Fluctuating hydrodynamics and the Rayleigh–Plateau instability

Significance

A cylindrical stream of water from a faucet breaks up into droplets due to the action of surface tension, a phenomenon known as the Rayleigh–Plateau instability. At nanometer scales, the random motion of the fluid molecules alters the instability. In this paper, we present a numerical method for studying these microscopic effects. The algorithm uses fluctuating hydrodynamics, an extension of conventional fluid dynamics that includes thermal fluctuations. Our simulations show that these fluctuations affect the shape of the tapering cylinder and hasten the pinching into droplets. We also find that short cylinders, which are stable in the absence of fluctuations, eventually also break into a droplet.

Abstract

The Rayleigh–Plateau instability occurs when surface tension makes a fluid column become unstable to small perturbations. At nanometer scales, thermal fluctuations are comparable to interfacial energy densities. Consequently, at these scales, thermal fluctuations play a significant role in the dynamics of the instability. These microscopic effects have previously been investigated numerically using particle-based simulations, such as molecular dynamics (MD), and stochastic partial differential equation–based hydrodynamic models, such as stochastic lubrication theory. In this paper, we present an incompressible fluctuating hydrodynamics model with a diffuse-interface formulation for binary fluid mixtures designed for the study of stochastic interfacial phenomena. An efficient numerical algorithm is outlined and validated in numerical simulations of stable equilibrium interfaces. We present results from simulations of the Rayleigh–Plateau instability for long cylinders pinching into droplets for Ohnesorge numbers of Oh = 0.5 and 5.0. Both stochastic and perturbed deterministic simulations are analyzed and ensemble results show significant differences in the temporal evolution of the minimum radius near pinching. Short cylinders, with lengths less than their circumference, were also investigated. As previously observed in MD simulations, we find that thermal fluctuations cause these to pinch in cases where a perturbed cylinder would be stable deterministically. Finally, we show that the fluctuating hydrodynamics model can be applied to study a broader range of surface tension–driven phenomena.

Liquid jets forming sprays are ubiquitous in nature and in industrial processes, a familiar example being a stream of water that breaks up into droplets. The 19th-century experiments of Plateau, Beer, and others showed that a long cylinder (length L, initial radius R₀) of fluid (density ρ) was unstable to variations that reduced its surface area (1, 2). Plateau predicted that perturbations are unstable for wavelengths λ ≥ 2πR₀ and Rayleigh derived that, in the inviscid limit, the fastest-growing wavelength is λ_p ≈ 9.01R₀. In the Stokes limit (negligible inertia), Tomotika (3) showed that λ_p ≈ 11.16R₀ for a fluid cylinder immersed in a similar fluid of equal viscosity.

There are several dimensionless numbers that characterize the dynamics of the Rayleigh–Plateau instability. The Ohnesorge number, $\text{Oh}=\eta /\sqrt{\rho R_0\gamma }$, compares viscous forces to inertial and surface tension forces, characterizing the relative importance of shear viscosity, η, to surface tension, γ. Other dimensionless quantities include the Weber, Bond, and Schmidt numbers that capture the effects of fluid velocity, gravity, and diffusion.

At microscopic scales (R₀≲ 10 nm), thermal fluctuations become important since the variance of velocity fluctuations in a volume V goes as k_BT/ρV where T is temperature, and k_B is the Boltzmann constant. For interfacial flows, this effect can be characterized by a stochastic Weber number We^⋆ = k_BT/γR₀² based on a thermal velocity $v^{\star }=\sqrt{k_{B}T/\rho {R_0}^3}$. Thermal energy becomes comparable to interfacial energy at length scale ${\ell }^{\star }=\sqrt{k_{B}T/\gamma }$ so We^⋆ = (ℓ^⋆/R₀)².

In molecular dynamics (MD) simulations by Moseler and Landman (We^⋆ ≈ 0.04), the fluid cylinder formed a double-cone (or hourglass) shape as it pinched, in contrast to the macroscopic predictions of an extended, thin liquid thread (4). Furthermore, their numerical solutions of a stochastic lubrication equation were in qualitative agreement with these MD results. For Oh ≫ 1, Eggers showed that near the pinching time, t_p, the minimum cylinder radius goes as (t_p − t)^α with α ≈ 0.4 when thermal fluctuations are significant and α = 1 when they are negligible (5). There have been other studies using MD simulations (6, 7), Lattice Boltzmann (8), and Dissipative Particle Dynamics simulations (9, 10). Recently, the group at Warwick has extensively analyzed both the Rayleigh–Plateau instability (11–13) and related thin film phenomena (14–18) with stochastic lubrication theory and MD simulations.

In this paper, we use a fluctuating hydrodynamic (FHD) model for numerical simulations of the Rayleigh–Plateau instability. The theoretical foundation of our model is the same as that of stochastic lubrication theory, namely the stochastic Navier–Stokes equations introduced by Landau and Lifshitz (19, 20). Since our multiphase FHD model (21–23) does not use the lubrication approximation, it has broader applicability, including modeling the instability past the initial pinching time and for a wider range of geometries and initial conditions. The next section outlines the model, followed by a description of the algorithm and its validation. Numerical results for the Rayleigh–Plateau instability are then presented for a variety of scenarios. We conclude with a summary of the current work and potential future studies.

Fluctuating Hydrodynamic Theory

We consider a binary mixture of similar species (molecule mass, m) at constant density and temperature. We model the specific free energy density of the mixture using the Cahn–Hilliard formalism with regular solution theory (24) and write

$$\begin{array}{c}\hfill \frac{\mathcal{G}}{\rho k_{B}T}=clnc+(1-c)ln(1-c)+\chi c(1-c)+\kappa {|\mathrm{\nabla }c|}^2,\end{array}$$ [1]

where c is the mass fraction of one of the species. The interaction coefficient is χ = 2T_c/T, where T_c is the critical temperature. For χ > 2, the mixture phase separates into concentrations c_{e, 1} and c_{e, 2} given by

$$\begin{array}{c}\hfill ln\left(\frac{c_e}{1-c_e}\right)=\chi (2c_e-1).\end{array}$$ [2]

The surface energy coefficie nt is κ, and the surface tension is

$$\begin{array}{c}\hfill \gamma =n{k}_{B}T\sqrt{2\chi \kappa }{\sigma }_r,\end{array}$$ [3]

where n = ρ/m is the number density, and

$$\begin{array}{c}\hfill {\sigma }_r={\int }_{c_{e,1}}^{c_{e,2}}dc{\left[\frac{2c}{\chi }ln\frac{c}{c_e}+\frac{2(1-c)}{\chi }ln\frac{1-c}{1-c_e}-2{(c-c_e)}^2\right]}^{1/2}.\end{array}$$ [4]

The expected surface interface thickness is

$$\begin{array}{c}{\mathcal{l}}_{\text{s}}=\sqrt{2}{\mathcal{l}}_{\text{c}}{\left(-1-\frac{2\log 4c_e(1-c_e)}{\chi {(1-2c_e)}^2}\right)}^{-1/2},\end{array}$$ [5]

where ${\ell }_{\mathrm{c}}=\sqrt{2\kappa /\chi }$ is a characteristic length scale for the interface. The characteristic length scale for capillary wave fluctuations is ${\ell }^{\star }=\sqrt{k_{B}T/\gamma }={(2\chi \kappa n^2{\sigma }_r^2)}^{-1/4}$.

For systems in which the characteristic fluid velocity is asymptotically small relative to the sound speed, we can obtain the low Mach number equations from the fully compressible equations by asymptotic analysis (25, 26). For constant density, this gives the incompressible flow equations

$$\begin{array}{cc}\hfill {(\rho c)}_t+\mathrm{\nabla }·(\rho uc)=& \mathrm{\nabla }·\mathcal{F}\hfill \\ \hfill {(\rho u)}_t+\mathrm{\nabla }·(\rho uu)+\mathrm{\nabla }\pi =& \mathrm{\nabla }·\mathit{\tau }+\mathrm{\nabla }·\mathcal{R}\hfill \\ \hfill \mathrm{\nabla }·u=& 0,\hfill \end{array}$$ [6]

where u is the fluid velocity, and π is a perturbational pressure. Here, ℱ, τ, and ℛ are the species flux, viscous stress tensor, and interfacial reversible stress, respectively.

In fluctuating hydrodynamics, the dissipative fluxes are written as the sum of deterministic and stochastic terms. The species flux is $\mathcal{F}=\overline{\mathcal{F}}+\tilde{\mathcal{F}}$ where the deterministic flux is

$$\begin{array}{c}\hfill \overline{\mathcal{F}}=\rho D\left(\mathrm{\nabla }c-2\chi c(1-c)\mathrm{\nabla }c+2c(1-c)\kappa \mathrm{\nabla }{\mathrm{\nabla }}^{2}c\right),\end{array}$$ [7]

and D is the diffusion coefficient. The stochastic flux is

$$\begin{array}{c}\hfill \tilde{\mathcal{F}}=\sqrt{2\rho mDc(1-c)}\mathcal{Z},\end{array}$$ [8]

where 𝒵(r, t) is a standard Gaussian white noise vector with uncorrelated components

$$\begin{array}{c}\hfill ⟨{\mathcal{Z}}_i(\mathbf{r},t){\mathcal{Z}}_j({\mathbf{r}}^{\prime },t^{\prime })⟩={\delta }_{i,j}\delta (\mathbf{r}-{\mathbf{r}}^{\prime })\delta (t-t^{\prime }).\end{array}$$ [9]

The viscous stress tensor is given by $\mathit{\tau }=\overline{\mathit{\tau }}+\tilde{\mathit{\tau }},$ where the deterministic component is

$$\begin{array}{c}\hfill \overline{\mathit{\tau }}=\eta [\mathrm{\nabla }u+{(\mathrm{\nabla }u)}^T].\end{array}$$ [10]

Here, bulk viscosity is neglected because it does not appear in the incompressible equations. The stochastic contribution to the viscous stress tensor is modeled as

$$\begin{array}{c}\hfill \tilde{\mathit{\tau }}=\sqrt{\eta k_{B}T}(\mathcal{W}+{\mathcal{W}}^T),\end{array}$$ [11]

where 𝒲(r, t) is a standard Gaussian white noise tensor with uncorrelated components. Finally, the interfacial reversible stress is

$$\begin{array}{c}\hfill \mathcal{R}=n{k}_{B}T\kappa \left[\frac{1}{2}{|\mathrm{\nabla }c|}^2\mathbb{I}-\mathrm{\nabla }c\otimes \mathrm{\nabla }c\right].\end{array}$$ [12]

Note that since ℛ is a nondissipative flux, there is no corresponding stochastic flux.

FHD Algorithm and Its Validation

The system of equations [6] is discretized using a structured-grid finite-volume approach with cell-averaged concentrations and face-averaged (staggered) velocities. The overall algorithm is based on methods introduced in refs. 22, 27, and 28. The algorithm uses an explicit discretization of concentration coupled to a semi-implicit discretization of velocity using a predictor-corrector scheme for second-order temporal accuracy.

The numerical method uses standard spatial discretization approaches. Details appear in SI Appendix, section 1. The basic time step algorithm consists of four steps:

Step 1: Compute the predicted velocity, u^{*, n + 1} and perturbational ${\pi }^{\ast ,n+\frac{1}{2}}$ by solving the Stokes system

$$\begin{array}{cc}\hfill \frac{\rho u^{\ast ,n+1}-\rho u^n}{\mathrm{\Delta }t}+\mathrm{\nabla }{\pi }^{\ast ,n+\frac{1}{2}}& =-\mathrm{\nabla }·{(\rho u{u}^T)}^n\hfill \\ \hfill & +\frac{1}{2}\left(\mathrm{\nabla }·{\overline{\mathit{\tau }}}^n+\mathrm{\nabla }·{\overline{\mathit{\tau }}}^{\ast ,n+1}\right)\hfill \\ \hfill & +\mathrm{\nabla }·{\tilde{\mathit{\tau }}}^n+\mathrm{\nabla }·{\mathcal{R}}^n,\hfill \end{array}$$ [13]

$$\begin{array}{cc}\hfill \mathrm{\nabla }·u^{\ast ,n+1}& =0.\hfill \end{array}$$ [14]

Step 2: Predict concentration at time n + 1/2 using

$$\begin{array}{cc}\hfill \rho c^{\ast ,n+\frac{1}{2}}& =\rho c^n-\frac{\mathrm{\Delta }t}{2}\mathrm{\nabla }·\left[\rho c^n\left(\frac{u^n+u^{\ast ,n+1}}{2}\right)\right]\hfill \\ \hfill & +\frac{\mathrm{\Delta }t}{2}\left(\mathrm{\nabla }·{\overline{\mathcal{F}}}^n+\mathrm{\nabla }·{\tilde{\mathcal{F}}}^n\right).\hfill \end{array}$$ [15]

Step 3: Compute concentration at time n + 1 using

$$\begin{array}{cc}\hfill \rho c^{n+1}& =\rho c^n-\mathrm{\Delta }t\mathrm{\nabla }·\left[\rho c^{\ast ,n+\frac{1}{2}}\left(\frac{u^n+u^{\ast ,n+1}}{2}\right)\right]\hfill \\ \hfill & +\mathrm{\Delta }t\left(\mathrm{\nabla }·{\overline{\mathcal{F}}}^{\ast ,n+\frac{1}{2}}+\mathrm{\nabla }·{\tilde{\mathcal{F}}}^{\ast ,n+\frac{1}{2}}\right).\hfill \end{array}$$ [16]

Step 4: Compute the corrected velocity u^{n + 1} and perturbational pressure ${\pi }^{n+\frac{1}{2}}$, by solving the Stokes system

$$\begin{array}{cc}\hfill \frac{\rho u^{n+1}-\rho u^n}{\mathrm{\Delta }t}+\mathrm{\nabla }{\pi }^{\ast ,n+\frac{1}{2}}& =-\frac{\left(\mathrm{\nabla }·{(\rho uu)}^n+\mathrm{\nabla }·{(\rho uu)}^{\ast ,n+1}\right)}{2}\hfill \\ \hfill & +\frac{1}{2}\left(\mathrm{\nabla }·{\overline{\mathit{\tau }}}^n+\mathrm{\nabla }·{\overline{\mathit{\tau }}}^{n+1}\right)\hfill \\ \hfill & +\mathrm{\nabla }·{\tilde{\mathit{\tau }}}^n+\mathrm{\nabla }·{\mathcal{R}}^{\ast ,n+\frac{1}{2}},\hfill \end{array}$$ [17]

$$\begin{array}{cc}\hfill \mathrm{\nabla }·u^{n+1}& =0.\hfill \end{array}$$ [18]

In both steps 1 and 4, the discretized Stokes system is solved by a generalized minimal residual (GMRES) method with a multigrid preconditioner, see ref. 29. The explicit treatment of the concentration equation introduces a stability limitation on the time step of

$$\begin{array}{c}\hfill D\left(\frac{12}{\mathrm{\Delta }{x}^2}+\frac{72\kappa }{\mathrm{\Delta }{x}^4}\right)\mathrm{\Delta }t\le 1,\end{array}$$ [19]

where Δx is the mesh spacing.

Unless otherwise specified, the physical parameters used in all simulations are mass density, ρ = 1.4 g/cm³, molecular mass, m = 6.0 × 10⁻²³ g, Boltzmann constant, k_B = 1.38 × 10⁻¹⁶ erg/K, temperature T = 84K, interaction parameter χ = 3.571, and surface energy coefficient κ = 2.7 × 10⁻¹⁴cm². These conditions are based on the model for liquid argon given in ref. 11, modified to increase both the interfacial tension and the interface thickness by a factor of 2. For these values, the equilibrium concentrations are c_{1, e} = 0.035 and c_{2, e} = 0.965. The surface tension is γ = 28.35 dyne/cm and ℓ^⋆ = 0.2 nm; for R₀ = 6 nm, the stochastic Weber number We^⋆ ≈ 10⁻³.

We considered two values for shear viscosity, η = 2.46 × 10⁻³ and 2.46 × 10⁻² g/cm s, for which the Ohnesorge numbers are Oh = 0.50 and 5.0, respectively. In general, the diffusion coefficient was D = η/ρ Sc with a Schmidt number of Sc = 35.1, the exception being a single run with Sc = 351.

In general, the simulations used periodic boundary conditions and cubic cells with mesh spacing Δx = 1.0 nm. With these parameters, each simulation cell represents roughly 23 fluid molecules. The time step was either Δt = 0.4 ps or 0.04 ps depending on the value of D, which corresponds to approximately one-quarter of the maximum stable time step (Eq. 19).

A variety of equilibrium systems were simulated to validate the algorithm. First, the interface thickness was measured in the simulations of a flat slab in a quasi-2D system (96 × 12 × 1 cells)^*; Fig. 1 shows that good agreement with Eq. 5 is found. The systematic shift in the predictions is a result of numerical error, which decreases with mesh spacing. Note that the surface interface thickness ℓ_s ≈ 3Δx ≈ 15ℓ^⋆.

Fig. 1.

Interface thickness, ℓ_s, in nm versus interaction coefficient χ as measured (markers) and as given by Eq. 5 (solid curve); large marker indicates the value of χ used in the Rayleigh–Plateau simulations. Note that ℓ_s → ∞ as χ → 2.

Next, the Laplace pressure, δp, was measured in a similar quasi-2D system (96 × 96 × 1 cells) with concentration c_{1, e} within a disk of radius R₀ = 6.0 nm and concentration c_{2, e} elsewhere.^† Fig. 2 shows that the surface tension, computed using γ = R₀δp, is in good agreement with the expected value given by Eq. 3.^‡ As part of this validation, we also considered disks of different radii and alternative parameters that resulted in a thinner interface. In all of these additional cases, the methodology continued to show excellent agreement with theory.

Fig. 2.

Surface tension, γ, in dyne/cm versus interaction coefficient χ as measured (markers) and as predicted by Eq. 3 (solid curve); large marker indicates value of χ used in the Rayleigh–Plateau simulations. Note that γ → 0 as χ → 2.

As a final validation test, the capillary wave spectrum (30–32) at thermal equilibrium was measured in a quasi-2D system (256 × 64 × 1 cells) with a flat slab of concentration c_{1, e} and concentration c_{2, e} elsewhere. As in ref. 21, the deviations in height from a flat interface, h(r, t), were measured and Fourier transformed to obtain $\widehat{h}(\mathbf{k},t)$. The temporal averaged spectrum, shown in Fig. 3, is in good agreement with the predicted result,

$$\begin{array}{c}\hfill ⟨|\widehat{h}(\mathbf{k}){|}^2⟩=\frac{k_{B}T}{A\gamma k^2},\end{array}$$ [20]

Fig. 3.

Capillary wave spectrum (in cm²) versus wavenumber (in cm⁻¹) as measured in the FHD simulation (points) and as predicted by Eq. 20 (solid line). The error bars represent ±3 SDs. The wavenumber is corrected to account for the discrete Laplacian (21).

where A is the surface area of the interface (here the simulation was two-dimensional with a cross-section of 5 nm, which sets the magnitude of the noise). The deviation from theory at large k is attributed to a wave vector–dependent surface tension. It is also observed in MD (33), and the deviation is sensitive to how the interface position is defined (34).

Rayleigh–Plateau Instability

The Rayleigh–Plateau instability in nanoscale systems was simulated and compared with earlier MD and stochastic lubrication calculations. Each simulation was initialized by starting with concentration c_{e, 1} inside a 2D disk of radius R₀ = 6.0 nm and concentration c_{e, 2} elsewhere. The system was then evolved deterministically until the interface had equilibrated. This initial slice was replicated to create a uniform cylinder of length L. Unless otherwise stated, the physical and numerical parameters are those used in the validation runs (see previous section). The characteristic time scale for capillary waves is ${\tau }_0=\sqrt{\rho {R_0}^3/\gamma }\approx 0.1$ ns. From linear stability theory, the growth rate for the fastest growing wavenumber (Rayleigh mode) is τ_inv ≈ 3 τ₀ in the inviscid limit; for uniform viscosity, it is τ_visc ≈ 28 Oh τ₀ (1, 35).

First, we consider “long” cylinders with L = 360.0 nm, so L ≫ λ_p ≈ 67 nm, the fastest growing wavelength. The domain of this 3D system has a cross-section of 48.0 nm by 48.0 nm in the x and y directions (48 × 48 × 360 cells) and is periodic in all directions. We ran both a “stochastic” and a “deterministic” version of the simulation. The former simply uses the FHD algorithm starting from the initial condition described above. The deterministic runs start the same way, but after a time t_init = 0.4 ns, the stochastic fluxes are set to zero. In both the stochastic and deterministic runs, the initial cylinders pinch into droplets, but there are qualitative differences, as seen in Fig. 4. In the stochastic case, the cylinders narrow into a double cone before pinching while in the deterministic case a filament forms, as seen by comparing stochastic simulation at 8.0 ns and deterministic simulation at 10.0 ns.

Fig. 4.

Center-line cross-section snapshots from stochastic (Left) and deterministic runs at t = 2, 4, 6, 8, and 10 ns for Oh = 0.5. Note the double cone shape in the stochastic case at 8 ns and the filament attached to a drop in the deterministic case at 10 ns.

From the simulation data, we calculate the cylinder radius, R(z, t), for each cross-section. The presence of thermal fluctuations introduces some difficulty in defining the radius, so we use a filter to sharpen the numerical interface.^§ Fig. 5 shows the minimum cylinder radius, R_min(t)=min_z{R(z, t)}, versus time for individual runs in ensembles of 10 runs. For the lower viscosity case (Oh = 0.5), the mean and SD for the pinch time in the stochastic runs were 6.93 ns and 0.156 ns; for the perturbed deterministic runs, they were 7.73 ns and 0.142 ns, so the fluctuations significantly hasten the breakup of the cylinder. For the higher viscosity (Oh = 5.0), stochastic runs mean and SD were 16.18 ns and 0.111 ns.

Fig. 5.

Minimum radius, R_min(t), versus time before pinching for each run (thin lines): (Top) stochastic runs, Oh = 0.5; (Middle) deterministic runs, Oh = 0.5; (Bottom) stochastic runs, Oh = 5.0. Average pinch times: (Top) 6.93 ns, (Middle) 7.73 ns, (Bottom) 16.18 ns. SDs: (Top) 0.156 ns, (Middle) 0.142 ns, (Bottom) 0.111 ns. Thick solid lines are power-law fits to the ensemble average; dotted lines are extrapolations. Fits give are α = 0.601 (Top), α = 0.675 (Middle), α = 0.243 (Bottom).

Theory, simulations, and experiments indicated that in general, R_min ∼ (t_p − t)^α, where t_p is the mean pinching time with the coefficient α depending on Oh, Sc, and We^⋆. Specifically, there are the inertia-dominated (small Oh), viscosity-dominated (large Oh, large Sc), and diffusion-dominated (large Oh, small Sc) regimes, which can be either deterministic (small We^⋆) or stochastic (large We^⋆). For example, α = 1 in the deterministic, viscosity-dominated regime (1, 36), α = 1/3 in the deterministic, diffusion-dominated regime (37), and α ≈ 0.412 in the stochastic, viscosity-dominated regime (5). The power-law fits to the simulation data are shown in Fig. 5 for the Oh = 0.5 and 5.0 runs with Sc = 35.1.^# As expected, the Oh = 0.5 are in an intermediate range between inertia-dominated and diffusion-dominated, similar to the MD simulations in ref. 12. Quantitative comparison was not possible since that work investigated cylinders of liquid in its own vapor while our current results are for two similar incompressible fluids.

A single run was performed using the higher viscosity (Oh = 5.0) with a Schmidt number of Sc = 351; all other physical parameters were unchanged. This run was computationally intensive because the time to pinching was nearly 50 ns. Fig. 6 shows that R_min(t) for this run is in good agreement with the prediction by Eggers (5) that R_min ∼ (t_p − t)^0.412 in the stochastic, viscous-dominated regime. A comparison with the lower Sc number simulations illustrates the importance of diffusion on the pinch-off dynamics.

Fig. 6.

Minimum cylinder radius, R_min(t), versus time before pinching for a high Schmidt number (Sc = 351), high Ohnesorge number (Oh = 5.0) run. The red solid line is power-law fit (α = 0.402); the dotted line is an extrapolation.

To quantify the dominant modes of growth leading to rupture, we Fourier transform R(z, t) to obtain $\widehat{R}(k,t)$ for an ensemble of 10 runs, similar to the analysis in ref. 11 (SI Appendix, section 6). Figs. 7 and 8 show the ensemble-averaged spectrum, $|\widehat{R}(k,t)|$, versus wave number at various times from simulations using the lower viscosity (η = 2.46 × 10⁻³ g/cm s, Oh = 0.5) and the higher viscosity (η = 2.46 × 10⁻² g/cm s, Oh = 5.0). In the former case, the results from an ensemble of deterministic runs are also shown in Fig. 7. These results are qualitatively similar to the MD measurements and stochastic lubrication prediction in ref. 11. Specifically, the spectrum is similar with the instability developing faster in the stochastic case but, again, that work is for a liquid/vapor system.

Fig. 7.

Averaged power spectrum for stochastic (solid lines) and deterministic (dashed lines) simulations with lower viscosity (Oh = 0.5) at t = 2, 4, and 6 ns (see legend).

Fig. 8.

Averaged power spectrum for stochastic simulations with higher viscosity (Oh = 5.0) at t = 4, 8, 12, and 15 ns (see legend).

We also investigated the breakup of classically stable cylinders, that is, cylinders with length L < L_c where the critical length, L_c, equals the circumference. Fig. 9 shows snapshots for cylinders of radius R₀ = 6.0 nm (L_c = 37.7 nm) for Oh = 0.5. Note that both stochastic cases pinch to form a droplet, while in the deterministic case, only the longer cylinder (L = 42 nm) forms a droplet. Fig. 10 shows the time to pinching for a range of cylinder lengths; comparable results have been reported for MD simulations (11). Interestingly, the deterministic case with L = 36 nm is unstable, which suggests that the effective critical length is slightly shorter (deterministic runs with L ≤ 34 nm did not pinch). This is less surprising when we recall that the diffuse interface is relatively thick (Fig. 1).

Fig. 9.

Snapshots of short cylinders with lengths of 32 nm (L < L_c) and 42 nm (L > L_c). The two columns on the left are stochastic runs, and the two on the right are deterministic runs; in all cases, Oh = 0.5.

Fig. 10.

Dimensionless pinch time, t_p/τ₀, for short cylinders versus L from ensembles of stochastic and deterministic runs. The dashed line marks the critical length (37.7 nm) based on the initial radius, and the dotted is the critical length (36.5 nm) based on the minimum radius when noises are turned off in the deterministic runs.

Finally, as an illustration of the capabilities of the algorithm, we performed simulations showing the Rayleigh–Plateau instability on a torus. In a 256.0 nm by 256.0 nm by 64.0 nm periodic system, a torus with a center-line radius of 86.0 nm and a cylindrical radius of 6.0 nm was initialized. The volumetric snapshots in Fig. 11 show the outer radius shrinking as the instability develops (38), which is observed in macroscopic experiments (39) and the appearance of satellite droplets, as predicted by theory (40). The impact of thermal fluctuations in this geometry remains an open question for future study.

Fig. 11.

Rayleigh–Plateau instability on a torus; snapshots show initial data and simulation results at t = 4.0, 6.0, 8.0, 10.0, and 12.0 ns.

Summary and Conclusions

The behavior of hydrodynamic instabilities plays a crucial role in determining the dynamics of fluid systems. At the nanoscale, the relative importance of various physical phenomena changes compared to macroscale systems. In particular, thermal fluctuations can significantly influence behavior. To assess the importance of thermal fluctuations in nanofluid systems, a additional dimensionless parameter, the stochastic Weber number, is required. The results in this paper demonstrate that thermal fluctuations can influence the growth and morphologies of nanostructures, specifically the pinching of liquid cylinders into droplets.

As discussed in the paper, this is not a new observation, and our results are in good agreement with previous nanofluidic studies of the Rayleigh–Plateau instabilities using stochastic lubrication theory and MD simulations. The significance of the present work is in developing a numerical method for using fluctuating hydrodynamics effectively without requiring the lubrication approximation. Fluctuating hydrodynamics offers several advantages compared with MD. Fluctuating hydrodynamics simulations allow us to set independently such physical parameters as surface tension and viscosity, which in MD are indirectly linked to the intermolecular potentials. Our earlier work demonstrates that fluctuating hydrodynamics calculations are also typically several orders of magnitude faster than MD simulations. Finally, in fluctuating hydrodynamics, the modeling of complex fluids (e.g., ionic liquids and reactive mixtures) is straightforward (22, 41, 42).

The methodology developed here can be used to more fully quantify the role of diffusion (37) and viscosity on the Rayleigh–Plateau instability. To facilitate this type of study, we are developing an implicit diffusion solver that will greatly improve the efficiency of the algorithm. The approach can also be extended to the case of fluids with dissimilar properties such as density, viscosity, and diffusivity. Fluctuating hydrodynamics can also be used to investigate the behavior of thin films and model the dynamics of contact lines (16–18). The overall approach can also be generalized to systems with more than two components including polymer mixtures, enabling the simulation of a wide range of multiphase phenomena at the nanoscale.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

The software used to generate the results in this manuscript is publicly available at: https://github.com/AMReX-FHD/FHDeX (43). This repository includes information about how to compile and run the software along with input files needed to run the simulations. A knowledgeable researcher should be able to recreate the simulations performed here. The full, raw simulation data are too large to feasibly store in a public data repository. However, these data will be available on request.)