Dynamic arrest during the spreading of a yield stress fluid drop

When a liquid drop is gently deposited on a wetting solid surface, it spreads due to capillary forces until it reaches a thermodynamical equilibrium set by the relative surface energies of the system. We investigate here experimentally the spreading ability of drops made of yield stress fluids, which flow only if the applied stress is above a finite value. We observe that in this case, after a spreading phase, the motion stops and a well-defined contact angle can be measured. This contact angle depends on the rheological properties of the fluid and in particular on its yield stress, on the drop radius and on the hydrodynamic boundary condition at the surface. These results are quantitatively compared to an analysis showing that, due to the yield stress of the fluid, a mechanical equilibrium is indeed reached which does not correspond to the thermodynamical equilibrium.

Figure 1

(a) Experimental setup. (b) Schematic and notation of a spreading drop: $\theta$ the contact angle, $R$ the radius, $\xi$ the height of the drop, and $z_y$ the thickness of the sheared zone inside the drop. (c) Pictures of drops of various YSFs after spreading on rough surface: (I) Carbopol gel (ETD at 0.3%) with a yield stress ${\sigma }_y=6$ Pa, (II) and (III) Carbopol gel (ETD at 2%) with ${\sigma }_y=35$ Pa, (IV) Carbopol gel (U10 at 0.5%) with ${\sigma }_y=85$ Pa. Fits of the drop shape assuming a spherical cap of a same volume as the drop are plotted in red. Scale bar corresponds to 1 mm.

Figure 2

(a) Final contact angle ${\theta }_f$ vs the final radius $R_f$ for three fluids defined in Table 1: Carbopol gel (ETD at 2%) with a yield stress ${\sigma }_y=35$ Pa (blue diamonds), Carbopol gel (U10 at 0.25%) with a yield stress ${\sigma }_y=29$ Pa (red squares) and PAA suspension (at 2%) with no yield stress (green circles), on smooth (open symbols) and rough substrates (plain symbols). (b) Final contact angle ${\theta }_f$ measured with YSFs on rough substrates as a function of the yield stress ${\sigma }_y$ for a given radius, $R_f=1.5\pm 0.2$ mm.

Figure 3

($cos{\theta }_e-cos{\theta }_f$) with ${\theta }_e={10}^{\circ }$ as a function of $\mathrm{Bc}=R_f{\sigma }_y/\mathrm{\Gamma }$ for two YSFs with various concentrations given in Table 1: Carbopol gel (ETD) (blue diamonds) and Carbopol gel (U10) (red squares), on smooth (open symbols) and rough substrates (plain symbols). Plain line corresponds to the numerical resolution assuming a no-slip boundary condition, black dotted line and red dashed line correspond to the analytical asymptotic limit, assuming a no-slip boundary condition [Eq. (3)] and a partial-slip boundary condition [Eq. (4) with $V_s=0.66\text{V}]$, respectively. A fitting numerical prefactor of $0.50\pm 0.03$ is obtained in both cases. Each data point corresponds to an average over experimental points in a range of 0.1 for the abscissa.

Figure 4

Schematic of the spreading regimes in the ($R_f-{\sigma }_y$) phase diagram for all YSFs. The dashed line corresponds to the frontier $R_f{\sigma }_y=\mathrm{\Gamma }$. The color scale corresponds to the deviation $\delta$ of the final drop shape from a spherical cap as defined in the text [Eq. (5)]. Roman numbers correspond to the position in this phase diagram of the four pictures from the Fig. 1.

Figure 5

Stress at the wall $\sigma (z=0)/{\sigma }_y$ as a function of the radial position for three velocities: ${10}^{-1}$ (blue), ${10}^{-3}$ (orange), and ${10}^{-5}$ (green). The velocity here is dimensionless, divided by $R_0{(K/{\sigma }_y)}^{1/n}$. As the velocity decreases, the wall stress tends to the yield stress. The data are calculated numerically (see Supplemental Material [26] for more details on the numerical resolution).

Figure 6

Final contact angles measured with three different Carbopol gels as a function of the final radius. Blue and red data have been obtained with YSF of similar ${\sigma }_y$ but different $G^{\prime }$. Blue and green data have been obtained with YSF of similar $G^{\prime }$ but different ${\sigma }_y$.