Statics and dynamics of polymeric droplets on chemically homogeneous and heterogeneous substrates

We present a molecular dynamics study of the motion of cylindrical polymer droplets on striped surfaces. We first consider the equilibrium properties of droplets on different surfaces, we show that for small stripes the Cassie-Baxter equation gives a good approximation of the equilibrium contact angle. As the stripe width becomes nonnegligible compared to the dimension of the droplet, it has to deform significantly to minimize its free energy; this results in a smaller value of the contact angle than the continuum model predicts. We then evaluate the slip length and thus the damping coefficient as a function of the stripe width. For very small stripes, the heterogeneous surface behaves as an effective surface, with the same damping as a homogeneous surface with the same contact angle. However, as the stripe width increases, damping at the surface increases until reaching a plateau. Afterwards, we study the dynamics of droplets under a bulk force. We show that if the stripes are large enough the droplets are pinned until a critical force. The critical force increases linearly with stripe width. For large enough forces, the average velocity increases linearly with the force, we show that it can then be predicted by a model depending only on droplet size, contact angle, viscosity, and slip length. We show that the velocity of the droplet varies sinusoidally as a function of its position on the substrate. However, for bulk forces just above the depinning force we observe a characteristic stick-slip motion, with successive pinnings and depinnings.

Figure 1

MD simulation snapshot of a droplet at equilibrium. The droplet of $N=50000$ atoms is on a striped surface having two types of atoms with $C_s=0.4$ (green) and $C_s=0.6$ (purple). The stripes have and equal widths of $w=7.56\sigma$. The atoms on the left-hand side of the figure are in the extremely low vapor phase of the polymeric fluid.

Figure 2

Density profiles for increasing number of monomers. The surface is homogeneous with hydrophobic parameter $C_s=0.5$. The average equilibrium contact angle of these droplets is ${\theta }_E={137}^{\circ }\pm 1^{\circ }$.

Figure 3

Density profiles of droplets with $N=30000$ atoms for increasing values of the hydrophobicity parameter $C_s$. The surfaces are chemically homogeneous substrates. The inset represents the equilibrium contact angle as a function of $C_s$. The contact angle is given in degrees.

Figure 4

(a) Density profiles of a droplet of $N=50000$ atoms on different striped substrates. The circles in the inset represent the Cassie-Baxter angle, and the squares the actual equilibrium contact angle as a function of the stripe width $w$. (b) Close up view of the contact line for the different droplets, the inset depicts the length of the droplet divided by the stripe width $w$ as a function of $w$.

Figure 5

(a) Slip length as a function of the equilibrium contact angle for homogeneous surfaces. (b) Slip length as a function of the stripe width. The solid lines are guides for the eyes.

Figure 6

Time-averaged velocity of the center of mass in the longitudinal direction for droplets of $N=50000$ monomers as a function of the force per atom for homogeneous surfaces. The circles are the results of the MD simulations and the plain lines are linear fits.

Figure 7

Time average of the velocity of the center of mass of droplets with $N=50000$ monomers as a function of the force per atom for the chemically heterogeneous surfaces. (a) $w=1.26\text{–}5.04\sigma$ and (b) $w=7.56\sigma$ and $w=10.1\sigma$. The symbols correspond to the results of the MD simulations and the dotted lines to linear fits. The arrows point to the depinning force.

Figure 8

The depinning force per atom as a function of the stripe width $w$. The circles are the results from the MD simulations and the solid line a linear fit on the nonzero values of the depinning force per atom.

Figure 9

Density fluctuations of a pinned droplet with a bulk force per atom $F=0.00001(m\sigma /{\tau }^2)$. (a) $w=5.04\sigma$, (b) $w=7.56\sigma$ and (c) $w=10.1\sigma$. The contour maps correspond to the standard deviation of the number density averaged over $2\times {10}^6$ simulation time steps.

Figure 10

An illustration for the depinning work $W_s$. A pinned droplet is in contact with $n$ hydrophobic stripes and $n+1$ hydrophilic stripes, the extremities must therefore be hydrophilic stripes.

Figure 11

(a) The surface energy per atom and (b) the averaged velocity of the center of mass as a function of the position of the center of mass. The energy was scaled by a factor ${10}^3$. The plain lines correspond to sinusoidal fits. The center of mass velocity for $w=3.78\sigma$ is multiplied by 5 for better visualization. (c) Velocity of the center of mass and position of the center of mass as a function of time for a typical trajectory in the stick-slip regime for a stripe width of $w=10.1\sigma$.