4.1. Experimental results

We perform a systematic study varying the particle radius $a$, the fibre radius $R$ and the viscosity of the interstitial fluid $\mu$. Here the capillary length is of the same order for all silicone oils considered and equal to $\ell _c \simeq 1.4\ \textrm {mm}$. Examples of experimental observations are reported in figure 3. For a given fibre and particle radii, there is a threshold velocity $U^*$ below which only a liquid film coats the fibre, and the particles remain trapped in the liquid bath. Beyond this threshold, particles start to cover the fibre. Therefore, three regimes are observed, similar to the 2-D situation with the withdrawal of a plate (Sauret et al. Reference Sauret, Gans, Colnet, Saingier, Bazant and Dressaire2019): a coating film (i) without particles as shown in the left-hand picture in figure 1(b); (ii) with clusters of particles formed in the meniscus and entrained in the coating film, visible on the middle picture in figure 1(b); and (iii) with individual particles as reported in the right-hand picture in figure 1(b). The different regimes are clearly visible in figure 1(b) where the volume fraction is slightly higher that what we considered for the systematic measurements of the entrainment threshold (figure 3). As shown by Sauret et al. (Reference Sauret, Gans, Colnet, Saingier, Bazant and Dressaire2019), the entrainment of clusters depends on the probability of forming a cluster of large enough size in the meniscus, whereas the entrainment of individual particles only requires that they be in close enough proximity to the fibre and overcome the resistive capillary force induced by the air–liquid meniscus. Therefore, at very small volume fraction, the range of existence of the cluster regime is very limited and makes it easier to estimate the entrainment threshold. For this reason, we consider only dilute suspensions in this work making it easier to focus on the threshold velocity $U^*$ for isolated particle entrainment.

Figure 3.

Examples of fibres after their withdrawal from a suspension: (a–e) $a=70\ \mathrm {\mu }\textrm {m}$ particles dispersed in AP 100, for a fibre of radius $R=1.2\ \textrm {mm}$ and increasing velocities $U=0.5,1,3,5,7\ \textrm {mm}\,\textrm {s}^{-1}$ (from left to right), scale bars are 1 mm; (f–j) $a=70\ \mathrm {\mu }\textrm {m}$ particles dispersed in AP 1000 for a fibre of radius $R=0.55\ \textrm {mm}$ and increasing velocities of $U=1,2,3,4,5\ \textrm {mm}\,\textrm {s}^{-1}$, scale bars are $500\ \mathrm {\mu }\textrm {m}$.

The entrainment threshold $U^*$ increases with the radius of the particles $a$ for a given fibre diameter and is not influenced by inertial effects. Indeed, the maximum value of the Reynolds number, defined as ${Re}=\rho h U / \mu$, is obtained for the thicker films corresponding to $\rho \simeq 1060\ \textrm {kg}\,\textrm {m}^{-3}$, $h\sim 50\ \mathrm {\mu }\textrm {m}$, $U \sim 10^{-2}\ \textrm {m}\,\textrm {s}^{-1}$ and $\mu =0.132\ \textrm {Pa}\,\textrm {s}$, and satisfies $Re < 10^{-2}$. We can thus consider that we are always in a viscous regime where ${Re} \ll 1$. The threshold is directly related to the thickness of the coating fluid on the fibre, which increases with the speed of withdrawal. Therefore, the entrainment of larger particles on a fibre of radius $R$ requires a larger withdrawal speed. The experimental trend observed here for the influence of the particle size is similar to the trend reported for 2-D plates (Sauret et al. Reference Sauret, Gans, Colnet, Saingier, Bazant and Dressaire2019).

The main difference brought by the cylindrical geometry is that the particle entrainment threshold not only depends on the size of the particles $a$ and the viscosity of the interstitial fluid $\mu$ but also on the radius of the fibre $R$. For fibres of different radii and the same interstitial fluid and particle size, figure 4(a) illustrates that the entrainment of particles does not occur at the same threshold velocity. The entrainment of particles in the liquid film occurs at a larger withdrawal speed for the smaller fibres. Therefore, the influence of the fibre radius provides a new degree of control of the entrainment threshold. We should also emphasize that when the diameter of the particle $2a$ becomes of the same order as the radius of the fibre $R$, the entrainment of particles will be difficult to achieve. Indeed, a coating film of the order of the particle radius would require much higher capillary number in this regime and our model for the coating film thickness would not be valid anymore.

Figure 4.

Withdrawal velocity threshold $U^*$ for particle entrainment in the coating film as a function of the radius of the fibre $R$ for various (a) particle radii $a$ dispersed in the silicone oil AP 100 and (b) different viscosity of the interstitial fluid for particles of radius $a=70\ \mathrm {\mu }\textrm {m}$ dispersed in silicone oils.

The influence of the viscosity is similar to the 2-D situation, as illustrated in figure 4(b). Higher fluid viscosity is associated with larger capillary numbers, $Ca=\mu U/\gamma$. For a constant withdrawal velocity and a fixed surface tension, the coating film thickness thus increases with the viscosity, and the entrainment threshold increases when decreasing the viscosity of the fluid at a fixed $R$.

Our experiments demonstrate that the larger the size of the fibre, the smaller the particle entrainment threshold. This effect can only be explained by considering the dependence of the film thickness on the fibre radius.

4.2. Stagnation point

The thickness at the stagnation point, $h^*$ (see figure 1), has been derived in the 2-D situation, i.e. for a plate exiting a liquid bath (Colosqui et al. Reference Colosqui, Morris and Stone2013) but has not been reported in the fibre configuration to the best of our knowledge.

We consider the withdrawal of a fibre of radius $R \ll \ell _c$. We use the cylindrical coordinates $r$ and $z$ (radial and along the axis of the fibre) and denote $p$ the pressure and $\boldsymbol {u}=(u_r,u_\theta ,u_z)$ the fluid velocity in these coordinates. In our experimental situation, we do not have surfactants, which could otherwise modify the interfacial tension and introduce Marangoni effects (Shen et al. Reference Shen, Gleason, McKinley and Stone2002). Zero normal velocity and zero normal and tangential stress balance boundary conditions need to be satisfied at the air–liquid interface, at $r=R+h(z)$. At the surface of the fibre, the no-slip boundary condition is $u_z(r=R)=0$. Dip-coating dynamics on a substrate lead to thin film, thus we use the standard lubrication equations. The velocity in the direction normal to the fibre $u_r$ is negligible compared to the velocity along the fibre $u_z$. The pressure $p$ is obtained from the Laplace equation in the axisymmetric meniscus. We define the position of the free surface of the liquid film as $\xi (z)=R+h(z)$. The pressure in the thin film approximation ($\textrm {d}_z\xi \ll 1$) is $p=\gamma [{1}/\xi -{\textrm {d}^2\xi }/{\textrm {d} z^2}]$. Using the boundary conditions, we obtain the velocity field

(4.1)\begin{equation} \frac{u_z}{U}=1-\frac{1}{4\,{Ca}}\left(\frac{\textrm{d}^3\xi}{\textrm{d} z^3}+\frac{1}{\xi^2}\,\frac{\textrm{d}\xi}{\textrm{d} z}\right)\left[r^2-R^2-2\xi^2\ln\left(\frac{r}{R}\right)\right]. \end{equation}

The corresponding flow rate $Q$ is constant and given by

(4.2)\begin{equation} \frac{Q}{U {\rm \pi}R^2}=\left(\frac{\xi^2}{R^2}-1\right)-\frac{R^2}{{ Ca}}\left(\frac{\textrm{d}^3\xi}{\textrm{d} z^3}+\frac{1}{\xi^2}\,\frac{\textrm{d}\xi}{\textrm{d} z}\right)\mathcal{F}\left(\frac{\xi}{R}\right), \end{equation}

where the function $\mathcal {F}(x)$ is defined as $\mathcal {F}(x)={1}/{8}+{3x^4}/{8}-{x^2}/{2}-x^4\ln (x)/2$. The stagnation point $S^*$ corresponds to the location of the interface where the surface velocity vanishes. The expression of the surface velocity is obtained from (4.1) evaluated at $r=\xi$. Since the thickness of the liquid film $h(z)$ is much smaller than the radius of the fibre, $h/R \ll 1$, the surface velocity $u_s$, corresponding to the velocity tangential at the interface, is given by

(4.3)\begin{equation} \frac{u_s(z)}{U}=1+\frac{h^2}{2\,{Ca}}\left(\frac{\textrm{d}^3h}{\textrm{d} z^3}+ \frac{1}{R^2}\,\frac{\textrm{d}h}{\textrm{d} z}\right). \end{equation}

The second term of this equation can be simplified by using the flow rate far from the meniscus, $Q_{\infty }=2{\rm \pi} UhR$. Using this expression with (4.2) allows us to simplify the expression (4.3) for the surface velocity

(4.4)\begin{equation} \frac{u_s}{U}=1-\frac{h^2\left({\xi_\infty}^2-\xi^2\right)}{2R^4\mathcal{F}(\xi/R)}. \end{equation}

Therefore, in the limit $h/R\ll 1$, the thickness at the stagnation point, i.e. where the surface velocity vanishes, is equal to ${h^*}={3}{h}$. The value obtained for a fibre is thus the same as the value obtained for a 2-D plate for $Ca\ll 1$ (Colosqui et al. Reference Colosqui, Morris and Stone2013).

4.3. Numerical simulation: coating thickness and stagnation point

The comparison of this calculation to experiments is challenging as it requires visualizing the streamlines in a very narrow region. We thus rely on numerical simulations to determine the position of the stagnation point. The numerical simulations are performed with the Basilisk open-source library (www.basilisk.fr), successor of Gerris (Popinet Reference Popinet2009), using an adaptive mesh and a volume-of-fluid method to describe the two phases, liquid and air, and their interface. We solve the unsteady Stokes equation with a homogenous surface tension, assuming the problem to be axisymmetric and the fluids to be incompressible and Newtonian. The ratio of density and viscosity between the two phases are fixed to 100 and 44.6, respectively. The surface tension and the gravity acceleration are implemented with the continuum surface force (known as CSF) method, as described for example by Popinet (Reference Popinet2018). For this implementation of the gravity acceleration, the hydrostatic pressure is analytically subtracted from the mechanical pressure, leading to an effective pressure and reducing the gravity to a surface force. The numerical domain is a square, whose expansion is chosen to be around three times the capillary length. It includes part of the bath from which the fibre is redrawn. On the left-hand side, we assume a no-slip condition of the fluids on the fibre, by imposing a homogeneous Dirichlet condition for the normal velocity and a Dirichlet condition equal to the velocity of the fibre for the tangential component. Free flow boundary conditions are assumed on the three other boundaries. Thanks to the reduced gravity formulation, these conditions are easily obtained, applying a homogeneous Dirichlet condition for the effective pressure and a homogeneous Neumann condition for the velocity field. The time step is determined by the Courant–Friedrichs–Lewy conditions (Popinet Reference Popinet2018). The simulation is initialized with a thin film on fibre linked to the bath by a meniscus. The simulation is stopped after the fibre has traveled two or three times the height of the numerical domain.

Numerically, the streamlines exhibit a clear stagnation point, as illustrated by the example in figure 5(a). The stagnation point is obtained by estimating the location where the surface velocity vanishes. At this location, the streamlines define the transition between the liquid that continues into the coating film, and the liquid that recirculates in the liquid bath. We report the evolution of the film thickness far from the meniscus $h$ and the thickness at the stagnation point $h^*$ for fibres associated with different Goucher numbers, $Go=0.032,0.1,0.5$, in figure 5(b). The results of the numerical simulations agree well (within 10 %) with the theoretical prediction of the thickness at the stagnation point $h^*/R=3h/R$ for a capillary number smaller than $Ca< 10^{-1}$.

Figure 5.

Numerical simulation of the dip-coating of a fibre. (a) Examples of streamlines observed around the meniscus. The colourbar indicates the dimensionless surface velocity. The stagnation point corresponds to the point where this surface velocity vanishes. (b) Evolution of the rescaled film thickness $h/R$ (filled symbols) and the rescaled thickness at the stagnation point $h^*/R$ (open symbols) for different values of the Goucher number Go. The dashed lines correspond to the theoretical expressions of the thickness given by (3.5) and the solid lines to the theoretical thickness at the stagnation point, $h^*/R=3h/R$.

4.4. Threshold for particle entrainment

For a particle to be entrained in the coating film, the thickness at the stagnation point $h^*$ must be larger than a fraction of the particle diameter $2a$, so that the viscous drag exerted on the particle overcomes the capillary resistance. The presence of the particle of the same size as the thickness at the stagnation point $h^*$ will disturb the flow. Nevertheless, comparing $h^*$ and the size of the particle have been shown to predict satisfactorily well the entrainment threshold (Colosqui et al. Reference Colosqui, Morris and Stone2013; Sauret et al. Reference Sauret, Gans, Colnet, Saingier, Bazant and Dressaire2019) and a similar criterion is used here which is expressed as

(4.5)\begin{equation} \alpha a \leq h^* \quad \textrm{where}\ 0<\alpha\leq 2. \end{equation}

Experiments performed with spherical particles entrained on a 2-D plate indicate that the prefactor $\alpha$ accounts for the complex shape of the meniscus around the particle, and reported $\alpha \simeq 1.1 \pm 0.1$. Such results are expected to hold for fibres in the limit of particle radii much smaller than the fibre radius, $a/R \ll 1$, since locally the particle sees the coating film as flat. But as the pure liquid film thickness analysis showed, the finite fibre radius must still be considered when estimating the fully developed film thickness around the fibre, and thus the thickness at the stagnation point.

We have demonstrated analytically in § 4.2 and confirmed through numerical simulation in § 4.3 that a sufficient estimate for the stagnation point thickness is $h^*=3h$. From this expression of $h^*$, together with the criterion of (4.5) leads to a critical condition on the fully developed film thickness $h=\alpha a/3$. This film thickness can be inserted into either (3.3) or (3.5) to obtain a relationship between particle's size $a$ and the threshold capillary number ${Ca}^*$ for entraining it.

The relationship according to the thin film limit (3.5) is

(4.6)\begin{equation} \frac{\alpha a}{3R}=\frac{1.34 \,{Ca^*}^{2/3}}{1+2.53\,{Go}^{1.85}/\left[1+1.79\, {Go}^{0.85}\right]}, \end{equation}

where $a/R$ is the ratio of the particle to fibre radius. Rearranging for the capillary number leads to

(4.7)\begin{equation} {Ca}^* \simeq 0.645 \left[\frac{\alpha a}{3R}\left(1+\frac{2.53\,{Go}^{1.85}}{1+1.79\, {Go}^{0.85}}\right)\right]^{3/2}. \end{equation}

In the limit of vanishing Goucher number, i.e. thin fibers, the threshold capillary number follows a simple power law in the particle size as follows:

(4.8)\begin{equation} Ca^*=0.645\left[\frac{\alpha a}{3 R}\right]^{3/2}. \end{equation}

The Goucher number dependence and particle size dependence in (4.7) are separable such that the threshold capillary number can be rescaled by the Goucher number dependence to follow the single power law on the right-hand side of (4.8).

In figure 6, we have reported the experimental results for different fibre radii, particle radii and viscosities. The results are expressed in terms of the threshold capillary number rescaled by the term containing the Goucher number dependence to emphasize that all the data are indeed described well by the simple power law dependence $(a/R)^{3/2}$.

Figure 6.

Capillary number threshold $Ca^*$ for particle entrainment as a function of the ratio $a/R$ for a fibre of various radii $R \in [165,1200]\ \mathrm {\mu }\textrm {m}$ leading to Goucher number $Go \in [0.12,0.84]$. The solid line is the expression (4.7). The experimental results are obtained for various particle radii $a=20\ \mathrm {\mu }\textrm {m}$, $40\ \mathrm {\mu }\textrm {m}$, $70\ \mathrm {\mu }\textrm {m}$, $125\ \mathrm {\mu }\textrm {m}$, $250\ \mathrm {\mu }\textrm {m}$ and viscosity of the interstitial fluid.

The solid line in figure 6 is the power law on the right-hand side of (4.8) with a ‘best fit’ value of $\alpha \sim 1.1$, which agrees well with the value obtained for a plate, $\alpha \simeq 1.1 \pm 0.1$ (Sauret et al. Reference Sauret, Gans, Colnet, Saingier, Bazant and Dressaire2019). This agreement suggests that the criterion of (4.5) holds despite moderately large particles, i.e. for particle to fibre aspect ratio $a/R$ of order 0.1, and that the influence of the cylindrical geometry of the fibre is well characterized by our approach. In particular, the limit of (4.7) as $Go\to \infty$ reduces after some simplification to $Ca^* \simeq 0.26\,(a/\ell _c)^{1/2}$, which is in agreement with previous experiments performed with a flat plate (Sauret et al. Reference Sauret, Gans, Colnet, Saingier, Bazant and Dressaire2019). White & Tallmadge (Reference White and Tallmadge1966) emphasized that the regime of large cylinders of flat plates is obtained for Goucher number $Go > 3$. According to (3.5), $Go>3$ means that the azimuthal curvature contributes less than 5 % deviation from the planar curvature. Therefore the expression proposed in this manuscript should be used for fibres of radius $R<3 \ell _c$, which corresponds to $R \sim 4.2\ \textrm {mm}$ for the fluids considered here.

We note finally that the full finite film thickness expression of (3.3) can also be inverted for an explicit form in threshold capillary number in cases where the thin film approximation is not appropriate.