Rupture process of liquid bridges: The effects of thermal fluctuations

Rupture of a liquid bridge is a complex dynamic process, which has attracted much attention over several decades. We numerically investigated the effects of the thermal fluctuations on the rupture process of liquid bridges by using a particle-based method know as many-body dissipative particle dynamics. After providing a comparison of growth rate with the classical linear stability theory, the complete process of thinning liquid bridges is captured. The transitions among the inertial regime (I), the viscous regime (V), and the viscous-inertial regime (VI) with different liquid properties are found in agreement with previous work. A detailed description of the thermal fluctuation regime (TF) and another regime, named the breakup regime, are proposed in the present study. The full trajectories of thinning liquid bridges are summarized as $\mathrm{I}\to \mathrm{V}\to \mathrm{VI}\to \mathrm{TF}\to$ breakup for low-Oh liquids and $\mathrm{V}\to \mathrm{I}\to$ Intermediate $\to \mathrm{V}\to \mathrm{VI}\to \mathrm{TF}\to$ breakup for high-Oh liquids, respectively. Moreover, the effects of the thermal fluctuations on the formation of satellite drops are also investigated. The distance between the peaks of axial velocity is believed to play an important role in forming satellite drops. The strong thermal fluctuations smooth the distribution of axial velocity and change the liquid bridge shape into a double cone without generating satellite drops for low-Oh liquids, while for high-Oh liquids, this distance is extended and a large satellite drop is formed after the breakup of the liquid filament occurs on both ends, which might be due to strong thermal fluctuations. This work can provide insights on the rupture mechanism of liquid bridges and be helpful for designing superfine nanoprinting.

Figure 1

Schematic of the rupture process of the liquid bridge. The periodic boundary condition is applied to mimic an infinitely long filament. The wavelength $\lambda$ is managed to get different reduced wave number $x$. The liquid bridge gradually shrinks at the position of the minimum radius and pinches off to form satellite drops.

Figure 2

(a) The variation of breakup time $t_0$ of liquid bridges with different $\mathrm{\Delta }t$, ranging from 0.0025 to 0.02. The error bars show the standard deviation among the five repeated cases. (b) The variation of $h_{min}$ with time under $\mathrm{\Delta }t$ from 0.01 to 0.0025.

Figure 3

The comparison of growth rate among MDPD results and three theories under $\mathrm{Oh}=0.359$. The sinusoidal perturbation is determined by the given $x$. According to Rayleigh instability, the liquid bridge begins to thin and break up eventually. For each value of $x$, the breakup time is monitored in order to get $\omega$. The initial perturbations of $x=0.2$ and 0.9 are also shown as snapshots.

Figure 4

Two Oh numbers are simulated to compare with Eq. (16). The black and red lines represent outcomes from Eq. (15) under the given Oh number. The discrete points are the growth rate obtained by MDPD simulations.

Figure 5

The comparison of maximal growth rate ${\omega }_{max}$ between MDPD results and the analytical solution. ${\omega }_{max}$ is normalized by ${\omega }_0$. The range of Oh numbers is from 0.311 to 1.214.

Figure 6

The evolution of $h_{min}$ with time under $\mathrm{Oh}=0.359$. The four regimes are observed according to scale fitting with self-similar theory. The TF regime is well described in the simulation and another regime, named the breakup regime, is captured.

Figure 7

(a) The variation of $w_{max}$ with $h_{min}$ for $\mathrm{Oh}=0.359$. The complete process is divided into four regimes where the transition from the V to the VI regime can be captured. (b) The variation of $R{e}_{\mathrm{local}}$ with $h_{min}$.

Figure 8

(a) The evolution of $L_{\mathrm{z}}$ with $h_{min}$ for $\mathrm{Oh}=0.359$. $L_{\mathrm{z}}$ first increases slowly in the period of thinning followed by $L_{\mathrm{z}}\propto h_{min}^{-0.18}$, and then quickly reduces with the power law of $L_{\mathrm{z}}\propto h_{min}^{2.78}$. (b) The variation of slenderness ratio $h_{min}/\phantom{h_{min}{L}_{\mathrm{z}}}{L}_{\mathrm{z}}$.

Figure 9

The evolution of $h_{min}$ with time to breakup under $\mathrm{Oh}=1.21$. The I regime is found in high-Oh liquids.

Figure 10

(a) The variation of $w_{max}$ with $h_{min}$ for $\mathrm{Oh}=1.21$. (b) The variation of ${\mathrm{Re}}_{\mathrm{local}}$ with $h_{min}$. The transition between different regimes can be captured depending on the value of ${\mathrm{Re}}_{\mathrm{local}}$.

Figure 11

The evolution of $L_{\mathrm{z}}$ with $h_{min}$ for $\mathrm{Oh}=1.21$. The increase near pinch-off captured in the fluid of $\mathrm{Oh}=0.359$ cannot be found here in the high-Oh situation.

Figure 12

The trajectory of $\alpha$ for low-Oh (blue line) and high-Oh (red line) liquids during thinning. The points without error bars mean that the error bar is smaller than the point.

Figure 13

The comparison between crossovers from the VI to the TF regime and thermal length $L_{\mathrm{T}}$ under different Oh numbers.

Figure 14

The comparison between $L_{\mathrm{T}}=0.227$ and 0.321 under $\mathrm{Oh}=0.359$. (a) $w_{max}$; (b) $L_{\mathrm{z}}$.

Figure 15

The axial velocity $w$ along the axis with different $L_{\mathrm{T}}$ and the snapshot near pinch-off. (a) $L_{\mathrm{T}}=0.227$. (b) $L_{\mathrm{T}}=0.321$. Three positions of the liquid bridge thin simultaneously to form satellite drops with weak thermal fluctuations. Only one position thins with strong thermal fluctuations to generate a double-cone shape without satellite drops.

Figure 16

The comparison of $w$ along the axial direction between different $L_{\mathrm{T}}$ when the liquid bridge approaches breakup under $\mathrm{Oh}=0.359$.

Figure 17

The comparison between $L_{\mathrm{T}}=0.145$ and 0.205 under $\mathrm{Oh}=1.21$. (a) $w_{max}$; (b) $L_{\mathrm{z}}$.

Figure 18

The axial velocity $w$ along the axis with different $L_{\mathrm{T}}$ and the snapshot near pinch-off. (a) $L_{\mathrm{T}}=0.145$. (b) $L_{\mathrm{T}}=0.205$. The strong thermal fluctuations increase the distance between peaks of $w$ and form a large satellite drop in the middle.

Figure 19

The comparison of $w$ along the axial direction between different $L_{\mathrm{T}}$ when the liquid bridge approaches breakup under $\mathrm{Oh}=1.21$.