Nucleation and growth of tightly packed droplets in fluids

Perspectives on a Few Puzzles in Phase Transformations: When Should the Farthest Reach the Earliest?

We briefly review the facts concerning two important aspects of phase transitions, namely, critical and coarsening phenomena. A discussion of the universal features, highlighting the current challenges, is provided. Following this, we elaborate on a topic of much recent interest, viz., the Mpemba effect, a puzzle that found mention even in the works of Aristotle. After a description of the debated case of faster freezing of a hotter sample of liquid water, into ice, than a colder one, when quenched to the same subzero temperature, we discuss more modern interest. There one asks, should a hotter body of a material equilibrate faster than a colder one when quenched to a common lower temperature? Within this broad scenario, we focus on magnetic systems. A surprising observation of the effect during the para- to ferromagnetic transition, in a simple model system, viz., the nearest-neighbor Ising model, without any built-in frustration, is described. Some associated future directions are pointed out. A discussion is provided by considering the effect as a kinetic outcome in the background of critical phenomena. A picture is drawn by putting emphasis on the role of spatial correlations in the initial configurations alongside discussing the importance of frustration and metastability in evolution from one state to another. In connection with dynamical freezing, concerning metastability, we have introduced the complex Ginzburg-Landau equation that has relevance in phase transitions, chemical oscillations, and elsewhere. For this model and a few other cases also, we have described how a lack of order or correlation in certain parameters can lead to quicker evolution.

Measuring the Transition Rates of Coalescence Events during Double Phase Separation in Microgravity

Phase transition is a ubiquitous phenomenon in nature, science and technology. In general, the phase separation from a homogeneous phase depends on the depth of the temperature quench into the two-phase region. Earth’s gravity masks the details of phase separation phenomena, which is why experiments were performed under weightlessness. Under such conditions, the pure fluid sulphur hexafluoride (SF6) near its critical point also benefits from the universality of phase separation behavior and critical slowing down of dynamics. Initially, the fluid was slightly below its critical temperature with the liquid matrix separated from the vapor phase. A 0.2 mK temperature quench further cooled down the fluid and produced a double phase separation with liquid droplets inside the vapor phase and vapor bubbles inside the liquid matrix, respectively. The liquid droplets and the vapor bubbles respective distributions were well fitted by a lognormal function. The evolution of discrete bins of different radii allowed the derivation of the transition rates for coalescence processes. Based on the largest transition rates, two main coalescence mechanisms were identified: (1) asymmetric coalescences between one small droplet of about 20 μm and a wide range of larger droplets; and (2) symmetric coalescences between droplets of large and similar radii. Both mechanisms lead to a continuous decline of the fraction of small radii droplets and an increase in the fraction of the large radii droplets. Similar coalescence mechanisms were observed for vapor bubbles. However, the mean radii of liquid droplets exhibits a t1/3 evolution, whereas the mean radii of the vapor bubbles exhibit a t1/2 evolution.

Pattern Evolution during Double Liquid-Vapor Phase Transitions under Weightlessness

Phase transition in fluids is ubiquitous in nature and has important applications in areas such as the food industry for volatile oils' extraction or in nuclear plants for heat transfer. Fundamentals are hampered by gravity effects on Earth. We used direct imaging to record snapshots of phase separation that takes place in sulfur hexafluoride, SF₆, under weightlessness conditions on the International Space Station (ISS). The system was already at liquid-vapor equilibrium slightly below the critical temperature and further cooled down by a 0.2-mK temperature quench that produced a new phase separation. Both full view and microscopic views of the direct observation cell were analyzed to determine the evolution of the radii distributions. We found that radii distributions could be well approximated by a lognormal function. The fraction of small radii droplets declined while the fraction of large radii droplets increased over time. Phase separation at the center of the sample cell was visualized using a 12× microscope objective, which corresponds to a depth of focus of about 5 μ m. We found that the mean radii of liquid droplets exhibit a t 1 / 3 evolution, in agreement with growth driven by Brownian coalescence. It was also found that the mean radii of the vapor bubbles inside the liquid majority phase exhibit a t 1 / 2 evolution, which suggest a possible directional motion of vapor bubbles due to the influence of weak remaining gravitational field and/or a composition Marangoni force.

Kinetics of Vapor-Solid Phase Transitions: Structure, Growth, and Mechanism

The kinetics of the separation between low and high density phases in a single component Lennard-Jones model is studied via molecular dynamics simulations, at very low temperatures, in the space dimension d=2. For densities close to the vapor branch of the coexistence curve, disconnected nanoscale clusters of the high density phase exhibit essentially ballistic motion. Starting from nearly circular shapes, at the time of nucleation, these clusters grow via sticky collisions, gaining filamentlike nonequilibrium structure at a later time, with a very low fractal dimensionality. The origin of the latter is shown to lie in the low mobility of the constituent particles, in the corresponding cluster reference frame, due to the (quasi-long-range) crystalline order. Standard self-similarity in the domain pattern, typically observed in the kinetics of phase transitions, is found to be absent. This invalidates the common method, that provides a growth law comparable to that in solid mixtures, of quantifying growth. An appropriate alternative approach, involving the fractality, quantifies the growth of the characteristic "length" to be a power law with time, the exponent being strongly temperature dependent. The observed growth law is in agreement with the outcome of a nonequilibrium kinetic theory.

Droplet growth during vapor-liquid transition in a 2D Lennard-Jones fluid

Results for the kinetics of vapor-liquid phase transition have been presented from the molecular dynamics simulations of a single component two-dimensional Lennard-Jones fluid. The phase diagram for the model, primary prerequisite for this purpose, has been obtained via the Monte Carlo simulations. Our focus is on the region very close to the vapor branch of the coexistence curve. Quenches to such region provide morphology that consists of disconnected circular clusters in the vapor background. We identified that these clusters exhibit diffusive motion and grow via sticky collisions among them. The growth follows power-law behavior with time, exponent of which is found to be in nice agreement with a theoretical prediction.

Ripening kinetics of bubbles: A molecular dynamics study

The ripening kinetics of bubbles is studied by performing molecular dynamics simulations. From the time evolution of a system, the growth rates of individual bubbles are determined. At low temperatures, the system exhibits a t^1/2 law and the growth rate is well described by classical Lifshitz-Slyozov-Wagner (LSW) theory for the reaction-limited case. This is direct evidence that the bubble coarsening at low temperatures is reaction-limited. At high temperatures, although the system exhibits a t^1/3 law, which suggests that it is diffusion-limited, the accuracy of the growth rate is insufficient to determine whether the form is consistent with the prediction of LSW theory for the diffusion-limited case. The gas volume fraction dependence of the coarsening behavior is also studied. Although the behavior of the system at low temperatures has little sensitivity to the gas volume fraction up to 10%, at high temperatures it deviates from the prediction of LSW theory for the diffusion-limited case as the gas volume fraction increases. These results show that the mean-field-like treatment is valid for a reaction-limited system even with a finite volume fraction, while it becomes inappropriate for a diffusion-limited system since classical LSW theory for the diffusion-limited case is valid at the dilute limit.

Percolation of Phase-Separating Polymer Mixtures

We study the percolation problem in a binary phase-separating polymer mixture. By well-designed experiments, we can delineate the percolation line on the phase diagram with sufficient accuracy. Our experiments show that the percolation thresholds start from the random percolation limit (Φ ∼ 0.15) located near spinodal point at T → T_c and then converge toward the geometric coalescence limit (Φ ∼ 0.36) with an increase in the quench depth. This apparent percolation difficulty comes about largely from the Rayleigh instability accompanied by large-amplitude, short-wavelength fluctuations during the spinodal decomposition at deeper quench depth. As a result, the broken "rigid" domains tend to pack closely, and the so-called droplet spinodal decomposition occurs. On the other hand, we observe that, between the selectively attractive walls, the surface-drying percolating phase will break up into droplets prematurely, thereby shifting its percolation line rather considerably. To our knowledge, such an effect is not yet predicted by theory or simulation.

Imaging critical fluctuations of pure fluids and binary mixtures

We use optical microscopy techniques to directly visualize the structures that emerge in binary mixtures and pure fluids near their respective critical points. We attempt to understand these structures by studying the image formation using both a phase contrast and a dark field filter to our microscope. We found that images of critical fluctuations for both liquid-liquid and liquid-gas critical systems have gray level intensity histograms with Gaussian shape. For all fluids investigated, the temperature-dependent standard deviation of the Gaussian histogram follows a power law with the same exponent. Since the image intensity fluctuations are determined by order parameter fluctuations, this direct imaging method allowed us to estimate the critical exponent of compressibility with very good accuracy.

Nucleation and growth of droplets in vapor-liquid transitions

Results for the kinetics of vapor-liquid transitions, following temperature quenches with different densities, are presented from molecular dynamics simulations of a Lennard-Jones system. For a critical density, bicontinuous liquid and vapor domains are observed which grow with time, obeying the predictions for the hydrodynamic mechanism. On the other hand, for quenches with density significantly below the critical one, phase separation progresses via nucleation and growth of liquid droplets. In the latter case, the Brownian diffusion and collision mechanism for the droplet growth is confirmed. We also discuss the possibility of interdroplet interaction leading to a different amplitude in the growth law. Arguments for faster growth, observed at early times, are also provided.

Dynamics of a wetting layer and Marangoni convection in microgravity

Near the liquid-vapor critical point in pure fluids, material and thermal properties vary considerably with temperature. In a series of microgravity experiments, sulfur hexafluoride (SF6) was heated ∼1 K above its critical temperature, then quenched below the critical temperature in order to form gas and liquid domains. We found a power law exponent of 0.389 ± 0.010 for the growth of the wetting layer thickness during the intermediate stage of phase separation. Full and microscopic view images of the sample cell unit were analyzed to determine the changes in the size distribution of liquid droplets inside the gas phase over time. We found that the distribution of diameters for liquid droplets always contains a fraction of very small droplets, presumably due to a continuous nucleation process. At the same time, the size distribution flattens over time and rapidly includes large-size droplets, presumably generated through a coalescence mechanism. By following both a large gas bubble over two hours of video recordings, we found periodic and synchronous motion of the gas bubble along both the x and y directions. By following a large liquid droplet embedded into the large gas bubble, we found periodic, out of phase motions, which we related to Marangoni convection. The experimentally measured velocity of the liquid droplet is in good agreement with the theoretical predicted velocity of ∼0.386 μm/s obtained from Young's thermocapillary effect.

Universality in early-stage growth of phase-separating domains near the critical point

We present both the experimental and computational methods and results of phase-separating experiments performed with sulfur hexafluoride (SF6) close to its critical density. These experiments were performed in microgravity to suppress buoyancy and convection-driven effects. Phase separation under reduced gravity is analyzed for both 0.3 mK and 3.6 mK temperature quenches in order to derive the early-stage growth law. We found a 1/3 growth law for early stages of phase separation for a volume fraction of minority domains of 50%. Our findings support the hypothesis of a crossover between Brownian motion and hydrodynamic effects in the early stages of phase separation. The temperature inside the bulk of the pure fluid was estimated using a proposed histogram method. Our histogram method allowed temperature estimation below thermistors' sensitivity and detected small temperature variations inside the bulk of the pure fluid.

Universal behavior of photochemical deposition in liquid solutions driven by a one-photon transition

Even if photochemical deposition of nearly all types of materials has been used for decades to pattern almost any kind of substrate for various applications (catalysis, chemical sensing, magnetic data storage, optoelectronics, spin-dependent electron transport, and solar cells), a rationalized description is still missing. This paper aims at fulfilling this lack by presenting a unified approach of the photodeposit growth initiated by a one-photon photochemical reaction. We experimentally investigate the robustness of growth scalings predicted for photochemical deposition driven by a continuous laser wave. Three types of one-photon photochemical reactions (photoexcitation of chromates, photodissociation of permanganates, and photocondensation of colloidal selenium) and three parameters (solvent p H variations, concentration in photoactive reagent, and influence of the exciting optical wavelength) were cross analyzed. In all the cases, including data taken from the literature, the same dynamic master behavior emerges from the data rescaling of measured deposit growth laws. The nice agreement observed between system-independent predictions and the whole data set strongly supports a universal description of the photodeposit growth whatever the photosensitive medium and the involved one-photon chemical reaction. Such an approach also points out the quantitative sorting of photochemical reactions in terms of deposition efficiency. This rationalization of the kinetics of photodeposition anticipates new methodologies to predict, design, and control substrate micropatterning for chemical, lithographic, and optoelectronic applications.

Experimentally determined growth exponents during the late stage of spinodal demixing in binary liquid mixtures

Spinodal demixing was initiated in two systems, with critical and off-critical compositions, using nanosecond pulsed laser-induced temperature jumps (T-jumps) of various magnitude. In this way, deep quenches could be imposed on the systems. One system was the simple triethylamine (TEA)/water mixture and the other was the ionic mixture of 2-butoxyethanol (2BE)/water/KCl. The demixing process was followed using the technique of nanosecond time-resolved microscopic shadowgraphy. The growth of the evolving phase-separated domains followed a simple power law with respect to time in every case. For a given composition, the magnitude of the T-jump had little effect on the growth exponent, however the composition was found to influence the rate of domain growth. At off-critical mole fractions of 0.2 with respect to TEA, the domains grew according to the following expression: L(t)=t(0.70) (where L(t)= the domain size) whereas at the critical TEA mole fraction of 0.08 the domains grew as L(t)=t(0.52). 2BE/water/KCl mixtures quenched at the just off-critical composition of fraction with respect to 2BE evolved as L(t)=t(0.63). These results will be compared to theoretical models and simulations and discussed in terms of estimated Reynolds numbers as well as the consumption and conversion of the available surface energy that fuels the demixing process.

Growth of monodisperse mesoscopic metal-oxide colloids under constant monomer supply

In closed systems, control over the size of monodisperse metal-oxide colloids is generally limited to submicrometric dimensions. To overcome this difficulty, we explore the formation and growth of silica particles under constant monomer supply. The monomer source is externally driven by the progressive addition into the system of one of the precursors. Monodisperse spherical particles are produced up to a mesoscopic size. We analyze their growth versus the monomer addition rate at different temperatures. Our results show that in the presence of a continuous monomer addition, growth is limited by diffusion over the investigated temporal window. Using the temperature variation of the growth rate, we prove that rescaling leads to a data reduction onto a single master curve. Contrary to the growth process, the final particle's size reached after the end of the reagent supply strongly depends on the addition rate. The variation of the final particle size versus addition rate can be deduced from an analogy with crystal formation in jet precipitation. Within this framework, and using the temperature dependences of both the particle growth law and the final size, we determine the value of the molecular heat of dissolution associated to the silica solubility. These observations support the fact that classical theories of phase-ordering dynamics can be extended to the synthesis of inorganic particles. The emergence of a master behavior in the presence of continuous monomer addition also suggests the extension of these theories to open systems.

Smart control of monodisperse Stöber silica particles: effect of reactant addition rate on growth process

Control over the synthesis of monodisperse silica particles up to mesoscopic scale is generally made difficult due to intrinsic limitation to submicrometric dimensions and secondary nucleation in seeded experiments. To investigate this issue and overcome these difficulties, we have implemented single step processing by quantifying the effects of the progressive addition of a diluted tetraethyl orthosilicate solution in ethanol on the size and monodispersity of silica particles. Contrary to particles grown in seeded polymerization, monodisperse particles with size up to 2 microm were synthesized. Moreover, the particles exhibit a final diameter (d(f)), which varies with V(-1/3) over more than 2 orders of magnitude in rate of addition (V). On the basis of a kinetic study in the presence of addition showing that particle growth is limited by the diffusion of monomer species, we developed a diffusion-limited growth model to theoretically explain the observed d(f)(V) behavior and quantitatively retrieve the measured amplitude and exponent. Using a single parameter procedure, we can therefore predict and generate in the room temperature range, monodisperse particles of a targeted size by simply adjusting the rate of addition.

Coarsening mechanism of phase separation caused by a double temperature quench in an off-symmetric binary mixture

We study phase-separation behavior of an off-symmetric fluid mixture induced by a "double temperature quench." We first quench a system into the unstable region. After a large phase-separated structure is formed, we again quench the system more deeply and follow the pattern-evolution process. The second quench makes the domains formed by the first quench unstable and leads to double phase separation; that is, small droplets are formed inside the large domains created by the first quench. The complex coarsening behavior of this hierarchic structure having two characteristic length scales is studied in detail by using the digital image analysis. We find three distinct time regimes in the time evolution of the structure factor of the system. In the first regime, small droplets coarsen with time inside large domains. There a large domain containing small droplets in it can be regarded as an isolated system. Later, however, the coarsening of small droplets stops when they start to interact via diffusion with the large domain containing them. Finally, small droplets disappear due to the Lifshitz-Slyozov mechanism. Thus the observed behavior can be explained by the crossover of the nature of a large domain from the isolated to the open system; this is a direct consequence of the existence of the two characteristic length scales.

Three-dimensional lattice-Boltzmann simulations of critical spinodal decomposition in binary immiscible fluids

We use a modified Shan-Chen, noiseless lattice-BGK model for binary immiscible, incompressible, athermal fluids in three dimensions to simulate the coarsening of domains following a deep quench below the spinodal point from a symmetric and homogeneous mixture into a two-phase configuration. The model is derivable from a continuous-time Boltzmann-BGK equation in the presence of an intercomponent body force. We find the average domain size grows with time as t(gamma), where gamma increases in the range 0.545+/-0.014<gamma<0.717+/-0.002, consistent with a crossover between diffusive t(1/3) and hydrodynamic viscous, t(1.0), behavior. We find good collapse onto a single scaling function, yet the domain growth exponents differ from previous results for similar values of the unique characteristic length L0 and time T0 that can be constructed out of the fluid's parameters. This rebuts claims of universality for the dynamical scaling hypothesis. For Re=2.7 and small wave numbers q we also find a q(2)<-->q(4) crossover in the scaled structure function, which disappears when the dynamical scaling reasonably improves at later stages (Re=37). This excludes noise as the cause for a q(2) behavior, as analytically derived from Yeung and proposed by Appert et al. and Love et al. on the basis of their lattice-gas simulations. We also observe exponential temporal growth of the structure function during the initial stages of the dynamics and for wave numbers less than a threshold value, in accordance with the diffusive Cahn-Hilliard Model B. However, this exponential growth is also present in regimes proscribed by that model. There is no evidence that regions of parameter space for which the scheme is numerically stable become unstable as the simulations proceed, in agreement with finite-difference relaxational models and in contradistinction with an unconditionally unstable lattice-BGK free-energy model previously reported. Those numerical instabilities that do arise in this model are the result of large intercomponent forces which turn the equilibrium distribution negative.

Scale invariance in coarsening of binary and ternary fluids

Phase separation in binary and ternary fluids is studied using a two-dimensional lattice gas automata. The lengths given by the the first zero crossing point of the correlation function and the total interface length is shown to exhibit power law dependence on time. In binary mixtures, our data clearly indicate the existence of a regime having more than one length scale, where the coarsening process proceeds through the rupture and reassociation of domains. In ternary fluids; in the case of symmetric mixtures there exists a regime with a single length scale having dynamic exponent 1/2, while in asymmetric mixtures our data establish the break down of scale invariance.

Critical properties and phase separation in lattice Boltzmann fluid mixtures

Basic equilibrium properties of lattice Boltzmann (LB) fluid mixtures (coexistence curve, surface tension, interfacial profile, correlation length) are calculated to characterize the critical phenomena occurring in these model liquids and to establish a reduced variable description allowing a comparison with real fluid mixtures. We observe mean-field critical exponents and amplitudes so that the LB model may be useful for modeling high molecular weight polymer blends and other fluid mixtures approximated over a wide temperature range by mean-field theory. We also briefly consider phase separation under quiescent and shearing conditions and point out the strong influence of interacting boundaries on the qualitative form of the late-stage phase-separation morphology.

Late stage kinetics of a phase separation induced by a cw laser wave in binary liquid mixtures

cw laser waves can be used to locally induce phase transitions. We investigate experimentally and theoretically a laser-driven liquid-liquid phase separation in a micellar phase of microemulsion and analyze its late stage kinetics. The medium is optically quenched in composition in the metastable region of the phase diagram. Two processes can lead to these concentration variations: electrostriction and thermodiffusion. The first originates from induced dipolar couplings in a field gradient. The second corresponds to a variation in concentration driven by a small thermal gradient. Since the nucleated droplets are optically trapped by the beam, we show that it becomes possible to experimentally analyze an academic situation, i.e., the diffusion-driven growth of a single droplet in compensated gravity. The late stage of this kinetics can be divided into two parts, a bulk behavior and a regime controlled by the finite transverse size of the beam. The bulk regime is totally analogous to that observed in classical situations (R proportional, variant t(1/3), where R is the droplet radius and t the time), and the scaling of the amplitudes in terms of reduced length and time scales is in total agreement with the expected behaviors for fluids belonging to the same Ising universality class. Moreover, the Gaussian beam behaves as an optical bottle with "soft walls" in which the absence of rigid boundaries, and thus of wetting couplings, allows an intrinsic description of the influence of finite-size effects on the kinetics. The beam size breaks the dynamic universality when the growing domains start to feel it. We experimentally investigate the resulting slowing down, and a diffusion-driven model of the growth inside a laser wave is built for comparison. The good agreement observed for the bulk regime and during its modification induced by finite-size effects opens a promising field for the development of this new application of laser waves to control out-of-equilibrium liquid mixtures.

Viscosity and elasticity during collagen assembly in vitro: relevance to matrix-driven translocation

In order to better understand the gelation process associated with collagen assembly, and the mechanism of the in vitro morphogenetic phenomenon of "matrix-driven translocation" [S.A. Newman et al. (1985) Science, 228, 885-889], the viscosity and elastic modulus of assembling collagen matrices in the presence and absence of polystyrene latex beads was investigated. Viscosity measurements at very low shear rates (0.016-0.0549 s(-1)) were performed over a range of temperatures (6.9-11.5 degrees C) in a Couette viscometer. A magnetic levitation sphere rheometer was used to measure the shear elastic modulus of the assembling matrices during the late phase of the gelation process. Gelation was detected by the rapid increase in viscosity that occurred after a lag time tL that varied between O and approximately 500 s. After a rise in viscosity that occurred over an additional approximately 500 s, the collagen matrix was characterized by an elastic modulus of the order of several Pa. The lag time of the assembly process was relatively insensitive to differences in shear rate within the variability of the sample preparation, but was inversely proportional to the time the sample spent on ice before being raised to the test temperature, for test temperatures > 9 degrees C. This suggests that structures important for fibrillogenesis are capable of forming at 0 degrees C. The time dependence of the gelation process is well-described by an exponential law with a rate constant K approximately 0.1 s(-1). Significantly, K was consistently larger in collagen preparations that contained cell-sized polystyrene beads. From these results, along with prior information on effective surface tension differences of bead-containing and bead-lacking collagen matrices, we conclude that changes in matrix organization contributing to matrix-driven translocation are initiated during the lag phase of fibrillogenesis when the viscosity is < or = 0.1 Poise. The phenomenon may make use of small differentials in viscosity and/or elasticity, resulting from the interaction of the beads with the assembling matrix. These properties are well described by standard models of concentrated solutions.