Withdrawal and dip coating of an object from a yield-stress reservoir

The dip-coating process consists of withdrawing immersed objects from a liquid reservoir. After withdrawal, a significant layer of liquid remains on the object. Various industrial processes (food and beverage industry, automotive industry) use this technique to coat or treat surfaces. Recent studies have shown that the thickness of deposit is determined by the flow inside the reservoir for yield-stress fluids. This is different from the behavior of simple liquids for which the coating thickness is solely determined by the flow inside the meniscus. In this work, we reexamine this question and propose a complete phase diagram linking the Newtonian case and the yield-stress fluid case. We provide asymptotic scaling laws for extreme cases. A good agreement with experiments is obtained.

Figure 1

(a) Schematic of the dip-coating experiment: a rod of radius $r_1$ is pulled out with velocity $V$ from a cylindrical liquid reservoir of radius $r_2$, which deposits a coating film of thickness $h$. (b) Typical thickness profile resulting from the dip coating. The coating profile can be divided into regions in a transient regime, a steady regime, and break-up regime (see text).

Figure 2

(a) Sequential pictures of a dip-coating experiment with a Carbopol gel with $n=0.35$ and $\text{Bm}=4$. Horizontal layers are colored by food coloring to indicate the fluid flow, which is vertical away from the meniscus and the end of the rod. (b) Schematic cross sections of a steady fluid flow with $n=0.35$ and $\text{Bm}=1$ for a cylindrical geometry with $r_1=r_2/4$. Inside the reservoir, the yielded regions are indicated light-shaded, and the unyielded regions are shown dark-shaded.

Figure 3

Normalized width of the inner liquid zone in steady flow for different power index $n$ and ratio $r_1/r_2$ as a function of Bingham number $\text{Bm}$. Dotted lines denote the low-$\text{Bm}$ limit for $n=1$ [Eq. (B2)]. Dashed lines denote the low-$\text{Bm}$ approximation for a reservoir with a narrow gap [Eq. (B4)]. Blue dotted-dashed lines denote the high-$\text{Bm}$ approximation for a large reservoir [Eq. (B14)]. Red dotted-dashed lines denote the high-$\text{Bm}$ approximation for a reservoir with a small gap [Eq. (B10)].

Figure 4

Asymptotic limits of the width of the liquid zone $\delta ={\lambda }_i-r_1$ (blue) and the reservoir-flow coating thickness $h$ (black) for $\frac{1}{3}<n<1$. The parameter space can be divided up into domains I–V (see Appendix pp2). Solid lines indicate the typical shape of the domain boundaries, whose nontrivial dependencies are given in gray. $c$ denotes a constant.

Figure 5

Asymptotic limits of the reservoir-flow coating thickness $\text{Co}$ (blue or black) in absence of surface tension. The parameter space can be divided up into three domains (IV, V, E). The boundary between region IV and V corresponds to $\overline{\mathrm{Y}}=\mathrm{Bm}$, the boundary between IV and E is given by $\overline{\mathrm{Y}}={\mathrm{Bm}}^{\frac{1}{2+n}}$, and the boundaries between E and V are given by $\overline{\mathrm{Y}}={\mathrm{Bm}}^{(3n-1)/(5n-1)}$ and $\mathrm{Bm}=1$.

Figure 6

The meniscus around a rod of radius $r_1$. The shape of the static meniscus that forms between the rod and the bath at rest is indicated by the blue dashed line. When the rod is withdrawn from the bath, a coating of thickness $h$ is formed. $l_x$ is the length of the dynamic meniscus, which marks the transition between the static meniscus and the coating film.

Figure 7

Asymptotic limits of the meniscus-driven coating thickness $h/r_1$ (blue). The parameter space can be divided up into domains A–E. The diagram corresponds to $n=0.5$ and $l_c/r_1=6.25$.

Figure 8

Asymptotic limits of the meniscus-driven coating thickness $h/r_1$ (blue). The parameter space can be divided up into domains A–E. This diagram corresponds to $n=0.5$ and $l_c/r_1=0.5$.

Figure 9

Asymptotic scaling limits of the meniscus-driven coating thickness $h/r_1$ (blue). The parameter space can be divided into three domains: A, B, and F. The diagram corresponds to $n=0.7, \text{Co}=0.1$, and $l_c/r_1=1.5$. $w=\frac{[{(Co+1)}^2-1]r_1^2}{l_c^2}$.

Figure 10

Asymptotic scaling limits of the meniscus-driven coating thickness in very wide geometry for $n=0.4, l_c/r_1=2$, and $\text{Co}=10$ ($h/r_1$ in blue).

Figure 11

The equivalent of Fig. 5 in the parameter plane $\{Y,{\text{Ca}}_f\}$.

Figure 12

Coating diagram in the parameter plane $\{Y,{\text{Ca}}_f\}$ for $l_c/r_1=2, \text{Co}=10$, and $n=0.4$.

Figure 13

Steady coating thickness for different ratio $r_1/r_2$ (color coding in legend) as a function of Bingham number Bm for Carbopol gels of ${\tau }_c=28$ Pa (sample 2, squares), ${\tau }_c=54$ Pa (sample 4, circles), and ${\tau }_c=67$ Pa (sample 3, triangles). The theoretical predictions (solid lines) follow from calculations of the flow inside the bath. Inset: normalized theoretical coating thickness $h_{\mathrm{th}}$ versus experimental values $h_{\exp }$.