,cr}}$
; dashed magenta line in figure 2) triggers an early onset of these sliding states due to the presence of slippage at the wall. Earlier studies reported such sliding states only at substantially larger coverage angles (Uspal et al. Reference Uspal, Popescu, Dietrich and Tasinkevych2015b
; Mozaffari et al. Reference Mozaffari, Sharifi-Mood, Koplik and Maldarelli2016). Thus, wall slip lowers the activity threshold required to achieve stable near-wall trapping, thereby expanding the operational parameter space without altering micromotor design.
The observed decrease in the longitudinal swimming speed
$V_x$
with increasing slip length
$l_s$
reveals a departure from the classical behaviour of passive particles near slippery walls, where drag reduction typically enhances translation (Loussaief et al. Reference Loussaief, Pasol and Feuillebois2015). For a self-diffusiophoretic micromotor, however, the swimming speed is governed not only by the hydrodynamic resistance associated with translation and rotation, but also by the phoretic thrust force and torque generated by surface activity. Since wall slip influences both the hydrodynamic resistance coefficients and the propulsive thrust on the swimmer, the resulting swimming speed reflects the balance among these contributions. At intermediate and high catalytic coverages, this balance can lead to a counterintuitive decrease in
$V_x$
with increasing
$l_s$
, despite the reduction in hydrodynamic drag.
Although wall slip does not induce any qualitative change in the stability behaviour at high catalytic coverage (
$\theta _{\textit{cap}} \ge 145^{\circ }$
), increasing the slip length
$l_s$
systematically shifts the hovering height
$h^*$
to larger wall–particle separations. At high slip lengths (e.g.
$l_s \gt 1$
) the effect saturates, leading to only marginal additional changes in the hovering height. As the particle approaches the wall, increasing
$l_s$
leads to a reduction in
$V_x$
, while both
$V_z$
and
$\varOmega _y$
increase. The enhancement in the wall-normal velocity
$V_z$
is the primary mechanism pushing the equilibrium hovering height upward. Once the particle orients to
$\theta _p = 180^{\circ }$
, both
$V_x$
and
$\varOmega _y$
vanish for all wall separations, even for
$l_s \gt 1$
, confirming that the hovering states remain stable in the presence of significant wall slip.
A distinguishing feature for the phase portraits depicting the dynamics is the absence of any limit cycle in the present scenario, indicating the absence of undamped oscillations of fixed amplitude. This is in stark contrast to the earlier squirmer studies (Poddar et al. Reference Poddar, Bandopadhyay and Chakraborty2020; Ghosh et al. Reference Ghosh, Ghoshal and Poddar2024), where undamped oscillations were widely observed. A regime map constructed in the (
$\theta _{\textit{cap}}$
–
$l_{s}$
) plane clearly delineates the transitions between reflection, sliding, collision and hovering trajectories. The boundaries between these regimes shift systematically with increasing slip, illustrating how interfacial slippage can be exploited to control swimmer behaviour near hydrophobic confinements. Slip effectively acts as a control parameter that governs the competition between the solute-induced diffusiophoretic thrust and the shear-modified hydrodynamic drag from the boundary. Slip weakens the near-wall CCW torque arising from solute redistribution and can even reverse its direction, thereby promoting enhanced trapping. Consequently, the interfacial slip length acts as a tunable control parameter for micromotor guidance in confined environments. Results also reveal that, depending on catalytic patterning and orientation, wall slip can either speed up or slow down micromotor sliding. This sensitivity offers a design pathway to intentionally tune transport speeds for specific microscale tasks.
Beyond advancing the fundamental understanding of swimmer-boundary interactions, the present results offer practical implications for both biophysical and engineered microscale environments. From an experimental perspective, the regimes identified in this work can be directly tested in microchannels with tunable wettability. By selecting appropriate substrate coatings, microtopographies or stimulus-responsive materials, one can steer, trap or release microswimmers solely through boundary design without modifying the swimmers themselves. Nevertheless, the results may not be immediately applicable for engineered surfaces where the microstructure of superhydrophobic surfaces can significantly influence the motion of nearby particles, including both passive tracers and active microswimmers (Pimponi et al. Reference Pimponi, Chinappi, Gualtieri and Casciola2014; Nizkaya et al. Reference Nizkaya, Asmolov, Zhou, Schmid and Vinogradova2015; Hu et al. Reference Hu, Wysocki, Winkler and Gompper2015; Nizkaya et al. Reference Nizkaya, Asmolov, Harting and Vinogradova2020). In those scenarios, characteristic asperity length scale (
$\widetilde {l}_a$
) is comparable to the slip length, i.e.
$\widetilde {l}_a \sim \widetilde {l}_S$
(Choi & Kim Reference Choi and Kim2006; Joseph et al. Reference Joseph, Cottin-Bizonne, Benoit, Ybert, Journet, Tabeling and Bocquet2006), and the assumption of a spatially uniform slip boundary condition may no longer be adequate (Ybert et al. Reference Zhu and Granick2007; Asmolov et al. Reference Asmolov, Zhou, Schmid and Vinogradova2013). This issue becomes especially relevant when the microswimmer is in close proximity to the surface, i.e. when
$\widetilde {\delta } \lt \widetilde {l}_s$
.
A more comprehensive investigation into microswimmer dynamics near superhydrophobic boundaries could incorporate the spatial variability of slip conditions along the confining walls (Ybert et al. Reference Zhu and Granick2007; Asmolov et al. Reference Asmolov, Zhou, Schmid and Vinogradova2013), which remains outside the scope of the present work. Despite these complexities, the current homogeneous-slip formulation remains highly relevant for atomically smooth hydrophobic surfaces coated with self-assembled monolayers, where intrinsic molecular slippage governs the near-wall fluid motion (Gentili et al. Reference Gentili, Bolognesi, Giacomello, Chinappi and Casciola2014). Therefore, the present findings provide a mechanistic basis for designing antifouling interfaces capable of reducing bacterial or colloidal deposition (Razavi et al. Reference Razavi, Oh, Haasch, Kim, Masoomi, Bagheri, Slauch and Miljkovic2019; Guo et al. Reference Guo, Tan, Fang, Li, Chen, Qin, Liu and Mo2024), targeted delivery platforms that exploit controlled near-wall retention or release of micromotors in microfluidic architectures (Kim et al. Reference Kim, Nicholas, Puigmartí-Luis, Nelson and Pané2025).
Funding
A.P. would like to acknowledge the support provided by the Anusandhan National Research Foundation, Government of India through the project: ANRF/ARG/2025/000835. S.M. gratefully acknowledges support from Saroj Poddar Trust.
Declaration of interests
The authors declare no conflict of interest.
Appendix
Range of number of terms (
$N_c$
) required for series truncation for the translational (a) and rotational (b) velocities. Other parameters are
$l_s=5$
,
$h=1.01$
,
$\theta _p=120^{\circ }$
and
$\theta _{\textit{cap}}=120^{\circ }$
.
∠AO′B=θcap
l~s
p
θp
θcap
ls
(h0,θp,0)
5∘≤θcap≤175∘
5∘
ls
0.01≤ls≤10
ls=0.01
θcap,cr
ls=0
ls=3.6
θcap=75∘
ls=0
ls=0.2
ls=1.35
θcap=105∘
ls=0
ls=1
θcap=130∘
θcap=75∘
θcap=105∘
θcap=130∘
(Vz)
(Ωy)
ΔVz|slip
ΔΩy|slip
θp=120∘
θp=105∘
h=2
Vx
Vz
Ωy
h−1
ls
θp=120∘
105∘
θcap=75∘
θcap=105∘
θcap=105∘
h=1.5
Vx
Vz
Ωy
h−1
ls
θcap=130∘
h=1.5
Vx
Vz
Ωy
h−1
ls
θcap=150∘
ls=1
h0=4
θp,0=115∘
h=1.5
h∗
ls
θcap=150∘
θcap=145∘
ls=0
1
5
θcap=150∘
θp=120∘
δ
θp=180∘
Vx
h
θp=120∘
θp=90∘
θcap=40∘
ls
Vx
ls
θcap
θp
Nc
ls=5
h=1.01
θp=120∘
θcap=120∘
