Droplet Scale Research Articles

Low-temperature dynamics of long-ranged spin-glasses: full hierarchy of relaxation times via real-space renormalization

We consider the long-ranged Ising spin-glass with random couplings decaying as a power-law of the distance, in the region of parameters where the spin-glass phase exists with a positive droplet exponent. For the Metropolis single-spin-flip dynamics near zero temperature, we construct via real-space renormalization the full hierarchy of relaxation times of the master equation for any given realization of the random couplings. We then analyze the probability distribution of dynamical barriers as a function of the spatial scale. This real-space renormalization procedure represents a simple explicit example of the droplet scaling theory, where the convergence towards local equilibrium on larger and larger scales is governed by a strong hierarchy of activated dynamical processes, with valleys within valleys.

Drop mass transfer in a microfluidic chip compared to a centrifugal contactor

A model system was developed for enabling a multiscale understanding of centrifugal‐contactor liquid–liquid extraction. The system consisted of Nd(III) + xylenol orange in the aqueous phase buffered to pH = 5.5 by KHP, and dodecane + thenoyltrifluroroacetone (HTTA) + tributyphosphate (TBP) in the organic phase. Diffusion constants were measured for neodymium in both the organic and aqueous phases, and the Nd(III) partition coefficients were measured at various HTTA and TBP concentrations. A microfluidic channel was used as a high‐shear model environment to observe mass transfer on a droplet scale with xylenol orange as the aqueous‐phase metal indicator; mass‐transfer rates were measured quantitatively in both diffusion and reaction limited regimes on the droplet scale. The microfluidic results were comparable to observations made for the same system in a laboratory scale liquid–liquid centrifugal contactor, indicating that single drop microfluidic experiments can provide information on mass transfer in complicated flows and geometries. © 2014 American Institute of Chemical Engineers AIChE J, 60: 3071–3078, 2014

Typical versus averaged overlap distribution in spin glasses: Evidence for droplet scaling theory

We consider the statistical properties over disordered samples of the overlap distribution $P_{\cal J}(q)$ which plays the role of an order parameter in spin-glasses. We show that near zero temperature (i) the {\it typical} overlap distribution is exponentially small in the central region of $-1<q<1$: $ P^{typ}(q) = e^{\bar{\ln P_{\cal J}(q)}} \sim e^{- \beta N^{\theta} \phi(q)} $, where $\theta$ is the droplet exponent defined here with respect to the total number $N$ of spins (in order to consider also fully connected models where the notion of length does not exist); (ii) the rescaled variable $v = - (\ln P_{\cal J}(q))/N^{\theta}$ remains an O(1) random positive variable describing sample-to sample fluctuations; (iii) the averaged distribution $\bar{P_{\cal J}(q)} $ is non-typical and dominated by rare anomalous samples. Similar statements hold for the cumulative overlap distribution $I_{\cal J}(q_0) \equiv \int_{0}^{q_0} dq P_{\cal J}(q) $. These results are derived explicitly for the spherical mean-field model with $\theta=1/3$, $\phi(q)=1-q^2 $, and the random variable $v$ corresponds to the rescaled difference between the two largest eigenvalues of GOE random matrices. Then we compare numerically the typical and averaged overlap distributions for the long-ranged one-dimensional Ising spin-glass with random couplings decaying as $J(r) \propto r^{-\sigma}$ for various values of the exponent $\sigma$, corresponding to various droplet exponents $\theta(\sigma)$, and for the mean-field SK-model (corresponding formally to the $\sigma=0$ limit of the previous model). Our conclusion is that future studies on spin-glasses should measure the {\it typical} values of the overlap distribution or of the cumulative overlap distribution to obtain clearer conclusions on the nature of the spin-glass phase.

Droplet Microfluidics to Prepare Magnetic Polymer Vesicles and to Confine the Heat in Magnetic Hyperthermia

In this work, we present two types of microfluidic chips involving magnetic nanoparticles dispersed in cyclohexane with oleic acid. In the first case, the hydrophobically coated nanoparticles are self-assembled with an amphiphilic diblock copolymer by a double-emulsion process in order to prepare giant magnetic vesicles (polymersomes) in one step and at a high throughput. It was shown in literature that such diblock copolymer W/O/W emulsion droplets can evolve into polymersomes made of a thin (nanometric) magnetic membrane through a dewetting transition of the oil phase from the aqueous internal cores usually leading to “acorn-like” structures (polymer excess) sticking to the membranes. To address this issue and greatly speed up the process, the solvent removal by evaporation was replaced by a “shearing-off” of the vesicles in a simple poly(dimethylsiloxane) chip designed to exert a balance between a magnetic gradient and viscous shear. In the second example, a simple oil-in-oil emulsion chip is used to obtain regular trains of magnetic droplets that circulate inside an inductor coil producing a radio-frequency magnetic field. We evidence that the heat produced by magnetic hyperthermia can be converted into a temperature rise even at the scale of nL droplets. The results are compared to heat transfer models in two limiting cases: adiabatic vs. dissipative. The aim is to decipher the delicate puzzle about the minimum size required for a tumor “phantom” to be heated by radio-frequency hyperthermia in a general scope of anticancer therapy.

Discussion of the Drop Rest Phenomenon at Millimeter Scale and Coalescence of Droplets at Micrometer Scale

Adsorption of surfactants at water-oil interfaces is of great importance in the coalescence of drops and stability of emulsions. In this work, we have studied the adsorption of nonionic surfactants Span 80 at water-oil interfaces and its influence on the drop rest phenomenon and W/O emulsion stability in a pulsed DC electrical field. The variation of interfacial tension with the concentration of surfactant was studied and the data were fitted using a surface equation of state derived from the Langmuir adsorption isotherm. A stochastic model for coalescence was used to fit the coalescence time distributions. The significance of the model parameters was discussed. The stability of the emulsion was evaluated by conductivity methods. The researches in this article indicated that both of the rest time distribution of the drops at the interface and stability of the emulsion in the electrical field was significantly affected by surfactant concentration.

Droplet formation and scaling in dense suspensions

When a dense suspension is squeezed from a nozzle, droplet detachment can occur similar to that of pure liquids. While in pure liquids the process of droplet detachment is well characterized through self-similar profiles and known scaling laws, we show here the simple presence of particles causes suspensions to break up in a new fashion. Using high-speed imaging, we find that detachment of a suspension drop is described by a power law; specifically we find the neck minimum radius, r(m), scales like near breakup at time τ = 0. We demonstrate data collapse in a variety of particle/liquid combinations, packing fractions, solvent viscosities, and initial conditions. We argue that this scaling is a consequence of particles deforming the neck surface, thereby creating a pressure that is balanced by inertia, and show how it emerges from topological constraints that relate particle configurations with macroscopic Gaussian curvature. This new type of scaling, uniquely enforced by geometry and regulated by the particles, displays memory of its initial conditions, fails to be self-similar, and has implications for the pressure given at generic suspension interfaces.

Droplet growth in warm turbulent clouds

AbstractIn this survey we consider the impact of turbulence on cloud formation from the cloud scale to the droplet scale. We assess progress in understanding the effect of turbulence on the condensational and collisional growth of droplets and the effect of entrainment and mixing on the droplet spectrum. The increasing power of computers and better experimental and observational techniques allow for a much more detailed study of these processes than was hitherto possible. However, much of the research necessarily remains idealized and we argue that it is those studies which include such fundamental characteristics of clouds as droplet sedimentation and latent heating that are most relevant to clouds. Nevertheless, the large body of research over the last decade is beginning to allow tentative conclusions to be made. For example, it is unlikely that small‐scale turbulent eddies (i.e. not the energy‐containing eddies) alone are responsible for broadening the droplet size spectrum during the initial stage of droplet growth due to condensation. It is likely, though, that small‐scale turbulence plays a significant role in the growth of droplets through collisions and coalescence. Moreover, it has been possible through detailed numerical simulations to assess the relative importance of different processes to the turbulent collision kernel and how this varies in the parameter space that is important to clouds. The focus of research on the role of turbulence in condensational and collisional growth has tended to ignore the effect of entrainment and mixing and it is arguable that they play at least as important a role in the evolution of the droplet spectrum. We consider the role of turbulence in the mixing of dry and cloudy air, methods of quantifying this mixing and the effect that it has on the droplet spectrum. Copyright © 2012 Royal Meteorological Society and British Crown Copyright, the Met Office

Photon Correlation Spectroscopy and SAXS Study of Mixture of NaCl with AOT Microemulsion at X = 6.7

Photon Correlation Spectroscopy is used to study the AOT microemulsion with and without NaCl. Collective diffusion coefficient was investigated by Photon Correlation Spectroscopy technique. We have studied effect of charge on dynamic of water-in-oil microemulsion (nano-droplet of water to the oil), which stabilized by AOT and dispersed in n-Decane at water/AOT with 6.7 molar ratio. The small angle X-ray scattering technique and hard sphere model were used to study the structural information of AOT microemulsion with and without NaCl. The structural investigation of samples shows a decrease of length scale of cylindrical droplets with increasing of NaCl concentration in AOT microemulsion.

Remote sensing the vertical profile of cloud droplet effective radius, thermodynamic phase, and temperature

Abstract. Cloud-aerosol interaction is a key issue in the climate system, affecting the water cycle, the weather, and the total energy balance including the spatial and temporal distribution of latent heat release. Information on the vertical distribution of cloud droplet microphysics and thermodynamic phase as a function of temperature or height, can be correlated with details of the aerosol field to provide insight on how these particles are affecting cloud properties and their consequences to cloud lifetime, precipitation, water cycle, and general energy balance. Unfortunately, today's experimental methods still lack the observational tools that can characterize the true evolution of the cloud microphysical, spatial and temporal structure in the cloud droplet scale, and then link these characteristics to environmental factors and properties of the cloud condensation nuclei. Here we propose and demonstrate a new experimental approach (the cloud scanner instrument) that provides the microphysical information missed in current experiments and remote sensing options. Cloud scanner measurements can be performed from aircraft, ground, or satellite by scanning the side of the clouds from the base to the top, providing us with the unique opportunity of obtaining snapshots of the cloud droplet microphysical and thermodynamic states as a function of height and brightness temperature in clouds at several development stages. The brightness temperature profile of the cloud side can be directly associated with the thermodynamic phase of the droplets to provide information on the glaciation temperature as a function of different ambient conditions, aerosol concentration, and type. An aircraft prototype of the cloud scanner was built and flew in a field campaign in Brazil. The CLAIM-3D (3-Dimensional Cloud Aerosol Interaction Mission) satellite concept proposed here combines several techniques to simultaneously measure the vertical profile of cloud microphysics, thermodynamic phase, brightness temperature, and aerosol amount and type in the neighborhood of the clouds. The wide wavelength range, and the use of multi-angle polarization measurements proposed for this mission allow us to estimate the availability and characteristics of aerosol particles acting as cloud condensation nuclei, and their effects on the cloud microphysical structure. These results can provide unprecedented details on the response of cloud droplet microphysics to natural and anthropogenic aerosols in the size scale where the interaction really happens.

Zero- and Low-Temperature Behavior of the Two-Dimensional±JIsing Spin Glass

Recently extended precise numerical methods and droplet scaling arguments allow for a coherent picture of the glassy states of two-dimensional Ising spin glasses to be assembled. The length scale at which entropy becomes important and produces "chaos", the extreme sensitivity of the state to temperature, is found to depend on the type of randomness. For the $\pm J$ model this length scale dominates the low-temperature specific heat. Although there is a type of universality, some critical exponents do depend on the distribution of disorder.

Random field Ising model: statistical properties of low-energy excitations and equilibriumavalanches

With respect to the usual thermal ferromagnetic transitions, thezero-temperature finite-disorder critical point of the random field Isingmodel (RFIM) has the peculiarity of involving some ‘droplet’ exponentθ that enters the generalized hyperscaling relation2 − α = ν(d − θ). In the present paper, to better understand the meaning of this droplet exponentθ beyond its role in the thermodynamics, we discuss the statistics of low-energyexcitations generated by an imposed single spin-flip with respect to the groundstate, as well as the statistics of equilibrium avalanches, i.e. the magnetizationjumps that occur in the sequence of ground states as a function of the externalmagnetic field. The droplet scaling theory predicts that the distributiondl/l1 + θ of thelinear size l of low-energy excitations transforms into the distributionds/s1 + θ/df for thesize s (number of spins) of excitations of fractal dimensiondf (s ∼ ldf). In the non-mean-field regiond < dc, droplets are compactdf = d, whereas in the mean-fieldregion d > dc, droplets havea fractal dimension df = 2θ leading to the well-known mean-field resultds/s3/2. Zero-field equilibrium avalanches are expected to display the same distributionds/s1 + θ/df. We also discuss the statistics of equilibrium avalanches integrated over the externalfield and finite-size behaviors. These expectations are checked numericallyfor the Dyson hierarchical version of the RFIM, where the droplet exponentθ(σ) can be varied as a function of the effective long-range interactionJ(r) ∼ 1/rd + σ ind = 1.

Gas-phase saturation and evaporative cooling effects during wet compression of a fuel aerosol under RCM conditions

Wet compression of a fuel aerosol has been proposed as a means of creating gas-phase mixtures of involatile diesel-representative fuels and oxidizer + diluent gases for rapid compression machine (RCM) experiments. The use of high concentration aerosols (e.g., ∼0.1 mL fuel/L gas, ∼1 × 10 9 droplets/L gas for stoichiometric fuel loading at ambient conditions) can result in droplet–droplet interactions which lead to significant gas-phase fuel saturation and evaporative cooling during the volumetric compression process. In addition, localized stratification (i.e., on the droplet scale) of the fuel vapor and of temperature can lead to non-homogeneous reaction and heat release processes – features which could prevent adequate segregation of the underlying chemical kinetic rates from rates of physical transport. These characteristics are dependent on many factors including physical parameters such as overall fuel loading and initial droplet size relative to the compression rate, as well as fuel and diluent properties such as the boiling curve, vaporization enthalpy, heat capacity, and mass and thermal diffusivities. This study investigates the physical issues, especially fuel saturation and evaporative cooling effects, using a spherically-symmetric, single-droplet wet compression model. n-Dodecane is used as the fuel with the gas containing 21% O 2 and 79% N 2. An overall compression time and compression ratio of 15.3 ms and 13.4 are used, respectively. It is found that smaller droplets ( d 0 ∼ 2–3 μm) are more affected by ‘far-field’ saturation and cooling effects, while larger droplets ( d 0 ∼ 14 μm) result in greater localized stratification of the gas-phase due to the larger diffusion distances for heat and mass transport. Vaporization of larger droplets is more affected by the volumetric compression process since evaporation requires more time to be completed even at the same overall fuel loading. All of the cases explored here yield greater compositional stratification than thermal stratification due to the high Lewis numbers of the fuel–air mixtures ( Le g ∼ 3.8).

Simulation of liquid jet primary breakup: Dynamics of ligament and droplet formation

Primary atomization of liquid injected at high speed into still air is investigated to elucidate physical processes by direct numerical simulation. With sufficient grid resolution, ligament and droplet formation can be captured in a physically sound way. Ligament formation is triggered by the liquid jet tip roll-up, and later ligaments are also produced from the disturbed liquid core surface in the upstream. Ligament production direction is affected by gas vortices. Disturbances are fed from the liquid jet tip toward upstream through vortices and droplet re-collision. When the local gas Weber number is O(1), ligaments are created, thus the ligament or droplet scale becomes smaller as the bulk Weber number increases. Observation of droplet formation from a ligament provides insights into the relevance between the actual droplet formation and pinch-off from a slow liquid jet in laboratory experiments. In the spray, the dominant mode is the short-wave mode driven by propagative capillary wave from the ligament tip. An injection nozzle that is necessary for a slow jet is absent for a ligament, thus the long-wave (Rayleigh) mode is basically not seen without the effect of stretch. By the present simulation, a series of physical processes have been revealed. The present result will be extended to LES modeling in the future.

The application of modeling techniques to film-coating processes

Background: The modeling of the processes for film coating of solid dosage forms can aid in the prediction and troubleshooting of coating operations. Aim: A review of the existing approaches to modeling the coating phenomena was undertaken with the aim of identifying key assumptions and limitations of each approach. Method: Models are categorized into macrolevel and microlevel approaches. Macrolevel models can predict mass coating uniformity for a batch of solids. On the other hand, microlevel models consider phenomena at the scale of the liquid droplets sprayed into the process. Macrolevel models consist either of phenomenological approaches, whereby the inherent variabilities in the coating process are described by probability distributions. Alternatively, first-principle models based on rigorous descriptions of solid and gas-phase motion can be used to describe the multiphase flow behavior in coating equipment. Result: The advantages of the rigorous macrolevel approach are that extrapolation of results for different operating conditions is more reliable than the phenomenological approach and much more information is available about the flow of both phases. However, these first-principle models are computationally intensive and cannot be used for real-time predictions. On the other hand, microlevel models describe chemical and physical phenomena occurring when droplets of coating solution impact the surfaces of the solid cores and are useful in determining the cause of morphological variations in coating properties. Conclusions: Both micro- and macro-level approaches can be used to predict and troubleshoot problems in coating processes. The use of such models in the control of coating operations is not widespread but offers significant potential to improve quality control.

Coexistence and transition between Cassie and Wenzel state on pillared hydrophobic surface

Water droplets on rugged hydrophobic surfaces typically exhibit one of the following two states: (i) the Wenzel state [Wenzel RN (1936) Ind Eng Chem 28:988-994] in which water droplets are in full contact with the rugged surface (referred as the wetted contact) or (ii) the Cassie state [Cassie, ABD, Baxter S (1944) Trans Faraday Soc 40:546-551] in which water droplets are in contact with peaks of the rugged surface as well as the "air pockets" trapped between surface grooves (the composite contact). Here, we show large-scale molecular dynamics simulation of transition between Wenzel state and Cassie state of water droplets on a periodic nanopillared hydrophobic surface. Physical conditions that can strongly affect the transition include the height of nanopillars, the spacing between pillars, the intrinsic contact angle, and the impinging velocity of water nanodroplet ("raining" simulation). There exists a critical pillar height beyond which water droplets on the pillared surface can be either in the Wenzel state or in the Cassie state, depending on their initial location. The free-energy barrier separating the Wenzel and Cassie state was computed on the basis of a statistical-mechanics method and kinetic raining simulation. The barrier ranges from a few tenths of k(B)T(0) (where k(B) is the Boltzmann constant, and T(0) is the ambient temperature) for a rugged surface at the critical pillar height to approximately 8 k(B)T(0) for the surface with pillar height greater than the length scale of water droplets. For a highly rugged surface, the barrier from the Wenzel-to-Cassie state is much higher than from Cassie-to-Wenzel state. Hence, once a droplet is trapped deeply inside the grooves, it would be much harder to relocate on top of high pillars.

Equilibrium of disordered systems: constructing the appropriate valleys in each sample via strong-disorder renormalization in configuration space

To describe the equilibrium properties of disordered systems and the possible emergence of various ‘phases’ at low temperature, we adopt here the ‘broken-ergodicity’ point of view advocated in particular by Palmer (1982 Adv. Phys. 31 669): the aim is then to construct the valleys of configurations that become separated by diverging barriers and to study their relative weights, as well as their internal properties. To characterize the slow non-equilibrium dynamics of disordered systems, we have recently introduced (Monthus and Garel 2008 J. Phys. A 41 255002, Preprint arXiv:0804.1847)) a strong-disorder renormalization (RG) procedure in configuration space, based on the iterative elimination of the smallest barrier remaining in the system. In the present paper, we show how this renormalization procedure allows us to construct the longest-lived valleys in each disordered sample, and to obtain their free-energies, energies and entropies. This explicit RG formulation is very general since it can be defined for any master equation, and it gives new insights into the main ingredients of the droplet scaling picture. As an application, we have followed numerically the RG flow for the case of a directed polymer in a two-dimensional random medium to obtain histograms of the free-energy, entropy and energy differences between the two longest-lived valleys in each sample.

Non-equilibrium dynamics of disordered systems: renormalization flow towards an ‘infinitedisorder’ fixed point at large times

To describe the non-equilibrium dynamics of random systems, we have recently introduced(Monthus and Garel, 2008 J. Phys. A: Math. Theor. 41 255002) a ‘strong disorderrenormalization’ (RG) procedure in configuration space that can be defined for any masterequation. In the present paper, we analyze in detail the properties of the large timedynamics whenever the RG flow is towards some ‘infinite disorder’ fixed point, where thewidth of the renormalized barrier distribution grows indefinitely upon iteration. Inparticular, we show how the strong disorder RG rules can be then simplified while keepingtheir asymptotic exactness, because the preferred exit channel out of a given renormalizedvalley typically dominates asymptotically over the other exit channels. We explain why thepresent approach is an explicit construction favoring the droplet scaling picturewhere the dynamics is governed by the logarithmic growth of the coherence lengthl(t)∼(lnt)1/ψ, and where the statistics of barriers corresponds to a very strong hierarchy of valleyswithin valleys. As an example of application, we have followed numerically the RG flow forthe case of a directed polymer in a two-dimensional random medium. The full RG rules areused to check that the RG flow is towards some infinite disorder fixed point, whereas thesimplified RG rules allow us to study bigger sizes and to estimate the barrier exponentψ of the fixed point.

Observations of initiation stage of spontaneous vapor explosions for droplet scale

AbstractIn this study, the initiation stage of spontaneous vapor explosions generated by single droplets of molten tin submerged in water was investigated using a high‐ speed video camera operated with a reflected light system. Photographs of the formation process of vapor film, the process of vapor film disturbance, and the initiation process of the vapor explosions for different masses of molten tin and different nozzle diameters were obtained. The results demonstrate that partial thermal interaction between tin and water does not cause a vapor explosion with fragmentation. The vapor film disappears locally during the formation of the vapor film around the hot liquid droplet. Direct contact between the hot molten tin surface and water is thereby generated. However, the local disappearance of the vapor film does not progress and the vapor film is reconstructed. A vapor explosion occurs when the vapor film collapses at the local area of the bottom or edge of the disk‐shaped droplet. © 2007 Wiley Periodicals, Inc. Heat Trans Asian Res, 37(1): 41–55, 2008; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/htj.20185

Production of Narrow‐Size‐Distribution Polymer‐Pigment‐Nanoparticle Composites via Electrohydrodynamic Atomization

AbstractNear‐monodisperse, size‐controllable, poly(methyl methacrylate)‐pigment nanoparticle composites were produced using electrohydrodynamic atomization (EHDA). The geometric mean diameters of the composite particles were in the 0.91 to 1.90 µm‐diameter range with geometric standard deviations of approximately 1.05 to 1.12. Increasing the polymer volume fraction and liquid flow‐rate resulted in an increase in the diameter of the composite particles, which agreed well with droplet scaling relations for EHDA. The results here demonstrate that EHDA can be used for polymer‐nanoparticle‐composite production and as an alternative to conventional inkjet printing.magnified image

Multi-scale simulation of oblique collisions of a droplet on a surface in the Leidenfrost regime

This work illustrates the phenomena of oblique impact of a water droplet on a hot solid surface in the Leidenfrost regime using a multi-scale model. The flow and heat transfer behavior on the droplet scale as described by the macroscopic model is solved using the finite volume method, together with the level set and the immersed boundary methods which quantify the variations of the gas–liquid and fluid–solid interfaces. A micro-scale vapor layer model is used to account for the resistance effect of the vapor layer generated by the film-boiling evaporation. These two models are coupled in each time step, and solved concurrently. Based on this multi-scale model, the effect of the impact velocity is investigated numerically. The droplet shape and impact parameters such as momentum loss and contact time calculated from the present model are compared with the experimental results obtained from the literature. A good agreement is seen between the simulated and the experimental results.