Critical Droplet Research Articles

Investigation of droplet dynamics in a convective cloud using a Lagrangian cloud model

A precipitating convective cloud is simulated successfully using the Lagrangian cloud model, in which the flow field is simulated by large eddy simulation and the droplets are treated as Lagrangian particles, and the results are analyzed to investigate precipitation initiation and to examine the parameterization of cloud microphysics. It is found that raindrops appear initially near the cloud top, in which strong turbulence and broadened droplet spectrum are induced by the entrainment of dry air, but high liquid–water mixing ratio is maintained within cloud parts because of insufficient mixing. Statistical analysis of the downward vertical velocity of a droplet W reveals that the transition from cloud droplets to raindrops occurs in the range 20 μm < r < 100 μm, while the variation of W depends on turbulence as well as the droplet radius r. The general pattern of the raindrop size distribution is found to be consistent with the Marshall–Palmer distribution. The precipitation flux can be underestimated substantially, if the terminal velocity \(w_{\text{s}}\) is used instead of W, but it is not sensitive to the choice of the critical droplet radius dividing cloud drops and raindrops. It is also found that precipitation starts earlier and becomes stronger if the effect of turbulence is included in the collection kernel.

Break-up of droplets in a concentrated emulsion flowing through a narrow constriction

This paper describes the break-up of droplets in a concentrated emulsion during its flow as a 2D monolayer in a microchannel consisting of a narrow constriction. Analysis of the behavior of a large number of drops (N > 4000) shows that the number of break-ups increases with increasing flow rate, entrance angle to the constriction, and size of the drops relative to the width of the constriction. As single drops do not break at the highest flow rate used in the system, break-ups arise primarily from droplet-droplet interactions. Analysis of droplet properties at a high temporal resolution of 10 microseconds makes it possible to relate droplet deformation with droplet break-up probability. Similar to previous studies on single drops, no break-up is observed below a set of critical flow rates and droplet deformations. Unlike previous studies, however, not all drops undergo break-up above the critical values. Instead, the probability of droplet break-up increases with flow rate and the deformation of the drops. The probabilistic nature of the break-up process arises from the stochastic variations in the packing configuration of the drops as they enter the constriction. Local break-up dynamics involves two primary drops. A close look at the interactions between the pair of drops reveals that the competing time scales of droplet rearrangement relative to the relaxation of the opposing drop govern whether break-up occurs or not. Practically, these results can be used to calculate the maximum throughput of the serial interrogation process often employed in droplet microfluidics. For 40 pL-drops, the highest throughput with less than 1% droplet break-up was measured to be approximately 7000 drops per second. In addition, the results presented are useful for understanding the behavior of concentrated emulsions in applications such as mobility control in enhanced oil recovery, and for extrapolating critical parameters such as injection rates to ensure the stability of the fluids going through small pore throats.

Ultra Fast Spray Cooling and Critical Droplet Daimeter Estimation from Cooling Rate

Spray cooling is an effective tool to dissipate high heat fluxes from hot surfaces. This paper thoroughly investigates the effects of spray parameters on the cooling time and cooling rate under varying inlet pressure using water as the coolant. Cylindrical samples of stainless steel with constant diameter, D = 25 mm, and thickness δ: 8.5 mm, 13 mm, 17.5 mm and 22 mm were investigated. Critical droplet diameter to achieve an ultrafast cooling rate of 300°C/s was estimated by using analytical model for samples of varying thickness. At an inlet pressure of 0.8 MPa, maximum cooling rates of 424.2°C/s, 502.81°C/s and 573.1°C/s were achieved for wall super heat ΔT = 600°C, 700°C and 800°C respectively.

Nucleation and growth for the Ising model in $d$ dimensions at very low temperatures

This work extends to dimension $d\geq3$ the main result of Dehghanpour and Schonmann. We consider the stochastic Ising model on ${\mathbb{Z}}^d$ evolving with the Metropolis dynamics under a fixed small positive magnetic field $h$ starting from the minus phase. When the inverse temperature $\beta$ goes to $\infty$, the relaxation time of the system, defined as the time when the plus phase has invaded the origin, behaves like $\exp({{\beta}{\kappa}_d})$. The value $\kappa_d$ is equal to \[{\kappa}_d=\frac{1}{d+1}({\Gamma}_1+\cdots+{\Gamma}_d),\] where ${\Gamma}_i$ is the energy of the $i$-dimensional critical droplet of the Ising model at zero temperature and magnetic field $h$.

Temperature of critical clusters in nucleation theory: Generalized Gibbs' approach

According to the classical Gibbs' approach to the description of thermodynamically heterogeneous systems, the temperature of the critical clusters in nucleation is the same as the temperature of the ambient phase, i.e., with respect to temperature the conventional macroscopic equilibrium conditions are assumed to be fulfilled. In contrast, the generalized Gibbs' approach [J. W. P. Schmelzer, G. Sh. Boltachev, and V. G. Baidakov, J. Chem. Phys. 119, 6166 (2003); and ibid. 124, 194503 (2006)] predicts that critical clusters (having commonly spatial dimensions in the nanometer range) have, as a rule, a different temperature as compared with the ambient phase. The existence of a curved interface may lead, consequently, to an equilibrium coexistence of different phases with different temperatures similar to differences in pressure as expressed by the well-known Laplace equation. Employing the generalized Gibbs' approach, it is demonstrated that, for the case of formation of droplets in a one-component vapor, the temperature of the critical droplets can be shown to be higher as compared to the vapor. In this way, temperature differences between critically sized droplets and ambient vapor phase, observed in recent molecular dynamics simulations of argon condensation by Wedekind et al. [J. Chem. Phys. 127, 064501 (2007)], can be given a straightforward theoretical interpretation. It is shown as well that - employing the same model assumptions concerning bulk and interfacial properties of the system under consideration - the temperature of critical bubbles in boiling is lower as compared to the bulk liquid.

Metastability for the Blume-Capel model with distribution of magnetic anisotropy using different dynamics

We investigate the relaxation time of magnetization or the lifetime of the metastable state for a spin S=1 square-lattice ferromagnetic Blume-Capel model with distribution of magnetic anisotropy (with small variances), using two different dynamics such as Glauber and phonon-assisted dynamics. At each lattice site, the Blume-Capel model allows three spin projections (+1, 0, -1) and a site-dependent magnetic anisotropy parameter. For each dynamic, we examine the low-temperature lifetime in two dynamic regions with different sizes of the critical droplet and at the boundary between the regions, within the single-droplet regime. We compute the average lifetime of the metastable state for a fixed lattice size, using both kinetic Monte Carlo simulations and the absorbing Markov chains method in the zero-temperature limit. We find that for both dynamics the lifetime obeys a modified Arrhenius-like law, where the energy barrier of the metastable state depends on the temperature and standard deviation of the distribution of magnetic anisotropy for a given field and magnetic anisotropy and that an explicit form of this dependence differs in different dynamic regions for different dynamics. Interestingly, the phonon-assisted dynamic prevents transitions between degenerate states, which results in a large increase in the energy barrier at the region boundary compared to that for the Glauber dynamic. However, the introduction of a small distribution of magnetic anisotropy allows the spin system to relax via lower-energy pathways such that the energy barrier greatly decreases. In addition, for the phonon-assisted dynamic, even the prefactor of the lifetime is substantially reduced for a broad distribution of magnetic anisotropy in both regions considered, in contrast to the Glauber dynamic. Our findings show that overall the phonon-assisted dynamic is more significantly affected by the distribution of magnetic anisotropy than the Glauber dynamic.

No‐slip consistent immersed boundary particle tracking to simulate impaction filtration in porous media

SUMMARYIn this paper, we present a new method for simulating the motion of a disperse particle phase in a carrier gas through porous media. We assume a sufficiently dilute particle‐laden flow and compute, independently of the disperse phase, the steady laminar fluid velocity using the immersed boundary method. Given the velocity of the carrier gas, the equations of motion for the particles experiencing the Stokes drag force are solved to determine their trajectories. The ‘no‐slip consistent’ particle tracking algorithm avoids possible numerical filtration of very small particles due to the nonzero velocity field at the solid–fluid interface introduced by the immersed boundary method. This physically consistent tracking allows a reliable estimation of the filtration efficiency of porous filters due to inertial impaction. We illustrate and test our new approach for model porous media consisting of a structured array of aligned rectangular fibers, arranged in line and staggered. In the staggered geometry, the effect of the residual velocity at the solid–fluid interface is significant for particles with low inertia. Without adopting the developed no‐slip consistent numerical method, an artificial numerical filtration is observed, which becomes dominant for small enough particles. For both the in line and the staggered geometries, the filtration rate depends quite strongly and non monotonically on the particle inertia. This is expressed most clearly in the staggered arrangement in which a very strong increase in the filtration efficiency is observed at a well‐defined critical droplet size, corresponding to a qualitative change in the dominant particle paths in the porous medium. Copyright © 2013 John Wiley &amp; Sons, Ltd.

Droplet sorting in a loop of flat microfluidic channels

Motivated by recent experiments, we numerically study the droplet traffic in microfluidic channels forming an asymmetric loop with a long and a short arm. The loop is connected to an inlet and an outlet channel by two right angled T-junctions. Assuming flat channels, we employ the boundary element method (BEM) to numerically solve the two-dimensional Darcy equation that governs two phase flow in the Hele-Shaw limit. The occurrence of different sorting regimes is summarized in sorting diagrams in terms of droplet size, distance between consecutive droplets in the inlet channel, and loop asymmetry for mobility ratios of the liquid phases larger and smaller than one. For large droplet distances, the traffic is regulated by the ratio of the total hydraulic resistances of the long and short arms. At high droplet densities and below a critical droplet size, droplet–droplet collisions are observed for both mobility ratios.

Monte Carlo tests of nucleation concepts in the lattice gas model

The conventional theory of homogeneous and heterogeneous nucleation in a supersaturated vapor is tested by Monte Carlo simulations of the lattice gas (Ising) model with nearest-neighbor attractive interactions on the simple cubic lattice. The theory considers the nucleation process as a slow (quasistatic) cluster (droplet) growth over a free energy barrier ΔF(*), constructed in terms of a balance of surface and bulk term of a critical droplet of radius R(*), implying that the rates of droplet growth and shrinking essentially balance each other for droplet radius R=R(*). For heterogeneous nucleation at surfaces, the barrier is reduced by a factor depending on the contact angle. Using the definition of physical clusters based on the Fortuin-Kasteleyn mapping, the time dependence of the cluster size distribution is studied for quenching experiments in the kinetic Ising model and the cluster size ℓ(*) where the cluster growth rate changes sign is estimated. These studies of nucleation kinetics are compared to studies where the relation between cluster size and supersaturation is estimated from equilibrium simulations of phase coexistence between droplet and vapor in the canonical ensemble. The chemical potential is estimated from a lattice version of the Widom particle insertion method. For large droplets it is shown that the physical clusters have a volume consistent with the estimates from the lever rule. Geometrical clusters (defined such that each site belonging to the cluster is occupied and has at least one occupied neighbor site) yield valid results only for temperatures less than 60% of the critical temperature, where the cluster shape is nonspherical. We show how the chemical potential can be used to numerically estimate ΔF(*) also for nonspherical cluster shapes.

Kinematic and dynamic collision statistics of cloud droplets from high-resolution simulations

We study the dynamic and kinematic collision statistics of cloud droplets for a range of flow Taylor microscale Reynolds numbers (up to 500), using a highly scalable hybrid direct numerical simulation approach. Accurate results of radial relative velocity (RRV) and radial distribution function (RDF) at contact have been obtained by taking advantage of their power-law scaling at short separation distances. Three specific but inter-related questions have been addressed in a systematic manner for geometric collisions of same-size droplets (of radius from 10 to 60 μm) in a typical cloud turbulence (dissipation rate at 400 cm2 s−3). Firstly, both deterministic and stochastic forcing schemes were employed to test the sensitivity of the simulation results on the large-scale driving mechanism. We found that, in general, the results are quantitatively similar, with the deterministic forcing giving a slightly larger RDF and collision kernel. This difference, however, is negligible for droplets of radius less than 30 μm. Secondly, we have shown that the dependence of pair statistics on the flow Reynolds number Rλ or larger scale fluid motion is of secondary importance, with a tendency for this effect to saturate at high enough Rλ leading to Rλ-independent results. Both DNS results and theoretical arguments show that the saturation happens at a smaller Rλ for smaller droplets. Finally, since most previous studies of turbulent collision of inertial particles concerned non-sedimenting particles, we have specifically addressed the role of gravity on collision statistics, by simultaneously simulating collision statistics with and without gravity. It is shown that the collision statistics is not affected by gravity when a < ac, where the critical droplet radius ac is found to be around 30 μm for the RRV, and around 20 μm for the RDF. For larger droplets, gravity alters the particle–eddy interaction time and significantly reduces the RRV. The effect of gravity on the RDF is rather complex: gravity reduces the RDF for intermediate-sized droplets but enhances the RDF for larger droplets. In addition, we have studied the scaling exponents of both RDF and RRV, and found that gravity modifies the RDF scaling exponents for both intermediate and large particles, in a manner very similar to the effect of gravity on the RDF at contact. Gravity is shown to cause the scaling exponents for RDF and RRV to level off for large droplets, in contrast to diminishing exponents for non-sedimenting particles.

Eliminating cracking during drying

When colloidal suspensions dry, stresses build up and cracks often occur -a phenomenon undesirable for important industries such as paint and ceramics. We demonstrate an effective method which can completely eliminate cracking during drying: by adding emulsion droplets into colloidal suspensions, we can systematically decrease the amount of cracking, and eliminate it completely above a critical droplet concentration. Since the emulsion droplets eventually also evaporate, our technique achieves an effective function while making little changes to the component of final product, and may therefore serve as a promising approach for cracking elimination. Furthermore, adding droplets also varies the speed of air invasion and provides a powerful method to adjust drying rate. With the effective control over cracking and drying rate, our study may find important applications in many drying- and cracking-related industrial processes.

Distribution of melting times and critical droplet in kinetic Monte Carlo and molecular dynamics

A kinetic Monte Carlo model on a lattice, based on a reaction-like mechanism, is used to investigate the microscopic properties of the homogeneous melting of a metastable crystal. The kinetic Monte Carlo model relies on nearest-neighbors interactions and a few relevant dynamical parameters. To examine the reliability of the model, careful comparison with molecular dynamics simulations of a hard sphere crystal is drawn. A criterion on the critical nature of a microscopic configuration is deduced from the bimodal character of the probability density function of melting time. For kinetic Monte Carlo simulations with dynamical parameter values which fit the molecular dynamics results, the number of liquid sites of the critical droplet is found to be smaller than 300 and the ability of the critical droplet to invade the entire system is shown to be independent of the droplet shape as long as this droplet remains compact. In kinetic Monte Carlo simulations, the size of the critical droplet is independent of the system size. Molecular dynamics evidences a more complex dependence of melting time on system size, which reveals non-trivial finite size effects.

Metastability in the dilute Ising model

Consider Glauber dynamics for the Ising model on the hypercubic lattice with a positive magnetic field. Starting from the minus configuration, the system initially settles into a metastable state with negative magnetization. Slowly the system relaxes to a stable state with positive magnetization. Schonmann and Shlosman showed that in the two dimensional case the relaxation time is a simple function of the energy required to create a critical Wulff droplet. The dilute Ising model is obtained from the regular Ising model by deleting a fraction of the edges of the underlying graph. In this paper we show that even an arbitrarily small dilution can dramatically reduce the relaxation time. This is because of a catalytic effect—rare regions of high dilution speed up the transition from minus phase to plus phase.

Critical Droplet Theory Explains the Glass Formability of Aqueous Solutions

When pure water is cooled at ~10(6) K / s, it forms an amorphous solid (glass) instead of the more familiar crystalline phase. The presence of solutes can reduce this required (or "critical") cooling rate by orders of magnitude. Here, we present critical cooling rates for a variety of solutes as a function of concentration and a theoretical framework for understanding these rates. For all solutes tested, the critical cooling rate is an exponential function of concentration. The exponential's characteristic concentration for each solute correlates with the solute's Stokes radius. A modification of critical droplet theory relates the characteristic concentration to the solute radius and the critical nucleation radius of ice in pure water. This simple theory of ice nucleation and glass formability in aqueous solutions has consequences for general glass-forming systems, and in cryobiology, cloud physics, and climate modeling.

Cascade partial coalescence phenomena at electrolyte–oil interfaces and determination of bounds for the surface potential

Small oil droplets (diameter <500 μm) in slow, steady, buoyancy-driven rise through pure or salty water at neutral pH exhibit a cascade partial-coalescence phenomenon upon soft collision with quasi-planar aqueous electrolyte solution–oil interfaces. For droplets of pure toluene, n-heptane, and their mixtures (heptol), and for moderate-strength electrolytes, the cascade partial-coalescence process continues until a critical droplet size is reached. We infer that this outcome is due to the oil–water interface being electrically charged and use the last partial-coalescence event in the sequence to estimate lower- and upper-bounds for the absolute value of the electrical potential of the interface.

Kawasaki Dynamics with Two Types of Particles: Critical Droplets

This is the third in a series of three papers in which we study a two-dimensional lattice gas consisting of two types of particles subject to Kawasaki dynamics at low temperature in a large finite box with an open boundary. Each pair of particles occupying neighboring sites has a negative binding energy provided their types are different, while each particle has a positive activation energy that depends on its type. There is no binding energy between particles of the same type. At the boundary of the box particles are created and annihilated in a way that represents the presence of an infinite gas reservoir. We start the dynamics from the empty box and are interested in the transition time to the full box. This transition is triggered by a critical droplet appearing somewhere in the box. In the first paper we identified the parameter range for which the system is metastable, showed that the first entrance distribution on the set of critical droplets is uniform, computed the expected transition time up to and including a multiplicative factor of order one, and proved that the nucleation time divided by its expectation is exponentially distributed, all in the limit of low temperature. These results were proved under three hypotheses, and involved three model-dependent quantities: the energy, the shape and the number of critical droplets. Here prove the third hypothesis and identify the shape and the number of critical droplets. The geometric properties of subcritical, critical and supercritical droplets, which are crucial in determining the metastable behavior of the system are identified. The geometry turns out to be considerably more complex than for Kawasaki dynamics with one type of particle, for which an extensive literature exists. The main motivation behind our work is to understand metastability of multi-type particle systems.

Analysis of a stagnation-point premixed flame influenced by inert spray, heat loss, and non-unity Lewis number

This study investigates the effects of the Lewis number, flow stretch, internal heat loss, and external heat loss on the extinction characteristics of propane/air flames using activation-energy asymptotics. A steady, planar, premixed flame generated in a stagnation-point, two-phase flow in which the dispersed phase is simulated by a monodisperse, dilute, and chemically inert spray (water spray) is considered. A completely prevaporized spray mode and a partially prevaporized spray mode of flame propagation are identified using a critical initial droplet size rc′ that is required for the droplet to achieve complete evaporation at the premixed flame front. The internal heat loss associated with water vaporization increases with increasing liquid loading or decreasing initial droplet size. The (positive) flow stretch coupled with the Lewis number (Le) respectively weakens and intensifies the burning intensity of a lean propane/air flame (Le>1) and a rich propane/air flame (Le<1). Additionally, the external heat loss weakens the burning intensity and has a significant influence on the flame behavior. The results show that a Le>1 flame can be extinguished with or without external heat loss. Flame extinction characterized by a C-shaped curve is dominated by external heat loss or the flow stretch. For a Le<1 flame without external heat loss, no extinction occurs under the influence of flow stretch. However, a Le<1 flame with completely prevaporized water sprays that is subjected to a small amount of flow stretch can be extinguished by external heat loss; this behavior is characterized by a C-shaped curve. A W-shaped extinction curve is mainly governed by internal heat loss. Flame extinction characterized by a W-shaped curve occurs when a Le<1 flame with external heat loss is subjected to a positive stretch and a partially prevaporized water spray with sufficiently large liquid loading and droplet size.

Confinement by Carbon Nanotubes Drastically Alters the Boiling and Critical Behavior of Water Droplets

Vapor pressure grows rapidly above the boiling temperature, and past the critical point liquid droplets disintegrate. Our atomistic simulations show that this sequence of events is reversed inside carbon nanotubes (CNT). Droplets disintegrate first and at low temperature, while pressure remains low. The droplet disintegration temperature is independent of the CNT diameter. In contrast, depending on CNT diameter, a temperature that is much higher than the bulk boiling temperature is required to raise the internal pressure. The control over pressure by CNT size can be useful for therapeutic drug delivery.

Numerical approaches to determine the interface tension of curved interfaces from free energy calculations

A recently proposed method to obtain the surface free energy σ(R) of spherical droplets and bubbles of fluids, using a thermodynamic analysis of two-phase coexistence in finite boxes at fixed total density, is reconsidered and extended. Building on a comprehensive review of the basic thermodynamic theory, it is shown that from this analysis one can extract both the equimolar radius R(e) as well as the radius R(s) of the surface of tension. Hence the free energy barrier that needs to be overcome in nucleation events where critical droplets and bubbles are formed can be reliably estimated for the range of radii that is of physical interest. It is found that the conventional theory of nucleation, where the interface tension of planar liquid-vapor interfaces is used to predict nucleation barriers, leads to a significant overestimation, and this failure is particularly large for bubbles. Furthermore, different routes to estimate the effective radius-dependent Tolman length δ(R(s)) from simulations in the canonical ensemble are discussed. Thus we obtain an instructive exemplification of the basic quantities and relations of the thermodynamic theory of metastable droplets/bubbles using simulations. However, the simulation results for δ(R(s)) employing a truncated Lennard-Jones system suffer to some extent from unexplained finite size effects, while no such finite size effects are found in corresponding density functional calculations. The numerical results are compatible with the expectation that δ(R(s) → ∞) is slightly negative and of the order of one tenth of a Lennard-Jones diameter, but much larger systems need to be simulated to allow more precise estimates of δ(R(s) → ∞).

Metastability for Kawasaki dynamics at low temperature with two types of particles

This is the first in a series of three papers in which we study a two-dimensional lattice gas consisting of two types of particles subject to Kawasaki dynamics atlow temperature in a large finite box with an open boundary. Each pair of particlesoccupying neighboring sites has a negative binding energy provided their types aredifferent, while each particle has a positive activation energy that depends onits type. There is no binding energy between neighboring particles of the same type.At the boundary of the box particles are created and annihilated in a way thatrepresents the presence of an infinite gas reservoir. We start the dynamics from the empty box and compute the transition time to the full box. This transition is triggered by a critical droplet appearing somewhere in the box. We identify the region of parameters for which the system is metastable. For thisregion, in the limit as the temperature tends to zero, we show that the firstentrance distribution on the set of critical droplets is uniform, compute theexpected transition time up to a multiplicative factor that tends to one, and prove that the transition time divided by its expectation is exponentially distributed. These results are derived under three hypotheses on the energy landscape, which are verified in the second and the third paper for a certain subregion of the metastable region. These hypotheses involve three model-dependent quantities - the energy, the shape and the number of the critical droplets - which are identified in the second and the third paper as well.