Size dependence of static polymer droplet behavior from many-body dissipative particle dynamics simulation

We used molecular simulation to study the static behavior of polymer droplets in vacuum and on solid surfaces, namely the size of the droplet and the contact angle, respectively. The effects of the polymer chain length and the total number of particles were calculated by the many-body dissipative particle dynamics method. For the spherical droplet containing the same number of particles, we show that its radius depends on the polymer chain length. The radius of the droplet is also proportional to one-third power of the total number of particles for all given chain lengths. For the hemispherical droplet, the contact angle increases with the number of particles in the droplet, and this effect is relatively strong, especially for longer polymer chains. The effect of wettability of the solid surface was also investigated by using polymerphobic (low-affinity) and polymerphilic (high-affinity) surfaces. As the chain length increases, the contact angle on the low-affinity surface decreases, while that on the hydrophilic surface increases. The simulation reveals that there is a critical affinity for the monomer on the solid surface; above and below which the wettability increases and decreases as the molecular length increases, respectively.

Figure 1

Snapshots of initial configuration of droplet systems ($\tau$ = 0). A spherical droplet system (a1) in the $x-y$ plane view (a2) and in the $x-z$ plane (a3). A hemispherical droplet on a solid surface system (b1) in the $x-y$ plane (b2) and in the $x-z$ plane (b3).

Figure 2

Verification of Laplace law for the our MDPD model. The vertical axis is a pressure difference ($p_{\mathrm{in}}-p_{\mathrm{out}}$) between inside and outside of the drop, and the horizontal axis is a reciprocal of the radius ($R$). The dashed line is the Laplace law prediction.

Figure 3

An outline of the computational procedure of the droplet radius. To calculate the density distribution, the concentric spherical shells around the center of mass of the droplet is considered. Here the distance between adjacent shells ($dr$) is set to 0.1. The density distribution at $N$ = 1920 and $N_c$ = 1. We defined $R$ as the radius when $\rho$ drops to half the value near the center of mass.

Figure 4

(a) Relationship between the droplet radius ($R$) and the number of the particles ($N$). The solid lines are the fitting results in the form of $R\left(N\right)$ = $C\times N^{1/3}$. (b) Equilibrium snapshots for $N$ = 30 720. ($\tau$ = 5000) (b1) $N_c$ = 1, (b2) $N_c$ = 2, (b3) $N_c$ = 4, (b4) $N_c$ = 8, (b5) $N_c$ = 16. (c) Radial distribution functions of (b).

Figure 5

(a) The radial distribution functions $g\left(r\right)$ depending on the chain length and (b) dependence of $g\left(r\right)$ on location in the droplet. ($N$ = 30 720 and $N_c$ = 1).

Figure 6

Number of particles dependence of (a) the mean-square radius of gyration $〈R_g〉$ and (b) the mean-square end-to-end distance $〈R_e〉$. Circles and squares represent spherical and hemispherical droplet system, respectively.

Figure 7

Dependencies of (a) the mean-square radius of gyration $〈R_g〉$ and (b) the mean-square end-to-end distance $〈R_e〉$ on location in the droplet ($N$ = 30 240 and $N_c$ = 16). Circles and squares represent spherical and hemispherical droplet system, respectively. The $〈R_g〉$ and the $〈R_e〉$ of bulk system are represented by dashed-dotted line.

Figure 8

Relationship between the radius of gyration $R_g$ and the fitting constant $C$.

Figure 9

An outline of the computational procedure of the contact angle between a solid surface and a droplet. The droplet is divided into thin layers in the $z$ direction to calculate a density distribution. Here the distance between adjacent layers ($dz$) is set to 0.5. (a) The density distributions at $N$ = 1920 and $N_c$ = 1. We determined a surface position in each layer when $\rho$ drops to half the value at $r$ = 0. (b) The contact angle is computed from the angle between the circle and the solid surface at their intersection.

Figure 10

(a) Relationship between the contact angle ($\theta$) and the number of the particles ($N$). (b) Relationship between $\theta$ and $N$ after a correction. (c, d) Equilibrium snapshots for droplet on the solid surface for $N$ = 30 720. ($\tau$ = 5000) (c) $N_c$ = 1 of view in the $x-z$ plane (c1) and in the $x-y$ plane (c2). (d) $N_c$ = 8 of view in the $x-z$ plane (d1) and in the $x-y$ plane (d2). (d3) and (d4) are enlarged figures of (d1) and (d2), respectively.

Figure 11

Relationship between the correction contact angle and the number of the particles when two types of an affinity with the solid surface such as a polymerphobic surface (a) and polymerphilic surface (b). Interaction parameters $a_{\mathrm{PS}}$ in polymerphobic and polymerphilic surfaces were set at $-15 k_{\mathrm{B}}T$ and $-35 k_{\mathrm{B}}T$, respectively.