^{1,a)}, Guoai Pan

^{2}and Daniel M. Kroll

^{2}

### Abstract

Capillary waves have been observed in systems ranging from the surfaces of ordinary fluids to interfaces in biological membranes and have been one of the most studied areas in the physics of fluids. Recent advances in fluorescence microscopy and imaging enabled quantitative measurements of thermally driven capillary waves in lipid monolayers and bilayers, which resulted in accurate measurements of the line tension in monolayer domains. Even though there has been a considerable amount of work on the statics and dynamics of capillary waves in three dimensions, to the best of our knowledge, there is no detailed theoretical analysis for two-dimensional droplet morphologies. In this paper, we derive the dynamic correlation function for two-dimensional fluid droplets using linear response theory and verify our results using a novel particle-based simulation technique for binary mixtures.

We thank Thomas Ihle and Benjamin Stottrup for helpful discussions. Support from the National Science Foundation under Grant No. DMR-0513393 is gratefully acknowledged. E.T. acknowledges support from an Institute for Mathematics and Its Applications postdoctoral fellowship and Worcester Polytechnic Institute for startup funds.

I. INTRODUCTION

II. CAPILLARY WAVE SPECTRUM

A. Statics

B. Linear response

C. Dynamics

III. NUMERICAL RESULTS

A. The simulation technique

B. Anisotropy

C. Statics

D. Dynamics

IV. CONCLUSIONS

### Key Topics

- Fluid drops
- 36.0
- Capillary waves
- 35.0
- Viscosity
- 21.0
- Interface dynamics
- 14.0
- Rheology and fluid dynamics
- 12.0

## Figures

A diagram showing a fluctuating droplet and its parametrization in terms of . The total area of the droplet is fixed and equal to .

A diagram showing a fluctuating droplet and its parametrization in terms of . The total area of the droplet is fixed and equal to .

Combined contour and velocity field plots showing the solutions for the velocity field given by Eqs. (34) and (35). The droplet contour is overlayed on the field plots for (a) standing and (b) traveling waves. The amplitude of the oscillations is chosen large enough so that the undulations can be observed. Parameters: and .

Combined contour and velocity field plots showing the solutions for the velocity field given by Eqs. (34) and (35). The droplet contour is overlayed on the field plots for (a) standing and (b) traveling waves. The amplitude of the oscillations is chosen large enough so that the undulations can be observed. Parameters: and .

The dimensionless frequency , obtained by taking the square root of Eq. (47), is plotted as a function of for (a) and (b) . The open circles (○), bullets (●), and open squares (◻) show the positive and negative real parts, and the imaginary part, respectively.

The dimensionless frequency , obtained by taking the square root of Eq. (47), is plotted as a function of for (a) and (b) . The open circles (○), bullets (●), and open squares (◻) show the positive and negative real parts, and the imaginary part, respectively.

The dynamic structure factor, , as a function of for . The solid (blue), dashed (red), dotted (green), and dashed-dotted (black) lines correspond to , 19.76, 0.1976, and 0.001 976, respectively. The dynamic structure factor scales as at intermediate frequencies for small and as at large angular frequencies.

The dynamic structure factor, , as a function of for . The solid (blue), dashed (red), dotted (green), and dashed-dotted (black) lines correspond to , 19.76, 0.1976, and 0.001 976, respectively. The dynamic structure factor scales as at intermediate frequencies for small and as at large angular frequencies.

Snapshot of a fluctuating droplet simulated using the particle-based simulation technique (Ref. 36) with 20% -80% particles. Average density in units of inverse . Parameters: , , and .

Snapshot of a fluctuating droplet simulated using the particle-based simulation technique (Ref. 36) with 20% -80% particles. Average density in units of inverse . Parameters: , , and .

The averaged Fourier coefficients, and , for and . The deviation at indicates the presence of fourfold anisotropy due to the cubic cell structure. Bullets (●, ○) and squares (◻, ◼) correspond to and, , respectively. The results are independent of the viscosity of the fluid as expected (filled symbols: SRD collisions every time step, open symbols: SRD collisions every tenth time step). Parameters: , , , and .

The averaged Fourier coefficients, and , for and . The deviation at indicates the presence of fourfold anisotropy due to the cubic cell structure. Bullets (●, ○) and squares (◻, ◼) correspond to and, , respectively. The results are independent of the viscosity of the fluid as expected (filled symbols: SRD collisions every time step, open symbols: SRD collisions every tenth time step). Parameters: , , , and .

Static structure factor, , as a function of mode number for and . The SRD collisions are performed at every time step [shown in bullets (●)], and at every tenth time step [shown in squares (◻)]. The corresponding average radii are given by and , respectively. The solid line is a fit to Eq. (16) which yields . Parameters: , , , and .

Static structure factor, , as a function of mode number for and . The SRD collisions are performed at every time step [shown in bullets (●)], and at every tenth time step [shown in squares (◻)]. The corresponding average radii are given by and , respectively. The solid line is a fit to Eq. (16) which yields . Parameters: , , , and .

Dynamic structure factor, , as a function of time for large damping . The bullets (●), open circles (○), and filled squares (◻) correspond to , 3, and 4, respectively. The solid lines are obtained by numerically integrating Eq. (48). The average droplet radius is , , and . Parameters: , , , , and .

Dynamic structure factor, , as a function of time for large damping . The bullets (●), open circles (○), and filled squares (◻) correspond to , 3, and 4, respectively. The solid lines are obtained by numerically integrating Eq. (48). The average droplet radius is , , and . Parameters: , , , , and .

Dynamic structure factor, , as a function of time for moderate damping . The bullets (●), open circles (○), and filled squares (◻) correspond to , 3, and 4, respectively. The solid lines are obtained by numerically integrating Eq. (48) using . The viscosity is lowered by performing SRD collisions every tenth time step. The average droplet radius is and . Parameters: , , , , and .

Dynamic structure factor, , as a function of time for moderate damping . The bullets (●), open circles (○), and filled squares (◻) correspond to , 3, and 4, respectively. The solid lines are obtained by numerically integrating Eq. (48) using . The viscosity is lowered by performing SRD collisions every tenth time step. The average droplet radius is and . Parameters: , , , , and .

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