Taming contact line instability for pattern formation

Abstract

Coating surfaces with different fluids is prone to instability producing inhomogeneous films and patterns. The contact line between the coating fluid and the surface to be coated is host to different instabilities, limiting the use of a variety of coating techniques. Here we take advantage of the instability of a receding contact line towards cusp and droplet formation to produce linear patterns of variable spacings. We stabilize the instability of the cusps towards droplet formation by using polymer solutions that inhibit this secondary instability and give rise to long slender cylindrical filaments. We vary the speed of deposition to change the spacing between these filaments. The combination of the two gives rise to linear patterns into which different colloidal particles can be embedded, long DNA molecules can be stretched and particles filtered by size. The technique is therefore suitable to prepare anisotropic structures with variable properties.

Figure 1. Deposition of a polymer solution.

(a) Schematic of the flexible blade set-up. A glass surface is translated below the blade, which is fixed to entrain fluid underneath the blade. The meniscus between the fluid and the surface resists wetting the latter and recedes as it is dragged by the bottom surface. (b) Photographs of filaments near the blade for different velocities (V=0.2, 0.5, 10 mm s⁻¹). Note the change in scale as the velocity is increased. (c) Wavelength versus deposition velocity for two different concentrations (3,000 and 10,000 p.p.m.) of PAM of molecular weight 18 M. The error bars in c represent the s.d. More details are shown in Supplementary Fig. 1.

Figure 2. Capillary number versus velocity.

(a) For Glycerine and for a PAM solution at 10,000 p.p.m. For Glycerine the capillary number Ca is linear versus velocity. It is not for PAM because of shear thinning. The grey area designates the region where cusps form as the meniscus is being dragged by the bottom plate. At higher velocities and therefore higher values of Ca, a film forms on the surface. For PAM, the values of Ca reside mainly in the grey region, giving rise to cusps and filaments. Left images are for Glycerine, right images are for PAM. Panel (b) shows a similar plot for another polymer solution PEO of molecular weight 4 M at two different concentrations. By increasing the concentration the Ca can be made to reside mainly in the grey region and cusps giving rise to filaments can be made stable. For the lower concentration (right images), the filaments break up into drops as the characteristic time for filament break-up is smaller than for the higher concentration (left images).

Figure 3. Filament characteristics.

(a) A schematic showing the distance between two filaments for V=0.25 mm s⁻¹. As this distance increases with time, a new filament emerges between the two. In another event depicted in this graph, as two filaments approach each other, they merge and give rise to a single filament. The images show the state of the filaments at the different instants denoted by a small numbering. (b) The shape of the base of the filaments is portrayed for different velocities. This shape, shown in the inset, can be made dimensionless (main figure) by rescaling the vertical and horizontal axes using characteristic scales L and l, respectively. (c) Both length scales vary linearly with velocity indicating a proportionality between the two lengths. The ratio V/L then defines a time scale found in good agreement with the stretching rate in the filament itself and measured using particle tracking shown in (d,e) where the velocity of tracer particles in the direction of the filaments v_z is plotted versus distance along the filament z. (f) The stretching rate obtained from dv_z/dz turns out to be constant versus plate velocity V and given by the characteristic time of the polymer solution measured using droplet pinch off experiments (see Supplementary Note 2 and Supplementary figures 6 & 7). The error bars in f represent the min and max values.

Figure 4. Useful patterns.

Photographs of the filaments (a) with embedded fluorescent colloidal particles of 1 μm in diameter (b). (c) Wavy filaments with the same fluorescent particles as in b obtained by translating the plate with a sinusoidal velocity in the plane of the substrate. (d,e) Stretched fluorescent DNA molecules, up to 1/3 of their total extension, embedded in the filaments of photo (a). (f) Filtering by size of 1 and 6 μm particles. The fluorescent 1 μm particles (such as particle 2) enter the thin filaments while the 6 μm (particles 1 and 3) do not. The three photographs at are times t₀, t₁=10.9 s, t₂=21.3 s.