3.1. The x–x, x–θ, θ–θ, coupling diffusivities

The HIs of ellipsoid pairs are characterized by the x–x, θ–θ and x–θ coupling diffusivities, as follows:

(3.1) \begin{equation}\left\{\begin{array}{@{}c} {D_C^{xx} = \langle \Delta {x_1}\Delta {x_2}\rangle /(2t),}\\ {D_C^{\theta \theta } = \langle \Delta {\theta_1}\Delta {\theta_2}\rangle /(2t),}\\ {D_C^{x\theta } = \langle \Delta {x_1}|{\Delta {\theta_2}} |\rangle /(2t) ={-} \langle |{\Delta {\theta_1}} |\Delta {x_2}\rangle /(2t) ={-} D_C^{\theta x},} \end{array}\right.\end{equation}

where $\Delta {x_i}$ is the CM displacement along the channel axis, and $\Delta {\theta _i}$ is the angular displacement with $- {\rm \pi} /2 \le \theta \le {\rm \pi} /2$ of ellipsoid i during time interval t (see figure 1b). Here $\langle \;\rangle$ is an average over trajectories of all isolated pairs at all times from experimental trials. The first equation in (3.1) has been used in previous studies about colloidal spheres (Cui et al. Reference Cui, Diamant and Lin2002, Reference Cui, Diamant, Lin and Rice2004; Mazo Reference Mazo2002; Valley et al. Reference Valley, Rice, Cui, Ho, Diamant and Lin2007; Diamant Reference Diamant2009; Novikov et al. Reference Novikov, Rice, Cui, Diamant and Lin2010), but the latter two about rotational motions have not been applied to colloidal experiments. In the diffusion matrix (Montgomery & Berne Reference Montgomery and Berne1977; Mazo Reference Mazo2002) $\boldsymbol{D}(X,Y)$ (see (1.1)), $D_C^{xx}$, $D_C^{\theta \theta }$ and $D_C^{x\theta }$ are the off-diagonal terms of the diffusion tensors ${\boldsymbol{D}^{\boldsymbol{xx}}}$, ${\boldsymbol{D}^{\boldsymbol{x\theta }}}$, and ${\boldsymbol{D}^{\boldsymbol{\theta \theta }}}$, while the diagonal terms are the self-diffusivities $D_S^{xx} \equiv \langle \Delta x_1^2\rangle /(2t)$, $D_S^{\theta \theta } \equiv \langle \Delta \theta _1^2\rangle /(2t)$ and $D_S^{x\theta } \equiv \langle \Delta {x_1}\Delta {\theta _1}\rangle /(2t)$.

The experimental results of $D_C^{xx}(X)$, $D_C^{\theta \theta }(X)$ and $D_C^{x\theta }(X)$ at different degrees of off-axial distance $Y/w$ are shown in figure 2 for different p and in figure 3 for different w. The effects of $(w,Y/w)$ have not been explored even for T–T HIs of two spheres. In our experiment, they all show strong couplings at small X which decay with increasing X and vanish beyond X_SC, reflecting a screening effect of the channel confinement on the HIs. The screening length ${X_{SC}}$ is defined as the minimal X where $|{{D_C}(X)} |\le 0.1{|{{D_C}(X)} |_{MAX}}$. The screening of HIs results in uncorrelated axial trajectories of ellipsoid pairs at $X > {X_{SC}}$ (see figure 1c,d). Interestingly, we find that $D_C^{xx}$ changes sign as $Y/w$ increases: $D_C^{xx} > 0$ at $Y/w < 0.5$ and X < X_SC for various p (see figure 2a–c) and various w (see figure 3a–c), whereas $D_C^{xx} < 0$ at $Y/w > 0.5$ (see figures 2d and 3d). Such positive and negative x–x couplings are directly observed from the concerted single-file movements of ellipsoid pairs that are in the same directions at $Y/w < 0.1$ (see inset of figure 1c) but in opposite directions at $Y/w > 0.5$ (see inset of figure 1d). We emphasize that the positive axial HI $(D_C^{xx} > 0)$ dominates in q1D confinements with $w/b < 4$ which is too strong to allow particles passing each other, while the anomalous negative axial HI $(D_C^{xx} < 0)$ occurs in weak channel confinement $(w/b > 4)$. By contrast, for spherical particles in q1D or wide channels, only positive x–x couplings were observed (Cui et al. Reference Cui, Diamant and Lin2002; Lin et al. Reference Lin, Cui, Lee and Yu2002; Novikov et al. Reference Novikov, Rice, Cui, Diamant and Lin2010; Misiunas et al. Reference Misiunas, Pagliara, Lauga, Lister and Keyser2015). It is well known that the single-file movements of the spheres occurred only in q1D confinements (Cui et al. Reference Cui, Diamant and Lin2002; Lin et al. Reference Lin, Cui, Lee and Yu2002; Misiunas et al. Reference Misiunas, Pagliara, Lauga, Lister and Keyser2015). By contrast, our result shows that single-file diffusions of ellipsoids can occur in wide channels because two off-axis ellipsoids tend to separate apart $(D_C^{xx} < 0)$ when they get close to each other.

Figure 2.

The experimental coupling diffusivities of ellipsoid pairs with different aspect ratio p as functions of interparticle distance $(X,Y)$ in a channel of width w = 6 μm. Here (a–d) $D_C^{xx}$, (e–h) $D_C^{\theta \theta }$ and (i–l) $D_C^{x\theta }$. The transverse interparticle separation $Y/w \in [0,0.1)$, $[0.1,0.3)$, $[0.3,0.5)$ and $[0.5,0.7)$ in (a,e,i), (b,f,j), (c,g,k) and (d,h,l), respectively.

Figure 3.

The experimental results of coupling diffusivities of ellipsoid pairs of aspect ratio p = 5.2 as functions of interparticle separation (X, Y) in channels of different width w. Here (a–d) $D_C^{xx}$, (e–h) $D_C^{\theta \theta }$ and (i–l) $D_C^{x\theta }$. The transverse interparticle separation $Y/w \in [0,0.1)$, $[0.1,0.3)$, $[0.3,0.5)$ and $[0.5,0.7)$ in (a,e,i), (b,f,j), (c,g,k) and (d,h,l), respectively.

The θ–θ coupling is negative $(D_C^{\theta \theta } < 0)$ at X < X_SC for various p, w and $Y/w$ (see figures 2e–h and 3e–h), indicating that an ellipsoid's clockwise rotation tends to induce its neighbour's anticlockwise rotation. The x–θ coupling is positive $(D_C^{x\theta } > 0)$ (see figures 2i–l and 3i–l), indicating that an ellipsoid tends to rotate away from the channel axis as a neighbour approaches, and vice versa.

The screening lengths measured from figures 2 and 3 for various p and w are shown in figure 4. We find that X_SC linearly increases with the long axis of ellipsoids (see figure 4a), leading to a rapid amplification of X_SC with particles’ shape anisotropy. For the same p, X_SC slowly increases with w (see figure 4b), reflecting a slightly stronger screening of HIs in a narrower channel. These results indicate that X_SC is much more sensitive to the particle elongation than the channel width in upper-side-open groove channels. By contrast, X_SC is on the length scale of w for colloidal spheres in the upper-side-open groove channels (Lin et al. Reference Lin, Cui, Lee and Yu2002; Valley et al. Reference Valley, Rice, Cui, Ho, Diamant and Lin2007). Therefore, particle elongation can extend the effective range of HIs exceeding the channel width $({X_{SC}} \gg w)$ as shown in figure 4(b). In addition, we find that ${X_{SC}}/a$ is the largest for $D_C^{xx}$ and the smallest for $D_C^{\theta \theta }$, indicating a stronger suppression of rotational HIs than axial translational HIs by the channel confinement. Particles at different $Y/w$ have similar behaviours of X_SC.

Figure 4.

The screening lengths X_SC (symbols) for $D_C^{xx}$, $D_C^{\theta \theta }$ and $D_C^{x\theta }$ of the coaxial ellipsoid pairs $({Y/w \in [0,0.1)} )$. Here (a) X_SC for different p and w = 6 μm and (b) X_SC for different w and p = 7.5. The lines are linear fittings.

The HIs can also be characterized by the Pearson correlation coefficient (Misiunas et al. Reference Misiunas, Pagliara, Lauga, Lister and Keyser2015): ${\rho ^{xx}} \equiv \textrm{cov(}\Delta {x_1},\Delta {x_2})/(\sigma (\Delta {x_1})\sigma (\Delta {x_2}))$ and similarly defined ${\rho ^{\theta \theta }}$ and ${\rho ^{x\theta }}$, where cov is the covariance, $\sigma $ is the standard deviation. They are equivalent to the normalized hydrodynamic coupling diffusivities ${\rho ^{xx}} = D_C^{xx}/D_S^{xx}$, ${\rho ^{\theta \theta }} = D_C^{\theta \theta }/D_S^{\theta \theta }$ and ${\rho ^{x\theta }} = D_C^{x\theta }/{(D_S^{xx}D_S^{\theta \theta })^{1/2}}$, which exclude the influence of channel confinement and particle anisotropy on self-diffusivities and enable a more precise comparison of HIs in different regimes of parameters $(p,w,Y/w)$ (Riley, Hobson & Bence Reference Riley, Hobson and Bence2006; Misiunas et al. Reference Misiunas, Pagliara, Lauga, Lister and Keyser2015). Figure 5 presents ${\rho ^{xx}}$, ${\rho ^{\theta \theta }}$ and ${\rho ^{x\theta }}$ as functions of the tip-to-tip distance of the ellipsoid pair. For the same tip-to-tip distance, the absolute value $|{{\rho^{xx}}} |$ increases more than one time as p increasing from 1.8 to 7.5 (see figure 5a–d), reflecting an enhanced T–T HIs for more anisotropic particles. Here $|{{\rho^{\theta \theta }}} |$ is non-monotonic with p and reaches the maximum at $p = 2.8$ (see figure 5e–h), reflecting the strongest R–R HIs at medium aspect ratios. We attribute the weak R–R HI as the small torque of the short moment arm for small p and the suppression from the channel confinement for large p. Here $|{{\rho^{xx}}} |$ and $|{{\rho^{\theta \theta }}} |$ decrease at larger $Y/w$, reflecting the suppression of T–T and R–R HIs for off-axis particle pairs which are closer to the channel walls. By contrast, figure 5(i–l) show that the maximum ${\rho ^{x\theta }}$ appears at middle $Y/w$, reflecting the strongest T–R HIs for moderate off-axis particles. The non-monotonic changes of ${\rho ^{\theta \theta }}$ with p (see figure 5e–h) and ${\rho ^{x\theta }}$ with $Y/w$ (see figure 5i–l) reflect non-trivial θ–θ and x–θ couplings. We explain these experimental results based on a theoretical model and low-Reynolds-number numerical simulation in the following sections.

Figure 5.

The Pearson correlation coefficients of coupling diffusivities of ellipsoid pairs as functions of the interparticle tip-to-tip distance $(X - 2a,Y)$ in a channel of width w = 6 μm. Here (a–d) ${\rho ^{xx}}$, (e–h) ${\rho ^{\theta \theta }}$ and (i–l) ${\rho ^{x\theta }}$. The transverse interparticle separations $Y/w \in [0,0.1)$, $[0.1,0.3)$, $[0.3,0.5)$ and $[0.5,0.7)$ in (a,e,i), (b,f,j), (c,g,k) and (d,h,l), respectively.

3.2. The y–y and y–θ coupling diffusivities

We next measure HIs about the transverse displacements by the y–y and y–θ coupling diffusivities as functions of (X, Y): $D_C^{yy} = \langle \Delta {y_1}\Delta {y_2}\rangle /(2t)$ and $D_C^{y\theta } = \langle \Delta {y_1}\Delta {\theta _2}\rangle /(2t) = \langle \Delta {\theta _1}\Delta {y_2}\rangle /(2t) = D_C^{\theta y}$ in figure 6, where $\Delta {y_i}$ is the CM displacement of ellipsoid i in the transverse direction of the channel during time interval t. Here $D_C^{xx}(X)$ and $D_C^{yy}(X)$ of two spheres have been measured in three dimensions (Meiners & Quake Reference Meiners and Quake1999) and quasi-two dimensions (Cui et al. Reference Cui, Diamant, Lin and Rice2004; Valley et al. Reference Valley, Rice, Cui, Ho, Diamant and Lin2007; Diamant Reference Diamant2009). In the present study, we comprehensively measure $D_C^{xx}$, $D_C^{\theta \theta }$ $D_C^{x\theta }$, $D_C^{yy}$ and $D_C^{y\theta }$ for ellipsoid pairs as a function of X and Y (see figures 2, 3 and 6).

Figure 6.

The experimental results of y–y and y–θ coupling diffusivities. Here (a,b) $D_C^{yy}$ and (c,d) $D_C^{y\theta }$ indicate no hydrodynamic coupling. Ellipsoid pairs with different p and at different axial interparticle distance X in a channel of w = 6 μm. (a,c) Coaxial $(Y/w \in [0,0.1))$ and (b,d) off-axis $(Y/w \in [0.5,0.7))$ ellipsoid pairs.

Figure 6 shows that $D_C^{yy}(X)$ and $D_C^{y\theta }(X)$ are nearly zero for different p and $Y/w$, reflecting vanishing y–y and y–θ couplings. This is attributed to the strong suppression of transverse motion in channel confinement as shown from the very small fluctuations in the transverse trajectories of ellipsoids (see figure 7a). In addition, we observe that two neighbouring ellipsoids with 2a < X < X_SC tend to locate symmetrically in the transverse direction about the central axis (y = 0) of the channel, that is, ${y_1}(t)\; = \; - {y_2}(t)$ (see figures 1a,b and 7); and the isolated single ellipsoids tend to equilibrate on the central axis, that is,$\; |{y(t)} |\; < \; 0.05w$ (see figure 7a). Note that X > 2a always hold since the ellipsoids cannot pass by each other in the narrow channel. Similar phenomena were observed in constant microfluidic flows in which the isolated polymers diffused towards the channel central axis (Usta, Ladd & Butler Reference Usta, Ladd and Butler2005) and the adjacent spherical droplets located symmetrically about the central axis and form zigzag patterns (Beatus, Tlusty, & Bar-Ziv Reference Beatus, Tlusty and Bar-Ziv2006; Fleury et al. Reference Fleury, Schiller, Thutupalli, Gompper and Seemann2014). These phenomena have been interpreted as resulting from the dipolar flow field around each particle in the channel confinement (Uspal & Doyle Reference Uspal and Doyle2014). However, the HIs and diffusion behaviours of particles due to the dipolar flow have not been measured. Our results indicate that the channel confinement suppresses the transverse motion much more than the rotational motion due to the large viscous drag along the ellipsoid short-axis. That is to say, the effect of the transverse component of a local dipolar flow created by a moving particle imposes a torque on the other particle and induces a rotational motion of that particle rather than a transverse motion.

Figure 7.

(a) Typical transverse positions $y(t)/w$ of an ellipsoid pair with 2a < X < X_SC (blue curves) and a single ellipsoid (black curve). The channel boundary is at $Y/w$ = ±0.5. (b) The corresponding histogram of the transverse positions $H({y_1}/w)$ and $H({y_2}/w)$ of the ellipsoid pair with 2a < X < X_SC (symbols) and the Gaussian fits (curves). The aspect ratio p = 7.5, and the channel width w = 6 μm.

4.1. Dumbbell-in-cylinder model

The flow field created by a moving particle is usually estimated using the point force (Stokeslet) approximation (Batchelor Reference Batchelor1970; Cui et al. Reference Cui, Diamant and Lin2002, Reference Cui, Diamant, Lin and Rice2004; Hernández-Ortiz, de Pablo & Graham Reference Hernández-Ortiz, de Pablo and Graham2006; Valley et al. Reference Valley, Rice, Cui, Ho, Diamant and Lin2007; Diamant Reference Diamant2009; Novikov et al. Reference Novikov, Rice, Cui, Diamant and Lin2010). For example, a spherical particle, an elongated particle and a long-thin rod or fibre are often modelled as a point force (Cui et al. Reference Cui, Diamant and Lin2002, Reference Cui, Diamant, Lin and Rice2004; Valley et al. Reference Valley, Rice, Cui, Ho, Diamant and Lin2007; Diamant Reference Diamant2009; Novikov et al. Reference Novikov, Rice, Cui, Diamant and Lin2010), a dumbbell of two point forces (Hernández-Ortiz et al. Reference Hernández-Ortiz, de Pablo and Graham2006; Bukowicki, Gruca, & Ekiel-Jeżewska Reference Bukowicki, Gruca and Ekiel-Jeżewska2014) and a line of point forces (Batchelor Reference Batchelor1970; Di Leonardo et al. Reference Di Leonardo, Cammarota, Bolognesi, Schäfer and Steinhart2011; de Graaf et al. Reference de Graaf, Menke, Mathijssen, Fabritius, Holm and Shendruk2016), respectively. The T–T hydrodynamic coupling of colloidal spheres has been described by treating the moving sphere as a point force at the sphere centre (Cui et al. Reference Cui, Diamant and Lin2002, Reference Cui, Diamant, Lin and Rice2004; Valley et al. Reference Valley, Rice, Cui, Ho, Diamant and Lin2007; Diamant Reference Diamant2009; Novikov et al. Reference Novikov, Rice, Cui, Diamant and Lin2010). The translation of a spheroid in unbounded flow has been described by a line distribution of Stokeslets between the foci (Chwang & Wu Reference Chwang and Wu1976). Here, we proposed a simplified dumbbell-in-cylinder model to estimate the T–T, R–R and T–R coupling of two ellipsoids in channels. This model is based on the following three approximations as illustrated in figure 8 and summarized in table 1. (i) The motion of an ellipsoid is treated as a dumbbell of point forces $(\boldsymbol{F},\boldsymbol{F^{\prime}})$, that is to say, the effect of the flow field induced by an ellipsoid on the other is equivalent to exerting $(\boldsymbol{F},\boldsymbol{F^{\prime}})$ on the fluid. The dumbbell length is 2X_OF along the long axis with the centre coinciding with the ellipsoid's CM. (ii) The HIs between two ellipsoids are dominated by the point-force pair F₁ and F₂ near the centre of the two ellipsoids. Here ${\boldsymbol{F_1^{\prime}}}$ and ${\boldsymbol{F}_2^{\prime}}$ at the far ends can be neglected due to the strong screening of HIs in the channel. (iii) The confinement effect of a rectangular groove with an open side is equivalent to that of a cylinder with a radius R > w as confirmed in Cui et al. (Reference Cui, Diamant and Lin2002), Diamant (Reference Diamant2009) and Novikov et al. (Reference Novikov, Rice, Cui, Diamant and Lin2010). These assumptions hold well under the low Reynolds number (~10⁻⁶), the no-slip boundary conditions (Happel & Brenner Reference Happel and Brenner1983), $b \ll R$, and $(a - {X_{OF}}) \ll {X_F}$.

Figure 8.

Schematic diagram of the dumbbell-in-cylinder model. (a) Rotational displacement $\Delta \theta $ and (b) axial displacement $\Delta x$ of an ellipsoid pair between time = 0 (solid ellipses with filled circles) and time = t (dashed ellipses with open circles) in the effective cylinder (green dashed contours) of diameter 2R which is equivalent to a channel (solid blue lines) of width w. The subscript represents ellipsoid 1 or 2. The dumbbell of point forces $\boldsymbol{F}({x_F},{y_F})$ and $\boldsymbol{F^{\prime}}({x_{\mathrm{F^{\prime}}}},{y_{\mathrm{F^{\prime}}}})$ at the red circles are symmetric about the ellipsoid's CM (black circle) at (x, y) along the long axis. The transverse force ${F^y} ={-} {F^{\prime y}}$ in (a) and axial force ${F^x} = {F^{\prime x}}$ in (b) are proportional to the translational and axial displacements (red arrows) $\Delta {y_F} ={-} \Delta {y_{F^{\prime}}}$ and $\Delta {x_F} = \Delta {x_{F^{\prime}}}$, respectively (Goldfriend et al. Reference Goldfriend, Diamant and Witten2015).

Table 1.

Summary of the parameters, coordinates and coupling diffusivities in the experiment and theoretical models with subscript 1 or 2 denoting the left or right ellipsoid in figure 8.

According to this theoretical model, the T–T, T–R and R–R HIs between ellipsoids are treated as derived from the x–x, x–y and y–y components of the Stokeslet tensor. The axial translational and rotational displacements $(\Delta x,\Delta \theta )$ of an ellipsoid can be represented by the axial and transverse displacements $(\Delta {x_F},\Delta {y_F})$ (see figure 8 and table 1). Consequently, ${D_C}(X,Y)$ of an ellipsoid pair in a channel can be approximately transformed into the dimensionless coupling diffusivities in an effective cylinder

(4.1)\begin{equation}\varDelta _F^{{\rm \mu} \nu } \equiv \langle \Delta {\boldsymbol{\mu} _{F1}}\Delta {\nu _{F2}}\rangle /(2t{k_B}T/6\boldsymbol{\pi} \eta R),\end{equation}

as functions of the rescaled axial and transverse separations ${X_F}/R = (X - 2{X_{OF}})/R$ and ${Y_F}/R \approx Y/R$ (see figure 8 and table 1), where μ, ν = x or y and η is the fluid viscosity. By substituting the Stokes-flow Oseen tensor in a cylinder derived from Liron & Shahar Reference Liron and Shahar1978 and Blake Reference Blake1979 into (4.1), we obtained the theoretical equations

(4.2) \begin{align} \varDelta _F^{{\rm \mu} \nu }({X_F}/R,{Y_F}/R) & = 6{\boldsymbol{\pi} ^2}\sum\limits_{k = 0}^3 \sum\limits_{n = 1}^3 \left\{ [a_{kn}^{{\rm \mu} \nu }({Y_F}/R)\,\textrm{cos}({\alpha _{kn}} \cdot {X_F}/R)\right.\nonumber\\ & \quad \left.+\, b_{kn}^{{\rm \mu} \nu }({Y_F}/R)\,\textrm{sin}({\alpha _{kn}} \cdot {X_F}/R)]\,{\textrm{e}^{ - {\beta _{kn}} \cdot ({X_F}/R)}} \right.\nonumber\\ & \left.\quad +\, c_{kn}^{{\rm \mu} \nu }({Y_F}/R)\,{\textrm{e}^{ - {\gamma _{kn}} \cdot ({X_F}/R)}}\right\},\end{align}

where the values of $({\alpha _{kn}},{\beta _{kn}},{\gamma _{kn}})$ and $(a_{kn}^{{\rm \mu} \nu },\; b_{kn}^{{\rm \mu} \nu },c_{kn}^{{\rm \mu} \nu })$ can be obtained from Liron & Shahar (Reference Liron and Shahar1978). Details about the theoretical expression are given in the Appendix (A). The theoretical results of $\varDelta _F^{{\rm \mu} \nu }$ at different $({X_F}/R,{Y_F}/R)$ in the dumbbell-in-cylinder model are shown in figure 9.

Figure 9.

Theoretical estimates of the (a) x–x, (b) y–y and (c) x–y coupling diffusivities of point-force pairs as functions of the rescaled axial distance ${X_F}/R$ and transverse distance ${Y_F}/R$ calculated from (4.2).

4.2. Comparisons between the experimental and theoretical results

We quantitatively compare the experimental ${D_C}$ with the theoretical prediction of the aforementioned model in figure 10. The two parameters X_OF and R in different $(p,w,Y/w)$ regimes can be fitted from two out of three couplings $\Delta _F^{xx},\;\Delta _F^{yy}$ and $\varDelta _F^{xy}$. For example, the fitted X_OF and R from $\varDelta _F^{xx},\;\varDelta _F^{yy}$ can describe $\varDelta _F^{xy}$ well, and have no discernable difference from their best fits from all the three couplings, indicating that the dumbbell-in-cylinder model is effective. Moreover, figure 11(a) shows that R fitted at a certain $(w,Y/w)$ is robust at different p, indicating that R is a good parameter for describing the effects of the channel without the effects of ellipsoids. Figure 11(d) shows that X_OF at a certain p is robust at different $(w,Y/w)$, indicating that X_OF is a good parameter for describing the effects of particles without the effects of the channel. We find that the theoretical model can describe the experimental data well by taking only the leading terms. As depicted in figure 10, the experimental ${\varDelta _F}({X_F}/R)$ (symbols) for all p and w at each $Y/w$ collapse onto the theoretical curves at the corresponding $Y/R$, suggesting a general mechanism of the Stokes-flow-induced hydrodynamic couplings. The experimental positive and negative x–x couplings of ellipsoid pairs at $Y/w < 0.5$ (see figures 2a–c and 3a–c) and $Y/w > 0.5$ (see figures 2d and 3d) are described well by the theoretical $\varDelta _F^{xx} > 0$ at ${Y_F}/R < 0.5$ (see figure 10a–c) and $\varDelta _F^{xx} < 0$ at ${Y_F}/R > 0.5$ (see figure 10d), respectively. The negative θ–θ (see figures 2e–h and 3e–h) and positive x–θ (see figures 2i–l and 3i–l) couplings of ellipsoid pairs agree with the theoretical $\varDelta _F^{yy} > 0$ (see figure 10e–h) and $\varDelta _F^{xy} > 0$ (see figure 10i–l). These consistent results reflect that the axial and transverse flows drive the axial and rotational motion of an ellipsoid, and T–T, R–R and T–R couplings reflect well the effect of HIs mediated by the x–x, y–y and x–y components of the induced flow. A reverse sign in $\varDelta _F^{yy}$ at ${X_F}/R\sim 1.0$ is predicted to arise from the alternating signed eddies of the Stokeslet along the channel (Liron & Shahar Reference Liron and Shahar1978; Blake Reference Blake1979), which is not observed experimentally here for ellipsoids nor for colloidal spheres previously (Novikov et al. Reference Novikov, Rice, Cui, Diamant and Lin2010). This discrepancy probably occurs because the surface of ellipsoid 1 between the point force and the tip, i.e. in the range of $({x_1} + {X_{OF}},{x_1} + a)$ near ellipsoid 2, produces positive $\varDelta _F^{yy}$ and vice versa, which superposes onto the negative $\varDelta _F^{yy}$ when the interparticle distance is  more than the screening distance ($X > {X_{SC}}$ in the experiment, i.e. ${X_F} > X_{SC}^F$ in the model). An accurate description needs an integration over the whole surface of the particle, see the finite element numerical simulations in § 5 which fully recover the experimental $\varDelta _F^{yy}$; however, our dumbbell model approximates each ellipsoid only by one point force which can describe the average effect within the screening distance (${X_F} < X_{SC}^F$ in figure 10), but deviates from the experimental $\varDelta _F^{yy}$ at ${X_F} > X_{SC}^F$. A more complicated model including multiple point forces may smear out the shallow negative $\varDelta _F^{yy}$ derived from our model with single Stokeslet approximation. Nevertheless, the agreement between the experimental and theoretical results in figure 10 ensures the validity and efficiency of this simple analytical model to describe the strongly screened HIs of ellipsoids in channel confinement.

Figure 10.

Rescaled coupling diffusivities (a–d) $\varDelta _F^{xx}$, (e–h) $\varDelta _F^{yy}$ and (i–l) $\varDelta _F^{xy}$ as functions of ${\xi ^x} = {X_F}/R$. The experimental data (symbols) for all p and w collapse onto the theoretical fittings of (4.2) (solid curves) except for the negative part of $\varDelta _F^{yy}$. The transverse separation of the ellipsoid pairs $Y/w \in [0,0.1)$, [0.1, 0.3), [0.3, 0.5) and [0.5, 0.7) in (a,e,i), (b,f,j), (c,g,k) and (d,h,l), respectively, with the fitted $Y/R = 0.04,\;0.14,\;0.32$ and 0.52. Vertical dashed lines mark the screening length $X_{SC}^F/R$.

Figure 11.

The effective cylinder radius R and semidumbbell length X_OF in the dumbbell-in-cylinder model. Here (a) R (symbols) versus $Y/w$ obtained from figure 10 with linear fittings (solid lines) for experiments at different w and for all p; (b) $2R/w$ versus $Y/w$ rescaled from (a) with the linear fittings (solid lines); (c) $Y/R$ versus $Y/w$ rescaled from (a) with power-law fittings (solid curves), in (b,c), the black fitting curves are for square and triangle symbols; (d) ${X_{OF}}/a$ versus p for experiment (black squares) obtained from figure 10 and for numerical simulation (red circles) fitted from figure 13(c,d) with the curve guiding for the eyes.

Here, $\varDelta _F^{xx},\;\varDelta _F^{yy}$ and $\varDelta _F^{xy}$ decay to zero at $X_{SC}^F/R \approx 1.5$, 0.5 and 0.8, respectively, for various p and w (vertical dashed lines in figure 10). Here $X_{SC}^F$ is the screening length of $\varDelta _F^{{\rm \mu} \nu }$. These different values of $X_{SC}^F/R$ indicate that the channel confinement suppresses the transverse local flow more than the axial flow. It explains the experimental phenomena in figure 4 that the screening length X_SC is shortest for R–R coupling, intermediate for T–R coupling and longest for T–T coupling of two ellipsoids.

The fittings in figures 10 and 11 show that the cylinder radius R is sensitive to w and $Y/w$ but not to p, while the dumbbell length X_OF is sensitive to p but not to w and $Y/w$. The effects of channel confinement, i.e. w and $Y/w$ and particle anisotropy, i.e. p, on HIs are captured separately and quantitatively by the cylinder radius R and the dumbbell length X_OF in our theoretical model. Figure 11(a) shows that R increases with w and linearly decreases as $Y/w$ increased. It reflects the stronger geometric confinement for particles closer to the channel wall can be linearly characterized by the smaller R. This results in that $R/w$ decreases and $Y/R$ increases with $Y/w$ (see figure 11b,c), respectively. Here $R/w$ or $Y/R$ show different trends as a function of $Y/w$ for channels with different cross-section shapes (figure 11b,c). Here, we discover that the fitted cylinder radius R is sensitive to both the degree of channel confinement $(w,Y/w)$ and the channel geometry $(w/d)$. As illustrated in figure 11(d), ${X_{OF}}/a$ increases with p and becomes asymptotic to 1 at large p, indicating that the effective point forces move toward the ellipsoid tips as the ellipsoid shape becomes more anisotropic. It predicts the increase of screening length ${X_{SC}} = X_{SC}^F + 2{X_{OF}}$ with a and p (see figure 4a). At small p, the logarithmic extrapolation of ${X_{OF}}/a$ in figure 11(d) correctly yields ${X_{OF}}/a = 0$ at p = 1 for spheres (Cui et al. Reference Cui, Diamant and Lin2002, Reference Cui, Diamant, Lin and Rice2004; Valley et al. Reference Valley, Rice, Cui, Ho, Diamant and Lin2007; Diamant Reference Diamant2009; Novikov et al. Reference Novikov, Rice, Cui, Diamant and Lin2010), reflecting the accuracy of the dumbbell model. As a result, ${X_{SC}} = X_{SC}^F$ for $D_C^{xx}$ of spheres in q1D groove channels (Cui et al. Reference Cui, Diamant and Lin2002) is much shorter than ${X_{SC}}$ for ellipsoids.

This model also explains the change in coupling strength at different parameter regimes in figures 2, 3 and 5. The more enhanced T–T coupling for ellipsoids of larger p (see figures 2a–d and 5a–d) arises from the larger $\varDelta _F^{xx}$ for closer point-force pair, i.e. at shorter ${X_F} = X - 2{X_{OF}}$ due to longer ${X_{OF}}$ (see figure 11d). The weak θ–θ coupling for ellipsoids with $p \le 2.2$ (see figures 2e–h and 5e–h) can be attributed to that the moment arm ${X_{OF}}/a < 0.5$ is too small for slightly anisotropic particles to induce sufficient torque against thermal fluctuations. The θ–θ coupling decreases at $p \ge 5.2$ in figures 2e–h and 5e–h can be regarded as a result of the accumulation of negative $\varDelta _F^{yy}$ (see figures 9b and 10e–h) along the surface of largely anisotropic ellipsoids. The strongest T–R coupling at middle $Y/w$ (see figures 2i–l, 3i–l and 5i–l) is consistent with the maximum $\varDelta _F^{xy}$ at middle $Y/R$ (see figures 9c and 10i–l).

5.1. Comparisons between the simulation, experimental and theoretical results

We further perform three-dimensional finite element numerical simulations to verify the theoretical model by comparing the experimental, simulation and theoretical results, and to interpret the experimental HIs by calculating the flow field. The finite element numerical simulations are carried out using the software Comsol Multiphysics v5.2. To connect between the experimental system and theoretical model in the simulation, we use the geometries of two ellipsoids in an effective cylinder in the theoretical model, and input the parameters of the size (a, b) and axial and angular velocities $(u,\omega )$ of the ellipsoids, and the fitted radius R of the cylinder for the groove channels with width w in the experiment. The two ellipsoids are separated with (X, Y) and symmetrically located about the central axis of the cylinder. We set no-slip boundary conditions on the surface of the ellipsoids and cylinder, and open boundary conditions at the two ends of the cylinder.

In order to simplify the calculation, we apply an instantaneous axial velocity $u = ({u_1} = \Delta {x_1}/\Delta t,{u_2} = 0)$ or an instantaneous angular velocity $\omega = ({\omega _1} = \Delta {\theta _1}/\Delta t,\; {\omega _2} = 0)$ separately on the ellipsoid pair at each (X, Y) in a cylindrical channel with radius R (see figure 8), i.e. let one ellipsoid move and the other be at rest. We set the typical velocities $({u_1} = 1\;{\rm \mu} \mathrm{m}\;{\textrm{s}^{ - 1}},{\omega _1} = 2^\circ \;{\textrm{s}^{ - 1}})$ and $\Delta t = 0.1\;\textrm{s}$ according to the ellipsoid motions and frame rate in the experiment. We solve the Stokes equation in the cylinder and yield F and L from integrating the force distribution density over the ellipsoid surface. Thus each component term of $\boldsymbol{\zeta }$ is directly calculated through (1.3), and the matrix of diffusion tensors $\boldsymbol{D}$ at each (X, Y) is obtained through (1.2) (Montgomery & Berne Reference Montgomery and Berne1977; Mazo Reference Mazo2002). Due to the velocity-independency of $\boldsymbol{\zeta }$, a different set of velocity and $\Delta t$ will not change the results as long as (i) the Reynolds number is approximately 10⁻⁶ for micron-sized particles and (ii) ${x_1}(0) - 0.1a < {x_1}(\Delta t) < {x_1}(0) + 0.1a$ for ellipsoid 1. Here $0.1a$ is the bin size in the analysis of experimental results. Such a numerical method in the Comsol software has been used in the study of HIs of two spheres in a cylindrical channel in Misiunas et al. (Reference Misiunas, Pagliara, Lauga, Lister and Keyser2015). More benchmarking studies for the numerical method can be found in Dettmer et al. (Reference Dettmer, Pagliara, Misiunas and Keyser2014), Misiunas et al. (Reference Misiunas, Pagliara, Lauga, Lister and Keyser2015) and Misiunas & Keyser (Reference Misiunas and Keyser2019). The numerical simulation results of hydrodynamic coupling diffusivities $D_C^{xx}(X)$, $D_C^{x\theta }(X)$ and $D_C^{\theta \theta }(X)$ for an ellipsoid pair in the effective cylinder are shown in figure 12, which reproduce the experimental results for the coaxial and most off-axis ellipsoid pairs in the groove channels in figure 2. The screening lengths ${X_{SC}}$ in the effective cylinder obtained in simulation (see figure 13) are consistent with those in the groove channels measured in experiments (see figure 5). Applying velocities in opposite directions gives the same quantitative results. The agreement between the experimental and simulation results confirm the validity of using the tube model to mimic the groove-like channel for the in-plane motions.

Figure 12.

Numerical simulation results of coupling diffusivities ${D_C}(X)$. (a,c,e) The coaxial (cylindrical radius R = 9.8 μm corresponding to $Y/w \in [0,0.1)$ and w = 6 μm in experiments) and (b,d,f) the off-axis (R = 6.5 μm, $Y/w \in [0.5,0.7)$ and w = 6 μm) ellipsoid pairs with different p. The vertical arrows mark the screening length X_SC. Here ${D_C}$ is calculated from the simulation result of $\boldsymbol{\zeta }$ based on (1.2) in order to compare with the experimental ${D_\textrm{C}}$ in figure 2.

We further transform the numerical simulation results of ${D_C}$ into $\varDelta _F^{{\rm \mu} \nu }$ and quantitatively compare them with the theoretical prediction by fitting the dumbbell length 2X_OF. Figure 14 shows that both the simulation and experimental results of $\varDelta _F^{xx},\;\varDelta _F^{yy}$ and $\varDelta _F^{xy}$ coincide with their theoretical curves. The fitted X_OF at each p from the simulation are the same in the three curves and consistent with those fitted from the experiment with only a 3%–5% systematic exceedance (see figure 11d). This exceedance probably reflects the larger hydrodynamic diameter than the geometrical diameter, which results in their effective aspect ratios slightly smaller than those used in the simulation.

Figure 13.

The numerical simulation counterpart of figure 4. The screening lengths X_SC (symbols) for $D_C^{xx}$, $D_C^{\theta \theta }$ and $D_C^{x\theta }$ of the coaxial ellipsoid pairs at $Y/w \in [0,0.1)$. Here (a) X_SC for different p and w = 6 μm and (b) X_SC for different w and p = 7.5. The lines are linear fittings.

Figure 14.

Rescaled coupling diffusivities $\varDelta _F^{{\rm \mu} \nu }({X_F}/R)$. The experimental data (symbols in a,b) and the numerical simulation data (symbols in c,d) for all p and w collapse onto the theoretical curves (solid curves) of (4.2) with parameters R and X_OF except for the negative part of $\varDelta _F^{yy}$. (a,c) The coaxial $({Y/w \in [0,0.1)} )$ and (b,d) the off-axis $({Y/w \in [0.5,0.7)} )$ ellipsoid pairs. The vertical arrows mark $X_{SC}^F/R$.

5.2. Flow field induced by a moving ellipsoid in a cylinder

We measure the flow field induced by the moving ellipsoid in the numerical simulation to illustrate the effects of channel confinement and particle anisotropy on x–x, θ–θ and x–θ HIs. These flow fields in figures 15–17 are more complicated than those of colloidal spheres in strong q1D confinements (Cui et al. Reference Cui, Diamant and Lin2002; Misiunas et al. Reference Misiunas, Pagliara, Lauga, Lister and Keyser2015) where the coaxial flow induced by an axially moving sphere dominated, and only a positive x–x coupling was observed. Here, we can see that the axial and rotational motion of an ellipsoid actually reflects the axial and transverse component of the local flow field. For the coaxial ellipsoid pair in figure 15(a,c,e), a rightward axial motion of the right ellipsoid induces a rightward axial flow pulling the left ellipsoid (see figure 15a), which leads to the positive x–x coupling. The axial flow turns transversely near the ellipsoid tips causing the rotation of the left ellipsoid towards the axis (see figure 15c), which leads to the positive x–θ coupling. A rotation of an ellipsoid induces a whirling flow imposing a torque on the other ellipsoid in an opposite direction (see figure 15e), which leads to the negative θ–θ coupling.

Figure 15.

Numerical simulation results of the flow fields created by the motion of an ellipsoid. (a–d) Translational and (e,f) rotational motions of the right ellipsoid. (a,c,e) The coaxial (cylinder radius R ₁ = 9.8 μm corresponding to $Y/w \in [0,0.1)$ and w = 6 μm in experiments) and (b,d,f) the off-axis (R ₂ =  6.5 μm, $Y/w \in [0.5,0.7)$ and w = 6 μm) ellipsoid pairs with p = 7.5. The length and direction of the black arrows describe the local flow velocities. The colour represents the local flow speed ${u^\nu }$ in the ν direction.

Figure 16.

Simulation results of the flow fields created by the motion of an ellipsoid in the opposite direction to that in figure 15. (a–d) Translational and (e,f) rotational motions of the right ellipsoid. (a,c,e) The coaxial (R ₁ = 9.8 μm corresponding to $Y/w \in [0,0.1)$ and w = 6 μm in experiments) and (b,d,f) the off-axis (R ₂ =  6.5 μm, $Y/w \in [0.5,0.7)$ and w = 6 μm) ellipsoid pairs with p = 7.5. The length and direction of the black arrows describe the local flow velocities. The colour represents the local flow speed ${u^\nu }$ in the ν direction.

Figure 17.

Numerical simulation results of the flow fields created by the motion of an ellipsoid. (a,b) Translational and (c,d) rotational motions of the right ellipsoid. The cylinder of R = 9.8 μm corresponding to the coaxial $({Y/w \in \; [0,0.1)} )$ ellipsoid pairs with (a,b) p = 2.2 and (c,d) p = 7.5 in a channel with w = 6 μm. The length and direction of the black arrows describe the local flow velocities. The colour represents the local flow speed ${u^\nu }$ in the ν direction.

By contrast, for the off-axis ellipsoid pair (see figure 15b,d,f), a rightward axial motion of the right ellipsoid induces a leftward backflow near the opposite sidewall repelling the left ellipsoid (see figure 14b), which leads to an anomalous negative x–x coupling as shown in the experiments, see figures 2(d) and 3(d). The weak channel confinement at $w/b > 4$ enables the propagation of the backflow which is necessary for the negative x–x coupling. This repulsive HI ensures the single-file motion of ellipsoids in opposite directions even in channels moderately wider than 4b. It is in contrast to the strong q1D confinement where the piston-like motion of a sphere can only push other particles move in the same directions (Cui et al. Reference Cui, Diamant and Lin2002; Misiunas et al. Reference Misiunas, Pagliara, Lauga, Lister and Keyser2015). The turning flow between the two ellipsoids’ tips in figure 15(d) is much stronger than that for the coaxial ellipsoids in figure 15(c), which results in the most enhanced x–θ coupling at middle $Y/w$ in figures 2(i–l), 3(i–l) and 5(i–l). The whirling flow in figure 15(f) leads to similar negative θ–θ coupling as that of coaxial ellipsoids in figure 15(e). The smaller radius of the effective cylinder for more off-axis ellipsoids corresponds to stronger screening effect of the channel. As a result, the fast decay of flow field near the cylinder wall predicts weaker couplings for more off-axis ellipsoids as shown in figures 2, 3 and 5. Figure 16 shows that an ellipsoid moving in directions opposite to that in figure 15 produces similar patterns of the induced flow fields with reverse signs, which leads to the same behaviour of hydrodynamic couplings.

We further compare the flow fields of ellipsoids with different aspect ratios. Figure 17 shows that a more anisotropic ellipsoid induces a stronger flow in a larger range, which explains the longer X_SC and stronger HIs for larger p (see figures 2–5). Especially, the weaker backflow at smaller p in figure 17(a,c) indicates that the backflow can be hardly probed by spheres, which may explain why the negative x–x coupling has not been observed before. The dipolar pattern of the local flow field in figures 14–16 confirms the validity of the dumbbell model for the ellipsoids. The transverse flow shows opposite directions at the two ends of an ellipsoid (see figure 15c–f). It verifies our interpretation for the suppressed transverse motion of the ellipsoids that the local flow induces mainly an effective torque rather than a transverse force. We find that the transverse flow in figure 15(e,f) decays much faster than the axial flow in figure 15(a,b), corresponding to the strongest screening of θ–θ coupling with shortest X_SC and the weakest screening of x–x coupling with longest X_SC by the channel confinement (see figures 2–5). Therefore, the effective region of the transverse flow is relatively shorter for longer ellipsoids. This screening effect on the anisotropic particles is featured by the longer X_OF for larger p in the dumbbell model (see figure 11d). It results in the effective point force very close to the ellipsoid tip at $p \ge 5.2$, which leads to a slight decrease in θ–θ coupling at $p \ge 5.2$ (see figures 2e–h and 5e–h).

The cylindrical tube with an appropriate diameter in our model reflects the average effects from all three walls and the open top of the groove. Particles have similar distances to the cylinder walls in the model, but a much closer distance to the bottom wall in the experiment. Such asymmetry with the lubrication flow in the bottom gap leads to a flow field distribution in the z direction different from that in the cylinder, which may make the flow field in the cylinder model quantitatively deviate from that in the experiment. Nevertheless, the HIs are still short-ranged due to the screening effect of the two sidewalls (figures 2 and 3) (Cui et al. Reference Cui, Diamant and Lin2002; Valley et al. Reference Valley, Rice, Cui, Ho, Diamant and Lin2007; Diamant Reference Diamant2009; Novikov et al. Reference Novikov, Rice, Cui, Diamant and Lin2010), thus whether the top of the groove is open or closed and whether the particles are closer to the bottom wall should not qualitatively change the results when the channel is not too narrow compared with the particle size. A qualitatively different case is a very narrow tube whose diameter is close to the particle size, where T–T HI is long-ranged due to the piston flow induced by the particle (Misiunas et al. Reference Misiunas, Pagliara, Lauga, Lister and Keyser2015). By contrast, for a cylinder with a radius much larger than the ellipsoid $(R \gg b)$, the short-ranged swirl and back flows induced by an ellipsoid (figures 15–17) are similar to those induced by a Stokeslet (point force) in the cylinder model (Liron & Shahar Reference Liron and Shahar1978; Blake Reference Blake1979). The open-top channels and the cylinder with comparable scale $(1 < R/w\mathrm{\ \mathbin{\lower.3ex\hbox{$\buildrel< \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}}\ }1.75)$ have similar confinement effects on HIs in the horizontal direction for ellipsoids much smaller than the channel width ($2.99 < w/b < 7.3$ in all cases) because $R/b$ and $w/b$ are of the same order. The lubrication flow in the narrow bottom gap does not affect the short-range nature of the HIs in the groove because the fluid can easily flow from the sidewalls or the open top. That is, the HIs are short-ranged and the similar swirl and back flows can be qualitatively expected for two ellipsoids either sediment to the bottom of an open-top channel or in the middle of a cylinder; the numerical results of two ellipsoids in a cylinder (figures 12 and 13) agree with the experimental results in the channel (figures 2 and 4); both experimental and numerical results are fitted by the model with consistent fitting parameters (figures 10, 11 and 14). Qualitatively, the decay length of flow field is comparable to w in the groove and comparable to R in the cylinder model, which is confirmed by the measured $0.8\mathrm{\ \mathbin{\lower.3ex\hbox{$\buildrel< \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}}\ }X_{SC}^F/w\mathrm{\ \mathbin{\lower.3ex\hbox{$\buildrel< \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}}\ }2$ and $0.5\mathrm{\ \mathbin{\lower.3ex\hbox{$\buildrel< \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}}\ }X_{SC}^F/R\mathrm{\ \mathbin{\lower.3ex\hbox{$\buildrel< \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}}\ }1.5$ for various p and w in figures 11 and 12. Therefore, the cylinder model with the Stokeslet approximation can capture the main features of the experimental T–T, T–R and R–R couplings (figures 10 and 11). Such a simple analytical cylinder model has successfully captured the main features of short-ranged T–T coupling between colloidal spheres in open-top grooves (Cui et al. Reference Cui, Diamant and Lin2002; Valley et al. Reference Valley, Rice, Cui, Ho, Diamant and Lin2007; Diamant Reference Diamant2009; Novikov et al. Reference Novikov, Rice, Cui, Diamant and Lin2010).