Critical Droplet Research Articles

MONTE CARLO SIMULATION OF NUCLEATION IN THE TWO-DIMENSIONAL POTTS MODEL

Nucleation in the two-dimensional q-state Potts model has been studied by means of Monte Carlo simulations using the heat-bath dynamics. The initial metastable state has been prepared by magnetic quench of the ordered low-temperature phase. The magnetic field dependence of the nucleation time has been measured as the function of the magnetic field for different q and lattice sizes at T = 0.5Tc. A size-dependent crossover from the coalescence to nucleation region is observed at all q. The magnetic field dependence of the nucleation time is roughly described by the classical nucleation theory. Our data show increase of the anisotropy in the shape of the critical droplets with increase of q.

Strangeness nucleation in neutron star matter

We study the transition from npe-type nuclear matter (consisting of neutrons, protons, and electrons) to matter containing strangeness, using a Walecka-type model predicting a first-order kaon-condensate phase transition. We examine the free energy of droplets of K matter as the density, temperature, and neutrino fraction are varied. Langer nucleation rate theory is then used to approximate the rate at which critical droplets of the new phase are produced by thermal fluctuations, thus giving an estimate of the time required for the new (mixed) phase to appear at various densities and various times in the cooling history of the protoneutron star. We also discuss the famous difficulty of ``simultaneous weak interactions'' which we connect to the literature on nontopological solitons. Finally, we discuss the implications of our results to several phenomenological issues involving neutron star phase transitions.

Nucleation in systems with elastic forces.

Systems with long-range interactions when quenched into a metastable state near the pseudospinodal exhibit nucleation processes that differ from classical nucleation seen near the coexistence curve. In systems with long-range elastic forces the description of the nucleation process can be quite subtle due to the presence of bulk and surface elastic compatibility constraints. We analyze the nucleation process in a simple 2D model with elastic forces and show that the nucleation process generates critical droplets with a different structure than the stable phase. This has implications for nucleation in many crystal-crystal transitions, specifically martensites and shape memory alloys, and for the structure of the final state.

Solution of Kramers' problem for a moderately to heavily damped elastic string.

We obtain the nucleation rate of critical droplets for an elastic string moving in a double-well potential and subject to noise and damping forces. We obtain this rate for a class of potentials that includes both the asymmetric straight phi(4) and the straight phi(6) potentials. The frequencies of small oscillations about the critical droplet are obtained from a Heun equation. We solve the Fokker-Planck equation for the phase-space probability density by projecting onto the eigenfunction basis. We present a comparison with simulations for the case of the asymmetric straight phi(4) potential.

Effect of Surface Elasticity on Ostwald Ripening in Emulsions

The coarsening of droplets in an emulsion with a size distribution that initially is given by the Lifshitz−Slyozov−Wagner (LSW) distribution is studied by means of numerical calculations taking into account elastic interfacial behavior. Droplets smaller than the critical radius will shrink while droplets larger than the critical radius will grow. For a zero interfacial elasticity the stationary LSW distribution is obtained and its coarsening rate matches theoretical values. The critical droplet radius, number-averaged droplet radius, and volume-surface-averaged droplet radius increase with time. Low interfacial elasticity with respect to initial interfacial tension causes the initial LSW distribution to become bimodal. The size distribution of the coarsening peak can still be described by the LSW distribution. The smaller peak that accumulates in time has an average radius that depends on the ratio between the interfacial elastic modulus and the interfacial tension. For large ratios (E/σ > 1), the system goes within a short time to a stable situation without changes in particle size with time.

Sterol Carrier Protein-2 Expression Modulates Protein and Lipid Composition of Lipid Droplets

Despite the critical role lipid droplets play in maintaining energy reserves and lipid stores for the cell, little is known about the regulation of the lipid or protein components within the lipid droplet. Although immunofluorescence of intact cells as well as Western analysis of isolated lipid droplets revealed that sterol carrier protein-2 (SCP-2) was not associated with lipid droplets, SCP-2 expression significantly altered the structure of the lipid droplet. First, the targeting of fatty acid and cholesterol to the lipid droplets was significantly decreased. Second, the content of several proteins important for lipid droplet function was differentially increased (perilipin A), reduced severalfold (adipose differentiation-related protein (ADRP), vimentin), or almost completely eliminated (hormone-sensitive lipase and proteins >93 kDa) in the isolated lipid droplet. Third, the distribution of lipids within the lipid droplets was significantly altered. Double labeling of cells with 12-(N-methyl)-N-[(7-nitrobenz-2-oxa-1,3-diazol-4-yl) amino]-octadecanoic acid (NBD-stearic acid) and antisera to ADRP showed that 70, 24, and 13% of lipid droplets contained ADRP, NBD-stearic acid, or both, respectively. SCP-2 expression decreased the level of ADRP in the lipid droplet but increased the proportion wherein ADRP and NBD-stearic acid colocalized by 3-fold. SCP-2 expression also decreased the lipid droplet fatty acid and cholesterol mass (nmol/mg protein) by 5.2- and 6.6-fold, respectively. Finally, SCP-2 expression selectively altered the pattern of esterified fatty acids in favor of polyunsaturated fatty acids within the lipid droplet. Displacement studies showed differential binding affinity of ADRP for cholesterol and fatty acids. These data suggested that SCP-2 and ADRP play a significant role in regulating fatty acid and cholesterol targeting to lipid droplets as well as in determining their lipid and protein components.

Schwarzschild black hole quantum statistics from Z(2) orientation degrees of freedom and its relations to Ising droplet nucleation

AbstractGeneralizing previous quantum gravity results for Schwarzschild black holes from 4 to D ≥ 4 space‐time dimensions yields an energy spectrum En = α n(D—3)/(D—2) EP,D n = 1, 2, …, α = O(1), where EP,D is the Planck energy in that space‐time. This energy spectrum means that the quantized area AD—2(n) of the D — 2 dimensional horizon has universally the form AD—2 = n aD—2, where aD—2 is essentially the (D — 2)th power of the Planck length in D dimensions. Assuming that the basic area quantum has a Z(2)‐degeneracy according to its two possible orientation degrees of freedom implies a degeneracy dn = 2n for the n‐th level. The energy spectrum with such a degeneracy leads to a quantum canonical partition function which is the same as the classical grand canonical potential of a primitive Ising droplet nucleation model for 1st‐order phase transitions in D — 2 spatial dimensions. The analogy to this model suggests that En represents the surface energy of a “bubble” of n horizon area quanta. Exploiting the well‐known properties of the so‐called critical droplets of that model immediately leads to the Hawking temperature and the Bekenstein‐Hawking entropy of Schwarzschild black holes. The values of temperature and entropy appear closely related to the imaginary part of the partition function which describes metastable states.

Patterned growth of individual and multiple vertically aligned carbon nanofibers

The results of studies of patterned growth of vertically aligned carbon nanofibers (VACNFs) prepared by plasma-enhanced chemical vapor deposition are reported. Nickel (Ni) dots of various diameters and Ni lines with variable widths and shapes were fabricated using electron beam lithography and evaporation, and served for catalytic growth of VACNFs whose structure was determined by high resolution transmission electron microscopy. It is found that upon plasma pre-etching and heating up to 600–700 °C, thin films of Ni break into droplets which initiate the growth of VACNFs. Above a critical dot size multiple droplets are formed, and consequently multiple VACNFs grow from a single evaporated dot. For dot sizes smaller than the critical size only one droplet is formed, resulting in a single VACNF. In the case of a patterned line, the growth mechanism is similar to that from a dot. VACNFs grow along the line, and above a critical linewidth multiple VACNFs are produced across the line. The mechanism of the formation of single and multiple catalyst droplets and subsequently of VACNFs is discussed.

Metastability and nucleation for conservative dynamics

In this paper we study metastability and nucleation for a local version of the two-dimensional lattice gas with Kawasaki dynamics at low temperature and low density. Let β>0 be the inverse temperature and let Λ̄⊂Λβ⊂Z2 be two finite boxes. Particles perform independent random walks on Λβ\Λ̄ and inside Λ̄ feel exclusion as well as a binding energy U>0 with particles at neighboring sites, i.e., inside Λ̄ the dynamics follows a Metropolis algorithm with an attractive lattice gas Hamiltonian. The initial configuration is chosen such that Λ̄ is empty, while a total of ρ|Λβ| particles is distributed randomly over Λβ\Λ̄ with no exclusion. That is to say, initially the system is in equilibrium with particle density ρ conditioned on Λ̄ being empty. For large β, the system in equilibrium has Λ̄ fully occupied because of the binding energy. We consider the case where ρ=e−Δβ for some Δ∈(U,2U) and investigate how the transition from empty to full takes place under the dynamics. In particular, we identify the size and shape of the critical droplet and the time of its creation in the limit as β→∞ for fixed Λ and limβ→∞(1/β) log|Λβ|=∞. In addition, we obtain some information on the typical trajectory of the system prior to the creation of the critical droplet. The choice Δ∈(U,2U) corresponds to the situation where the critical droplet has side length lc∈(1,∞), i.e., the system is metastable. The side length of Λ̄ must be much larger than lc and independent of β, but is otherwise arbitrary. Because particles are conserved under Kawasaki dynamics, the analysis of metastability and nucleation is more difficult than for Ising spins under Glauber dynamics. The key point is to show that at low density the gas in Λβ\Λ̄ can be treated as a reservoir that creates particles with rate ρ at sites on the interior boundary of Λ̄ and annihilates particles with rate 1 at sites on the exterior boundary of Λ̄. Once this approximation has been achieved, the problem reduces to understanding the local metastable behavior inside Λ̄ in the presence of a nonconservative boundary. The dynamics inside Λ̄ is still conservative and this difficulty has to be handled via local geometric arguments. Here it turns out that the Kawasaki dynamics has its own peculiarities. For instance, rectangular droplets tend to become square through a movement of particles along the border of the droplet. This is different from the behavior under the Glauber dynamics, where subcritical rectangular droplets are attracted by the maximal square contained in the interior, while supercritical rectangular droplets tend to grow uniformly in all directions (at least for not too long a time) without being attracted by a square.

Critical cluster size and droplet nucleation rate from growth and decay simulations of Lennard-Jones clusters

We study a single cluster of Lennard-Jones atoms using a novel and physically transparent Monte Carlo simulation technique. We compute the canonical ensemble averages of the grand canonical growth and decay probabilities of the cluster, and identify the critical cluster, the size for which the growth and decay probabilities are equal. The size and internal energy of the critical cluster, for different values of the temperature and chemical potential, are used together with the nucleation theorems to predict the behavior of the nucleation rate as a function of these parameters. Our results agree with those found in the literature, and roughly correspond to the predictions of classical theory. In contrast to most other simulation studies, we are able to concentrate on the properties of the clusters which are most important to the process of nucleation, namely those around the critical size. This makes our simulations computationally more efficient.

Thermodynamic expansion of nucleation free-energy barrier and size of critical nucleus near the vapor-liquid coexistence

Nucleation free-energy barrier height and size of the critical nucleus are expanded in powers of the chemical potential difference between the supersaturated vapor (or expanded liquid) in the metastable state and the saturated vapor-liquid system in the stable equilibrium state at the same temperature. The coefficients in the expansions are expressed in terms of the thermodynamic properties at the stable equilibrium state. Comparisons with the results obtained from the density-functional calculation for nucleation of the Lennard–Jones fluid show that systematic improvement in predicting properties of the critical nucleus, either liquid droplet or vapor cavity, is achieved by adding the higher order terms in the expansions. The scaling relations proposed by McGraw and Laaksonen are found to be good approximations to the general expansion; in particular, the barrier height displacement appearing in these scaling relations is naturally given as the second order coefficient in the expansion of the barrier height.

Semiempirical van der Waals/Cahn–Hilliard theory: Size dependence of the Tolman length

The interfacial density profiles, pressure tensor, and Tolman-length δT are determined for the critical droplets of nonane in the framework of the semiempirical van der Waals/Cahn–Hilliard theory. It is shown that the Tolman’s length is strongly size dependent, and tends to a negative value for large drops; indicating that Tolman’s approximation for the curvature dependence of the interfacial free-energy γ≈γ∞/(1+2δT,eq/R) based on the equilibrium values δT,eq, and γ∞, is inaccurate for typical liquid nuclei.

Kinetics of phase change in binary mixtures with complete wetting interfaces

Some properties are described of the nucleation and the spinodal decomposition of a binary mixture in the neighbourhood of a three-phase coexistence state that exhibits complete wetting of one phase at the interface of the other two. A square-gradient density functional approach to inhomogeneities in only one spatial direction is applied, and an analysis is attempted with the help of the equivalent nonlinear dynamic system. The different possible types of behaviour are visualized in terms of orbits in the corresponding phase portraits. There are some exceptional structural, prewetting and wetting film, features both for critical nucleation droplets and for spatially periodic stationary states of spinodal decomposition.

Critical clusters and droplet nucleation rates using Monte Carlo simulation and the second nucleation theorem

Critical clusters and droplet nucleation rates using Monte Carlo simulation and the second nucleation theorem

Fluctuations and instabilities of curved interfaces

We study the stability of cylindrical and spherical interfaces with respect to density fluctuations within the square-gradient approximation. That is, we determine the stability matrix (of the second derivatives of the free energy functional with respect to the density) when the stationary state is a cylindrical or spherical droplet of a stable phase embedded in the metastable phase. For these geometries the stationary states are unstable, some of the eigenvalues are negative and their eigenfunctions represent those fluctuations that are amplified in a process where the equilibrium state is reached. At early times, in a simple model A kinetics, the eigenfunctions represent the fluctuations that grow or decay with a simple exponential law and with a characteristic time that is proportional to the inverse of the eigenvalues. Our results agree with the stability criteria obtained from the Laplace equation, that is, the nucleation of critical droplets and in the case of cylinders also the Rayleigh instability. In the limit of infinite radius we recover the known results for the planar interface between two stable phases. The modes with lowest energy correspond to the customary capillary waves, while other modes with higher energy associated to changes in the interfacial width are shown to be related to a novel interfacial coefficient.

A physically based scheme for the treatment of stratiform clouds and precipitation in large‐scale models. II: Comparison of modelled and observed climatological fields

AbstractFields from two experiments performed with 18‐level versions of the Commonwealth Scientific and Industrial Organization (CSIRO) global climate model (GCM) are compared with observed fields, focusing on quantities related to clouds and precipitation. The first experiment (denoted by PROG) employed a new prognostic treatment of stratiform clouds and precipitation, while the second experiment (denoted by DIAG18) employed a diagnostic treatment, similar to that used in the standard 9‐level CSIRO GCM. The main findings are as follows. Global‐mean quantities agree well with observations, although the global cloudiness in both runs is a little lower than observed values. Zonal‐mean fields generally show good to very good agreement with observations, particularly in the PROG run, where marked improvements in the cloudiness and long‐wave cloud radiative forcing (LWCF) at high latitudes are noted. The PROG run has cloud‐liquid‐water paths (LWPs) that are larger over mid‐latitude oceans than those from satellite retrievals. Geographical distributions of precipitation, cloudiness, LWCF and SWCF (short‐wave cloud radiative forcing) from both runs are generally in reasonable agreement with observations. Overall, the cloudiness and LWCF are somewhat more realistic in the PROG run, the SWCF is slightly more realistic in the DIAG18 run, and the precipitation is not greatly affected by the change of cloud scheme. Problems affecting both runs to some degree are: deficient cloudiness in the subtropics, and to a lesser extent in mid‐latitudes;deficient SWCF in mid‐latitudes, with a tendency towards excessive SWCF at low latitudes;deficient LWCF over land, mainly in the tropics and northern mid‐latitudes;excessive precipitation, cloudiness and cloud radiative forcing in the tropical western Pacific Ocean in July.The reasons for the above findings are investigated, in part, via the use of sensitivity tests. The improved high‐latitude cloudiness in the PROG run results from (a) replacement of a cloudiness parametrization based on relative humidity with one based on a generalized relative humidity that includes the contribution from cloud water, and (b) inclusion of the effect of frozen precipitation processes on the cloud fraction. The improved LWCF is primarily the result of more realistic treatment of cloud emissivity in the prognostic cloud scheme. The excessive LWPs over mid‐latitude oceans in the PROG run can be corrected by a modest reduction in the critical cloud droplet radius that controls the onset of autoconversion. The deficient cloudiness in the subtropics and mid‐latitudes (typical of current GCMs) can be improved by simple changes to the critical relative humidities used to control the onset of cloud formation or by an increase of vertical resolution, but this improvement comes at the cost of excessive cloudiness in the tropics. The errors in the modelled SWCF (also typical of current GCMs) suggest that there is a systematic latitudinal bias in the calculation of cloud–radiation interactions, such as the effect of solar zenith angle. The deficient LWCF over land is related to a deficiency of high cloud. The vigorous circulation in July over the tropical western Pacific is much more prominent in these 18‐level simulations than in the standard 9‐level version of the model, and is related to aspects of the model other than the cloud treatment.

Strongly interacting combustion of two miscible binary-fuel droplets at high pressure in microgravity

The interactive combustion of two miscible binary-fuel droplets at elevated pressures, up to the values above the critical pressure of the fuel, was investigated experimentally. Heptane and hexadecane were selected as the fuel mixture, to study the occurrence of staged combustion, and a nitrogen-oxygen atmosphere with a relatively low oxygen concentration was employed to facilitate observation of droplets during combustion at elevated pressures. The effects of fuel composition, ambient pressure, and droplet spacing were investigated. Results show that staged combustion of binary-fuel droplets still exists for interacting droplets at high pressure and that burning-rate constants of the first and the third stages, as well as the transition droplet volume, are not strongly affected by interaction. The increase in droplet lifetime by strong interaction is demonstrated and explained. It is shown that the lifetime achieves a minimum value at a critical droplet spacing even for binary-fuel droplets that undergo staged combustion at high pressure without buoyancy, a result which is explained by influences of radiant energy transfer. It is also demonstrated experimentally that flame lifetimes exceed droplet lifetimes at high pressure both for single droplets and strongly interacting droplet pairs, which is explained by vapor presence after liquid disappearance and reduced tendencies toward flame extinction at elevated pressures.

The metastable behavior of the three-dimensional stochastic Ising model I

The metastable behavior of the stochastic Ising model is studied in a finite three-dimensional torus, in the limit as the temperature goes to zero. The so-called critical droplet is determined, a clear view of the passage from the configuration that all spins are down (-1) to the configuration that all spins axe up (+1) is given and the logarithmic asymptotics of the hitting time of+1 starting at -1 orvice versa is calculated. The proof uses large deviation estimates of a family of exponentially perturbed Markov chains.

Transient nucleation in metastable nuclear matter

We present an approximate analytical solution of the kinetic equation describing the formation of droplets of the real nuclei (fragments) within metastable nucleonic gas produced by intermediate-energy, heavy-ion collisions. It is found that for a finite-range, Brink - Boeker effective interaction the production of large supercritical fragments is highly improbable above a certain temperature because this process requires times comparable to the lifetime of the fireball. Critical droplet size and the difference in chemical potentials of nucleonic vapour and the fragments are determined from the kinetic Fokker - Planck equation. On the other hand, the repulsive Coulomb force might cause a conversion via spinodal decomposition in small systems.

Viscosity effects on nebulisation of aqueous solutions

While the viscosity of sucrose and sodium citrate solutions increased with increasing percentage (w/v) of solute, the surface tension largely remained consistent. Water, sucrose (10–60% w/v) and sodium citrate (7–36% w/v) solutions were nebulised in two air-jet nebulisers (Pari LC; Medix A II) operated at 6 1 min −1 and in an ultrasonic device (Medix Electronic) operated at mid-power setting. A wide variation in nebuliser size and output characteristics existed between the different commercially available models studied. Contrary to atomisation theories for air-jet nebulisation, droplet size was inversely related to solution viscosity (over 1–6 cP). Beyond this critical value droplet size increased as viscosity increased. This anomaly may be due to viscosity acting either directly or through liquid flow-rates. By contrast the ultrasonic device generated droplets of size proportional to viscosity but was unable to nebulise high viscosity fluids (i.e. > 6 cP). The low viscosity solutions offered less resistance to the integral fountain disintegration process, thereby producing smaller droplets and higher outputs.